The Elementary Theory of the Frobenius Automorphisms

Ehud Hrushovski ^{∗}^{†}

January 2006

Abstract

We lay down some elements of a geometry based on difference equations. Various constructions of algebraic geometry are shown to have meaningful analogs: dimensions, blowingup, moving lemmas.

Analogy aside, the geometry of difference equations has two quite different functorial connections with ordinary algebraic geometry. On the one hand, a difference scheme is determined by geometric data, including principally a pro-algebraic scheme. On the other hand, for each prime power p^{m}, one has a functor into algebraic schemes over 𝔽_{p}, where the structure endomorphism becomes Frobenius.

Transformal zero-cycles have a rich structure in the new geometry. In particular, the Frobenius reduction functors show that they encapsulate data described in classical cases by zeta or L-functions. A theory of rational and algebraic equivalence of 0-cycles is initiated, via a study of the transformal analog of discrete valuation rings.

The central application and motivation is the determination of the elementary theory of the class of Frobenius difference fields (algebraically closed fields of characteristic p > 0, enriched with a symbol for x ↦ x^{p^{m}}. It coincides with the model companion ACFA of the theory of difference fields. In particular, a sentence P holds in (𝔽_{p}^{a}, x ↦ x^{p}) for almost all primes p if and only if it is true for a generic automorphism of a field of characteristic 0; i.e. true in (L, σ) for a co-meager set of σ ∈ Aut(L), where L is the field of algebraic functions in denumerably many variables over ℚ.

The proof requires a twisted version of the Lang-Weil estimates, related to Deligne’s conjecture but less precise and valid more generally. It is proved on the basis of the preceding work on difference geometry.

Some applications are given, in particular to finite simple groups, and to the Jacobi bound for difference equations.

∗Department of Mathematics, Hebrew University, Jerusalem, Israel; ehud@math.huji.ac.il †Research supported in part by the Israel Science Foundation grant no. 244/03

Contents

1 Introduction 4 1.1 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Difference algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Description of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

I Difference schemes 12

2 Well-mixed rings 12

3 Definition of difference schemes 16 3.1 Localization, rings of sections . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Some functorial constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Dimensions 26 4.1 Total dimension and transformal dimension . . . . . . . . . . . . . . . . . . . 26 4.2 Dimensions of generic specializations . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Difference schemes and pro-algebraic varieties . . . . . . . . . . . . . . . . . . 32 4.4 Transformal degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Finite presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.6 A chain condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.7 Directly presented difference schemes . . . . . . . . . . . . . . . . . . . . . . . 41

5 Transformal multiplicity 45 5.1 Transformally separable extensions . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 The reduction sequence of a difference scheme . . . . . . . . . . . . . . . . . . 46 5.3 σ-primary ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Transformal valuation rings in transformal dimension one 51 6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.2 Transformal discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . 53 6.3 The value group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.4 The residue field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.5 Valued field lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.6 Structure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Transformal valuation rings and analyzability 70 7.1 The residue map on difference varieties . . . . . . . . . . . . . . . . . . . . . . 70 7.2 Sets of finite total dimension are residually analyzable . . . . . . . . . . . . . 72 7.3 Analyzability and inertial dimension . . . . . . . . . . . . . . . . . . . . . . . 73 7.4 Direct generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8 Transformal specialization 79

8.1 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.2 Boolean-valued valued difference fields . . . . . . . . . . . . . . . . . . . . . . 82 8.3 Pathwise specialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9 Frobenius reduction 86 9.1 The functors M_{q} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 9.2 Dimensions and Frobenius reduction . . . . . . . . . . . . . . . . . . . . . . . 87 9.3 Multiplicities and Frobenius reduction . . . . . . . . . . . . . . . . . . . . . . 89

9.3.1 An explicit example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 9.3.2 Algebraic lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 9.3.3 Relative reduction multiplicity . . . . . . . . . . . . . . . . . . . . . . 92 9.3.4 Transformal multiplicity and reduction multiplicity . . . . . . . . . . . 94

II Intersections with Frobenius 96

10 Geometric Preliminaries 96 10.1 Proper intersections and moving lemmas . . . . . . . . . . . . . . . . . . . . . 96 10.2 Degrees of cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10.3 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10.4 Geometric statement of the uniformity . . . . . . . . . . . . . . . . . . . . . . 104 10.5 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 10.6 Rough bound (Bezout methods) . . . . . . . . . . . . . . . . . . . . . . . . . 105 10.7 Finite covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.8 Reduction to smooth varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

11 The virtual intersection number 110

12 Equivalence of the infinite components: asymptotic estimate 116

13 Proofs and applications 120 13.1 Uniformity and Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 13.2 Finite simple groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 13.3 Nonstandard powers and a suggestion of Voloch’s . . . . . . . . . . . . . . . . 123 13.4 Consequences of the trichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . 124

14 Jacobi’s bound for difference equations 127

15 Complements 129 15.1 Transformal dimension and degree of directly presented difference schemes . . 129 15.2 Projective difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 15.3 Blowing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 15.4 Semi-valuative difference rings . . . . . . . . . . . . . . . . . . . . . . . . . . 139

1 Introduction

1.1 The main theorem

A classical result of algebraic model theory ([Ax],[Fried-Jarden86]) shows that all elementary statements about finite fields are determined by three facts from geometry. The fundamental fact is Weil’s Riemann Hypothesis for curves, entering via the Lang-Weil estimates [Lang-Weil]. The auxiliary statements are the Cebotarev density theorem, and the cyclic nature of the Galois group. All three facts are proved, within algebraic geometry, number theory and Galois theory respectively, by viewing a finite field as the fixed field of the Frobenius automorphism. One might therefore hope that the pertinent geometry could be used directly to derive the full elementary theory of the Frobenius maps. The elementary theory of finite fields would follow as a special case.

This hope turns out to be correct, except that the Lang-Weil estimates do not suffice, and need to be replaced by a more general principle. Let k be an algebraically closed field. If V is a variety over k and σ is an automorphism of k, we denote by V^{σ} the variety obtained from V by applying σ to the defining parameters. (In terms of schemes, if f : V → Spec(K) is a scheme over Spec(K), then V^{σ} is the same scheme, but with f replaced by f ∘ σ^{−1}. ) ϕ_{q} denotes the Frobenius automorphism x ↦ x^{q}. Φ_{q} ⊂ (X × X^{ϕ_{q}}) denotes the graph of ϕ_{q}, viewed as a subvariety.

Theorem 1.1 (1) Let X be an affine variety over k, and let S ⊂ (X × X^{ϕ_{q}}) be an irreducible subvariety. Assume dim (S) = dim (X) = d, the maps S → X, S → X′ are dominant, and one is quasi-finite. Let a = [S : X]/[S : X′]_{insep}. Then there exists a constant c depending only on the degrees of X, S such that for q > c,

|S(k) ∩ Φ_{q}(k)| = aq^{d} + e with |e| ≤ cq^{d− \frac{1}{2}}

The numbers [S : X],[S : X′]_{insep} refer to the degree (respectively purely inseparable degree) of the field extensions k(S)/k(X), k(S)/k(X′). In particular if S is the graph of a separable morphism X → X, we have a = 1. The degrees of X, S referred to can be made explicit as follows. If X ⊂ 𝔸^{m}, let S̄ be the closure of S ⊂ (𝔸^{m} × 𝔸^{m}) ⊂ ℙ^{m} × ℙ^{m}; then c depends only on m and the bidegrees of S̄. In particular, if q is large compared to these degrees, the theorem implies that S(k) ∩ Φ_{q}(k) ≠ ∅.

We will give other versions of the statement, better suited to the inherent uniformity of the situation. Below (1.1A) we state it in the language of difference algebra. A algebro-geometric version will be given later (§10.4.)

Observe that X need not be defined over the fixed field of ϕ_{q} (or indeed over any finite field.) If X is defined over GF (q), then X = X^{ϕ_{q}}, and the diagonal (or the graph any other dominant morphism X → X) becomes a possible choice of S. In the case of the diagonal, one obtains the Lang-Weil estimates.

When X and S descend to a finite field, the projection S → X is proper, and S → X′ is quasi-finite, Theorem 1.1 follows from Deligne’s conjecture ([Fujiwara97],[Pink92]) together with his theorem on eigenvalues of Frobenius. In general these last two assumptions ([Pink92] 7.1.1) cannot be simultaneously obtained in our context, as far as I can see.

Since X is affine, an arbitrary proper divisor can be removed, preserving the same estimate. We thus find that the set of points of S ∩(X ×X^{ϕ_{q}}) is “asymptotically Zariski dense”. Let us state a special case of this (when S is a morphism, it was used by Borisov and Sapir for a group theoretic application.)

Corollary 1.2 (1c) Let X be an affine variety over k = GF (q), and let S ⊂ X^{2} be an irreducible subvariety over K^{a}. Assume the two projections S → X are dominant. Then for any proper subvariety W of X, for large enough m, there exist x ∈ X(k^{a}) with (x, ϕ_{q}^{m}(x)) ∈ S and x ∉ W.

(k^{a} denotes the algebraic closure of k.)

Difference algebraic statement Here is the same theorem in the language of difference algebra, that will be used in most of this paper, followed by a completely algebraic corollary; we do not know any purely algebraic proof of this corollary.

A difference field is a field K with a distinguished endomorphism σ. K is inversive if σ is an automorphism. A subring of K, closed under σ, is called a difference domain. A difference ring R is simple if every difference ring homomorphism is injective or zero on R.

If q is a power of a prime p, let K_{q} be the difference field consisting of an algebraically closed field of characteristic p, together with the q-Frobenius automorphism ϕ_{q}(x) = x^{q}. K_{q} will be called a Frobenius difference field. Theorem 1.1A Let D ⊂ R be finitely generated difference domains. Assume there exists an embedding of D into an algebraically closed, inversive difference field L, such that R⊗_{D}L is a difference domain, of transcendence degree d over L. Then there exist b, b′ ∈ ℕ, 0 < c ∈ ℚ, and d ∈ D such that for any prime power q ≥ b, and any homomorphism of difference rings h : D → K_{q} with h(d) ≠ 0, h extends to a homomorphism h̄ : R → K_{q}. Moreover, the number of different h̄ is cq^{d} + e, where e ≤ b′q^{d−1/2}.

Corollary 1.3 Let D be a finitely generated difference ring. Assume D is simple, and has no zero divisors. Then D is a finite field, with a power of Frobenius.

The assumption that D is a domain can be weakened to the condition: ab = 0 implies aσ(b) = 0. A difference ring satisfying this condition will be called well-mixed.

Model theoretic consequences With this in hand, the methods of Ax easily generalize to yield the first order theory of the Frobenius difference fields. Let T_{∞} be the set of all firstorder sentences θ such that for all sufficiently large q, K_{q} ⊧ θ.

Theorem 1.4 (dec) T_{∞} is decidable. It coincides with the model companion “ACFA” of difference fields.

Here is a more precise presentation of the theorem (restricted to primes), that does not explicitly mention the axioms of ACFA. Fix a countable algebraically closed field L of infinite transcendence degree over ℚ. Let G = Aut(L); for σ ∈ Aut(ℚ^{a}), let G_{σ} = {τ ∈ G : τ|ℚ^{a} = σ}. G_{σ} is a Polish space, and it makes sense to talk of meager and co-meager sets; we will mean “almost every” in this sense.

Let ϕ be a first order sentence in the language of difference rings. The following numbers are equal:

1) The Dirichlet density of the set of primes p such that K_{p} ⊧ ϕ.

2) The Haar measure of the set of σ ∈ Aut(ℚ^{a}), such that, for almost every τ ∈ G_{σ}, (L, τ) ⊧ ϕ.

The theory ACFA is described and studied in [Chatzidakis-Hrushovski]. The axioms state that the field is algebraically closed, and:

Let V be an absolutely irreducible variety over K, and let S be an irreducible subvariety of V × V^{σ} projecting dominantly onto V and onto V^{σ}. Then there exists a ∈ V (K) with (a, σ(a)) ∈ S.

With different axioms, the existence of a model companion for difference fields was discovered earlier by Angus Macintyre and Lou Van den Dries, precisely in connection with the search for the theory of the Frobenius. See [Macintyre95]. In positive characteristic, we can also conclude: Corollary 1.5 (1cor) Let F = GF (p)^{alg} . For almost every automorphism σ of F (in the sense of Baire category,) (F, σ) ⊧ ACF A.

I.e. the set of exceptional automorphisms is meager.

A similar result holds, much more trivially, if GF (p)^{alg} is replaced by an algebraically closed field of infinitely countable transcendence degree over the prime field. However, an example of Cherlin and Jarden shows that a generic automorphism of ℚ̄ does not yield a model of ACFA, nor is this true for any other field of finite transcendence degree.

We can also consider the theory T_{Frobenius} consisting of those sentences true in all K_{q}, or in all K_{p} with p prime. The completions of this theory are just the completions of ACF A together with the complete theories of the individual K_{q}. With some extra care, we can also conclude: Theorem 1.6 (4) The theory of all K_{p}, p prime, is decidable.

A similar result holds for the theory of all K_{q}, q an arbitrary prime power; or with q ranging over all powers of a given prime; etc.

1.2 Some applications

Finite simple groups The theory of definable groups has been worked out for ACF A, and has some suggestive corollaries for finite simple groups. This work has not yet been published; but an important special case (groups over the fixed field) contains the main ideas, see [Hrushovski-Pillay]. Here we give a preview.

We say that a family of finite groups indexed by prime powers q is uniformly definable if there exist first order formulas ϕ, ψ such that ϕ defines a finite subset of each K_{q}, ψ defines a group operation on ϕ, and the family consists of these groups for the various K_{q}. Examples of such families include G_{n}(q), where n is fixed, and G_{n}(q) is one of the families of finite simple groups (e.g. P SL_{n}(q)). All but two of these families are already definable over finite fields. However, the Ree and Suzuki families are not. Instead they are defined by the formula:

σ(x) = ϕ(x), x ∈ G

where G is a certain algebraic group, and ϕ is an algebraic automorphism whose square is the Frobenius ϕ_{2} or ϕ_{3}.

From Theorem 1.4, we obtain immediately:

Theorem 1.7 For each fixed n, the first order theory of each of the classes G_{n}(q) of finite simple groups is decidable.

The remaining results use the theory of groups of finite S1-rank.

Theorem 1.8 Every uniformly definable family of finite simple groups is (up to a finite set) contained in a finite union of families G_{n}(q)

More precisely, for each q, there is a definable isomorphism in K_{q} between the group ϕ(K_{q}), and some G_{n}(q); only finitely many values of n occur, and finitely many definable isomorphisms. The theorem is proved without using the classification of the finite simple groups. Note that it puts the Ree and Suzuki groups into a natural general context. (See also [Bombieri] in this connection.)

Theorem 1.9 Let G_{n}(q) be one of the families of finite simple groups. Then there exists an integer r = r(n) such that for any q, every nontrivial conjugacy class of G_{n}(q) generates G_{n}(q) in at most r steps.

This is a special case of more general results on generation of subgroups by definable subsets. One can indeed take r(n) = 2dim H_{n}, where H_{n} is the associated algebraic group.

Jacobi’s bound for difference equations A linear difference or differential equation of order h, it is well-known, has a solution set whose dimension is at most h. The same is true for a nonlinear equation, once the appropriate definitions of dimension are made. Jacobi, in [Jacobi], proposed a generalization to systems of n differential equations in n variables. The statement is still conjectural today; an analogous conjecture for difference equations was formulated by Cohn. We prove this in §14. Our method of proof illustrates theorem 1.1: we show that our analog of the Lefschetz principle translates Cohn’s conjecture to a very strong, but known, form of Bezout’s theorem.

See §13 for proofs and some other applications.

1.3 Difference algebraic geometry

Take a system X of difference equations: to fix ideas, take the equations of the unitary group, σ(u)u^{t} = 1, where u is an n × n -matrix variable; or in one variable, take σ(x) = x^{2} + c, for some constant c. Consider four approaches to the study of X.

It may be viewed as part of the category of objects defined by difference equations; i.e. part of an autonomous geometry of difference equations. See the discussion and examples in [Hrushovski01]. Here X is identified, in the first instance, with the set of solutions of the defining equations in a difference field. The rough geography of the category of definable sets and maps is worked out in [Chatzidakis-Hrushovski], [Chatzidakis-Hrushovski-Peterzil]. A great deal remains to be done, for example with respect to topology. The definition of difference schemes in Part I can be viewed as a small step in a related direction.

Alternatively, it is possible to view a difference equation in terms of the algebro-geometry data defining it. One can often represent the system as a correspondence: a pair (V, S), where V is an algebraic variety over a difference field k, S a subvariety of V × V^{σ}; the equation is meant to be X = {x : (x, σ(x)) ∈ S}. See 4.36. The great strength of this approach is that results of algebraic geometry become directly accessible, especially when V is be smooth and complete, and the fibers of S → V, S → V^{σ} of equal dimensions. It is generally not possible to meet these desiderata. Moreover, given a system (V, S) in this form, simple difference-theoretic questions (such as irreducibility) are not geometrically evident; and simple operations require radical changes in V. Nevertheless, we are often obliged to take this approach, in order to use techniques that are not available directly for difference varieties or schemes; especially the cohomological results used in §11. We will in fact stay with this formalism for as long as possible, particularly in §10, even when a shorter difference-theoretic treatment is visible; cf. the proof of the moving lemmas in §10.1. One obtains more explicit statements, and a contrast with those points that seem to really require a difference-theoretic treatment.

The third and most classical approach is that of dynamics. See Gromov’s [Gromov] for a combination of dynamics with algebraic geometry in a wider context, replacing our single endomorphism with a finitely generated group; it also contains, in a different way than ours, reductions to finite objects. Here the space corresponding to (V, S) is

Y = {(a_{0}, a_{1}, …) : (a_{n}, a_{n+1}) ∈ S}

This is a pro-algebraic variety; we can topologize it by the coarsest topology making the projections to algebraic varieties continuous, if the latter are given the discrete topology. Note that the points of X and Y are quite different; in particular it makes sense to talk of points of Y over a field (rather than a difference field.)

For the time being the work described here barely touches upon dynamical issues; though just under the surface, the proofs in [Chatzidakis-Hrushovski-Peterzil] and in §11, the definition of transformal degree, and other points do strongly involve iteration. It is also true that by showing density of Frobenius difference fields among all difference fields, we show density of periodic points (at least if one is willing to change characteristics); raising hopes for future contact.

A fourth point of view, introduced in §9, is to view the symbol σ as standing for a variable Frobenius. For each prime power q = p^{m} we obtain an algebraic variety (or scheme) X_{q}, by interpreting σ as the Frobenius x ↦ x^{q} over a field of characteristic p. The main theorem can be read as saying that this approach is equivalent to the others.

It can be quite curious to see information, hidden in (V, S), released by the two apparently disparate processes of iteration, and intersection with Frobenius. Even simple dimensiontheoretic facts can sometimes be seen more readily in this way. Cf. 15.1, or the proof of the Jacobi bound for difference equations in §14 (where I know no other way.)

Here is a the simplest example of a definition and a number from the various points of view. The following are essentially equivalent:

• X has transformal dimension 0, or equivalently finite SU-rank in the sense of [Chatzidakis-Hrushovski], or finite total order (§4); • The map S → V is quasi-finite;

• The space Y is locally compact;

• Almost all X_{q} are finite. If this is the case, and X is irreducible, the following numbers are essentially equal:

• the degree of the correspondence S (adjusted, in positive characteristic, by the purely inseparable degrees)

• the entropy of Y;

• the number c such that X_{q} has asymptotic size cd^{q}, where d = dim (V).

The associated invariant of X needs to be defined in terms of difference schemes, rather than varieties, and will be described in §5.2.

”Essentially” indicates that these equivalences are not tautologies, and require certain conditions; for instance it suffices for S → V to be generically quasi-finite, if the exceptional infinite fibers do not meet X. The various technicalities will be taken up in §2- 6.

1.4 Description of the paper

§2 - 6 introduce a theory of difference schemes. As in algebraic geometry, they are obtained by gluing together basic schemes of the form Spec^{σ} A, with A a difference ring. The Spec^{σ} A are more general than difference varieties in that A may have nilpotents, both in the usual sense, and elements a such that σ(a) = 0 while a ≠ 0.

§2 develops the basic properties of a well-mixed difference ring. Any difference ring without zero-divisors, as well as any difference ring whose structure endomorphism is Frobenius, is well-mixed. Well-mixed rings admit a reasonable theory of localization; in particular, Proposition 3.8 shows that a difference scheme based on well-mixed rings makes sense intrinsically, without including the generating difference rings explicitly in the data.

In §4, the dimension and degree of a difference scheme is defined. In fact there are two dimensions: transformal dimension, and when that vanishes, total dimension. The transformal affine line has transformal dimension one, and contains many difference subschemes of transformal dimension zero and finite total dimension, notably the “short” affine line over the fixed field, of total dimension one. Here too, for well-mixed schemes we obtain a smoother theory. We improve a theorem of Ritt and Raudenbush by proving Noetherianity of finitely generated difference rings with respect to radical, well-mixed ideals, in finite total dimension. We do not know whether the same theorem holds for all well-mixed ideals. (Ritt-Raudenbush proved it for perfect ideals; cf. [Cohn].)

§5 is concerned with the kernel of the endomorphism σ on the local rings, or with transformal nilpotents; it is necessary to show how their effect on total dimension is distributed across a difference scheme. §4.7 deals with the presentation of a difference scheme as a correspondence (V, S), and how in simple case one may test for e.g. irreduciblity.

Further basic material, not used in the main line of the paper, is relegated to §15. In §15.1, the transformal dimension of the difference scheme corresponding to (V, S) is determined in general; I do not know a proof that does not go through the main theorem of the paper.

§15.2 looks at transformal projective space. §15.3 introduces two transformal versions of the blowing-up construction. They are shown to essentially coincide; as one (based on the short line, over the fixed field) has no Frobenius analog, this may be geometrically interesting.

The idea of a moving family of varieties, and of a limit of such a family, is characteristic of geometry. It is poorly understood model-theoretically, and has not so far been studied in difference algebraic geometry. In algebraic geometry, it is possible to give a completely synthetic treatment: given a family V_{t} of varieties over the affine t-line (except over 0), take the closure and let the limit be the fiber V_{0}. This works in part since an irreducible family over the line is automatically flat. In difference algebraic geometry, this still works over the short line, but not over the transformal affine line. It is possible to use blowing ups to remedy the situation, but the use of valuation theory appears to be much clearer.

Transformal valued fields are studied in §6. They differ from the valued difference fields of [Scanlon], primarily in that the endomorphism does not fix the value group: on the contrary, it acts as a rapidly increasing map. We are not concerned with the first order theory here, but rather with the structure of certain analogues of discrete valuation rings. Their value group is not ℤ but ℤ[σ], the polynomial ring over ℤ in a variable σ, with σ > ℤ. A satisfactory structure theory can be obtained, especially for the completions. Extensions of such fields of finite transcendence degree can be described in terms of inertial and totally ramified extensions, and the completion process; there are no other immediate extensions. As a result, the sum of the inertial transcendence degree extension and the ramification dimension is a good invariant; cf. 6.39. This valuative dimension will be used in §7.

Given a family X_{t} of difference varieties, we define in §8 the specialization X_{→0}, a difference subvariety of the “naive” special fiber X_{0}. The total dimension of X_{→0} is at most that of the generic fiber X_{t}; this need not be the case for X_{0}. Together with the material in §7, this lays the basis for a theory of rational and algebraic equivalence of transformal cycles. (It will be made explicit in a future work.)

The main discovery allowing the description of specializations in terms of transformal valuation rings is this: definable sets of finite total dimension are analyzable in terms of the residue field. The simplest case is of a definable set X_{t} in K, contained entirely in the valuation ring, and such that the residue map is injective on X_{t}. The equation σ(x) = x for is immediately seen to have this property. In general, for an equation of transformal dimension zero, X_{t} is analyzable over the residue field (§7.2.) This model-theoretic notion is explained in §7.3. It means that sets of finite total dimension over the generic fiber can actually be viewed as belonging to the residue field; but in a somewhat sophisticated sense, that will be treated separately. Here we will content ourselves with the numerical consequence for Frobenius fields, bounding the number of points specializing to a given point of X_{0} in terms of the valuative dimension. With this relation in mind, §7.4 contains a somewhat technical result, needed in the main estimates, bounding the valuative dimension in terms of directly visible data, when a difference scheme is presented geometrically as (V, S).

The proof of the main theorem begins in §10. In this section, it is reduced to an intersection problem on a smooth, complete algebraic variety V: one has correspondence S ≤ V ×V^{σ}, and wants to estimate the number of points of S ∩ Φ_{q}, where Φ_{q} is a graph of Frobenius; or rather those points lying outside a proper subvariety W of V. The main difficulty is that, away from W, the projection S → V need not be quasi-finite. For instance, S may contain a “square” C × C, and then any Frobenius will meet S in an infinite set. A moving lemma in §10.1 shows that, while this can perhaps not be avoided, when viewed as a cycle S can be moved to another, S′, whose components do not raise a similar difficulty.

In §11 we obtain an estimate for the intersection number S ·Φ_{q} in the sense of intersection theory. See 11.3 for a quick proof in the case of projective space, and 11.4 for a proof for curves: indeed Weil’s proof, using positivity in the intersection product on a surface, works in our case too.

For general varieties, in the absence of the standard conjectures, we use the cohomological representation and Deligne’s theorem. According to the Lefschetz fixed point formula, the question amounts to a question on the eigenvalues of the composed correspondence Φ^{−1} ∘ S, acting on the cohomology H^{∗}(V) of V. When V = V^{σ} and S, Φ commute, one can apply Deligne’s theorem on the eigenvalues of Φ. But in general, Φ induces a map between two distinct spaces H^{∗}(V) and H^{∗}(V^{σ}); so one cannot speak of eigenvalues. Some trickery is therefore needed , in order to apply the results of Deligne. It is in these manipulations that the precision of Deligne’s theorem is needed, rather than just the Lang-Weil estimates. It may suffice to know that every eigenvalue on H^{i}, i < 2n, has absolute value ≤ q^{2n−1/2}; but because of possible cancellations in the trace (when e.g. all m’th roots of unity are eigenvalues for some m), this still seems beyond Lang-Weil.

It would be interesting to sharpen the statement of the theorem, so as to give a precise rather than asymptotic cohomological account of the size of Frobenius specializations of zero-dimensional difference schemes.

Having found the intersection number, we still do not know the number of (isolated) points of the intersection S ∩ Φ_{q}(\W); this is due to the possible existence of infinite components, mentioned above. Cf. [Fulton], and Kleiman’s essay in [Seidenberg1980]. §12 is devoted to estimating the “equivalence” of these components. A purely geometric proof using the theory of [Fulton] appears to be difficult; the main trouble lies in telling apart the 0-dimensional distinguished subvarieties of the intersection, from the true isolated points (not embedded in larger components) that we actually wish to count. Instead, we use the methods of specialization of difference schemes, developed above. By the moving lemma, there exists a better-behaved cycle S_{t} on V ×V^{σ}, specializing to S. The intersection numbers corresponding to S_{t}, S are the same; and we can assume that the equivalence has been bounded for S_{t}. The problem is to bound the number of points of S_{t} ∩ Φ_{q} specializing into W; as well as the number (with multiplicities) of points of S_{0} ∩ Φ_{q} to which no point of S_{t} ∩ Φ_{q} specializes. The inverse image of W under the residue map is shown to be analyzable over the residue field, of dimension smaller than dim (V), proving this point.

This use of intersection theory does not flow smoothly, since the difference varieties do not really wish to remain restricted to V × V^{σ}. They are often more naturally represented by subvarieties of, say, V × V^{σ} × V^{σ^{2}}. Even the decomposition into irreducible components of distinct transformal dimensions cannot be carried out in V × V^{σ}; e.g. above, the closure of {x ∈ V \ W : (x, σ(x)) ∈ S} is always a difference scheme of finite total dimension; but S is irreducible, and the closure cannot be represented on V × V^{σ}. However, the intersectiontheoretic and cohomological methods that we use do not seem to make sense when one intersects two dim (V)-dimensional subvarieties of V × V^{σ} × V^{σ^{2}}; and so we must strain to push the data into V ×V^{σ} (as in 7.4.) This is one of a number of places leading one to dream of a broader formalism.

The proofs of the various forms of the main theorem, the applications mentioned above, and a few others, are gathered in §13 and §14.

Thanks to Dan Abramovich, Phyllis Cassidy and Ron Livne for useful discussions of this problem, and to Gabriel Carlyle, Zoé Chatzidakis, Emmanuel Kowalski, Yves Laszlo, and Thomas Scanlon for their comments at various stages. Warm thanks to Richard Cohn for information about the Jacobi bound problem.

Part I

Difference schemes

2 Well-mixed rings

The basic reference for difference algebra is [Cohn].

A difference ring is a commutative ring R with 1 and with a distinguished ring homomorphism σ : R → R. We write a^{σ} = σ(a), and a^{∑ m_{i}σ^{i}} = Π_{i}σ^{i}(a)^{m_{i}}.

A well-mixed ring is a difference ring satisfying:

ab = 0 ⇒ ab^{σ} = 0

Given a difference ring R, a difference ideal is an ideal such that x ∈ I implies σ(x) ∈ I. If in addition R/I is well-mixed we say that I is well-mixed. A transformally prime ideal is a prime ideal, such that, moreover, x ∈ I ⇔ σ(x) ∈ I. Every transformally prime ideal is well-mixed.

Any difference ring R has a smallest difference ideal I = rad_{wm}(R) such that R/I is well-mixed.

A difference ring is a difference domain if the zero-ideal is a transformally prime ideal. ^{1} A difference field is a difference domain, that is a field. (An endomorphism of a field is

automatically injective.)

Lemma 2.1 (alg-h) Let R be a difference ring, P an algebraically prime difference ideal, K the fraction field of R/P, h : R → K the natural map.

1Ritt and Cohn ([Cohn]) use the term prime difference ideal for what we call a transformally prime ideal. When as we will one considers difference ideals that are prime, but not transformally prime, this terminology becomes awkward. To avoid confusion, we will speak of algebraically prime difference ideal to mean difference ideals that are prime. We kept the use of difference domain, as being less confusing; though transformally integral might be better. At all events, difference rings with the weaker property of having no zero-divisors will be referred to as as algebraically integral. Cohn calls an ideal mixed if the quotient is well-mixed.

1. P is transformally prime iff there exists a difference field structure on K, extending the natural quotient structure on R/P.

2. Let R′ ≤ R be a difference subring, P′ = (P ∩R′). Let K′ be the fraction field of R′/P′, and suppose K is algebraic over K′. Then P is transformally prime iff P′ is.

Proof (1) If P is transformally prime, R/P is a difference domain, and σ extends naturally to the field of fractions: if b ≠ 0 then σ(b) ≠ 0, and one can define σ(a/b) = σ(a)/σ(b). Conversely, if L is a difference field, and R →_{h} R/P ⊂ L is a homomorphism, if a ∉ P then h(a) ≠ 0 so h(σ(a)) = σ(h(a)) ≠ 0, and thus σ(a) ∉ P.

(2) If P is transformally prime, clearly P′ is too. If P′ is transformally prime, then the multiplicative set h(R′ \ P′) ⊆ K′ \ (0) ⊂ K \ (0) is closed under σ, so σ extends to the localization of h(R) by this set, i.e. to M = K′(R/P). But since K′ ⊆ M ⊆ K and K is algebraic over K′, M is a field, hence M = K. □

Notation 2.2 Let I be a difference ideal.

σ \sqrt{I} = \{a ∈ R : a^{∑_{0≤i≤n} m_{i}σ^{i}} ∈ I, some m_{0}, m_{1}, …, m_{n} ≥ 0\}
\sqrt{I} = \{a ∈ R : a^{n} ∈ I, some n ∈ ℕ\}

An ideal of R is perfect if xσ(x) ∈ I ⇒ x, σ(x) ∈ I.

Lemma 2.3 (well-mixed1) Let R be a well-mixed ring. Then \sqrt{0} is a radical, well-mixed ideal. ^{σ} \sqrt{0} coincides with I = \{a ∈ R : a^{mσ^{i}} = 0 for some m ≥ 1, i ≥ 0\} = \{a ∈ R : a^{σ^{i}} ∈ \sqrt{0} for some i ≥ 0\} and is a perfect ideal.

Proof The statement regarding \sqrt{0} is left to the reader. If a^{mσ^{i}} = 0, and m ≤ m′, i ≤ i′, then a^{m′σ^{i′}} = 0. It follows that I is closed under σ and under multiplication, and addition (if a^{mσ^{i}} = 0 and b^{mσ^{i}} = 0 then (a + b)^{2mσ^{i}} = 0.) Moreover if aa^{σ} ∈ I, then a^{(σ+1)mσ^{i}} = 0. So bb^{σ} = 0 where b = a^{mσ^{i}}. As R is well-mixed, b^{σ}b^{σ} = 0. Thus a^{2mσ^{i+1}} = 0, so a ∈ I. □

Lemma 2.4 (well-mixed1.4) Let R be a difference ring. Any intersection of well-mixed ideals is well-mixed.

Proof : Clear.

We write Ann(a/I) for {b ∈ R : ab ∈ I}, Ann(a) = {b ∈ R : ab = 0}.

Lemma 2.5 (well-mixed1.5) Let R be a well-mixed ring, a ∈ R, I(a) the smallest wellmixed ideal containing a. Then Ann(a) is a well-mixed ideal. If b ∈ I(a) ∩ Ann(a), then b^{2} = 0.

Proof Let b ∈ Ann(a). Then σ(b) ∈ Ann(a) since R is well-mixed. Thus Ann(a) is a difference ideal. Suppose cb ∈ Ann(a). Then abc = 0. So abc^{σ} = 0, as R is well-mixed. Thus bc^{σ} ∈ Ann(a). So Ann(a) is well-mixed.

Let c ∈ Ann(a). Then a ∈ Ann(c). Since Ann(c) is an well-mixed ideal, I(a) ⊂ Ann(c). Thus if at the same time c ∈ I(a), we have c ∈ Ann(c), so c^{2} = 0.

Lemma 2.6 (well-mixed1.6) Let R be a well-mixed domain, a ∈ R, J(a) the smallest algebraically radical, well-mixed ideal containing a. Then Ann(a) is an algebraically radical, well-mixed ideal; and J(a) ∩ Ann(a) = (0).

Proof In 2.5 it was shown that Ann(a) is a well-mixed ideal. If b^{m} ∈ Ann(a), then b^{m}a = 0, so (ba)^{m} = 0, so ba = 0. Thus Ann(a) is algebraically radical. Let c ∈ Ann(a). Then a ∈ Ann(c). Since Ann(c) is a well-mixed, algebraically radical ideal, J(a) ⊂ Ann(c). Thus if at the same time c ∈ I(a), we have c ∈ Ann(c), so c^{2} = 0, hence c = 0.

Let Spec^{σ} R denote the set of prime ideals P of R with σ^{−1}(P) = P. Later, we will give this set more structure.

Lemma 2.7 (well-mixed1.7) Let R be a well-mixed ring, 0 ≠ a ∈ R. For p ∈ Spec^{σ} R, let a_{p} denote the image of a in the local ring R_{p}. Then {p ∈ Spec^{σ} R : a_{p} ≠ 0} is a nonempty closed subset of Spec^{σ} R.

Proof By 2.6, Ann(a) is a well-mixed ideal. Let p be a maximal well-mixed ideal containing Ann(a). Then p is a prime ideal. For if cd ∈ p, then p ⊂ Ann(c/p), so p = Ann(c/p) (and then d ∈ p) or R = Ann(c/p) (and then c ∈ p). Moreover as p is well-mixed, it is a difference ideal, so p ⊂ σ^{−1}(p); again by maximality of p, p = σ^{−1}(p). Thus p is a transformally prime ideal. If d ∉ p, then ad ≠ 0 since d ∉ Ann(a). So a_{p} ≠ 0. □

Lemma 2.8 (well-mixed1.8) Let R be a well-mixed difference ring, S a subring.

1. Let b ∈ R, q = Ann(b) ∩ S. If a ∈ q, σ(a) ∈ S then σ(a) ∈ q. Same for Ann(b/\sqrt{0}) ∩ S.

2. If S is Noetherian and q is a minimal prime of S, the same conclusion holds.

3. Call an ideal p of R cofinally minimal if for any finite F ⊂ R there exists a Noetherian subring S of R with F ⊂ S and with p ∩ S a minimal prime of S. Then any cofinally minimal ideal is a difference ideal.

Proof

1. By 2.3, 2.5, Ann(b) and Ann(b/\sqrt{0}) are difference ideals of R. The statement about the intersection with S is therefore obvious.

2. Let S be a Noetherian subring of R. Then \sqrt{0}_{S} = q_{1} ∩ … ∩ q_{l} for some minimal primes q_{1} ∩ … ∩ q_{l}; and q = q_{i} for some i, say q = q_{1}. Let b_{i} ∈ q_{i}, b_{i} ∉ q for i > 1, and let b = b_{2} · . . . · b_{l}. Then q = Ann(b/\sqrt{0}) ∩ S.

3. Let p be cofinally minimal, a ∈ p. Let F = {a, σ(a)}, and let S be a Noetherian subring

of R containing F with p ∩ S a minimal prime of S. By (2), σ(a) ∈ (p ∩ S), so σ(a) ∈ p. □

Remark 2.9 (well-mixed1.8R) Let k be a Noetherian commutative ring, R be a countably generated k-algebra. Then cofinally minimal ideals exist. Indeed if S is a finitely generated k-subalgebra of R, p_{S} any minimal prime ideal of S, then p_{S} extends to a cofinally minimal p with p_{S} = p ∩ S.

Proof Find a sequence S = S_{1} ⊂ S_{2} ⊂ … of finitely generated k-subalgebras of R, with R = ∪_{n}S_{n}. Find inductively minimal prime ideals p_{n} of S_{n} with p_{n+1} ∩ S_{n} = p_{n}, p_{1} = p_{S}; then let p = ∪_{n}p_{n}. Given p_{n}, we must find a minimal prime p_{n+1} of S_{n+1} with p_{n+1}∩S_{n} = p_{n}. Let r_{1}, …, r_{k} be the minimal primes of S_{n+1}; then ∩_{i}r_{i} is a nil ideal; so ∩_{i}(r_{i} ∩ S_{n}) is a nil ideal; thus ∩_{i}(r_{i} ∩ S_{n}) ⊂ p_{n}. As p_{n} is prime, for some i, r_{i} ∩ S_{n} ⊂ p_{n}; as p_{n} is minimal, r_{i} ∩ S_{n} = p_{n}. □

Lemma 2.10 (wm1.11) Let R be a difference ring, I = \sqrt{I} a well-mixed ideal. Then I is the intersection of algebraically prime difference ideals.

Proof We may assume I = 0. If a ≠ 0, we must find an algebraically prime difference ideal q with a ∉ q. By compactness (see below), we may assume here that R is finitely generated. Find using 2.9 a cofinally minimal prime q with a ∉ q. By 2.8 it is an algebraically prime difference ideal. (Here is the model theoretic compactness argument. Assume for each finitely generated diff. subring S of R we are given an alg. prime diff. ideal q_{S} with a ∉ q_{s}. One can then choose an ultrafilter on the set of such subrings, concentrating for each finite F on {S : F ⊆ S}. Then R embeds into the ultraproduct of the S. But they come with an alg. prime diff. ideal q = the ultraproduct of the q_{S}, and the pullback is as required.) □

Lemma 2.11 (wm1.12) Let R be a well-mixed ring, a ∈ R. Then a ∈ p for all p ∈ Spec^{σ} R iff a^{n} = 0, some n ∈ ℕ[σ].

Proof One direction is immediate. For the other, assume a^{n} ≠ 0 for n ∈ ℕ[σ]. Let I be a maximal well-mixed ideal with a^{n} ∉ I, n ∈ ℕ[σ]. By maximality, and using Lemma 2.3, σ \sqrt{I} = I. So I = ∩p_{j}, p_{j} transformally prime ideals. Since a ∉ I, a ∉ p_{j} for some j. □

Difference polynomial rings Let R be a difference ring. A difference monomial over R is an expression rX^{ν}, where ν = ∑_{i=0}^{m} m_{i}σ^{i}, m_{i} ∈ ℕ. The order of the monomial is the highest i with m_{i} ≠ 0. A difference polynomial in one variable X (of order ≤ M) is a formal sum of difference monomials over R (of order ≤ M.) The difference polynomials form a difference R-algebra R[X]_{σ}. If k is a field, k(X)_{σ} denotes the field of fractions of k[X]_{σ}.

Transformal derivatives Let F ∈ K[X]_{σ} be a difference polynomial in one variable. We may write F (X) = ∑_{ν} c_{ν}X^{ν}, ν ∈ ℕ[σ]; where {ν : c_{ν} ≠ 0}, the support of F, is assumed finite.

Clearly F (X + U) = ∑_{ν} F_{ν}(X)U^{ν}, where F_{ν} are certain (uniquely defined) polynomials.

Definition 2.12 (trans-der) The F_{ν} will be called the transformal derivatives of F, and denoted ∂_{ν}xF = F_{ν}.

Clearly ∂_{ν}xF = 0 for all but finitely many ν. The ∂_{ν}xF satisfy rules analogous to those written down by Hasse. In particular (as can be verified at the level of monomials.)

∂_{0}(F) = F

∂_{ν}(F + G) = ∂_{ν}xF + ∂_{ν}xG ∂_{ν}(F G) = ∑_{μ+μ′=ν} F_{μ}(X)F_{μ′}(Y) ∂_{σν}σ(F) = σ(∂_{ν}(F)) The definitions in several variables are analogous.

3 Definition of difference schemes

3.1 Localization, rings of sections

(localization)

Localization If R is a difference ring, X ⊂ R, σ(X) ⊂ X, XX ⊆ X, and 0 ∉ X, we consider the localization of R by X, and write

R[X^{−1}] = \{ \frac{a}{b} : a, b ∈ R, b ∈ X\}

It admits a natural difference ring structure, with σ( \frac{a}{b}) = \frac{σa}{σb}. (This abuses notation slightly, since R need not inject into R[X^{−1}]; \frac{a}{b} is taken to denote an element of R[X^{−1}], the ratio of the images of a, b.)

An element a ∈ R is said to be σ-nilpotent if 0 lies in the smallest X ⊆ R with a ∈ X, σ(X) ⊂ X, XX ⊆ X. If a is not σ-nilpotent, we denote R[a^{−1}]_{σ} = R[X^{−1}] for this X.

If p is a transformally prime ideal, then the localization R_{p} is defined to be

R[(R \ p)^{−1}] = \{ \frac{a}{b} : a, b ∈ R, b ∉ p\}

The difference spectrum, Spec^{σ} (R), is defined to be the set of transformally prime ideals. It is made into a topological space in the following way: a closed subset of Spec^{σ} (R) is the set of elements of Spec^{σ} (R) extending a given ideal I.

We note immediately that Spec^{σ} (R) is compact (though rarely Hasudorff.)

Recall that an ideal of R is perfect if xσ(x) ∈ I ⇒ x, σ(x) ∈ I. A perfect ideal is an intersection of transformally prime ideals. There is a bijective correspondence between closed sets in Spec^{σ} (R) and perfect ideals of R. (cf. [Cohn].)

A perfect ideal is well-mixed: modulo a perfect ideal, ab = 0 ⇒ (ab^{σ})(ab^{σ})^{σ} = 0 ⇒ ab^{σ} = 0.

We define a sheaf of difference rings on Spec^{σ} (R), called the structure sheaf and denoted O_{Spec^{σ} R}, as follows.

If U is an open subset of Spec^{σ} R, a section of O_{Spec^{σ} R}, by definition, is a function f on U such that f(p) ∈ R_{p}, and such that for any p ∈ U, for some b ∉ p and a ∈ R, f(q) = \frac{a}{b} for any q ∈ U, b ∉ q.

We let O_{Spec^{σ} R}(U) be the collection of all such functions; it is a difference ring with the pointwise operations. One verifies immediately that this gives a sheaf O_{Spec^{σ} R}.

The space Spec^{σ} (R) together with the sheaf O_{Spec^{σ} R} is called the affine difference scheme determined by R.

If h : R′ → R is a difference ring homomorphism, X = Spec^{σ} R, X′ = Spec^{σ} R′, we have a natural map h^{∗} : X → X′ of difference-ringed spaces. At the level of topological spaces, for p ∈ Spec^{σ} R, let H(p) = h^{−1}(p). This permits us to define a sheaf on X′, U ↦ O_{X}(H^{−1}(U)), and a morphism of sheaves, O_{X′}(U) → O_{X}(H^{−1}(U)), by f ↦ h^{∗}f, h^{∗}f(p) = h_{p}f(q), where q = h^{−1}(p), and h_{p} : (R′)_{q} → R_{p} is the natural difference ring homomorphism.

Note that if U is an open subset of X = Spec^{σ} R, p ∈ U, f ∈ O_{X}(U), then for any sufficiently large finitely generated difference subring R′ of R, there exists an open U′ ⊆ X′ = Spec^{σ} R′, and f′ ∈ O_{X′}(U′) such that if i : R′ → R is the inclusion, then p ∈ (i^{∗})^{−1}(U′) ⊆ U, and i^{∗}f′ = f|(i^{∗})^{−1}(U′).

Moreover, if f ∈ O_{X}(X), we can find a finitely generated R′ and f′ ∈ O_{X′}(X′) with f = i^{∗}f′. In other words, each f is determined by finite data. For each p ∈ X, find b_{p} ∉ p and a_{p} ∈ R such that for p′ ∈ Spec^{σ} X with b_{p} ∉ p, f(p′) = a_{p}/b_{p} ∈ R_{p′}. Then no prime difference ideal of R can contain all b_{p}. So the perfect ideal generated by some finite set b_{p_{1}}, …, b_{p_{n}} is improper. Also given i, j we have b_{p_{i}}a_{p_{j}} − a_{p_{i}}b_{p_{j}} = 0 in R_{q}, for any q such that b_{p_{i}}, b_{p_{j}} ∉ q. So every prime difference ideal containing A(i, j) = Ann(b_{p_{i}}a_{p_{j}} − a_{p_{i}}b_{p_{j}}) includes b_{p_{i}} or b_{p_{j}}. Thus b_{p_{i}}b_{p_{j}} lies in the perfect ideal generated by A(i, j). Some finite set F of elements of R is involved in these various generation processes. Now let R′ be a subring with each b_{p_{i}} and a_{p_{i}} in R′, and containing F. Then any prime difference ideal p′ of R′ avoids some b_{p_{i}}, and we can define f′(p′) = a_{p_{i}}/b_{p_{i}} for this i. The statement is then clear.

If Y = Spec^{σ} R, a ring of the form O_{Y} (U) will be called an affine ring.

Affine rings of global functions .

c ∈ R is a σ-unit if c belongs to no transformally prime ideal. More generally, c|^{σ}d if for every transformally prime ideal p, for some b ∉ p, c|db.

It is easy to verify that a difference ring R without zero-divisors is an affine ring iff for all c, d ∈ R, if c|^{σ}d then c|d.

Each of the two properties: well-mixed, affine, imply a property that one might call ”residually local”: if a ∈ R, and the image of a in every localization R_{p} at a transformally prime ideal vanishes, then a = 0. (For the well-mixed case, cf. 2.7.)

A difference scheme is a topological space X together with a sheaf O_{X} of well-mixed rings, locally modeled on affine difference schemes of well-mixed rings. In other words X has an covering by open sets U_{i}; and there are isomorphisms f_{i} : Spec^{σ} (R_{i}) → (U_{i}, O_{X}|U_{i}) for some family of well-mixed rings R_{i}.

A morphism of difference schemes is a morphism of locally ringed spaces, preserving the difference ring structure on the local rings.

If Y is a difference scheme, a difference scheme over Y is a difference scheme X together with a morphism f : X → Y. If Y = Spec^{σ} D, we also say that X is over D.

A pre- difference scheme is defined in the same way, but with general difference rings in place of well-mixed rings.

Gluing Assume given a space X with a covering by open sets U_{i} ; a family of well-mixed rings R_{i}, and homeomorphisms f_{i} : Spec^{σ} (R_{i}) → U_{i}; such that for any i, j, the map

f_{j}^{−1}f_{i} : f_{i}^{−1}(U_{j}) ⊂ Spec^{σ} (R_{i}) → f_{j}^{−1}(U_{i}) ⊂ Spec^{σ} (R_{j})

is induced by difference ring isomorphisms between the appropriate localizations of R_{i}, R_{j}. In this situation there is a unique difference scheme structure on X, such that f_{i} : Spec^{σ} (R_{i}) → U_{i} is an isomorphism of difference schemes for each i (with U_{i} given the open subscheme structure.)

If there are finitely many U_{i}, each of the form Spec^{σ} (R_{i}) with R_{i} a finitely generated difference ring, (or difference D-algebra) we will say that X is of finite type (over D).

Remarks and definitions.

3.1

We will usually consider difference rings R that are finitely generated over a difference field, or over ℤ, or localizations of such rings. Now by [Cohn], Chapter 3, Theorem V (p. 89), such rings have no ascending chains of perfect ideals. So their difference spectra are Noetherian as topological spaces. (Their Cantor-Bendixon rank will be < ω^{2}, not in general < ω as in the case of algebras.)

3.2

A special place is held by difference rings in positive characteristic p whose distinguished homomorphism is the Frobenius x ↦ x^{q}, q a positive power of p. Note that for such rings, σ is injective iff the ring has no nilpotents, and surjective iff it is perfect. They are always well-mixed.

3.3

Spec^{σ} (R) may be empty: e.g. R = ℚ[X]/(X^{2} − 1), σ(X) = −X.

However, this does not happen if R is well-mixed: by 2.3, R has a perfect ideal I ≠ R; by [Cohn] (or cf. 2.8) a maximal proper perfect difference ideal is prime.

3.4

Let R be a ring, σ a ring endomorphism of R. Then σ acts on Spec R, taking a prime p to σ^{∗}(p) = σ^{−1}(p). The transformally prime ideals are precisely the points of Spec R fixed by this map. (Note that though σ^{∗} is continuous on Spec R, the fixed point set is rarely closed.)

In some contexts it may be useful to consider not only points of Spec R fixed by σ, but also those with finite orbits, or even topologically recurrent orbits. (Cf. [Chatzidakis-Hrushovski-Peterzil]). For instance, when R is Noetherian, the closed subscheme of Spec^{σ} R corresponding to a difference ideal J may be empty even if J ≠ R, while this is avoided if the wider definition is taken. In this paper we take a different approach, replacing arbitrary difference ideals by well-mixed ideals.

3.5 (wm12)

Let R be a difference ring, R̄ be the ring of global sections of Spec^{σ} (R). There is a natural map i : R → R̄ It induces a map i^{∗} : Spec^{σ} R̄ → Spec^{σ} R.

There is also a natural map in the opposite direction, Spec^{σ} R → Spec^{σ} R̄:

p ↦ p^{∗} = {F ∈ R̄ : F (p) ∈ pR_{p}}

And R̄_{p^{∗}} → R_{p} is defined by : F/G ↦ F (p)/G(p)

p^{∗} is the largest prime ideal of R̄ restricting to p. For if q ∩ R = p, and F ∈ q and F (p) ∉ p, say F (p) = c/d with d ∉ p, dF ∈ q, dF (p) = c, c ∉ pR_{p}, but c ∈ q ∩ R.

The image of the map ^{∗} is a closed subset of Spec^{σ} R̄; it consists of the transformally prime ideals containing bF whenever F ∈ R^{∗} vanishes on every p ∈ Spec^{σ} R with b ∉ p. I do not know whether the additional primes are of interest (or if they exist) in general. In the well-mixed context, they do not:

Lemma 3.6 (wm13) Let R be a well-mixed ring, X = Spec^{σ} R, f ∈ O_{X}(X), p, q ∈ Spec^{σ} R. Suppose f(p) = 0 ∈ R_{p}. Then there exists b ∈ R, b ∉ p, with bf(q) = 0 ∈ R_{q}.

Proof Assume first that X is Noetherian (as a topological space). Say f(q) = \frac{c}{d} ∈ R_{q}. If ce = 0, e ∉ q, then f(q) = 0 ∈ R_{q} and we are done. If bc = 0, b ∉ p, then bf(q) = 0 ∈ R_{q} and we are done again. Thus we may assume Ann(c) ⊂ p ∩ q.

Since R is well-mixed, so is Ann(c); by 2.3, I = ^{σ} \sqrt{A}nn(c) is perfect. Thus I = p_{1}∩…∩p_{n} for some transformally prime ideals p_{i}.

If p_{i} ⊂ p ∩ q, then f(p_{i}) = 0 ∈ R_{p_{i}} (since p_{i} ⊂ p) and f(p_{i}) = c/d ∈ R_{p_{i}} (since p_{i} ⊂ q), hence c = 0 ∈ R_{p_{i}}, contradicting Ann(c) ⊂ p_{i}. Thus for each i, there exists a_{i} ∈ p_{i}, a_{i} ∉ q or else b_{i} ∈ p_{i}, b_{i} ∉ p. Let a be the product of the a_{i}, b the product of the b_{i}. So a ∉ q, b ∉ p, ab ∈ I; thus (ab)^{n} ∈ Ann(c), some n ∈ ℕ[σ]f. Replacing a, b by a^{n}, b^{n}, we obtain a ∉ q, b ∉ p, ab ∈ Ann(c). So abc = 0, hence bc = 0 ∈ R_{q}, and thus bf(q) = 0 ∈ R_{q}. □

In general, let R′ be a difference subring of R (hence also well-mixed), X′ = Spec^{σ} R′, f′ ∈ O_{X′}(X′) such that f is the pullback of f′ under the natural map X → X′ (See §3.1, end of the first paragraph.) Let p′ = p ∩ R′, q′ = q ∩ R′. By the Ritt-Raudenbsh theorem and the Noetherian case, there exists b ∈ R′, b ∉ p′, with bf′(q′) = 0 ∈ R_{q′}. It follows that b ∈ R, b ∉ p, bf(q) = 0 ∈ R_{q}. □

Lemma 3.7 (wm14) Let R be a well-mixed ring, X = Spec^{σ} R f ∈ O_{X}(X), p ∈ Spec^{σ} R. Then there exist a, b ∈ R, b ∉ p, with bf − a = 0. (I.e. bf(a) − a = 0 ∈ R_{q} for all q ∈ Spec^{σ} R.)

Proof By definition of O_{X}(X) for some a, b ∈ R, b ∉ p, we have: f(p) = \frac{a}{b} ∈ R_{p}. If bf − a satisfies the lemma, then so does f. So we may assume f(p) = 0 ∈ R_{p}. By 3.6, for each q ∈ Spec^{σ} R there exists b_{q} ∈ R, b_{q} ∉ p, with b_{q}f(q) = 0 ∈ R_{q}. But by definition of O_{X}(X) again, for some c, d, d′ with d, d′ ∉ q we have f(q′) = \frac{c}{d} ∈ R_{q′} whenever d′ ∉ q′. The equation b_{q}f(q) = 0 ∈ R_{q} means that there exists d′′ ∉ q with d′′b_{q}c = 0. So b_{q}f(q′) = 0 ∈ R_{q′} whenever dd′d′′ ∉ q′ Let U_{q} = {q′ : dd′d′′ ∉ q′}. By compactness of X, finitely many open sets U_{q} cover X; say for q_{1}, …, q_{n}. Let b = b_{q_{1}} · . . . b_{q_{n}}. Then bf(q′) = 0 ∈ R_{q′} whenever q′ ∈ X. □

If R is affine, without 0-divisors, then i : R → R̄ is an isomorphism; so X ≃ Spec^{σ} R̄. This latter statement is true more generally:

Proposition 3.8 (wm-localization) Let R be a well-mixed ring, X = Spec^{σ} R, R̄ = O_{X}(X). Then i^{∗} : Spec^{σ} R̄ → X is an isomorphism of difference schemes.

Proof By 2.7, i : R → R̄ is an embedding. Let p̄ ∈ Spec^{σ} R̄, p = p̄ ∩ R. By 3.7, there exist a, b ∈ R, b ∉ p, with bf − a = 0 ∈ R̄. So f ∈ p̄ iff a ∈ p. This shows that i^{∗} is injective, and so induces a bijection of points; similarly 3.7, 3.5 show that i induces an isomorphism R_{p} → R̄p̄. □

3.2 Some functorial constructions

3.9

Let R be a finitely generated difference ring extension of an existentially closed difference field U. The closed points of Spec^{σ} R are the kernels of the difference ring homomorphisms f : R → U; if R is provided with generators (a_{1}, ..., a_{n}), then the closed point ker f corresponds to the point (f(a_{1}), …, f(a_{n})) ∈ U^{n}. For instance if say R = U[X, σX, …], then the closed points of Spec^{σ} (R) become identified with the points of U (viewed as the affine line over U.)

In general, if X,Y are difference schemes, we define a point of X with values in Y be a morphism Y → X. The set of Y -valued points of X is denoted X(Y).

The action of σ on Spec^{σ} U induces an action on the U-valued points of R “by composition”. This should not be confused with the action of σ on Spec R. In particular, Spec^{σ} R can be viewed as the set of points of Spec R fixed by σ. But this just means we are looking at difference ideals, and has nothing to do with the fixed field of U.

This observation yields a construction of difference schemes, starting from an action on an ordinary (but non-Noetherian) scheme. I do not know whether or not every difference scheme is obtained in this way; at least it seems not to be the case via a natural functor adjoint to Fix^{σ} .

Definition 3.10 (fixed) Let Y be a scheme, not necessarily Noetherian, and let ψ : Y → Y be a morphism of schemes. Let Y^{ψ} be the subspace of the topological space underlying Y, whose set of points is {p ∈ Y : ψ(p) = p}. For an open W ⊆ Y^{ψ}, let F_{′}(W) = lim_{W ⊆U}O_{Y} (U) be the direct limit of O_{Y} (U) over all open subsets U of Y containing W. Let F be the sheafification of the presheaf F_{′}.

Notes:

(1) Y is not assumed to be Noetherian.

(2) In general, Y^{ψ} is neither open nor closed in Y.

(3) For each open W ⊆ Y^{ψ}, F_{′}(W) has a natural difference ring structure: each O_{Y} (U) is a ring, and the ring homomorphisms

ψ_{U}^{∗} : O_{Y} (U) → O_{Y} (ψ^{−1}(U))

induce first, by going to the limit on the right, maps O_{Y} (U) → lim_{W ⊆U′} O_{Y} (U′) = F_{′}(W) (since W ⊆ ψ^{−1}(U)) and secondly, taking the limit on the left, a ring homomorphism σ_{W} : F_{′}(W) → F_{′}(W).

(4) By (3), F is a sheaf of difference rings. Given an open W ⊆ Y, F_{W} denotes the restriction of F to W.

Lemma 3.11 (fixed1) (1) Y^{ψ} has a basis of open sets W such that (W, F_{W}) is an affine difference scheme.

Proof

(1) To verify that Y^{ψ} is locallly modeled on affine difference scheme, let p ∈ Y^{ψ}, and let W_{0} = Spec R be an open affine neighborhood of p in Y. Then ψ^{−1}W_{0} ∩ W_{0} contains an open affine W_{1} = Spec R[f^{−1}] of Spec R, with p ∈ W_{1}. The scheme morphism ψ restricts to ψ : W_{1} → W_{0}, corresponding to a ring homomorphism ϕ_{1} : R → R[f^{−1}]. Let R_{1} = R[f^{−1}], f_{1} = ϕ_{1}(f).

Localizing ϕ_{1} yields a homomorphism ϕ_{2} : R[f^{−1}] → R[f^{−1}, ϕ_{1}(f)^{−1}], i.e. ϕ_{2} : R_{1} →

R_{1}[f_{1}^{−1}]. Let R_{2} = R_{1}[f_{1}^{−1}], f_{2} = ϕ_{2}(f_{1}), etc. Let R′ = R[f^{−1}][f1^{−1}][f_{2}^{−1}] · · ·. The ϕ_{i}

converge to an endomorphism σ′ of R′.

On the scheme side, for n ≥ 1 let W_{n+1} = W_{n} ∩ ψ^{−1}(W_{n}). Now f is a regular function on W_{0}, and W_{1} = {q ∈ W_{0} : f ∉ q}. If θ : X → W_{0} is any morphism of schemes, it follows that θ^{∗}(f) is a regular function on X, and θ^{−1}(W_{1}) = {q : θ^{∗}(f) ∉ q}. In particular with X = W_{1}, θ = ψ|W_{1} we see that W_{2} = {q : θ^{∗}(f) ∉ q} = Spec R_{2}. Continuing this way we obtain W_{n} = Spec (R_{n}) for n ≥ 1. So ∩_{n}W_{n} = Spec (R′) as subsets of Y, and in fact the induced topology on ∩_{n}W_{n} as a subspace of Y agrees with the topology of Spec (R′)

Let W = W_{0} ∩ Y^{ψ}; then inductively, W ⊆ W_{n} for each n. So W is an open neighborhood of p in Y^{ψ}, and a subspace of ∩_{n}W_{n} = Spec (R′). But as a subspace of Spec (R′), W is identified with the prime difference ideals. Hence topologically W ≅ Spec^{σ} (R′). The isomorphism of sheaves can be seen by looking at stalks.

Most of the difference schemes we will encounter will admit projective embeddings; hence they can all be constructed as Fix^{σ} (Y) for some scheme Y.

A modification of the above approach does yield all difference schemes. Given a difference ring R, let Spec′R be the set of prime ideals of R, with the following topology: a basic open set has the form ∩_{n}σ^{−n}G, where G is a Zariski open set. Equivalently, a basic open set is the image of Spec′R′, where R′ is a localization of R by finitely many elements as a difference ring. Define a structure sheaf, and gluing; obtain a category that one might call transformation schemes. Extend the functor Fix^{σ} to this category, and obtain all difference schemes in the image (but a Noetherian difference scheme need not be the image of a Noetherian transformation scheme.)

Products, pullbacks, fiber products

3.12 Let X , Y be difference schemes, with underlying spaces X, Y and sheaves of rings O_{X}, O_{Y} . Let O_{X,p}, O_{Y,q} denote the stalks at points p, q. Let R_{p,q} = O_{X,p}⊗O_{Y,q} be the tensor product, with the natural difference ring structure. Let i_{X}, i_{Y} denote the maps (Id⊗1) : O_{X,p} → R_{p,q}, resp. (1⊗Id) : O_{Y,q} → R_{p,q}.

We define the product Z = X × Y. As a set, we let Z be the disjoint union over (p, q) ∈ X × Y of Z_{p,q} = {r ∈ Spec^{σ} R_{p,q} : (i_{X})^{−1}(r) = pO_{X,p}, (i_{Y})^{−1}(r) = qO_{Y,p}} If r ∈ Z_{p,q}, we let pr_{X}(r) = p, pr_{Y} (r) = q, pr(r) = (p, q). Let R_{r} be the localization of R_{p,q} at r.

Begin with open U ⊂ X and V ⊂ Y, and f ∈ O_{X}(U)⊗O_{Y} (V) f defines a function F on pr^{−1}(U × V); F (r) ∈ R_{r} is the image of f under the natural homomorphism O_{X}(U)⊗O_{Y} (V) → R_{r}. A function such as F will be called a basic regular function. A set such as W_{F} = {r ∈ pr^{−1}(U × V) : F (r) ∉ rR_{r}} will be called a basic open set. Topologize Z using the sets W_{F} as a basis for the topology.

Given an open W ⊂ Z, we let O_{Z}(W) be the set of functions on W that agree at a neighborhood W of each point with a quotient F′/F, where F, F′ are basic regular functions and W ⊂ W_{F}.

It is easy to check that if X = Spec^{σ} R, Y = Spec^{σ} S, then Z is isomorphic to Spec^{σ} (R⊗S). Similarly, if X , Y are difference schemes over U, we can define X ×_{U} Y. Alternatively it

can be defined as a closed difference subscheme of X × Y, see (13) below. If ambiguity can arise as to the map h : Y → U, we will write X ×_{U,h} Y.

Notation 3.13 (Xy) Let X be a difference scheme over Y. If y is a point of Y with values in a difference field L, y : Spec^{σ} L → Y, we let X_{y} = X ×_{y} Spec^{σ} L

3.14

Let R be a difference ring. A difference-module is a module A together with an additive σ : A → A, such that σ(r)(σ(a)) = σ(ra) for r ∈ R, a ∈ A. A sheaf of difference modules over a sheaf of difference rings is defined as in [Hartshorne], II.5, adding the condition that the sheaf maps respect σ. If X is a difference scheme, a sub(pre)sheaf of O_{X}, viewed as a (pre)sheaf of difference modules over itself, is called a difference ideal (pre)sheaf.

We similarly take over the definition of a quasi-coherent sheaf.

Given a difference ideal sheaf I, the stalk I_{p} of I at a point p ∈ X is a difference ideal of the local difference ring O_{X,p}. define the associated closed subscheme Y as follows. The underlying set is Y = {p ∈ X : I_{p} ≠ O_{X,p}} with the topology induced from X. Let O_{Y} (U) be the ring of maps F on U such that F (p) ∈ O_{X,p}/I_{p}, and U admits a covering by open sets U′ such that F |U′ is represented by an element of O_{X}(U′) .

Note that I can be retrieved from Y as a subscheme of X; I(U) is the kernel of the map O_{X}(U) → O_{Y} (U).

3.15

Let f : X → Y be a morphism of difference schemes, and let J be a difference ideal sheaf on Y. Then f^{∗}J defined as in [Hartshorne] II.5 is a difference ideal sheaf on X. The closed subscheme associated with f^{∗}J is the pullback of the closed subscheme associated with J.

3.16 Let f : X → Z and g : Y → Z be morphisms of difference schemes. Then we obtain a map (f, g) : X × Y → (Z × Z). Let Z′ be the diagonal closed subscheme of Z × Z (defined by the obvious equations). The pullback of Z′ via (f, g) is called the fiber product and denoted X ×_{Z} Y.

3.17

Let X be a difference scheme, W an open subset of the set of points of X. Define O_{W} to be the restriction of O_{X} to W. Then (W, O_{W}) is a difference scheme. (The question is local, so we may assume X = Spec^{σ} R; and also, since W is the union of open sets of the form {p ∈ X : a ∉ p}, we may assume W is of this form. But then (W, O_{W}) = Spec^{σ} R[a^{−1}, a^{−σ}, …].)

Difference scheme associated to an algebraic scheme For any commutative ring

R, there exists a ring homomorphism h : R → S into a difference ring S, with the universal property for such morphisms: any ring homomorphism on R into a difference ring factors uniquely as gh, with g a difference ring homomorphism. This universal ring is denoted by S = [σ]R. If D is a difference domain, the same construction in the category of D-algebras is denoted [σ]_{D}R. For instance, [σ]ℚ[X_{0}] = U[X_{0}, X_{1}, …] with σ(X_{i}) = X_{i+1}.

Let C be the category of reduced, irreducible affine schemes over k, C′ the category of reduced, irreducible affine schemes over k with a distinguished endomorphism compatible with that of k.

If k is an existentially closed difference field (a model of ACFA), the map Spec^{σ} [σ]_{k}R → Spec R induces a homeomorphism on the closed points.

This gives a functor from C to C′, adjoint to the forgetful functor C′ → C. Composing with the Spec^{σ} functor from C′ to difference schemes, we obtain a natural functor from affine varieties over D to affine difference varieties over D. Call Spec^{σ} [σ]_{D}R the difference scheme associated with Spec R/Spec D.

The functor C → C′ above does not naturally extend to a similar functor on irreducible schemes to schemes with endomorphisms. (In the affine category, the image of an open subset of a scheme may not be an open subset of the image; rather a countable intersection of open subsets.) Similarly, while irreducible affine difference schemes are the same as irreducible affine schemes with endomorphisms, the notion of gluing is different, reflecting the fact that the endomorphism is viewed as part of the algebraic structure.

These differences cancel out; the composed functor, taking an affine variety to the associated affine difference scheme, does extend to a functor on schemes. (By gluing.) For projective schemes, we will see the associated difference scheme can also be described in another way, defining the difference analog of the functor Proj.

3.3 Components

(irred)

For ordinary algebraic schemes, we have two notions, of a reduced and an irreducible scheme. For difference schemes a third type of reducibility arises.

A σ-nilpotent is an element a of a difference ring R such that some product of elements b_{i} = σ^{m(i)}(a) vanishes. R is perfectly reduced if it has no σ-nilpotents. Equivalently, 0 is a perfect ideal.

R is transformally reduced if σ(a) = 0 implies a = 0 in R.

By way of contrast, if a difference ring R, viewed simply as a ring, has no nilpotent elements (is an integral domain, has Spec R irreducible) we will say that R is algebraically reduced ( algebraically integral, algebraically irreducible.)

Lemma 3.18 (red-localize) Let R be a difference ring, S is a subset closed under multiplication and under σ. Let R[S^{−1}] be the localization. If R is well-mixed / perfectly reduced / algebraically reduced / algebraically irreducible, then so is R[S^{−1}].

Proof Straightforward verification.

Definition 3.19 A difference scheme X is irreducible if the underlying topological space is irreducible, i.e. it is not the union of two proper closed subsets. X is well-mixed / perfectly reduced/ algebraically reduced / algebraically irreducible if for any open U, the rings O_{X}(U) have the property. X is transformally integral if X is perfectly reduced and irreducible.

Notation 3.20 If X is a difference scheme, then the ideals red_{wm}(R_{p}) (smallest well-mixed ideals of the local rings) generate an ideal sheaf on the structure sheaf O_{X}; it defines a closed subscheme, the well-mixed subscheme X_{wm} of X. We may similarly defined the underlying perfectly reduced subscheme, and the somewhat thicker X_{wm,red} (underlying algebraically reduced, well-mixed subscheme.)

Every Noetherian topological space X is a union of finitely many irreducible components X_{i}. The X_{i} are defined by the following: they are closed subsets, no one contained in another, and X = ∪_{i}X_{i}.

Definition 3.21 (component)

Let X be a difference scheme, and let Z be an irreducible component of X, defined by a prime ideal sheaf P ⊂ O_{X}. Define an ideal sheaf I as follows:

I(U) = {a ∈ O_{X}(U) : ab = 0, some b ∉ P(U)}

The sub-difference scheme of X supported on Z is defined to have underlying space Z, and structure sheaf (O_{X}/I)|Z.

If Y is the sub-difference scheme of X supported on Z, then the corresponding perfectly reduced scheme will be called the perfectly reduced subscheme supported on Z; and similarly for other reduction notions (such as well-mixed, defined below.)

Definition 3.22 Let X be a difference scheme over a difference field k . We will say that a property holds of X absolutely if it holds of X ×_{Spec^{σ} k} Spec^{σ} K for any difference field extension K of k.

We attach here some lemmas on analogs of separability.

Definition 3.23 1. A difference domain D is inversive if σ_{D} is an automorphism of D.

2. Let D be a difference domain. Up to isomorphism over D, there exists a unique inversive difference domain D^{inv} containing D, and such that D^{inv} = ∪_{m}D^{σ^{−m}}. It is called the inversive hull of D.

3. Let K ⊂ L be difference fields. L is transformally separable over K if L is linearly disjoint over K from K^{inv}.

4. Recall also the classical definition of Weil: Let K ⊂ L be fields. L/K is a regular extension if L is linearly disjoint from K^{alg} over K.

Lemma 3.24 (inv-ext) Let D be a difference domain, p a transformally prime ideal of D. Then there exists a unique prime ideal p^{inv} of D^{inv} such that p^{inv} ∩ D = p. The natural map Spec^{σ} D^{inv} → Spec^{σ} D is a bijection at the level of points.

Proof If p^{inv} exists, then clearly a ∈ p^{inv} iff σ^{n}(a) ∈ p for some n. So define p′ in this way: p′ = ∪_{n∈ℕ}σ^{−n}(p). It is easy to check that p′ is prime, σ^{−1}(p′) = p′, and p′ ∩ D = p.

Lemma 3.25 Let X bean algebraically integral difference scheme over a difference field k

1. If for some algebraically closed field L ⊃ k, X⊗_{k}L is an integral domain , then X is absolutely algebraically integral.

2. Let K be the inversive hull of k. If X⊗_{k}K is transformally reduced, then X is absolutely transformally reduced.

3. If for some algebraically closed difference field L ⊃ k, X⊗_{k}L is an integral domain, then X is absolutely transformally integral.

Proof The question reduces to the case X = Spec^{σ} D, D a k-difference algebra and a domain. (1) Let D′ = D⊗_{k}K. Whether or not D′ is a domain clearly depends only on the k-algebra structures of D and K, and in particular the choice of difference structure on K is irrelevant. Moreover for any field extension L of k, we may find L′ extending K and containing a copy of L; then by algebra D⊗_{k}L′ is a domain, hence so is D⊗_{k}L.

(2) Let D′ = D⊗_{k}K. Let L′ be a difference field extending K. Let e = ∑ d_{i}⊗c_{i} be a nonzero element of D′⊗_{K}L′, with d_{i} ∈ D, c_{i} ∈ L′. We may choose the c_{i} linearly independent over K′. It follows (using inversivity) that the σ(c_{i}) are linearly independent over K′ . But then σ(e) = ∑ σ(d_{i})⊗σ(c_{i}) ≠ 0. Now if L is a difference field extension of k, the inversive hull L′ of L extends K. If 0 ≠ e ∈ D⊗_{k}L, then the image of e in D⊗_{k}L′ is nonzero, since we are dealing with vector spaces. Thus σ(e) ≠ 0 as required.

(3) Follows from (1) and (2).

Remark 3.26

Note that [σ]ℚ[\sqrt{2}] is not a domain. However let R be a k- algebra and assume R⊗_{k}k^{alg} is a domain; then Then [σ]_{k}R is also a domain. More generally, suppose D is a difference domain, subring of an algebraically closed difference field L. Let R be a D-algebra, and suppose R⊗_{D}L is a domain. Then R^{∗} = [σ]_{D}R is a domain.

Definition 3.27

Let k be a difference field. An affine difference variety over k is a transformally integral affine difference scheme of finite type over k.

4 Dimensions

(dimensions)

4.1 Total dimension and transformal dimension

Two types of dimensions are naturally associated with difference equations. If one thinks of sequences (a_{i}) with σ(a_{i}) = a_{i+1}, the transformal dimension measures, intuitively,the eventual number of degrees of freedom in choosing a_{i+1}, given the previous elements of the sequence. The total dimension measures the sum of all degrees of freedom in all stages. These correspond to transformal transcendence degree and order in [Cohn], but we need to generalize them to difference schemes (i.e. mostly from difference fields to difference rings.)

The dimensions we consider here and later will take extended natural number values (0, 1, …, ∞). We will sometimes define the dimension as the the maximal integer with some property; meaning ∞ if no maximum exists.

Since these dimensions are defined in terms of integral domains, they give the same value to a difference ring R and to R/I, where I is the smallest well-mixed ideal of R.

If k is a difference field and K is a difference field extension, a subset B = {b_{1}, …, b_{m}} of K is transformally independent over k if b_{1}, σ(b_{1}), …, b_{2}, σ(b_{2}), …, b_{m}, σ(b_{m}), … are algebraically independent over k. The size of a maximal k- transformally independent subset of K is called the transformal dimension.

Let k be a difference field, and let R be a difference k-algebra. Consider the set Ξ of triples (h, D, L), with D an algebraically integral difference k- algebra, h : R → D is a surjective homomorphism of difference k-algebras, and L is the field of fractions of D. Let Ξ′ be the set of (h, D, L) ∈ Ξ where in addition D is a difference domain. In general, L is only a field; but if (h, D, L) ∈ Ξ′, then L carries a canonical difference field structure.

The transformal dimension of R over k is the supremum over all (h, D, L) ∈ Ξ′ of the transformal dimension of L over k.

The reduced total dimension of R over k is the supremum over all (h, D, L) ∈ Ξ′ of the transcendence degree of L over k.

When R is a finitely generated difference k-algebra, the total dimension of R over k is the supremum over (h, D, L) ∈ Ξ of the transcendence degree of L over k.

Lemma 4.1 (dim-trans) Let k be a difference field, R a well-mixed difference k-algebra. Then the transformal dimension of R over k is the maximal n such that the transformal polynomial ring in n-variables k[X_{1}, …, X_{n}]_{σ} embeds into R over k.

Proof If the transformal dimension is ≥ n, let (h, D, L) ∈ Ξ′ show it; then D contains a copy of k[X_{1}, …, X_{n}]_{σ}. We have X_{i} = h(Y_{i}) for some Y_{i} ∈ R; and k[Y_{1}, …, Y_{n}]_{σ} must also be a copy of the transformal polynomial ring. Conversely, assume k[X_{1}, …, X_{n}]_{σ} ≃_{k} S ≤ R. Let I be a maximal well-mixed ideal with I ∩ S = (0) (note the ideal (0) has this property.) Claim I is an algebraically prime ideal: if ab ∈ I, b ∉ I, then a ∈ I. Proof First suppose a ∈ S. Then Ann(a/I) is a well-mixed ideal (2.5), and Ann(a/I)∩S = (0) since S is an integral domain. As b ∉ I, Ann(a/I) ≠ I. By maximality, 1 ∈ Ann(a/I), i.e. a ∈ I.

Now in general: Ann(b/I) is a well-mixed ideal. If c ∈ Ann(b/I) ∩ S, then bc ∈ I, so by the case just covered, as c ∈ S, c ∈ I; so c = 0. Thus Ann(b/I) is a well-mixed ideal meeting S trivially, and 1 ∉ Ann(b/I); so again by maximality, Ann(b/I) = I; thus a ∈ I. Claim I is transformally prime. Proof Let J = σ^{−1}(I). Then J is well-mixed (if σ(ab) ∈ I, then σ(aσ(b)) = σ(a)σ^{2}(b) ∈ I. ) Also J ∩ S = (0). As I ⊂ J, we have I = J.

Now R/I shows that the transformal dimension of R is ≥ n. □

Lemma 4.2 (dim-alg) Let k be a difference field k, R a difference k-algebra. Then R, R/rad_{wm}(R) have the same total dimension. If R is well-mixed and finitely generated, then the following numbers are equal:

1. The total dimension t.dim(R) of R.

2. The maximal transcendence degree t_{2} over k of the fraction field of a quotient of R by a prime ideal.

3. t_{3} = the maximal n such that the polynomial ring over k in n variables embeds into R.

4. t_{4} = the maximal Krull dimension of a finitely generated k-subalgebra of R.

Proof Any difference ring homomorphism of R into a difference ring without zero divisors factors through R/rad_{wm}(R), so the first point is clear. Now assume R is well-mixed, of total dimension n. Clearly n ≤ t_{2} ≤ t_{4}, while t_{4} ≤ t_{3} since one can lift a a transcendence basis from the quotient ring to a set of elements of R, necessarily independent over k. Finally, t_{3} ≤ t.dim(R): R contains a free polynomial ring R_{1} of rank t_{3}. By 2.9, there exists a cofinally minimal (prime) ideal p of R with p ∩ R_{1} = (0); by 2.8, p is a difference ideal. The quotient R/p clearly has field of fractions with transcendence degree ≥ n. □

If R′ is a finitely generated difference subalgebra of a well-mixed difference k-algebra R, by 4.2 (3), t.dim(R′) ≤ t.dim(R). In what follows, the total dimension over k of a well-mixed difference k-algebra R is defined to be the supremum over all R′ of the total dimension of R′, where R′ is a finitely generated difference k-subalgebra of R. Observe that R has the same total dimension as R/\sqrt{0}. If R is not well-mixed, define the total dimension to be that of R/J, with J = rad_{wm}(J) the smallest well-mixed ideal of B.

Lemma 4.3 (tdim-prep) Let R be a finitely generated difference k-algebra. Then t.dim(R) = sup_{p∈Spec^{σ} R} t.dim(R_{p}). Hence t.dim(R) ≥ t.dim(R[a^{−1}]_{σ}) when a ∈ R

Proof We may assume R is well-mixed. If t.dim(R_{p}) ≥ n, let h′ : R_{p} → D′ be surjective, D′ an algebraically integral difference k-algebra, L the field of fractions fo D, tr.deg._{k}(L) = n. Let h be the composition R → R_{p} → D′, and let D = h(R). Any element of D′ has the form h′(a/b) for some a, b ∈ R, b ∉ p. Since b becomes invertible in R_{p}, h(b) ≠ 0, and h′(a/b) = h(a)/h(b). Thus L is also the field of fractions of D, so t.dim(R) ≥ n. Conversely suppose t.dim(R) ≥ n, and let (h, D, L) ∈ Ξ show it. Let P = ker(h), an algebraically prime difference ideal of R. Let p = ^{σ} \sqrt{P} = ∪_{m≥1}(σ^{m})^{−1}(P). Then p ∈ Spec^{σ} R. Since R → D is surjective, and P ⊆ p, we have p = h^{−1}(q) for some transformally prime ideal q of D. If a ∉ p then h(a) ∉ q; so h extends to a surjective homomorphism R_{p} → D_{q}. Since D_{q} has the same field of fractions as D, this shows that t.dim(R_{p}) ≥ n.

Let X be a difference scheme over Spec^{σ} k. The transformal (resp. total, reduced total) dimension of X over k is the maximum transformal (total, reduced total) dimension of O_{X}(U) over k, where U is an open affine subset of X.

If R is a finitely generated well-mixed difference k-algebra, X = Spec^{σ} R, then the total dimension of R over k equals the total dimension of X. Indeed let R̄ = O_{X}(X). By Proposition 3.8, the sets of local difference k-algebras {R̄_{q} : q ∈ Spec^{σ} R̄}, {R_{p} : p ∈ X} coincide. So by Lemma 4.3, t.dim(R̄) = t.dim(R); whereas any localization (and hence the ring of global sections) has smaller or equal dimension, so it does not change the supremum.

If f : X → Y is a morphism of difference schemes, the transformal (resp. total) dimension of X over Y is the supremum of the corresponding dimension of X ×_{Y} Spec^{σ} k over Spec^{σ} k, where k is a difference field and Spec^{σ} k → Y is a k-valued point of Y. (cf. 4.10 for the coherence of these definitions, when Y = Spec^{σ} k.

Proposition 4.4 (trans-0) Let k be a difference field, and X a well-mixed difference scheme of finite type over k. The following conditions are equivalent:

1. X has finite total dimension over Spec^{σ} k

2. X has finite reduced total dimension over Spec^{σ} k

3. X has transformal dimension zero over Spec^{σ} k

Proof Evidently (1) ⇒ (2) ⇒ (3) . To prove that (3) ⇒ (1) , we may assume X = Spec^{σ} R, R a finitely generated difference k-algebra.

By 4.1, R does not contain a copy of the difference-polynomial ring in one variable k[t, t^{σ}, …].

Let r_{1}, …, r_{n} be generators for R, r_{ij} = σ^{j}(r_{i}). Let Ξ be the set of triples (h, D, L) as in the definition of total dimension. For each i, and each (h, D, L) ∈ Ξ, for some m, there exists a nontrivial polynomial relation F (h(r_{i}), …, h(r_{i,m})) = 0, F ∈ k[X_{0}, …, X_{m}].

By compactness, there exists m and finitely many F_{1}, …, F_{r} ∈ k[X_{0}, …, X_{m}] such that for all i and all (h, D, L) ∈ Ξ, for some j ≤ r, F_{j}(h(r_{i}), …, h(r_{i,m})) = 0. By the claim below, tr.deg._{k}(L) ≤ mn, finishing the proof. Claim Let k be a difference field, R = k[a, a^{σ}, …] a difference k-algebra with no zero divisors, L the field of fractions of R. Write a_{i} = σ^{i}(a), and assume F (a_{0}, …, a_{m}) = 0, 0 ≠ F ∈ k[X_{0}, …, X_{m}]. Then tr.deg._{k}(L) ≤ (m). Proof It suffices to show that a_{k} ∈ k(a_{0}, …, a_{k−1})^{a} for all k ≥ m. We have F^{σ^{k−m}}(a_{k−m}, …, a_{k}) = 0. Let l be least such that for some 0 ≠ G ∈ k[X_{0}, …, X_{l}] we have G(a_{k−l}, …, a_{k}) =

0. Then a_{k−l}, …, a_{k−1} are algebraically independent over k. (If H(a_{k−l}, …, a_{k−1}) = 0 then H^{σ}(a_{k−(l−1)}, …, a_{k}) = 0, lowering the value of l.) So a_{k} ∈ k(a_{k−l}, …, a_{k−1})^{a} ⊂ k(a_{0}, …, , a_{k−1})^{a}. □

Corollary 4.5 (trans0-r) If R is a k-difference algebra with generators a_{0}, …, a_{n}, and R is an integral domain with field of fractions L, L_{0} = k({σ^{j}(a_{i}) : i ≤ n, j ≤ d}), and tr.deg._{k}L_{0} ≤ d , then tr.deg._{k}L ≤ d.

Proof By the Claim of 4.4, we have σ^{j}(a_{i}) ∈ (L_{0})^{a} for each i and all j. □

Corollary 4.6 (total-krull-0) Let k be a difference field, R be a well-mixed difference k-algebra. Assume R is finitely generated as a k-algebra. Then the total dimension of R (as a difference k-algebra) equals the Krull dimension of R. If this dimension is 0, then dim_{k}R < ∞.

Proof As R is a finitely generated k-algebra, the equality follows follows from Lemma 4.2. A finitely generated k-algebra of Krull dimension 0 is finite dimensional over k (explicitly: let I be the nil ideal of R; I = ∩_{i=1}^{n}p_{i}, with p_{i} prime. R/p_{i} is a finite field extension of k; so dim_{k}R/p_{i} < ∞, hence also dim_{k}R/I < ∞. Now R is Noetherian, so I is finitely generated; thus I^{l}/I^{l+1} is a finite-dimensional R/I-space for each k; so dim_{k}I^{l}/I^{l+1} < ∞ for each l ≤ r. It follows that dim_{k}R < ∞.) □

Lemma 4.7 (dim-cl) Let X be a well-mixed difference scheme of finite type over a difference field k. Let W be a subscheme of X, W̄ the closure of W in X. Let Z = W̄ \ W. Then the transformal (total) dimension of Z and of W is at most that of X. If W has finite total dimension, then Z has smaller total dimension. (Hence W, W̄ have the same dimension.)

Here W is a closed subscheme of an open subscheme of X. The closure of W in X is the smallest well-mixed subscheme of X containing W. Proof The statement regarding transformal dimension is obvious; the one for total dimension follows, say, from 4.2 (4). (A similar statement is true for pro-algebraic varieties.)

Example 4.8 One cannot expect a strict inequality for transformal dimension. Consider the subschemes of 𝔸^{3} defined by xy = σ(x)z vs. x = 0, or just any 0-dimensional scheme vs. a point.

Remark 4.9 (obviousmonotonicity)

If X is a difference scheme over Y, Ȳ is a closed subscheme of Y, and X̄ is the pullback to X, then the transformal (total) dimension of X̄ over Ȳ is bounded by that of X over Y. Any closed subscheme of X will also obviously have dimension ≤ m over Y.

Lemma 4.10 (total-dim-base-change) Let k be a difference field, R a difference k-algebra, of total dimension l. Let K be a difference field extension of k. Then R⊗_{k}K has total dimension ≤ l over K.

Proof Let (h, D, L) be: an algebraically integral difference K-algebra D, a surjective homomorphism h : R⊗_{k}K → D, with L the field of fractions of D. We must show that tr.deg._{K}(L) ≤ l. Let h′ = h|R, D′ = h′(R), L′ = the field of fractions of D′ within L. Then in L, L is the field amalgam of K,L′. Thus tr.deg._{K}(L) ≤ tr.deg._{k}(L′) ≤ l. □

Remark 4.11 (total-dim-base-change-rem)

If K/k is a regular field extension, or if R/k is a contained in a regular extension, equality holds in 4.10.

In general it may not, because of possible incompatibility within k^{alg}: if k = ℚ, a ∈ R, b ∈ K with a^{2} = 2, σ(a) = a, b^{2} = 2, σ(b) = −b, then regardless of the total dimension of R, the total dimension of R⊗ℚK equals zero.

If one counts total dimension only with respect to difference domains lying within a given universal domain (model of ACFA), and this universal domain contains K, then again this dimension is base-change invariant.

Lemma 4.12 (inv-base-change) Let k be a difference field, R a difference k-algebra of total dimension l. Then:

(1) There exists a difference field k̄ extending k, with underlying field structure equal to k^{alg}, such that R⊗_{k}k̄ has total dimension l over k̄.

(2) R⊗_{k}k^{inv} has total dimension l over k^{inv}.

Proof In both cases, the total dimension is at most l by Lemma 4.10. Thus we must show that it equals at least l. We may pass to to a quotient of R demonstrating that R has total dimension ≥ l; i.e. we may assume R is an integral domain, whose field of fractions K has k-transcendence degree ≥ l.

To prove (1), let R′ be the ring generated by R and k^{alg} within K^{alg}. Then R′ is an integral extension of R. By the lemma on extending homomorphisms, the homomorphism σ : R → R ⊂ K admits an extension σ̄ : R′ → K^{alg}. But σ̄(R) ⊆ R, σ̄(k^{alg}) ⊆ k^{alg}, so σ̄(R′) ⊆ R′. Let k̄ = (k^{alg}, σ̄). Then R′ is a difference ring of total dimension l over k̄. Since R′ is a homomorphic image of R⊗_{k}k̄, the same is true of the latter.

(2) If K/k is a regular field extension, then R⊗_{k}k^{inv} ≤ K⊗_{k}k^{inv} is an integral domain, and the assertion is clear. In particular, (2) is true when k is algebraically closed. Thus t.dim(R⊗_{k}k̄)⊗k̄k̄^{inv} = t.dim(R⊗_{k}k̄) = l. But (R⊗_{k}k̄)⊗k̄k̄^{inv} = R⊗_{k}k̄^{inv} = (R⊗_{k}k^{inv})⊗_{k^{inv}} k̄^{inv}. Hence by Lemma 4.10, t.dim(R⊗_{k}k^{inv}) ≥ l.

Note that the reduced total dimension can certainly go down upon base change to K^{inv}, even if k = k^{alg}; e.g. R = k(t), K = k(σ(t)) ⊂ R.

Embedding in the transformal line A difference domain R is twisted-periodic if for some n, p,m, R ⊧ (∀x)σ^{n}(x) = x^{p^{m}}. If a subring R of a difference field K is not twistedperiodic, then R^{n} is Ritt-Cohn dense in K^{n}, i.e. every K-difference variety containing R^{n} equals K^{n} (cf. [Cohn]. Note that if every element of R satisfies a difference equation of total dimension n, then every element of K satisfies one of total dimension ≤ 2n. In one dimension, polarize.)

Below, a morphism f : X → Y of difference schemes over a field K will be said to be point- injective if for every difference field extension K′ of K, f induces an injective map X(K′) → Y (K′).

Lemma 4.13 (tpl) Let R be a difference domain, with field of fractions K; let r : R → k be a surjective homomorphism into a difference field k, with k not twisted-periodic. Assume R is a valuation ring. Let X ⊂ 𝔸^{n} be an difference scheme over R, of finite total dimension.

Then there exist c_{1}, …, c_{n} ∈ R, such that if H(x) = ∑ c_{i}x_{i}, h(x) = ∑ c̄_{i}x_{i}, then H and h are point- injective.

Proof If ∑ c_{i}x_{i} = ∑ c_{i}y_{i}, x, y ∈ X distinct, then c = (c_{1}, …, c_{n}) has transformal dimension < n over K(x, y). Thus {c : (∃x ≠ y ∈ X)c · x = c · y} has transformal dimension ≤ (n − 1) + 2trans.dim(X) = n − 1. So for any c, outside a proper difference subscheme of 𝔸^{n}, H(x) = ∑ c_{i}x_{i} is injective. □

Corollary 4.14 (fin-dim-1) Let X be a projective difference scheme of finite total dimension over a field k, with k not twisted-periodic. Then there exists a linear projection to ℙ^{1}, defined over k, point- injective on X.

Proof Say X ⊂ ℙ^{n}. The set of hyperplanes passing through some point of X is contained in a difference scheme of transformal dimension n − 1. So there exists a hyperplane defined over k and avoiding X. Thus we may assume X ⊂ 𝔸^{n}. Now 4.13 applies.

(Improve this to: X is isomorphic to a difference subscheme of ℙ^{1}.)

4.2 Dimensions of generic specializations

Definition 4.15 Let X be a difference scheme over a difference domain D with field of fractions K. A property will be said to hold generically of X if it is true of X ×_{Spec^{σ} D} Spec^{σ} K.

For example, if R is a difference D-algebra, the generic transformal (total) dimension of R over D is the transformal (total) dimension of R⊗_{D}K over K.

Lemma 4.16 (5.11) Let D be a difference domain, R a finitely generated difference D-algebra, of generic transformal (total) dimension d over D. Then for some dense open U ⊂ Spec^{σ} (D), for all p ∈ U, if h : D → K is a map into a difference field with kernel p, then R⊗_{D}K has transformal (total) dimension ≤ d over K.

Proof It is easy to give an effective argument (cf. 9.8.) We give a qualitative one here. Suppose the lemma is false; then there exist difference ring homomorphisms h_{i} : R → D_{i} ⊂ L_{i} such that h_{i}(R) = D_{i} is a domain with field of fractions L_{i}, h_{i}(D) has field of fractions K_{i}, and L_{i} is a difference field of transformal dimension > d over K_{i}, (respectively, tr.deg._{K_{i}}L_{i} > d); and such that p_{i} = ker(h_{i}|D) approaches the generic point of D. Let F be a finite set of generators of R as a D-algebra. Then for each i, for some r_{0}, …, r_{d} ∈ F (respectively: F ∪ … ∪ σ^{d}(F)) the images h_{i}(r_{0}), …, h_{i}(r_{d}) are transformally independent over K_{i}. (In the total dimension case, use 4.5.) We may assume it is always the same sequence r_{0}, …, r_{d} (by refining the limit.) Taking an ultraproduct we obtain h_{∗} : R → D_{∗} ⊂ L_{∗}, injective on D, with the analogs of the above properties; in particular h_{∗}(r_{0}), …, h_{∗}(r_{d}) are transformally (resp. algebraically) independent over K_{∗}, contradicting the assumption on generic dimension. □

Remark 4.17

(1) One can find a difference domain D′, D ⊂ D′, D′ a finite integral extension of D, and a dense open U ⊂ Spec^{σ} (D), such that for all p ∈ U, if h : D → L is a map into a difference field with kernel p, and h extends to a difference ring homomorphism h′ : D′ → L, then R⊗_{D}L has transformal dimension precisely d over L. The proof of this fact, that we will not require, uses 4.43 and 4.36 below.

(2) The base extension D′ in (1) cannot be avoided:

for instance let D = ℤ, R = D[X, σ(X), …]/(X^{2} + σ(X)^{2}, σ^{2}(X) − X). Then R has transformal dimension 1 over the field of fractions of D. But if p is any rational prime with p = 1 mod 4, and h : R → L is a map into a difference field of characteristic p, then the fixed field of (L, σ) has an element i with i^{2} = −1, so h(X) = ±ih(σX). Applying σ, h(σX) = ±ih(σ^{2}X) so hσ^{2}(X) = −hX, yielding X = 0. It follows that R⊗_{D}(Z/pZ) has transformal dimension 0.

Lemma 4.18 (wta) Let D be a difference domain, with field of fractions of D of transcendence degree ≥ k over a difference field k. Let R be a D-difference algebra, of generic total dimension m. Then R has total dimension ≥ k + m over k.

Proof There exists a surjective homomorphism of transformal K-algebras h : R⊗_{D}K → D′, D′ has field of fractions K′, tr.deg._{K}K′ = m. So tr.deg_{k}K′ ≥ k + m, and h(R) generates K′ as a field. □

4.3 Difference schemes and pro-algebraic varieties

(pro-alg-sec)

Let X be a difference subscheme of an algebraic variety V over a difference field k; more precisely, of the difference scheme [σ]_{k}V. Let V_{n} = V × … × … × V^{σ^{n}}. (So V = V_{0}.)

We associate with X a sequence of algebraic subschemes X[n] of V_{n}; X[n] will be called the n’th -order weak Zariski closure of X. We describe these locally on V. Let U be an open affine subset of V. So U can be identified with Spec A, with A = O_{V} (U), a k-algebra. The inclusion X → [σ]_{k}V gives a k-algebra homomorphism [σ]_{k}A → X (U), with kernel I (so I is a difference ideal of Spec^{σ} [σ]_{k}A, and X ∩ U ≃ Spec^{σ} ([σ]_{k}A)/I.) Let A_{n} be the sub k-algebra of A generated by A ∪ … ∪ σ^{n}(A); so Spec A_{n} = U × … × … × U^{σ^{n}}. Finally define (X ∩ U)_{n} = Spec A_{n}/(I ∩ A_{n}).

We also let X_{ω} be the projective limit of the X[n]; it can be viewed as a scheme, or as a pro-(scheme of finite type). In particular, [σ]_{k}V_{ω} = Π_{n≥0}V^{σ^{n}}. Note that [σ]_{k}V_{ω} is isomorphic to the scheme Π_{n≥1}V^{σ^{n}} (by an isomorphism intertwining with σ : k → σ(k)). At the same time we have the projection r : Π_{n≥0}V^{σ^{n}} → Π_{n≥1}V^{σ^{n}}.

Remark 4.19 (pro-alg) A subscheme Y of Π_{n≥0}V^{σ^{n}} has the form X_{ω} for some difference subscheme X of [σ]_{k}V iff Y contains rY^{σ^{−1}} as a scheme

Proof Locally, Y is defined by an ideal I. I is a difference ideal iff I ⊂ σ^{−1}(I ∩ R^{σ}).

Definition 4.20 X is weakly Zariski dense in V if the 0-th-order weak Zariski closure of X equals V.

We will also say that X is Zariski dense in V if for some difference field extension K of k, X(K) is Zariski dense in V. (It is safer to use this only for k = k^{alg} and X/k irreducible.)

It is quite possible that X be weakly Zariski dense in V, while the set of points of X is not Zariski dense in V. For instance, when V = 𝔸^{1}, the subscheme X cut out by σ(x) = 0 has this property.

If X is algebraically integral, each X[n] is an irreducible algebraic variety over k, of dimension ≤ ndim V. The natural map X[n + 1] → X[n] is dominant.

4.4 Transformal degree

Let notation be as above: k is a difference field, X[n] is the n’th-order weak Zariski closure of X.

Lemma and Definition 4.21 (dimgrowth-int) Assume X is an algebraically integral difference subscheme of an algebraic variety V over k. There exist integers a, b such that for all sufficiently large n ∈ ℕ,

dim (X[n]) = a(n + 1) + b

a is the transformal dimension of X, while b is called the dimension growth degree or transformal degree. If a = 0 then b is the total dimension of X.

Proof One proof uses intersections with generic hyperplanes (a Bertini argument.) We can assume V is projective. We define the notion of a generic hyperplane section X′; it is the intersection of X with a linear equation, whose coefficients are generic in the transformal sense. Show that if X has positive transformal dimension, then dim X′[n] = dim X[n] − 1 for large n. In this way reduce to the case of transformal dimension 0, where we must show that dim (X_{n}) is bounded (hence eventually constant), with eventual value equal to the total dimension. This is clear by Proposition 4.4 and Lemma 4.2.

Here is a more direct proof. Let (c_{0}, c_{1}, …) be such that (c_{0}, …, c_{n}) is a generic point over k of X[n], in some field extension of k. (This makes sense since X is algebraically integral; so X[n] is an irreducible variety over k; and X[n + 1] projects dominantly to X[n].) So (c_{0}, …, c_{n}) → (c_{l}, …, c_{n+l}) is a (Weil) specialization over k, and thus (with k[l, n] = k(c_{l}, …, c_{n+l}))

a(l, n) = tr.deg._{k}k[l, n] is non-increasing with l. So for each n, for some β(n), for l ≥ β(n), a(l, n) = a(l + 1, n) = … =_{def} e(n). Take the least possible value of β(n + 1), subject to β(n + 1) ≥ β(n).

Now ϵ(n) = e(n + 1) − e(n) is non-increasing: for large enough l, e(n + 1) − e(n) = tr.deg._{k[l,n]}k[l, n + 1]; so e(n + 2) − e(n + 1) = tr.deg._{k[l,n+1]}k[l, n + 2] ≤ tr.deg._{k[l+1,n]}k[l + 1, n + 1] = e(n + 1) − e(n)

Thus ϵ(n) is eventually constant, with value a. So for all n ≥ γ, for all l ≥ β(n), a(l, n) = α(n) + an for some α(n).

Note that β(n) = β(n + 1) unless e(n) > e(n + 1). Thus β(n) also stabilizes at some maximal β.

It follows that for l ≥ β, n ≥ γ, a(l, n) = α + an.

Finally, tr.deg._{k(c_{β},c_{β+1},...)}k(c_{0}, c_{1}, …) = δ = tr.deg._{k}(c_{β}, c_{β+1}, …, c_{L})k(c_{0}, …, c_{L}), for some sufficiently large L ≥ β + γ.

So for n ≥ L, tr.deg._{k}k(c_{0}, …, c_{n}) = α + δ + an.

For the identification of a with the transformal dimension, we argue as follows.

Let R be an integral domain, and a finitely generated difference k-algebra. If B is a finite set of generators of R, let R[n; B] be the k-algebra generated by B ∪ … ∪ σ^{n}(B), and let a(n, R, B) = dim_{k}R, the Krull dimension of R[n; B]. We have seen that lim_{n→∞}(1/n)a(n, R, B) exists; denote this a(R, B).

We will use two statements about the dimension of finitely generated k-algebras S, S′ without zero-divisors, with fields of fractions F, F′:

(*) If S ≤ S′ then dim_{k}(S) ≤ dim_{k}(S′).

(**) If in addition F′ is algebraic over F, then dim_{k}(S) = dim_{k}(S′).

(In terms of varieties, if V′ → V is dominant then dim (V) ≤ dim (V′), and if in addition V′ → V is generically finite, then equality holds.)

Now:

1) If R ⊆ R′, and B ⊆ B′, then R[n; B] ⊆ R′[n; B′]. By (*), a(n, R, B) ≤ a(n, R′, B′) and in particular a(R, B) ≤ a(R′, B′).

2) If B ⊆ B′ ⊆ R (B′ finite), then B′ ⊆ R[k; B] for some k, σ^{l}(B′) ⊆ R[k + l, B], and so R[n; B] ⊆ R[n; B′] ⊆ R[n + k; B]. Thus a(n, R, B) ≤ a(n, R, B′) ≤ a(n + k, R, B). So a(R, B) = a(R, B′), and a(R, B) does not depend on the choice of finite generating set; we can write a(R) for this.

3) Let D be a difference domain, and a finitely generated difference k-algebra. Let R be a finitely generated difference D-algebra, not embedding D[X]_{σ} (the difference polynomial ring in one variable over D.) Then a(D) = a(R). Indeed, a(D) ≤ a(R) by (1). For the other inequality let F be the field of fractions of D; it is a difference field. Let B_{0} be a finite set of generators for R as a difference D-algebra. For b ∈ B_{0}, for some n = n(b), σ^{n}(b) ∈ F (b, …, σ^{n−1}(b))^{alg} (otherwise the homomorphism D[X]_{σ} → R, X ↦ b, is an embedding.) Let B_{1} = {σ^{i}(b) : b ∈ B_{0}, i < n(b)}. Then B_{1} is a finite set of generators for R as D-algebra, and σ(B_{1}) ⊆ F (B_{1})^{alg}. Let B′ be a finite subset of D contains the coefficients of the relevant polynomials, so that if F′ is the field of fractions of the k-algebra generated by B′, then σ(B_{1}) ⊆ F′(B_{1})^{alg}). Finally let B = B_{1} ∪ B′. Then the field of fractions of R[n, B] is algebraic over the field of fractions of D[n, B′][B_{1}]. (For n = 1 by definition; for n+1, since σ^{n+1}(B) = σ^{n+1}(B_{1}) ∪ σ^{n+1}(B′), and σ^{n+1}(B_{1}) ⊆ σ^{n}(F′(B_{1})^{alg}), and induction.) Thus by (**), a(n, R, B) = dim_{k}D[n, B′][B_{1}] ≤ a(n, D, B′) + e where e is the number of points in B_{1}. Since e is constant, e/n →_{n→∞} 0, and a(R) = a(R, B) = a(D, B′) = a(D).

Now assume R is well-mixed, and use Lemma 4.2; Applying (3) to D = k[X_{1}, …, X_{t}]_{σ} with t the transformal dimension of R, we see that a(R) = s. □

Example 4.22 (growth-exmpl) The eventual dimension growth formula of 4.21 need not hold for all n; the differences dim (X[n + 1]) − dim (X[n]) need not be monotone.

Let D = ℚ[z]_{σ}, R = D[x] with σ(x) = 0, and let y = xz^{σ}. Let R[0] = ℚ[x, y, z] ≤ R, V = 𝔸^{3} = Spec R[0], X = Spec^{σ} R embedded into V dually to R[0] → R. Then X[n] = Spec ℚ[x, y, z, z^{σ}, …, z^{σ^{n}}]; letting a_{n} = dim X[n] we have (a_{0}, a_{1}, a_{2}, a_{3}, …) = (3, 3, 4, 5, …). Taking the product of two copies of this with one (such as σ^{3}(t) = t) of growth (1, 2, 3, 3, ...), we obtain the dimension growth sequence (7, 8, 11, 13, 15, ...) with differences (1, 3, 2, 2, ...).

Remark 4.23 (dimdeg) Assume X ⊂ Y are two algebraically integral difference subschemes of the algebraic variety V. If X, Y have the same transformal dimension and dimension growth degree, then X = Y.

Proof In fact, for every sufficiently large n, X[n] ⊂ Y [n], and dim (X[n]) = dim (Y [n]); so X[n] = Y [n]; and so X = Y. □

Corollary 4.24 (rm1) Let k be a difference field, D an algebraically integral finitely generated k-difference algebra. Then for some n, σ^{n}(D) is a difference domain.

In fact if R is a finitely generated k-subalgebra of D, generating D as a difference ring, and b is the dimension growth degree of Spec^{σ} (D) as a difference subscheme of Spec (R), then one can take n ≤ b.

Proof Let a be the transformal dimension of Spec^{σ} D over k; this equals the transformal dimension of Spec^{σ} σ^{n}(D) over σ^{n}(k), for any n. Let b_{n} be the dimension growth degree of Spec^{σ} σ^{n}(D) as a difference subscheme of Spec σ^{n}(R), a variety over σ^{n}(k). In other words, let R_{n,m} be the subring of D generated by σ^{n}(R) ∪ … ∪ σ^{n+m−1}(R), K_{n,m} the field of fractions of R_{n,m}, c(n, m) the transcendence degree of K_{n,m} over σ^{n}(k). Then for any n, for large enough m, c(n, m) = am + b_{n}. We have b = b_{0} ≥ b_{1} ≥ …, so b_{i} = b_{i+1} for some i ≤ b. The lemma now follows by applying the following claim to σ^{i}(D). Claim If b = b_{1}, then D is a difference domain. Proof Pick any c ∈ D, c ≠ 0, in order to show that σ(c) ≠ 0. Let m be large enough so that c ∈ R_{0,m}, that c(0, m) = am + b and c(1, m) = am + b_{1}. As b = b_{1}, c(0, m) = c(1, m), i.e. tr.deg._{k}K_{0,m} = tr.deg._{σ(k)}K_{1,m}. Thus by Krull’s theorem, the Krull dimensions of R_{0,m} and of R_{1,m} are equal. Now the surjective homomorphism σ : R_{0,m} → R_{1,m} has a prime ideal p for kernel (as R_{1,m} is an integral domain.) . Thus R_{0,m}/p ≅ R_{1,m} has the same Krull dimension as R_{0,m}. As 0 is also a a prime ideal of R_{0,m}, this forces 0 = p. So c ∉ p, and thus σ(c) ≠ 0. □

4.5 Finite presentation

We will need a certain uniform version of Hilbert’s basis theorem; for lack of reference we give the proof. Let S be a family of subrings of some commutative ring S (with S ∈ S). Call S uniformly Noetherian if for any ideal I of S, for some finite F ⊆ S, for any R ∈ S containing F, F ∩ I generates I ∩ R as an ideal of R.

Lemma 4.25 (hilbert) Let K be a field, or K = ℤ, n ∈ ℕ, and let S(K, n) be the family of polynomial rings R[X_{1}, …, X_{n}], with R a subring of K. Then S(K, n) is uniformly Noetherian.

Proof Note that the lemma is true for n = 0 ( for a field K, S(K, 0) is the family of subrings of K; one can always take F = {0, 1}.) Thus it suffices to show that if S is a uniformly Noetherian family of subrings of a ring S, then S[X] := {R[X] : R ∈ S} is a uniformly Noetherian family of subrings of S[X]. The usual proof of the Hilbert basis theorem shows this: Let I be an ideal of S[X], and let J_{k} = {a ∈ S : (∃f)(deg(f) < k, aX^{k} + f ∈ I}, an ideal of S. We have J_{0} ⊆ J_{1} ⊆ … J_{m} = J_{m+1} = … for some m. Let F_{k} be a uniform set of generators for J_{k}, i.e. F_{k} ∩J_{k} generates J_{k} ∩R for any R ∈ S containing F_{k}. For each a ∈ F_{k}, find f = f_{a}^{k} ∈ I of the form aX^{k} + g, deg(g) < k. Let F be the union of all f_{a}, with a ∈ F_{k} , for k ≤ m. So F is finite. Let R ∈ S with F ⊆ R[X]. Then all coefficients of all f ∈ F are in R; in particular each F_{k} ⊆ R. Hence F_{k} ∩ J_{k} generates J_{k} ∩ R for k ≤ m. Let I′ be the ideal of R[X] generated by F ∩ I. To show that I′ = I ∩ R[X], let h ∈ I ∩ R[X], deg(h) = l; use induction on l. Write h = aX^{l} + g, deg(g) < l. Then a ∈ J_{l}, so a ∈ J_{k} with k = min(l, m). By assumption, a = ∑ r_{i}a_{i} for some r_{i} ∈ R and a_{i} ∈ F_{k}. So h′ = h − X^{l−k}(∑ r_{i}f_{a_{i}}^{k}) has degree < l. By induction h′ ∈ I′; so h ∈ I′.

Corollary 4.26 (hilbert-c) Let D be an integral domain, R a finitely generated D-algebra. Then for some a ∈ D, a ≠ 0, R[a^{−1}] is a finitely presented D[a^{−1}]-algebra.

In fact, the presentation is uniform over all sufficiently large localizations of subrings of D, in the following sense. Let K be the field of fractions of D, and let h : D[X] → R be a surjective homomorphism of D-algebras, D[X] a polynomial ring; extend naturally to h_{K} : K[X] → R_{K} = R⊗_{D}K. Then for some finitely generated R_{0} ⊆ D and 0 ≠ a ∈ R_{0}, for some finite F_{0} ⊆ R_{0}[a^{−1}][X], for any subring R′ of K with R_{0}[a^{−1}] ⊆ R′, the restriction of h_{K} to R′[X] has kernel generated by F_{0}.

Proof Let F be a uniform set of generators for ker h_{K} in the sense of Lemma 4.25; so that F ∩ ker h_{K} generates R′[X] ∩ ker h_{K} whenever F ⊆ R′ ⊆ K. Find R_{0}, a such that F ⊆ R_{0}[a^{−1}]; let F_{0} = F ∩ ker h_{K} ⊆ R_{0}[a^{−1}]. If R′ contains R_{0}[a^{−1}], let h′ = h_{K}|R′. Then h′ vanishes on F_{0}. Conversely, if f ∈ ker h′ then f ∈ ker h_{K} ∩ R′[X], so f ∈ R′[X]F_{0}. So ker h′ is generated by F_{0}.

Corollary 4.27 (hilbert-c-c) Let D, R be as in 4.26, and let I be an ideal of R. Then for some d ∈ D, d ≠ 0, and some finitely generated I_{0} ⊆ I, for any r ∈ I, for some N ∈ ℕ, rd^{N} ∈ I_{0}.

Proof Applying 4.26 in R/I, we find d, I_{0} such that IR[d^{−1}] = I_{0}R[d^{−1}]. But I_{0}R[d^{−1}] = {a/d^{n} : a ∈ I_{0}}. So for any a ∈ I, for some b ∈ I_{0}, a = b/d^{n}; ad^{n} − b = 0; ad^{n+m} − bd^{m} = 0; ad^{N} ∈ I_{0}. □

Corollary 4.28 (noether-1) Let k be a difference field, or ℤ. D be a difference domain, finitely generated as a difference k-algebra. Then there exists 0 ≠ d ∈ D such that D[d^{−1}]_{σ} is finitely presented as a difference k-algebra.

Proof Let K be the field of fractions of D, K^{inv} the inversive hull of K. Let a_{1}, …, a_{r} be generators of D, a = (a_{1}, …, a_{r}). In this proof, for Y ⊆ K^{inv}, write k[Y] for the k-algebra of K^{inv} generated by Y, k[Y]_{σ} = k[Y, σ(Y), …]. Let D^{−} = k[{σ^{n}(a) : n < 0}]. By Corollary 4.26, there exists 0 ≠ e ∈ D^{−} and a finite f ⊂ D^{−}[e^{−1}][X] such that the natural map induces an isomorphism

(*) D^{−}[e^{−1}, a] ≅ D^{−}[e^{−1}][X]/(f),

and indeed R′[Y^{−1}][a] ≅ R′[Y^{−1}][X]/(f) whenever R ⊆ R′ ⊆ D and e ∈ Y ⊆ R′ \ (0).

The element e as well as all coefficients of the polynomials in f lie in k[σ^{−1}(a), …, σ^{−l}(a)] for some l. Let d = σ^{l}(e). For n ≥ 0, let R_{−n} = k[a^{σ^{−1}}, …, a^{σ^{−l−n}}, e^{−1}, …, σ^{−n}(e^{−1})]. By (*),

R_{−n}[a] ≅ R_{−n}[X]/(f)

Note that σ(e) ∈ R_{−l}[a]; say σ(e) = h(a) with h ∈ R_{−l}[T]. Let c = (σ(e))^{−1}, and let g = (f, Y h(X) − 1). Then

(∗∗) R_{−n}[a][c] ≅ R_{−n}[X, Y]/(g)

Let

R_{n} = σ^{l+n}(R_{−n}) = k[a, …, a^{σ^{l+n−1}}, d^{−1}, …, σ^{n}(d)^{−1}]

Applying σ^{n+l} to (**), we see that R_{n+1} ≅ R_{n}[X, Y]/(σ^{n+l}(g)) naturally. Let H : k[X, Y]_{σ} → D[d^{−1}]_{σ} be the difference ring homomorphism with X ↦ a, Y ↦ d^{−1}. Then the kernel of H is generated by the kernel of H|k[X, Y, …, X^{σ^{l}}, Y^{σ^{l}}] together with σ^{l}(g), σ(σ^{l}(g)), . . .. As a difference ideal, this is finitely generated. □

Corollary 4.29 (hilbert-c-1) Let R be a finitely generated difference k-algebra, with k a difference field or ℤ. Let P be a transformally prime ideal of R. Then there exists a ∉ P such that with R̄ = R[a^{−1}]_{σ}, P R̄ is finitely generated as a difference ideal in R̄.

Proof This follows from 4.28, applied to R/P. □

Definition 4.30 (radicial) Let f : R → S be a morphism of difference rings. S is transformally radicial over R if for any s ∈ S, for some m ≥ 0, s^{σ^{m}} ∈ f(R).

Lemma 4.31 (rad-fg) If S is a finitely generated R-difference algebra and is transformally radicial over R, then S is a finitely generated R-algebra.

If in addition R is a difference domain, then for some 0 ≠ a ∈ R, S[a^{−1}]_{σ} is a finitely presented R[a^{−1}]_{σ}-algebra, hence a finitely presented R[a^{−1}]_{σ}- difference algebra.

Proof Let H be a finite set of generators of R as a difference k-algebra. For a ∈ H, σ^{m}(a) ∈ k · 1 for some m; let T (a) = {a, σ(a), …, σ^{m−1}(a)}. Then H′ = ∪_{a∈H}T (a) is a finite set of generators of R over as a k- algebra.

Now assume R is a difference domain. By Corollary 4.26, for some 0 ≠ a ∈ R, S[a^{−1}]_{σ} is a finitely presented R[a^{−1}]_{σ}-algebra.

Now in general, if S′′ is a difference R′′-algebra, and a finitely presented R′′-algebra, then it is a finitely presented R′′-difference algebra: if S′′ = R′′[b], b = (b_{1}, …, b_{n}), then σ(b) = f(b) for some f ∈ R[X], and it suffices to add to the R′′-algebra presentation the relation σ(b) = f(b). □

Lemma 4.32 (noether-2) Let R be a finitely generated algebraically reduced, well-mixed difference k-algebra, with k a difference field or ℤ. Assume R has a unique minimal transformally prime ideal P. Then there exists a ∉ P such that R̄ = R[a^{−1}]_{σ} is a finitely presented difference k-algebra. In fact for some n, D = σ^{n}(R̄) is a difference domain, and R̄_{σ} is a finitely presented difference D-algebra.

Proof

By Corollary 4.29, for some a ∉ P, P R[a^{−1}]_{σ} is a finitely generated difference ideal. Passing from R to R[a^{−1}]_{σ}, we may assume P is finitely generated. By Lemma 2.3 ^{σ} \sqrt{0} is a perfect ideal by , hence an intersection of transformally prime ideals; by assumption on P, ^{σ} \sqrt{0} = P. Since P is finitely generated, and ^{σ} \sqrt{(}0) = ∪_{n}σ^{−n}(0), we have σ^{−n}(0) = P for some n. Thus if a ∈ P then σ^{n}(a) = 0. Claim D = σ^{n}(R) is a difference domain.

Indeed if a, b ∈ σ^{n}(R) and ab = 0, write a = σ^{n}(c), b = σ^{n}(d); then σ^{n}(cd) = 0, so c ∈ P or d ∈ P; thus a = 0 or b = 0. Similarly if σ(a) = 0 then a = 0. Thus σ^{n}(R) is a difference domain.

Now R is a finitely generated radicial D-difference algebra. By Lemma 4.28, for some 0 ≠ a ∈ D, D[a^{−1}]_{σ} is finitely presented as a difference k-algebra. By Lemma 4.31, for some 0 ≠ b ∈ D (with a|b), R[b^{−1}]_{σ} is a finitely presented difference D[b^{−1}]_{σ}-algebra. Thus R[a^{−1}]_{σ} is finitely presented as a difference k-algebra. □

Lemma 4.32 applies in particular to algebraically integral finitely generated k-algebras; since in this case ^{σ} \sqrt{0} is the unique minimal transformally prime ideal. Hence 4.29 extends to:

Corollary 4.33 (hilbert-c-2) Let R be a finitely generated difference k-algebra, with k a difference field or ℤ. Let P be an algebraically prime difference ideal of R. Then there exists a ∉ ^{σ} \sqrt{P} such that with R̄ = R[a^{−1}]_{σ}, P R̄ is finitely generated as a difference ideal in R̄. □

4.6 A chain condition

We present two variants of the Ritt-Raudenbush finite basis theorem for perfect difference ideals; one (4.34) for algebraically integral well-mixed ideals, the other (4.35) for algebraically radical ones in finite total dimension. Yves Laszlo found a mistake in an earlier proof of 4.35, stated without the finite total dimension assumption; I do not know if the statement is valid without it.

Lemma 4.34 (prime-noether) There is no strictly descending infinite chain of algebraically integral difference subschemes of an algebraic variety V

Proof In such a descending chain, the transformal dimension must be non-increasing, and eventually stabilize; after that point, the dimension growth degree cannot increase, and eventually stabilizes too. By 4.23, if X ⊂ Y ⊂ V are algebraically integral, and have the same transformal dimension and dimension growth degree, then X = Y. □

Let WM_{rad}(R) be the set of an algebraically radical, well-mixed ideals of a difference ring R. When X ⊆ R, let < X >=< X >_{R} be denote the smallest I ∈ WM_{rad}(R) containing X. I ∈ WM_{rad}(R) is finitely generated if I =< X > for some finite X. Let k be a difference field, or ℤ, and let R be any algebraically reduced, well-mixed k- algebra. For the purposes of the present proof, we call R finitely presented if R ≃ k[X]_{σ}/I for some finitely generated I ∈ WM_{rad}(k[X]_{σ}), where k[X]_{σ} is the difference polynomial algebra over k in finitely many variables. ℕ[σ] denotes the ring of expressions ∑_{i=0}^{m} m_{i}σ^{i}, with m_{i} ∈ ℤ, m_{i} ≥ 0.

Lemma 4.35 (rad-noether-0) Let R be a finitely generated well-mixed k-algebra. Assume R has finite total dimension, or the weaker assumption:

(α) For any well-mixed quotient R̄ of R, there is a bound m on the transformal degree of R̄/p, as p ranges over the minimal algebraically prime difference ideals of R̄.

Then every element of WM_{rad}(R) is finitely generated.

The first part of the proof (through Claim 2) just recovers the Ritt-Raudenbush basis theorem, and does not make use of finite total dimension. We may factor out \sqrt{0}, so assume R is algebraically reduced.

If the union of a increasing chain of WM_{rad}(R) is finitely generated, then cofinally many elements of the chain are already finitely generated. Thus if not every element of WM_{rad}(R) is finitely generated, by Zorn’s lemma, there exists a maximal non-finitely-generated I ∈ WM_{rad}(R). We will reach a contradiction from this assumption. Note that every element of WM_{rad}(R/I) is finitely generated (either it is 0, or it lifts to a finitely generated element of WM_{rad}(R).) Thus every increasing chain of elements of WM_{rad}(R/I) is eventually constant. Claim 1 I is algebraically prime. Hence (2.3) P = ^{σ} \sqrt{I} is transformally prime. Proof If ab ∈ I, let I_{1} = Ann(a/I); then I_{1} is well-mixed and radical. Let I_{2} = Ann(I_{1}/I); then I_{2} is well-mixed and radical, and I^{2} ⊆ I_{1}I_{2} ⊆ I. If I ≠ I_{1} and I ≠ I_{2} then I_{1}, I_{2} are finitely generated as well-mixed radical ideals, say by X_{1}, X_{2}. Any algebraically prime difference ideal containing X_{1}X_{2} must contain I_{1} or I_{2}; hence also I. Thus any radical well-mixed difference ideal containing X_{1}X_{2} must contain I. So I is finitely generated, a contradiction. Thus I = I_{1} or I = I_{2}; so b ∈ I or a ∈ I. Thus I is prime. □

Since σ^{−n}(I) stabilizes, σ^{−n}(I) = P for some n.

By Lemma 4.28, there exists d ∈ R \ P and a finite X_{0} ⊆ R such that IR[d^{−1}]_{σ} ∈ WM_{rad}(R[d^{−1}]_{σ}) is generated by X_{0}. But < X_{0} >_{R[d^{−1}]_{σ}} = {a/d^{n} : a ∈< X_{0} >, n ∈ ℕ[σ]}. It follows that for any a ∈ I, for some n ∈ ℕ[σ], d^{n}a ∈< X_{0} >. So d ∈ ^{σ} \sqrt{Ann(a/ < X_{0} >)}.

But by the maximality of I, < I, d > is finitely generated; so < I, d >=< X_{1} ∪ {d} > for some finite X_{1} ⊆ I.

Replacing R by R/ < X_{0} ∪ X_{1} >, we may assume:

(*) For any a ∈ I, d ∈ ^{σ} \sqrt{Ann(a)}

(**) < I, d >=< d >.

By (*) and (**), for any a ∈ I, < I, d >⊆ ^{σ} \sqrt{Ann(a)}; in particular a ∈ ^{σ} \sqrt{Ann(a)}; so a^{σ^{n}} ∈ Ann(a) for some n, i.e. a^{σ^{n}}a = 0, as R is well-mixed and reduced, a^{σ^{m}} = 0 for some m.

We thus have: Claim 2 For any a ∈ I, for some n ∈ ℕ, a^{σ^{n}} = 0. □

Claim 3 Say R/I has transformal dimension n. Then there exists a finitely generated I_{0} ∈ WM_{rad}R, I_{0} ⊆ I such that R/I_{0} has transformal dimension n. Proof Let a_{1}, …, a_{N} be generators of R; for each subset s of {X_{1}, …, X_{N}} of size n+1, there exists a nonzero f_{s} ∈ k[X_{1}, …, X_{N}] using the variables in s only, with f_{s}(a_{1}, …, a_{N}) ∈ I. let I_{0} =< {f_{s}(a_{1}, …, a_{N}) : s} >. Then in R/I_{0} the image of a_{1}, …, a_{N} is a set of generators, with no transformally independent subset of size n + 1; so the transformal dimension is at most n, hence n. □

Again we may replace R by R/I_{0}, so we may assume R has transformal dimension n. Let Q be the set of cofinally minimal prime ideal of R. We now use the assumption (α); so for each p ∈ Q, R/p has transformal degree ≤ m. By Lemma 4.24, σ^{−m}(p) is transformally prime. By Claim 2, P ⊆ σ^{−m}(p).

Now using 2.10, σ^{−m}0 = σ^{−m}(∩_{p∈Q}p) = ∩_{p∈Q}σ^{−m}(p) ⊇ P.

So P = σ^{−m}(0). Let D = σ^{m}(R). D ≃ R/ ker(σ^{m}) = R/P, so D is a difference domain. By Lemmas 4.31 and 4.27, for some finitely generated I_{1} ∈ WM_{rad}R, and some d ∉ P, for

any a ∈ I we have ad^{n} ∈ I_{1} for some n ∈ ℕ. But I_{1} = \sqrt{I_{1}}, so ad^{n} ∈ I_{1} implies (ad)^{n} ∈ I_{1} and ad ∈ I_{1}. Thus, replacing R by R/I_{1}, we have: Claim 3 For any a ∈ I, ad = 0.

By Claim 3, for any a ∈ I, d ∈ Ann(a). On the other hand by (**), a ∈< d > . So a ∈ Ann(a). \sqrt{A}nn(a) is a radical well-mixed ideal containing d, so a ∈< d >⊆ \sqrt{A}nn(a), so a^{n}a = 0 for some n ≥ 1; hence a = 0. So I = 0. This contradicts the assumption that I is not finitely generated. □

Corollary 4.36 (direct-ritt-0) Let X be an affine or projective difference scheme over Spec^{σ} k, k = ℤ or a difference field, of finite total dimension. Then X may be embedded as a closed subscheme of a directly presented (over k) difference scheme X̃. Moreover, X, X̃ have the same topological space and the same underlying algebraically reduced well-mixed scheme X_{wm,red} = X̃_{wm,red}.

Proof (Affine case.) We may write X = Spec^{σ} ([σ]_{D}D[X]/J), where D[X] is a polynomial ring over D in finitely many variables, and J a prime ideal of [σ]_{D}D[X]. By Proposition 4.35, J is finitely generated as a well-mixed, algebraically reduced ideal. Adding variables, we may assume J is so generated by J_{0} = J ∩D[X, σ(X)]. Let X̃ = Spec^{σ} ([σ]_{D}D[X]/J_{0}). □ .

Corollary 4.37 (rad-prime-0) Let k be a difference field, R a finitely generated difference k-algebra of finite total dimension. If p is a well-mixed ideal and is radical, then there exists a finite number of algebraically prime difference ideals whose intersection is p.

Proof Using Lemma 4.35, we can prove the statement by Noetherian induction on wellmixed, radical ideals. Let p be such an ideal. If p is algebraically prime, we are done; otherwise, say ab ∈ p, a, b ∉ p; let p_{1} = Ann(a/p), p_{2} = Ann(p_{1}/p); then p_{1}, p_{2} are well-mixed, radical ideals bigger than p, with p_{1}p_{2} ⊆ p ⊆ p_{1} ∩ p_{2}. Let q = p_{1} ∩ p_{2}; then q^{2} ⊆ p_{1}p_{2} ⊆ p, so as p is radical, q ⊆ p; and thus p = p_{1} ∩ p_{2}. Now by induction, each p_{i} is the intersection of finitely many algebraically prime difference ideals; hence so is p. □

The corollary can be restated as follows: S = R/p has a finite number of minimal algebraically prime difference ideals; their intersection equals 0 in this ring. (Indeed if p_{1}, …, p_{m} are algebraically prime difference ideal of the difference ring S, no one contained in another, and with p_{1} ∩ … ∩ p_{m} = (0) then each is minimal.)

Let P_{i} = {a : σ^{m}(a) ∈ p_{i}, some m}. By Corollary 4.24 applied to R/p_{i}, for some fixed m, P_{i} = {a : σ^{m}(a) ∈ p_{i}}. Thus P_{i} is a transformally prime ideal. Every transformally prime ideal contains some p_{i} and hence some P_{i}.

Remark 4.38 (t-emb)

Every minimal transformally prime ideal must therefore equal some P_{i}; but not every P_{i} must be minimal. (Consider ℚ(x, y)_{σ}/(xx^{σ}, xy^{σ}).)

The prime ideals of R[a^{−1}]_{σ} correspond to primes p of R with a^{−σ^{n}} ∉ p for all n. This correspondence preserves inclusion, and also preserves the set of difference ideals. The minimal algebraically prime difference ideals of R[a^{−1}]_{σ} are thus precisely the proper p′_{i} = p_{i}R[a^{−1}]_{σ}, and p′_{i} ∩ R = p_{i}. Thus for any irreducible difference scheme X of finite total dimension, one obtains canonically a finite set of algebraically prime difference ideal sheaves; the corresponding closed subschemes are called the algebraic components of X.

Definition 4.39 (wct) Let X be a difference scheme. An algebraic component of X is a maximal algebraically integral subscheme Y of X.

From the discussion above, we obtain:

Lemma 4.40 (power) Let X be a difference scheme of finite type, and of finite total dimension. Then every algebraically integral subscheme of X is contained in some algebraic component. If C is an algebraic component of X, it has a unique maximal transformally integral difference subscheme C̄. For some n, for any algebraically integral difference ring D, if a ∈ C(D) then σ^{n}(a) ∈ C̄(D). □

(In the language of the next section, 5.2, we can say that B_{n}(C) is a component of B_{n}(X).)

4.7 Directly presented difference schemes

(dirpres)

Definition 4.41

Let D be a difference (well-mixed) ring. Let F = D[X_{1}, …, X_{n}]_{σ} be the difference polynomial ring over D in variables X_{1}, …, X_{n}. Let I be a difference (well-mixed) ideal of F. I is directly generated if I is generated as a difference (resp. well-mixed) ideal by I ∩ F_{1}. A surjective difference homomorphism h : F → R is said to be a direct presentation of R as a difference (resp. well-mixed) D-algebra via R_{1} = h(F_{1}) if Ker(h) is directly generated.

In other words, R is generated by R_{1}, and the relations between the generators can all be deduced from relations between R_{1} and σ(R_{1}). Every finitely presented difference Dalgebra admits a direct presentation. We will say that the direct total dimension of R (for this presentation) is the Krull dimension over D of the ring R_{2} generated by R_{1} ∪ σ(R_{1}), and that R is directly reduced (irreducible, absolutely irreducible) if f = 0 whenever f ∈ R_{1} and f^{n}σ(f)^{m} = 0, n, m > 0 (respectively, if R_{2} has no zero-divisors, R_{2}⊗_{D}L has no zero-divisors for some algebraically closed field L containing D.)

A similar definition can be made for difference schemes. Let V be an algebraic scheme over a difference ring D, S a subscheme of Y = (V × V^{σ}). We let Σ denote the graph of σ on [σ]_{D}V ×_{Spec^{σ} D} [σ]_{D}V^{σ}; more precisely, in any affine open neighborhood where V = Spec^{σ} R, R a difference D-algebra, we let I_{σ} be the ideal in R⊗_{D}R generated by the elements σ(r)⊗1−1⊗r; Σ is the corresponding closed difference subscheme. We also let S ⋆Σ be the projection to [σ]_{D}V of the difference scheme [σ]_{D}S ∩ Σ. It is isomorphic to [σ]_{D}S ∩ Σ.

A direct presentation of X is an embedding of X into [σ]_{D}V, V an algebraic scheme of finite type over D, such that the image of X has the form S ⋆ Σ as above.

Definition 4.42 (The limit degree, cf. [Cohn]). Let Z be an irreducible difference variety over a difference field K, of transformal dimension 0. Let a be a generic point of Z over K. Then K_{n} = K(a, σ(a), …, σ^{n}(a)) is a field; for large n, K_{n+1} is a finite algebraic extension of K_{n}; and the degree [K_{n+1} : K_{n}] is non-increasing with n. Let deg_{lim}(Z) be the eventual value of this degree.

Lemma 4.43 (direct2) Let V be a smooth algebraic variety over a difference field K, dim (V) = d. Let S ⊂ V × V^{σ} be an absolutely irreducible subvariety, and assume either the projection S → V or S → V^{σ} is smooth, of relative dimension e. Let Z = S ⋆ Σ ⊂ [σ]_{K}V be the difference subscheme of [σ]_{K}V cut out by: (x, x^{σ}) ∈ S. Then

1. Z is a disjoint union of components.

2. Each component W is actually a component of the pro-algebraic variety

S[∞] = S ×_{V^{σ}} S^{σ} ×_{V^{σ^{2}}} × …

in the strong sense that W [n] is a component of

S[n] = S ×_{V^{σ}} S^{σ} ×_{V^{σ^{2}}} × … S^{σ^{n−1}}

3. Each component W is Zariski dense in V, and has dim (W [n]) = d + ne for all n ≥ 1.

4. If e = 0,

deg_{cor} S = ∑_{W} deg_{lim} W

where the sum is taken over all components W of Z.

5. Z is perfectly reduced.

Proof We take the case: S → V is smooth (the other being similar.)

Since S^{σ^{n}} → V^{σ^{n}} is smooth, and smooth morphisms are preserved under base change, S[n + 1] → S[n] is also smooth. Compositions of smooth morphisms are also smooth, so for each n, S[n] → V is a smooth morphism, and indeed S[n] is smooth over K.

It follows that each S[n] is a disjoint union of irreducible components. By the dimension theorem (in the smooth variety V ×V^{σ} ×…×V^{σ^{n}}), each component has dimension ≥ d+ne. On the other hand by adding up relative dimensions we see that dim S[n] ≤ d+ne, so equality holds.

Since the components of the S[n] are disjoint, letting S[∞] be the projective limit of the S[n], we see that the pro-algebraic variety S[∞] is a disjoint union of irreducible pro-algebraic varieties C; C is an inverse limit of components C[n] of S[n]. Now by definition, C[n + 1] projects to C[n]; but one can also project to V^{σ} × … × V^{σ^{n}}; the image is a component which is either contained in, or disjoint from, C[n]^{σ}. In the former case, for any difference subvariety W of Z, W [n + 1] ∩ C[n + 1] = ∅; call such a pro-component empty of difference points. If the latter case holds for each n, then C defines a perfect difference scheme, and so a component W (C) of Z. Thus Z is the disjoint union of the various (nonempty) W (C). It follows that Z is perfectly reduced.

Finally, assume e = 0. The additivity of degree may be proved by induction on deg_{cor}(S). If for all m, the S[m + 1] is absolutely irreducible, then Z is transformally integral, and deg_{lim}(Z) = deg_{cor}(S). Otherwise, consider the minimal m such that S[m + 1] is reducible. Let Y = S[m], and let T = ((a_{0}, …, a_{m}), (a_{1}, …, a_{m+1})) ∈ (Y × Y^{σ}) : (a_{0}, …, a_{m+1}) ∈ S[m + 1]} Let {T_{j}} be the components of T projecting dominantly to Y. They map finite-to-one to Y^{σ}, hence dominantly. Then

∑_{j} deg_{cor}(T_{j}) = deg_{cor}(T)

Since deg_{cor}(T_{j}) < deg_{lim}(Z) for each j, deg_{cor}(T_{j}) = ∑ deg_{lim}(Z′) where the Z′ here range over the components of Z , Zariski dense in X, such that for a ∈ Z′ (a, σ(a), …, σ^{m+1}(a)) ∈ T_{j} Since each Zariski dense Z′ falls into a unique T_{j} in this sense, the required equation follows.

□

Let f[1](S) = {a : dim (f^{−1}(a) ∩ S) ≥ 1}. Note also, in a converse direction to 4.43:

Lemma 4.44 (zd) Let S ⊂ V × V^{σ} be absolutely irreducible k-varieties of dimension d, Z = [σ]_{k}pr_{0}[1](S). Let X be a difference subscheme of S ⋆ Σ; [X] is the set of points of X.

(1) If [X] ∩ Z = ∅ or σ[X] ∩ pr_{1}[1](S) = ∅, then X has total dimension ≤ d.

Assume X is algebraically integral.

(2) If X is weakly Zariski dense in V, then X is a component of S ⋆ Σ, and X has total dimension d.

(3) If t.dim(X) = d, and [X] ⊈ [Z], then [X] is Zariski dense in V.

Proof (1) If b ∈ X(D), D = k[b]_{σ} an algebraically integral difference k- algebra, with fraction field K, then (b, b^{σ}) ∈ S, b ∉ pr_{0}[1](S), so b^{σ} ∈ k(b)^{alg}. Inductively, k[b]_{σ} ⊆ k(b)^{alg}. Thus tr.deg._{k}K = tr.deg._{k}k(b) ≤ dim (V) = d. This shows that X has total dimension ≤ d.

If b^{σ} ∉ pr_{1}[1](S), then b ∈ k(b^{s}i)^{a}lg. But b^{σ} is a specalization of b. So tr.deg._{k}k(b) ≥ tr.deg._{k}k(b^{σ}). It follows that b^{σ} ∈ k(b)^{alg}, and we continue as before.

For (2,3) assume X is algebraically integral, and let (a_{0}, a_{1}, …) be a generic point of the pro-algebraic variety corresponding to X, i.e. (a_{0}, …, a_{n}) is a generic point of X[n] for each n.

(2) AssumeX is weakly Zariski dense in V, i.e. X[0] = V. Then X[1] → X[0] is dominant, so d = dim (S) ≥ dim X[1] ≥ X[0] = d. So (a_{0}, a_{1}) is a generic point of S, and thus a_{0} ∈ k(a_{1})^{alg}. So the specialization

(a_{0}, …, a_{n}) → (a_{1}, …, a_{n+1})

does not lower transcendence degree, hence it is an isomorphism. Thus k(a_{0}, …) is a difference field, and hence the prime ideal corresponding to X is a transformally prime ideal, and X is a component. Also, since k(a_{0}), k(a_{1}) have the same transcendence degree and a_{0} ∈ k(a_{1})^{alg}, we have a_{1} ∈ k(a_{0})^{alg}, and since the displayed specialization is an isomorphism, a_{n+1} ∈ k(a_{n})^{alg}; so tr.deg._{k}k(a_{0}, a_{1}, …) = d. As this is a generic point, the algebraically integral difference scheme X has total dimension d.

(3) The point a_{n} cannot be in the closed set σ^{n}(pr_{0}[1](S)): if this were the case for some n, it would be true for all larger values of n. But by 4.40, for sufficiently large n, a_{n} ∈ σ^{n}([X]); and σ^{n}([X]) ∩ σ^{n}(pr_{0}[1](S)) = ∅.) Now (a_{n}, a_{n+1}) ∈ S^{σ^{n}}, so a_{n+1} ∈ k(a_{n})^{alg} for each n. Thus tr.deg._{k}k(a_{0}) = d. In other words, (2) holds; and (3) follows.

Lemma 4.45 (dim-const) Let Y be a difference scheme with no strictly decreasing chain of perfectly reduced subschemes, and let X be a difference scheme of finite type over Y (via f : X → Y.) Let m be an integer. Then:

1. there exists a perfectly reduced difference subscheme D_{m}(f) = D_{m}(X/Y) of Y, such that X_{a} has total dimension ≥ m if a is a generic point of (any component of) D_{m}(X/Y), and has total dimension < m (or is empty) if a ∉ D_{m}(X/Y).

2. The above properties characterize D_{m}(X/Y) uniquely.

3. If X has finite total dimension d, then D_{m}(X/Y) has total dimension ≤ d − m.

4. If equality holds in (3), then f^{−1}D_{m}(X/Y) contains an algebraic component of X. (cf. 4.39).

Proof The uniqueness (2) is clear, since if Z, Z′ are two candidates, each generic point of Z must lie on Z′, and vice versa. To show existence (1), and (3), we use Noetherian induction on Y. We may assume Y is perfectly reduced. We may also assume Y is irreducible, since if it has a number of components Y_{j}, then we can let D_{m}(X/Y) = ∪_{j}D_{m}(f^{−1}Y_{j}/Y_{j}). If X/Y has generic total dimension ≥ m, let D_{m}(X/Y) = Y. By 4.18, Y has total dimension ≤ d−m in this case. Otherwise, Y has total dimension ≤ m−1. By 4.16, there exists a proper closed difference subscheme Y′ of Y such that for a ∈ Y \Y′, X_{a} has total dimension ≤ m−1. Now D_{m}(f^{−1}Y′/Y′) exists by Noetherian induction, and we can let D_{m}(X/Y) = D_{m}(f^{−1}Y′/Y′).

Finally, (4) is clear since f^{−1}D_{m}(X/Y) is a difference subscheme of X of the same total dimension d (by the first part of (1).) □

Caution: This is not preserved under base change Y′ → Y. D_{m}(X′/Y′) = ∅, D_{m}(X/Y) = Y is possible.

From a logical point of view, ACFA does not eliminate quantifiers. D_{0}(X/Y) for instance is therefore not the projection, but rather a difference-scheme theoretic closure of the projection.

5 Transformal multiplicity

(trans-mult)

Recall Definition 4.30: if f : R → S is a morphism of difference rings, S is transformally radicial over R if for any s ∈ S, for some m ≥ 0, s^{σ^{m}} ∈ f(R). We will extend this to difference schemes, and analyze transformally radicial maps X → Y by dividing Y according to the relative total dimension of X.

Remark 5.1 (red-rad) Let k be an inversive difference field, let R ⊂ S be difference kalgebras, and assume S is transformally radicial over R. Then R,S have the same reduced total dimension over k.

Proof Let I = {r ∈ R : (∃m ≥ 1)σ^{m}(r) = 0}, J = {s ∈ S : (∃m ≥ 1)σ^{m}(s) = 0}. Any difference ring homomorphism on R into a difference domain must factor through R/I; thus R,R/I have the same reduced total dimension, and similarly so do S, S/J. But I = J ∩ R, and R/I ≃_{k} S/J. □

Definition 5.2 A difference scheme X over a difference scheme Y is transformally radicial if there exists an open affine covering {U_{j}} of Y and {V_{ij}} of X, such that O_{X}(V_{ij}) is transformally radicial over O_{Y} (U_{j})

If a difference scheme X of finite type over Y is transformally radicial over Y, then X has finite total dimension over Y; this follows from the first statement of Lemma 4.31.

We begin by a look at the generic behavior; for this purpose we may pass to the fields of fractions.

5.1 Transformally separable extensions

We define an invariant ι dual to the limit degree, and a purely inseparable variant ι′. The invariant ι′ of 5.4 below figure in the asymptotics in q of the multiplicity of the points of M_{q}(X), the reduction of X to a difference scheme with structure endomorphism the q-Frobenius. Most likely these lemmas appear in [Cohn].

Lemma 5.3 (inv-fin) Let F be an inversive difference field, K a difference field extension of F with tr.deg._{F} (K) < ∞. Then K^{a} is inversive.

Proof F ⊂ σ(K) ⊂ K, and tr.deg._{F} (σ(K)) = tr.deg._{F} (K). □

Lemma 5.4 (towers) Let K ⊂ L ⊂ M be difference fields, with M of finite transcendence degree over K. Consider L, K, M as subfields of M^{inv}.

1. L^{inv} is an algebraic extension of K^{inv}L. Thus if L is finitely generated over K as a difference field, then LK^{inv} is a finite extension of σ(L)K^{inv}. Write ι(L/K) = [LK^{inv} : σ(L)K^{inv}], ι′(L/K) = [LK^{inv} : σ(L)K^{inv}]_{insep}

2. Assume M/L is a regular field extension. Then M/K is transformally separable iff M/L and L/K are transformally separable.

3. Assume M is finitely generated and transformally separable over K. Then ι(M/K) = ι(M/L)ι(L/K), and similarly for ι′.

Proof

By Lemma 5.3, we have: Claim Let K be an inversive difference field, L a difference field extension of K of finite transcendence degree. Then L^{inv} is an algebraic extension of L.

(1) follows: since the transcendence degree of L over K is finite, so is that of LK^{inv} over K^{inv}, and the claim applies.

(2) The ”if” direction follows from the transitivity properties of linear disjointness. Assume M/K is transformally separable. Then L/K is a fortiori transformally separable. We need to prove that M/L is transformally separable. M is linearly disjoint from the compositum K^{inv}L over L. We must show that K^{inv}M is linearly disjoint from L^{inv} over K^{inv}L. Since M/L is a regular field extension , and M is linearly disjoint from LK^{inv} over L, MK^{inv}/LK^{inv} is a regular extension. In other words MK^{inv} is linearly disjoint over LK^{inv} from the algebraic closure of LK^{inv}. By (1), L^{inv} ⊂ (LK^{inv})^{alg}.

(3) We may assume here that K is inversive. We have

ι(M/K) = [M : M^{σ}] = [M : M^{σ}L][M^{σ}L : M^{σ}]

Yet M is linearly disjoint from L^{σ^{−1}} over L, so M^{σ} is linearly disjoint over L^{σ} from L, and

[M^{σ}L : M^{σ}] = [L : L^{σ}] = ι(L/K)

Similarly

[M : M^{σ}L] = [ML^{inv} : M^{σ}L^{inv}] = ι(M/L)
□

5.2 The reduction sequence of a difference scheme

(redseq) Let X be a well-mixed scheme. We define a sequence of functors B_{n}, and maps giving a sequence X → B_{1}X → B_{2}X ↦ . . .. Denote B_{n}X by X_{n} in this section. We will also define functorially maps r_{n} : X → X_{n} as well as i_{n} : X_{n} → X. The maps i_{n} will allow us to identify X_{n} with a subscheme of X, but it is the sequence of maps r_{n} that will really interest us.

When X = Spec^{σ} R is affine, we let X_{n} = Spec^{σ} σ^{n}(R). Let r_{n} be induced by the inclusion σ^{n}(R) ⊂ R. Let i_{n} be induced by the surjective homomorphism σ^{n} : R → σ^{n}(R).

If f : R → S is a ring homomorphism,, we let B_{n}(f) : σ^{n}(R) → σ^{n}(S) be the restriction of f; and with f^{∗} : Spec^{σ} S → Spec^{σ} R the corresponding map of difference schemes, B_{n}(f^{∗}) = B_{n}(f)^{∗}.

If X is a multiplicatively and transformally closed subset of R, 0 ∉ X, let S = R[X^{−1}]. We compare σ^{n}(S) to the localization R′[σ^{n}(X)^{−1}], where R′ = σ^{n}(R). The natural map R′[σ^{n}(X)^{−1}] → σ^{n}(S) is clearly surjective, and since R is assumed to be well-mixed, injective too. Thus B_{n}(j) : B_{n}(Spec^{σ} S) → B_{n}(Spec^{σ} R) is compatible with localizations. By gluing we obtain a functor B_{n} on well-mixed schemes.

Observe that R[X^{−1}] and R[σ^{n}(X)^{−1}] may not be the same ring, but they have the same affinization, i.e. the same difference spectrum. Indeed an element a ∈ X may not be invertible in R[σ^{n}(X)^{−1}], but σ^{n}(a) will be invertible, and therefore a will be a σ-unit in this ring.

The maps i_{n} : Spec^{σ} R → B_{n}(Spec^{σ} R) and r_{n} : B_{n}(Spec^{σ} R) → Spec^{σ} R, also globalize.

Remark 5.5

If k is inversive, B_{n}(k) = k, and B_{n} induces a functor on difference schemes over k. The maps r_{n} are maps of k-difference schemes, but the maps i_{n} are not (so that the identification of X_{n} with a subscheme of X involves a twisting vis a vis k.)

Let X be a well-mixed scheme of finite type (or, of finite type over an inversive difference field K.) It follows from Lemma 4.31 that each ring O_{X}(U) is finitely generated over O_{B_{n}(X)}(U). In particular it has finite relative total dimension τ_{n}. If X has finite total dimension, then all the τ_{n} are bounded by this dimension.

Definition 5.6 The transformal multiplicity of a difference scheme X is the supremum of the total dimensions of the morphisms X → B_{n}(X).

If X is a well-mixed scheme of finite type over a difference field k, we define the transformal multiplicity of X over k to be the transformal multiplicity of the difference scheme X⊗_{k}K , where K is the inversive closure of k.

If X is a well-mixed scheme of finite type over a difference scheme Y, define the relative transformal multiplicity of X over Y to be the supremum of the transformal multiplicity of X_{y} over L, where L is a difference field and y is an L-valued point of Y.

Observe that as a map of points, Spec^{σ} X → Spec^{σ} B_{n}(X) is bijective. (Every transformally prime ideal p of σ^{n}(R) extends uniquely to a transformally prime ideal of R, namely σ^{−n}(p).) While B_{n}(X) may not be Noetherian as a scheme, X → B_{n}(X) is a map of finite type of schemes, and by Lemma 4.6, the transformal multiplicity is the maximum dimension of a fiber of this map (an algebraic scheme) over a point p ∈ Spec^{σ} B_{n}(X).

Example 5.7 Consider the subscheme X of 𝔸^{1} defined over an algebraically closed difference field K by f(X^{σ}, …, X^{σ^{m+1}}) = 0, f an irreducible polynomial. If K is inversive, then we may write f = g^{σ}, and the corresponding reduced scheme will be given by g(X, …, X^{si^{m}}) = 0; the transformal multiplicity is 1. In general X may be transformally reduced; but it becomes reducible after base change, and the transformal multiplicity is at all events equal to 1.

Proposition 5.8 (si-dim3.2) Let K be an inversive difference field, X/K a well-mixed difference scheme of finite type and of total dimension d. Let k ≥ 1. There exist canonically defined closed subschemes Mlt_{k}X of X such that:

1. X \ Mlt_{k}X has transformal multiplicity < k over K.

2. Mlt_{k}X has reduced total dimension ≤ d − k over K.

3. If Mlt_{k}X has reduced total dimension d − k, then it contains an algebraic component of X. In this case, X has transformal multiplicity ≥ k.

Proof Let Y = B_{k+1}X, and let r = r_{k+1} : X → Y be the reduction sequence map. Let Y_{k} = D_{k}(r); cf. 4.45. Let Mlt_{k}X = r^{−1}Y_{k}. (Or the underlying reduced scheme).

(1) By Lemma 5.12 below, we have to show that if L′ is an inversive difference ring and y is an L′-valued point of B_{k+1}(X \ Mlt_{k}X), then X_{y} has total dimension < k over L′. Now y ∉ Y_{k}; by definition of Mlt_{k}(Y), and 4.45, the total dimension of X_{y} is < k.

(2),(3) come from 4.45 (3), (4).

□

Notation 5.9 (z-mlt-not) Z_{0}X = X \ Mlt_{1}X. For k ≥, Z_{k}X = Mlt_{k}X \ Mlt_{k+1}X.

Remark 5.10 (si-dim3.2R) The assumption in 5.8 that K is inversive is not necessary.

Proof Let X′ = X ×_{K} K′, and let X′_{k} = Mlt_{k}(X′), satisfying the conclusion of Proposition 5.8. Let j : X′ → X be the natural map of difference schemes. Then the radicial map j induces a bijection between the points of X′ and of X, or between the perfectly reduced subschemes of X′ and of X, preserving reduced total dimension. Let Mlt_{k}X be the perfectly reduced subscheme of X corresponding to Mlt_{k}X′. Then (1),(2),(3) are clear. (cf. 4.12). □

The next lemma, 5.11, falls a little short of concluding, when X itself has transformal multiplicity ≤ n, that B_{n}(X) must have transformal multiplicity 0. The obstacle can be explained in terms of sheaves of difference algebras; (B_{n})_{∗}(O_{X}) need not coincide with O_{B_{n}(X)}.

Lemma 5.11 (trans-mult-n) Let X = Spec^{σ} R be a well-mixed difference scheme, and assume X → B_{n′}(X) has total dimension ≤ n < n′. Then for any difference field L and any L-valued point y_{n′} of B_{n′}(X), and y ∈ X(L) lifting y_{n′}, σ^{n}(O_{X,y} ×_{B_{n′} (X),y_{n′}} L) has total dimension 0.

Proof By 4.12, we may take L to be inversive.

We may assume X = Spec^{σ} R, R a well-mixed difference ring. Let R_{n} = σ^{n}(R). Let y be the unique extension of y_{n′} to a difference ring morphism y : R → L; let y_{m} = y|R_{m}. Let ⊗′ denote the tensor product in the well-mixed category; A⊗′_{B}C is the quotient of A⊗_{B}C by the smallest well-mixed ideal of that ring. Let S = R⊗′_{R_{n′} ,y_{n′}} L, z : S → L the induced map, S_{n} = σ^{n}(S), z_{n} = z|S_{n}.

We have to show that S_{n} has total dimension 0. Now S is a well-mixed L-algebra, and σ^{n′}(S) ⊂ L. By assumption, S has total dimension ≤ n. We are reduced to showing:

Claim Let L be an inversive difference field, S a finitely generated well-mixed difference L-algebra, with σ^{n′}(S) ⊂ L. Assume S has total dimension n < n′. Then σ^{n}(S) has total dimension 0.

There is no harm factoring out the nil ideal of S (as a k-algebra), as this is a difference ideal, and the total dimension of S_{n} will not be effected. As S is well-mixed, and Noetherian, 0 is the intersection of finitely many prime ideals p_{1}, …, p_{r}, and they are difference ideals. We have ∩_{i=1}^{r}(p_{i} ∩ S_{n}) = 0, so it suffices to show that S_{n}/(p_{i} ∩ S_{n}) has Krull dimension 0 for each i; for this we may work with S/p_{i}. Thus we may assume S is an integral domain. At this point we will show that S = L. Otherwise let a ∈ S \ L. As L is inversive, σ^{n′}(a) = σ^{n′}(b) for some b ∈ L, so a−b ∉ L, and σ^{n′}(a−b) = 0. Thus it suffices to show that for n ≤ m < n′, if a ∈ S and σ^{m+1}(a) = 0 then σ^{m}(a) = 0. S is a f.g. domain of Krull dimension ≤ n. If a ∈ S, then a, σ(a), …, σ^{n}(a) are algebraically dependent over L in the field of fractions of S; so there exists an F ≠ 0 ∈ L[X, X_{1}, …, X_{n}] with F (a, …, σ^{n}(a)) = 0. Take 0 ≠ F ∈ L[X, X_{1}, …, X_{m}] of smallest number of monomials, and then least degree, such that F (a, …, σ^{m}(a)) = 0. Let i_{0} be least such that F has a monomial from L[X, X_{1}, …, X_{i_{0}}]. Apply σ^{m−i_{0}} to F. The monomial involving X_{i} for some i > i_{0} disappear when applied to a (as σ^{m+1}(a) = 0); deleting them, we obtain a shorter polynomial (though in higher -indexed variables ) vanishing on (a, …, σ^{m}(a)). This is impossible; so no monomials disappear; i.e. all monomials are from L[X, X_{1}, …, X_{i_{0}}]; but none are from L[X, X_{1}, …, X_{i_{0}−1}]. So F has the form X_{i_{0}}F′; and being irreducible, it is just a multiple of X_{i_{0}}. So σ^{i_{0}}(a) = 0. □

Corollary 5.12 (trans-mult-0) A difference scheme X has transformal multiplicity ≤ n iff X → B_{n+1}X has total dimension ≤ n.

Proof It suffices to show that if n < n′ and X → B_{n′}(X) has total dimension ≤ n, then so does X → B_{n′+1}(X). Let k be a difference field, y an k-valued point of B_{n′+1}. In terms of local rings, we have R ⊃ σ^{n′}(R) ⊃ σ^{n′+1}(R), and z : σ^{n′+1}(R) → k. We have to show that S = R⊗_{σ^{n′+1}(R),z}k has total dimension ≤ n. Let (h, D, L) be as in the definition of total dimension. Let S_{m} denote the image of σ^{m}(R)⊗_{σ^{n′+1}(R),z}k in S.

Observe in general that if X →_{f} Y has relative total dimension ≤ n, then so does the induced map BX → BY. Indeed, by Remark 4.9, the pullback f^{−1}BY → BY has total dimension ≤ n, and BX is a subscheme of f^{−1}BY, so 4.9 applies again. In particular, as X → B_{n+1}X has total dimension ≤ n, so does B_{n′−n}X → B_{n′+1}X .

By lemma 5.11, σ^{n}(σ^{n′−n}R⊗_{σ^{n′+1}(R),z}k) has total dimension 0. Thus h(S_{n′}) is finite dimensional over k, so it is a difference subfield of the domain D, call it k′. So h(σ^{n′}(R)) ⊂ k′. As X → B_{n′}(X) has total dimension ≤ n, h(R) is contained in a field F of k′-transcendence degree ≤ n. But [k′ : k] < ∞, so tr.deg._{k}(F) = tr.deg_{k′}(F) ≤ n. □

The reduction sequence: two side remarks

Lemma 5.13 (rm2) If X is algebraically integral,of finite type over a field K, then the X_{n} stabilize as subschemes of X: for some n, X_{n} = X_{n+1} = . . .. Proof This reduces to the affine case by taking an open affine cover; there it follows from Corollary 4.24. □

( What can be said without the algebraic integrality assumption?

If X is transformally integral, the schemes X_{n} are all isomorphic to X. The maps r_{n} : X → X_{n} need not be isomorphisms however.

If X is a scheme over an inversive difference field K, then X_{n} is not necessarily isomorphic to X over K, but is the transform of X under σ^{n}. In this case we can also define X_{n} for negative values of n, obtaining a sequence … X_{−2} → X_{−1} → X → X_{1} → . . ..)

5.3 σ-primary ideals

Definition 5.14 Let R be a difference ring. A well-mixed ideal I is σ-primary if whenever a, b ∈ R and ab ∈ I, either a ∈ I or b^{σ^{n}} ∈ I for some n ∈ ℕ[σ].

We have: if ab ∈ I, b ∉ P then a ∈ I. If ^{σ} \sqrt{I} = P and this condition holds, we say that I is P -σ-primary. Any finite intersection of P -σ-primary ideals is again P -σ-primary.

If P is a transformal prime, we say that a well-mixed ideal I is P -σ-primary if P = ^{σ} \sqrt{I} and whenever ab ∈ I, b ∉ P we have a ∈ I.

Lemma 5.15 (primary) Let I be a well-mixed ideal, and let P = ^{σ} \sqrt{I}, and R_{P} the localization of R outside P. Let ι : R → R_{P} be the natural map. The following conditions are equivalent:

1. I is σ-primary

2. P is a transformal prime, and I is P -σ-primary

3. P is a transformal prime, and I = ι^{−1}(I′) for some well-mixed ideal I′ of R_{P}.

Proof Since I is a well-mixed ideal, ^{σ} \sqrt{I} is a perfect ideal (Lemma 2.3.) If I is σ-primary, then P is a prime ideal, hence a transformal prime. Thus (1) implies (2). Assume (2). Let I′ = {a/b : a ∈ I, b ∉ P }. This is a well-mixed ideal of R′: if (a/b)(a′/b′) ∈ I′, then (a/b)(a′/b′) = a′′/b′′ with a′′ ∈ I, b′′ ∉ P. So aa′b′′ = a′′bb′ in R′; upon multipliying denominators and numerators by an appropriate element outside P, we may assume aa′b′′ = a′′bb′ in R. As I is P -primary, and a′′ ∈ I, we have aa′ ∈ I; as I is well-mixed, a^{σ}a′ ∈ I. Now (a/b)^{σ}(a′/b′) = (a^{σ}a′)/(b^{σ}b′) ∈ I′. So I′ is well-mixed. We have I ⊆ ι^{−1}(I′), and if a/1 ∈ I′, then a = a′/b for some a′ ∈ I, b ∉ P, so (ab − a′)b′ = 0 for some b′ ∉ P, thus ab − a′ ∈ I, so ab ∈ I, so a ∈ I. This proves (3). Now assume (3). If ab ∈ I, and b ∉ P, then ι(b) is invertible; so a/1 ∈ I′, thus a ∈ I. □

Lemma 5.16 Let R be a finitely generated difference k-algebra, k a difference field. Then there is no strictly increasing chain of algebraically radical σ-primary well-mixed ideals. Proof Suppose I_{1} ⊂ I_{2} ⊂ … ⊂ P is such a chain. Let P_{k} = ^{σ} \sqrt{I_{k}}. Then P_{1} ⊆ P_{2} ⊆ …, so by Ritt-Raudenbush the P_{k} are eventually equal, and we may assume ^{σ} \sqrt{I_{k}} = P. Let I = ∪_{k}I_{k}. If b ∈ R \ P, R′ = R[b^{−1}]_{σ}, I′_{k} = I_{k}R[b^{−1}]_{σ}, then the I′_{k} form a strictly ascending chain, by Lemma 5.15 (3). Thus, replacing R by an appropriate R′ using Lemma 4.32, we may assume I = ∪_{k}I_{k} is finitely generated; but it follows that I = I_{k} for large enough k, a contradiction. □

6 Transformal valuation rings in transformal dimen-

sion one

(td1)

6.1 Definitions

Notation 6.1 When K is a valued field, O_{K}, M_{K}, val (K), K_{res} will denote the valuation ring, maximal ideal; value group, residue field of K; but we will often denote R = O_{K}, M = M_{K}, K̄ = K_{res} , Γ = val (K). Definition 6.2 (tvr-def)

1. A transformal valuation ring (domain) is a valuation ring R that is also a difference ring, such that σ(M) ⊂ M (and σ injective.)

2. The valuation v will be said to be m-increasing (resp. strictly increasing) if for all a ∈ R with v(a) > 0, v(σ(a)) ≥ m · v(a) (resp. v(σ(a)) > v(a)). It is ω-increasing if m-increasing for all m.

3. A weakly transformal valued field is an octuple (K, R, M, σ, Γ, K̄, val , res ) such that (R, M, σ) is a transformal valuation ring, K is the field of fractions of R, res : K → K̄ is the residue map, val : K^{∗} → Γ = G_{m}(K)/G_{m}(R) is the valuation map. ^{2} We will also apply the term to parts of the data. If σ extends to an endomorphism of K, we denote it too by σ, and we say that K is a transformal valued field.

4. Assume given a distinguished t ∈ R, or at least a distinguished τ ∈ val (R) (τ = val (t)).

Let

Δ = {v ∈ val (K) : −τ < nv < τ, n = 1, 2, …}

Write a << b if ma < b for all m ∈ ℕ. K has transformal ramification dimension rk_{ram}(K) = r (over τ) if r is the length of a maximal chain v_{0}, …, v_{r} ∈ val (R) with 0 = v_{0} << v_{1} << … << v_{r} << τ. When K is ω- increasing, rk_{ram}(K) ≤ dim ℚ(ℚ⊗Δ). (In transformal dimension one, equality holds; cf 6.7.) When K extends k(t)_{σ}, taking val (t) distinguished, we define the valuative rank rk_{val} (K/k(t)_{σ}) of K over k(t)_{σ} to be rk_{ram}(K) + tr.deg._{k}K̄.

2G_{m}(R) is the group of invertible elements of R.

Weakly transformal valuation rings will not be used in the proof of the main results.

Notation 6.3 (not-moving)

A ℤ-polynomial in one variable F (X) is said to be positive at ∞ if F (t) > 0 for sufficiently large real t.

Given a difference ring k, let k[t, t^{−1}]_{σ} be the transformal localization by t of the transformal polynomial ring k[t]_{σ}, and let k[t̆]_{σ}′ be the sub-difference ring

k[t̆]_{σ}′ = k[t^{F (σ)} : F ∈ ℤ[X], F (∞) > 0] ≤ k[t, t^{−1}]_{σ}

Write t_{n} = σ^{n}(t). Note the homomorphism k[t̆]_{σ} → k, with kernel generated by the (t_{n}).

Alternative description: Let k be a difference field. Let k(t)_{σ} = k(t_{0}, t_{1}, …), with σ(t_{n}) = t_{n+1}, t = t_{0}. Then k(t)_{σ} admits unique a k-valuation, with 0 < val(t_{i}) << val(t_{j}) whenever i < j ∈ ℕ. (Here α << β means: mα < β for all m ∈ ℤ.) k[t̆]_{σ} is the associated valuation ring, and Ă_{k} = Spec^{σ} k[t̆]_{σ}. Note that k[t̆]_{σ} is the localization of k[t̆]_{σ}′ at the prime t = 0.

Lemma 6.4 (tvrl) Let (K, R, M, σ, Γ, K̄, val) be a weakly transformal valued field.

1. σ^{−1}(M) = M. σ(M) = M ∩ σ(R). If K is 1-increasing, then every ideal of R is a well-mixed difference ideal.

2. There exists a unique difference field structure on K̄, such that the residue map R → K̄ is a morphism of difference rings.

3. There exits a convex subgroup Γ′ of Γ and a homomorphism σ_{Γ} : Γ′ → Γ of ordered Abelian groups, such val(a) ∈ Γ′ and σ_{Γ}(val(a)) = val(σ(a)) when a ∈ R, σ(a) ≠ 0. If the valuation is k-increasing, then kv ≤ σ_{Γ}(v) for v ∈ Γ, v ≥ 0.

4. Let L be a subfield of K. Then σ(L^{a} ∩ R) ⊂ (σ(L ∩ R))^{a}. (where S^{a} denotes the algebraic closure of the field of fractions of S.)

5. If L is a subfield of K, and σ(L ∩ R) ⊂ L^{a}, then L^{a} ∩ R is a difference subring of R.

6. Let Δ be a convex subgroup of Γ with σ_{Γ}^{−1}(Δ) ⊂ Δ. (An automatic condition for 1increasing valuations.) Let v̂ be the valuation val(a)+Δ; with value group Γ/Δ, residue field K̂. Then a weakly transformal valued field structure is induced on K̂, with residue field K̄, value group Δ.

Proof

1. σ(M) ⊂ M by assumption, while σ^{−1}(M) ⊂ M since M is the unique maximal ideal of R. Thus σ^{−1}(M) = M. If σ(r) ∉ σ(M), then r ∉ M, so r is a unit of R, rr′ = 1, so σ(r)σ(r′) = 1, and thus σ(r) ∉ M.

2. σ induces an endomorphism σ : K̄ = R/M → σ(R)/σ(M) = σ(R)/(M ∩ σ(R)) ⊂ K̄ of K̄.

3. Let R′ = {r ∈ R : σ(r) ≠ 0}, Γ′ = {±val(a) : a ∈ R, a, σ(a) ≠ 0}. If a, b ∈ R and ab ∈ R′ then a ∈ R′, so Γ′ is convex. If a, b ∈ R′ and val(a) = val(b), then a = cb, b = da for some b, d ∈ R, so σ(a) = σ(c)σ(b), σ(b) = σ(d)σ(a), and thus valσ(a) = valσ(b). Define σ_{Γ}(val(a)) = val(σ(a)), and observe that σ : val(R′) → Γ is a homomorphism of ordered semi-groups. Extend it to an ordered group homomorphism Γ′ → Γ.

4. Let b ∈ L^{a} ∩ R. There is a nonzero polynomial P ∈ L[Y] such that P (b) = 0. Dividing by the coefficient of lowest value, we can assume the coefficients of P are in R, and at least one of them has value 0. By (1), it follows that P^{σ} ≠ 0 ∈ σ((L ∩ R))[X], and P^{σ}(b^{σ}) = 0.

5. From (4): σ(L^{a} ∩ R) ⊂ (σ(L ∩ R))^{a} ⊂ L^{a}, so σ(L^{a} ∩ R) ⊂ (L^{a} ∩ R).

6. Let R_{Δ} = {a ∈ K : (∃δ ∈ Δ)(δ ≤ val(a))}, M_{>Δ} = {r ∈ K : (∀δ ∈ Δ) δ < val(r)}, π : R_{Δ} → K̂ the natural map. Let R̂ = π(R); it is a valuation ring of K̂, with maximal ideal M̂ = π(M). If r ∈ M_{>Δ}, then σ(r) = 0, or val(σ(r)) = σ_{Γ}(val(r)) ∉ Δ (otherwise val(r) ∈ Δ.) So σ(M_{>Δ}) ⊂ M_{>Δ}, and σ induces an endomorphism σ : R̂ → R̂. π is a morphism of difference rings, and σ(M̂) = σ(π(M)) = π(σ(M)) ⊂ π(M) = M̂. We have val(σ(r)) ≥ val(r), implying in particular when val(r) ∈ Δ that val(σr̂) ≥ val_{K}(r̂), so that (R̂, σ) is 1-increasing if (R, σ) is.

6.2 Transformal discrete valuation rings

We will be interested in ω-increasing transformal valued fields L, finitely generated and of transformal dimension one over trivially valued difference subfield F. We will call these transformal discrete valuation rings. Picking any t ∈ L, val (t) > 0, we can view L as an extension of F (t)_{σ} of finite transcendence degree.

Let ℤ_{σ} = ℤ[σ], ℚ_{σ} = ℚ[σ], viewed as ordered ℤ[σ]-modules, (with ℚ < ℚσ < . . .. ). ℤ_{σ},ℚ_{σ} are the value groups of F (t)_{σ}, (F (t)_{σ})^{a}.

In this section, all transformal valued fields are assumed to be ω-increasing, with value group contained in ℚ_{σ}.

Let L be a transformal valued field. .

Lemma 6.5 (hensel-alg) Let L^{h} be the Henselization of L as a valued field. The endomorphism σ of L lifts uniquely to a valued field endomorphism of L^{h}. If L is ω-increasing, so is L^{h}.

Proof σ : L → σ(L) is an isomorphism of valued fields; by the universal property of the Henselization, for any Henselian valued field M containing σ(L), σ extends uniquely to an embedding L^{h} → M; in particular, with M = L^{h}, σ extends to σ : L^{h} → L^{h}. The property of being ω-increasing depends on the value group, which does not change. □

One can also canonically define a transformal Henselization of a transformal valued field, using difference polynomials and their derivatives; cf. the remarks following 6.29, as well as 7.4. Since we will only work with the transformal analogue of discrete valuation rings, we will be able to use the somewhat softer notion of topological closure.

Completion and closure Let L be a transformal valued field, with value group contained in ℚ_{σ}. We define a topology on L^{h}, a basic open set being a ball of nonzero radius. This topology is in general incompatible with valued field extensions. However, according to lemma 6.6, all the valued field extensions we will consider here have cofinal value groups; so the inclusions of valued fields are continuous, and the induced topology on a subfield coincides with the intrinsic one.

We can construct the completion L̂ of L^{h} for this topology; an element of L̂ is represented by a sequence a_{n} of elements of L^{h}, with val (a_{n+1} − a_{n}) ≥ σ^{n}(v) for some v ∈ val , v > 0. This completion carries a natural ω-increasing transformal valued field structure.

If K ≤ L are transformal discrete valuation rings over F, the topological closure of K^{h} within L̂ can be identified with K̂.

The Henselization and completion processes do not change the value group or residue field. By Lemmas 6.6, 6.14, the residue field is an extension of F of finite transcendence degree, and the value group is a ℤ[σ]-submodule of ℤ_{σ}.

6.3 The value group

Lemma 6.6 (vt1) Let L be an ω-increasing transformal valued field of transformal dimension 1 over F. Then the value group val (L^{a}) of L^{a} is isomorphic to ℚ_{σ}, as an ordered ℤ[σ]-module. Let K ≤ L be a difference subfield, nontrivially valued. Then val (K) is cofinal in val (L). val (L^{a})/ val (K^{a}) is a principal, torsion ℚ[σ]-module. If L is a finitely generated difference field extension of F, then val (L) is isomorphic to a ℤ[σ]-submodule of ℤ_{σ}.

Proof Pick t ∈ K, val (t) > 0. Since tr.deg._{F (t)_{σ}} L < ∞, val (L^{a})/ val (F (t)_{σ}^{a}) is a finitedimensional ℚ-space. Thus val (L^{a}) is a finitely generated ℚ[σ]-module. Since ℚ[σ] is a principal ideal domain, and val (L^{a}) is torsion free, val (L^{a}) is a free ℚ[σ]-module. Now the quotient val (L^{a})/ val (F (t)_{σ}^{a}) is finite-dimensional, so the rank of val (L^{a}) must equal one. Let h : ℚ_{σ} → val (L^{a}) be an isomorphism of ℤ[σ]-modules. Replacing h by −h if necessary, we may assume h(1) > 0. It follows that h is order preserving. Any nonzero ℤ[σ] - submodule of ℚ_{σ} is cofinal, and co-torsion.

There remains the last statement, concerning finite generation. Let t ∈ L, val (t) > 0, K = F (t)_{σ}. Then val K = ℤ[σ], and L is a finitely generated difference field extension of K of finite transcendence degree. Let L_{0} be a subfield of L, finitely generated over K as a field, generating L as a difference field, and of the same K-transcendence degree as L. Let L_{n} be the subfield of L generated by ∪_{k≤n}σ^{k}(L_{0}). Then [L_{n+1} : L_{n}] ≤ [L_{1} : L_{0}] < ∞. Let A_{n} = val L_{n}. Then A_{0} ⊂ A_{1} ⊂ … ⊂ val L^{a} ≃ ℚ[σ], A_{0} is finitely generated, and A_{n+1}/A_{n} is bounded. By lemma 6.8 below, val L is a finitely generated ℤ[σ]-module. And by 6.9, every finitely generated ℤ[σ] submodule of ℚ[σ] is contained in a free ℤ[σ]-module of rank one. □

In particular, since val (L) ≤ ℚ_{σ} as an ordered Abelian group, there are no ”irrational” values in val (L); for any 0 < u < v ∈ val (L), for some (unique) 0 ≤ q ∈ ℚ, |qv − u| << v. This follows from the fact that the divisible ordered Abelian group ℚ − σ, is a lexicographic direct sum of copies of ℚ.

Thus:

Corollary 6.7 (vt1c) Let L be an ω-increasing transformal valued field of transformal dimension 1 over F, t ∈ L, val (t) > 0. Then rk_{ram}(L/F (t)_{σ}) = rkℚ( val (L)/ val F (t)_{σ}).

Proof This is in fact a statement about the ordered Abelian group A = ℚ_{σ}. Let 0 < τ ∈ A, so that B = ℚ[σ]τ is a copy of ℚ_{σ} embedded in A; and let 0 << a_{1} << … << a_{n} << τ with n maximal possible. We have to show that rkℚ(B/A) = n. Clearly a_{1}, …, a_{n} are ℚ-linearly independent over B, so n ≤ rkℚ(B/A). Factoring out the (convex) ℚ-space generated by a_{1}, …, a_{n}, we may assume n = 0. In this case we have to show that A = B. Indeed otherwise let A ∈ A, a ∉ B, 0 < a. As in Lemma 6.6, B is cofinal in A. Let m be least such that for some a ∉ B, 0 < a < σ^{m}(τ). Then for some rational q ≥ 0, a′ = a − qσ^{m}(τ) << σ^{m}(τ); but a′ ∉ B, contradicting the minimality of m.

Lemma 6.8 (vt1-0) Let A_{0} ⊆ A_{1} ⊆ … be finitely generated subgroups of ℚ[T] containing ℤ[T], T A_{n} ⊆ A_{n+1}, with A_{n+1}/A_{n} finite and bounded. Then A = ∪_{n}A_{n} is a finitely generated ℤ[T]-module.

Proof If A/ℤ[T] is finitely generated, so is A, thus we may replace A_{i} by A_{i}/ℤ[T] ≤ ℚ[T]/ℤ[T].

We use induction on m = lim sup_{n} |A_{n+1}/A_{n}|. If A ≠ 0, there exists c ∈ A, c ≠ 0, lc = 0, l prime. Say c ∈ A_{n_{0}}. Claim Let E = E_{l} = {x : lx = 0}. Then for n ≥ n_{0}, E ∩ A_{n+1} ⊈ A_{n}.

Suppose the claim is false; then any x ∈ E ∩ A_{n} satisfies T x ∈ E ∩ A_{n+1} = E ∩ A_{n}. Thus ℤ[T]c ⊂ A_{n}. But A_{n} is a finitely generated group, while ℤ[T]c is not: c, T c, … are ℤ/lℤ - linearly independent. A contradiction.

Thus the natural surjective map e_{l} : A_{n+1}/A_{n} → lA_{n+1}/lA_{n}, e_{l}(x) = lx, is not injective, for n ≥ n_{0}. So lim sup_{n} |lA_{n+1}/lA_{n}| < m. By induction, lA is a finitely generated ℤ[T]- module.

Now E_{l}/(ℤ[T]c) is finite; indeed E_{l} ≃ (ℤ/lℤ)[T], so E_{l}/(ℤ[T]c) ≃ (ℤ/lℤ)[T]/f for some nonzero polynomial f(T). Thus (A ∩ E_{l})/(ℤ[T]c) is finite. Since ℤ[T]c is a finitely generated ℤ[T]-module, so is A∩E_{l} = ker(x ↦ lx). We saw lA was finitely generated; hence so is A. □

Lemma 6.9 (vt1-2) Let M be a finitely generated ℤ[σ]-submodule of ℚ_{σ}. Let M̃ be the union of all submodules N of ℚ_{σ} with N/M finite. Then M̃ is 1-generated.

Proof If M is generated by a_{1}, …, a_{n}, m_{i}a_{i} ∈ ℤ_{σ} for some 0 < m_{i} ∈ ℕ; so each a_{i} ∈ ℤ_{σ}[1/m], where m = Π_{i}m_{i}. Thus M ⊂ N for some principal N ≤ ℚ_{σ}. Claim For any ideal J of ℤ[σ], there exists a principal ideal J′ containing J with J′/J finite. Proof J is k-generated for some finite k. The claim reduces inductively to the case k = 2. So say J = ℤ[σ](a, b). Since ℤ[σ] is a unique factorization domain, we may write a = a′c, b = b′c with no prime element of ℤ[σ] dividing both a′ and b′. As ℤ[σ] has Krull dimension 2, ℤ[σ]/(a′, b′) must be finite.

Let J′ = ℤ[σ]c. Then a′, b′ annihilate J′/J, so that J′/J is a quotient of ℤ[σ]/(a′, b′), and hence is also finite.

Actually, J′ is unique: Claim If N ≤ ℚ_{σ} is principal, then ℚ_{σ} has no ℤ[σ]-submodules N′ such that N ⊆ N′, and N′/N is finite and nonzero.

For otherwise it would have a finitely generated one, so again one contained in a principal ℤ[σ]-module. Thus it suffices to show that if J is a submodule of ℤ[σ] containing a principal module K, then J/K is zero or infinite. If J/K is nonzero and finite, so is J′/K, where J′/J is finite and J′ is principal. So ℤ[σ]/fℤ[σ] is finite nonzero; but this is clearly absurd.

Thus M̃ ⊂ N. Let I = {r ∈ ℤ[σ] : rN ⊂ M̃}. So M̃ = IN. By definition of M̃, if I′/I is finite then I′ = I. Thus by the Claim I is principal. So M̃ is 1-generated.

□

Example 6.10 val (L) need not itself be free. Take F (t^{σ}, t^{2})_{σ} ≤ F (t)_{σ}; the value group is M = ℤ[2v, σ(v)] ⊂ ℤ[v, σ(v), …]. We have M/σ(M) ≃ ℤ ⊕ (ℤ/2ℤ)[σ]. □

At all events, when L is finitely generated, of limit degree d, ℤ[1/d!] val (L) is already free on one generator.

This suggests that L may have an algebraic extension whose value group is a free ℤ[σ]- module of rank one. We prove such a statement in characteristic 0; in positive characteristic p > 0, the proof works up to localization at p, and we did not investigate it further. Proposition 6.11 Let F be a difference field of characteristic 0, with F = F^{n}. Let L be a transformal discrete valuation ring over F, Henselian as a valued field. Then L has a finite σ-invariant extension M, whose value group is isomorphic to ℤ[σ]. Proof

By 6.9, val L ⊂ M̃ for some free 1-generated ℤ[σ]-module M̃ with M̃/ val L finite; we have val L = IM̃ for some ideal I, and necessarily n ∈ I for some n > 0.

We may assume F has n’th roots of unity; adding them will not change the value groups. Consider the surjection

val : L^{∗}→> val L

It induces a surjection

(L^{∗})^{n}→>n val L

Let H be the pullback of nM̃ ⊂ val L. We have

H→>nM̃

and σ(H) ⊂ H. Also H/(L^{∗})^{n} ≃ nM̃/n val L ≃ M̃/ val (L). So H/(L^{∗})^{n} is finite. By Kummer theory, there exists a (unique) Galois extension M of L, such that (M^{∗})^{n} ∩ L = H, and [M : L] = [H : (L^{∗})^{n}] = M̃/ val L. Tracing back the isomorphisms we see that M̃ ⊂ val (M), so M̃ = val (M). □

Notation 6.12 For a linear ordering L and X ⊆ L, let H(X) = {y : (∃x ∈ X)(y ≤ x)} Remark 6.13 (vt1c2) Let M ⊂ ℤ_{σ} be a ℤ[σ]-module. Let H refer to the induced ordering on M. Let Y = H(Y) ⊂ M, 0 ∈ Y. Then for some nonzero convex subgroup S of M, and 0 ≤ a ∈ M,

Y = H(a + S) = {y : (∃c ∈ S)y ≤ a + c}

The nonzero convex subgroups form a single σ-orbit {E_{n} : n = 1, 2, …} (in the sense that E_{n+1} is the convex hull of σ(E_{n}); E_{n} = σ^{−1}(E_{n+1}).)

Proof Let (0) = C_{0} ⊂ C_{1} ⊂ … be the proper convex subgroups of ℤ_{σ} (with σ^{−1}(C_{n+1}) = C_{n}). Then for some m, M ∩ C_{m+1} ≠ (0), M ∩ C_{m} = (0). So for n ≥ 0, E_{n} = C_{n+m} ∩ M are convex subgroups of M. If E is any proper convex subgroup of M, then the convex hull of E in ℤ_{σ} must be some C_{n+m}, and it follows that E = E_{n}.

If c ∈ E_{1} \ E_{0}, then σ^{n}(c) ∈ E_{n+1} \ E_{n}. Now C_{n+1}/C_{n} ≃ ℤ, so E_{n+1}/E_{n} is isomorphic to a nonzero subgroup of ℤ, hence also isomorphic to ℤ.

Let S = {a : a + Y = Y }. This is a convex subgroup of M; so S = E_{n} for some n ≥ 0.

There exists an E_{n+1}-coset Z with Z ∩Y ≠ ∅ and Z ⊈ Y. Now Z/E_{n} is order-isomorphic to ℤ. If (Z ∩ Y)/E_{n} has no greatest element, then Z ∩ Y is cofinal in Z so Z ⊂ Y, a contradiction. Similarly, Y can have no element above Z ∩ Y. Thus (Z ∩ Y)/E_{n} has a greatest element a/E_{n}, and this is also a greatest element of Y/E_{n}. It follows that Y = H(a+E_{n}), □

6.4 The residue field

Lemma 6.14 (vt2) Let L be an ω-increasing transformal valued field of transformal dimension 1 over F. The residue field res (L) of L has finite transcendence degree over F.

Proof The residue field of F (t)_{σ} is F itself, and L has finite transcendence degree over F (t)_{σ}. □

The analogy with 6.6 raises the question, that I did not look into: If in 6.14 L/F (t)_{σ} is finitely generated, is res L finitely generated over res (K) as a difference field (up to purely inseparable extensions )?

6.5 Valued field lemmas

We will need an observation from the theory of valued fields. (Compare [Haskell-Hrushovski-Macpherson] Part I, §2.5 ( ”Independence and orthogonality for unary types”.) )

Let K ≤ L be an inclusion of valued fields, c ∈ L. We let

T (c/K) = {val(c − b) : b ∈ K}

Also let E(c/K) be the stabilizer of T = T (c/K) in val (K), i.e.

E = {e ∈ val K : e + T = T }

If T (c/K) has no greatest element, then (6.15) val K(c) = val K. In this case T (c/K) is a downwards-closed subset of val K. Hence E is a convex subgroup of val K.

Lemma 6.15 (val-lem) Let K be an algebraically closed valued field, L an extension of transcendence degree 1. Let c ∈ L \ K. Then the following conditions are equivalent:

1. res (K) ≠ res (L) or val (K) ≠ val (L)

2. res (K(c)) ≠ res (K) or val (K(c)) ≠ val (K)

3. T (c/K) = {val(c − b) : b ∈ K} has a maximal element. In particular, the third condition depends on K, L alone and not on the choice of c.

T (c/K) = val K iff L embeds into the completion K̂ as a valued field.

Proof : Let O_{L} be the valuation ring of L. (1) implies (2) since if res (K(c)) = res (K), and val (K(c)) = val (K), then res (K(c)) is algebraically closed and val (K(c)) is divisible; so they cannot change under algebraic field extensions. Now assume val (K(c)) ≠ val (K). Then val f(c) ∉ val (K) for some f ∈ K[X]. Splitting f into linear factors, wee see that we can take it to be linear. So val (c − b) ∉ val (K) for some b ∈ K. It follows that val (c − b′) ≤ val (c − b) for all b′ ∈ K. (Otherwise val (c − b) = val (b − b′).) Next suppose val (K(c)) = val (K), but res f(c) ∉ res (K) for some f ∈ K[X] with f(c) ∈ O_{L}. Taking into account val (K(c)) = val (K), we can split f into linear factors f_{i} such that each f_{i}(c) ∈ O_{L}. Thus ac − b ∈ O_{L}, res (ac − b) ∉ res (K) for some a, b ∈ K. It follows that val (ac − b′) ≤ 0 for all b′ ∈ K. So val (c − b/a) ≥ val (c − b′′) for all b′′ ∈ K. This proves that (2) implies (3).

Now assume (3): val (c − b) ≥ val (c − b′) for all b′ ∈ K. If val (c − b) ∉ val (K), (2) holds. If val (c − b) = val (d), d ∈ K, then (c − b)/d ∈ O_{L}, and val ((c − b)/d − d′) ≤ 0 for all d′; so res ((c − b)/d) ∉ res (K). Thus (1). □

Example 6.16 (vt4-e) Let F be a field, K = F (t_{0}, t_{1}, …)^{a}, valued over F with 0 < val (t_{0}) << val (t_{1}) << · · ·. Let K′ = F (t_{e}, t_{e+1}, …)^{a}, and let c ∈ K̂ \ K̂′. Then T (c/K′) = { val (c − b) : b ∈ K′} has a maximal element.

Proof If T (c/K′) is unbounded in val (K′), then c ∈ K̂′. Otherwise, for all b ∈ K′, val (c − b) ≤ α. Find c′ ∈ K with val (c − c′) > α. Then T (c′/K′) = T (c/K′). Now K^{a}/(K′)^{a} is generated (as an algebraically closed field) by e elements, whose values are ℚ-linearly independent over val (K′). Thus val K′(c′) ≠ val K′; so 6.15 applies, and shows T (c′/K′) has a maximal element. □

Lemma 6.17 (val-ort) Let K be an algebraically closed valued field, L an extension of transcendence degree 1, c, d ∈ L \ K. Then E(c/K) = E(d/K).

Proof

If one of T (c/K), T (d/K) has a last element, then by 6.15 so does the other, and E(c/K) = (0) = E(d/K). Thus we may assume T (c/K), T (d/K) have no last element, so that val K = val L =: Γ; and that E(c/K) ⊆ E := E(d/K).

Special Case E = Γ.

In this case, by the last statement of 6.15, L embeds into K̂, and hence E(c/K) = Γ too. In general, let Γ′ = Γ/E, r : Γ → Γ′ the quotient map, and let val′ : L → Γ′ be the

induced valuation. Note that val′(L) = val′(K) = Γ′. Call the residue fields K′, L′. Let T′(y/K) = {val′(y − b) : b ∈ K}.

If E(c/K) ≠ E(d/K), then T′(c/K) has a last element. (Indeed let γ ∈ E(d/K) with γ > 0 and γ ∉ E(c/K). Then there exists α ∈ T (c/K) with α + γ ∉ T (c/K). Let α′ be the common image of α, α + γ in Γ′. Then clearly α′ is the greatest element of T′(c/K).) By 6.15, T′(d/K) has a greatest element too. Effecting additive and multiplicative translations, we may assume these greatest elements are val′(c) = 0, val′(d) = 0. Let c′, d′ be the residues of c, d.

Since val′(L) = val′(K) = Γ′, by 6.15 we must have K′ ≠ L′. We have an induced valuation val′′ : L′ → E, val′′(res′(a)) = val (a) for a with val (a) ∈ E. Clearly T′′(c′/K′) = { val′′(c′ − b′ : b′ ∈ K′)} = E ∩ T (c/K), and similarly for d; thus E′′(c′/K′) = E(c/K), E′′(d′/K′) = E(d/K). But now T′′(d/K) is the entire value group E(d/K), so by the special case, E′′(c′/K′) = E′′(d′/K′). □

Corollary 6.18 (vt1r1) Let K′ = K(c)^{a}, K′′ = K(d)^{a} be two valued field extensions of an algebraically closed valued field K, with E(c/K) ≠ E(d/K). Then K′, K′′ have a unique valued field amalgam.

We have T (c/K′′) = HT (c/K).

In particular if val K′′ = val K then T (c/K′′) = T (c/K), and E(c/K′′) = E(c/K).

Proof Assume K′, K′′ are embedded in L = K′K′′. Let a ∈ K′, b ∈ K′′. Then E(a/K) = E(c/K) ≠ E(d/K) = E(b/K), so T (a/K) ≠ T (b/K). One of T (a/K), T (b/K) is bigger, say T (a/K) ⊂ T (b/K). Find e ∈ K with val (b − e) > T (a/K). Then val (a − b) = val (a − e). This shows that the values of elements a − b are all determined. In particular the values val (c − b) for b ∈ K(d) are determined; this determines tp(c/K′′) and hence tp(K′/K′′).

Moreover we see that every element val (c − a) of T (c/K′′) either equals val (c − e) for some e ∈ K, so that it lies in T (c/K), or else equals val (a−e) where val (c−e) > val (a−e); so in either case it lies in HT (c/K). □

This proof uses the stationarity theorem of [Haskell-Hrushovski-Macpherson] (cf. Theorem 2.11); alternatively one can prove the corollary directly, along the lines of the proof of 6.17.

Proposition 6.19 (vt1r3) Let L be a valued field extension of an algebraically closed valued field K. Let a_{i} ∈ L \ K,

T_{i} = T (a_{i}/K) = {val(a_{i} − b) : b ∈ K}

Assume that the convex subgroups

E_{i} = E(a_{i}/K) = {e ∈ val K : e + T_{i} = T_{i}}

are distinct. Then a_{1}, …, a_{n} are algebraically independent over K. The valued field structure of K(a_{1}, …, a_{n}) is uniquely determined by the T_{i}.

Proof By assumption, a_{i} ∉ K^{alg}. Use induction on n. Some E_{i} is nonzero; say E_{1} ≠ (0). Let K′ = K(a_{1})^{a}. By 6.15, val K′ = val K. By 6.17, a_{i}, a_{1} are algebraically independent over K for i ≠ 1. By 6.18, the type of (a_{i}, a_{1})/K is determined, and E(a_{i}/K′) = E(a_{i}/K). By induction, a_{2}, …, a_{n} are independent over K′, and their type over K′ is determined. The conclusion follows. □

Let L be a model, and let Let F ≤ K_{1}, …, K_{n} ≤ L. We say that K_{1}, …, K_{n} are almost orthogonal over F if for any a = (a_{1}, …, a_{n}), a_{i} a tuple of elements of K_{i}, ∪_{i}tp(a_{i}/F) implies tp(a/F).

In the case of an algebraically closed valued field F, K_{1}, K_{2} are almost orthogonal over F if whenever f_{i} : K_{i} → M are embeddings of valued fields, with f_{1}|F = F_{2}|F, there exists a valued field embedding f : K → M, with f|K_{i} = f_{i}.

In this language, 6.19 asserts the almost-orthogonality of certain unary types.

Let F, K, L, K′, L denote algebraically closed valued fields. F̂ denotes the completion of F.

When K ≤ L is an extension of valued fields, K = K^{a}, and tr.deg._{K}L = tr.deg._{K_{res}} L_{res} , we will say that L/K is purely inertial. If instead tr.deg._{K}L = rkℚ( val (L)/ val (K), we will say that L/K is purely ramified.

The following lemma will use Theorem 2.11 of [Haskell-Hrushovski-Macpherson], or rather the following corollary: if F (a), K are almost orthogonal over F = F^{a}, then so are F (a)^{a}, K. (Note that this is immediate for perfect Henselian closure in place of algebraic closure.)

Lemma 6.20 (acf-acvf-comp)

Let F ≤ K_{1}, K_{2} be algebraically closed valued fields. Assume K_{1} can be embedded into K̂_{2} over F; and that in any embedding of K_{1}, K_{2} into a valued field L, the images are linearly disjoint over F. Then K_{1}, K_{2} are almost-orthogonal over F.

In particular, if F ≤ K_{1}, K_{2} ≤ F̂, then K_{1}, K_{2} are almost-orthogonal over F if and only if they are linearly disjoint over F.

Proof We may assume here that tr.deg._{F} K_{1} < ∞. If F ≤ F′ ≤ K_{1} then the hypotheses hold for F, F′, K_{2} and then also for F′, K_{1}, F′K_{2}; so using transitivity of almost orthogonality the lemma reduces to the case tr.deg._{F} K_{1} = 1. Fix some embedding of K_{1}, K_{2} into a valued field L. Let a ∈ K_{1} \F. Then a ∉ K_{2}, by the linear disjointness assumption. Let B be a K_{2}-definable ball with a ∈ B. So B = B_{ρ}(b) = {y : val (y − b) ≥ ρ}, ρ ∈ val (K_{2}), b ∈ K_{2}. As a ∉ K_{2}, ρ < ∞. Since b ∈ F̂ and ρ ∈ val (F̂) = val (F), there exists b′ ∈ F with val (b − b′) > ρ. So B is F -definable. Thus tp(a/F) implies a ∈ B. Since B was arbitrary, and K_{2} is algebraically closed, the formulas x ∈ B of this kind generate tp(a/K_{2}). Thus tp(a/F) implies tp(a/K_{2}). So F (a), K_{2} are almost orthogonal over F. By Theorem 2.11 of [Haskell-Hrushovski-Macpherson], K_{1}, K_{2} are almost-orthogonal over F. □

Lemma 6.21 (complete-max1) Let F be a complete valued field, K an immediate extension of F, and a_{1}, …, a_{n} ∈ K linearly independent over F. Then there exists λ = λ(a_{1}, …, a_{n}) ∈ Γ(F) such that if b_{1}, …, b_{n} ∈ F and min_{i} val (b_{i}) = 0 then val (∑ b_{i}a_{i}) < λ.

Proof By induction on n. The case n = 1 is evident, with λ(a_{1}) = val (a_{1}). Suppose for contradiction that for each λ ∈ Γ there exist b_{1}^{λ}, …, b_{n}^{λ} ∈ F with min_{i} val (b_{i}^{λ}) = 0 and val (∑ b_{i}^{λ}a_{i}) > λ. For some i ≤ n, for cofinally many λ we have val (b_{i}^{λ}) = 0; so we may assume val (b_{n}^{λ}) = 0; and so in fact we may assume b_{n} = −1. Now for λ′ > λ we have val (∑_{i=1}^{n−1}(b_{i}^{λ′} − b_{i}^{λ})a_{i}) > λ. Write (b_{i}^{λ′} − b_{i}^{λ}) = c_{i}(λ, λ′)e(λ, λ′) with min_{i} c_{i}(λ, λ′) = 0.

Then λ(a_{1}, …, a_{n−1}) > ∑ c_{i}(λ, λ′)a_{i} > λ − val (e(λ, λ′)) so

val (b_{i}^{λ′} − b_{i}^{λ}) ≥ val (e(λ, λ′)) > λ − λ(a_{1}, …, a_{n−1})

Hence b_{i}^{λ} convereges; let b_{i}^{∞} = lim_{λ→∞} b_{i}^{λ}. Clearly ∑_{i=1}^{n} b_{i}^{∞}a_{i} = 0. But this contradicts the F linear independence of a_{1}, …, a_{n}. Lemma 6.22 (complete-max) Let F = F̂ be an algebraically closed valued field. Let K_{1} be an immediate extension of F, and let K_{2} be another extension of F, almost orthogonal to K_{1} over F. Then K_{1}, K̂_{2} are almost orthogonal over F. Moreover the compositum K_{1}K̂_{2} within any common valued field extension is an immediate extension of K̂_{2} Proof Just as in Lemma 6.20, it suffices to show that for any valued field L ⊃ F and valued field embeddings j_{1} : K_{1} → L, j_{2} : K̂_{2} → L over F, the images j_{i}(K_{i}) are linearly disjoint over F. Since no embeddings were given in advance, we may take j_{1} = j_{2} = Id, and prove based on the given assumptions that K_{1}, K̂_{2} are linearly disjoint.

Let a_{1}, …, a_{n} ∈ K_{1} be linearly independent over F. Let λ = λ(a_{1}, …, a_{n}) from Lemma 6.21. Suppose, for contradiction, that b′′_{1}, …, b′′_{n} ∈ K̂_{2} and ∑ b′′_{i}a_{i} = 0. Then we can approximate b′′_{i} by b′_{i} ∈ K_{2} to within valuation > λ − val (a_{i}), so that val (∑ b′_{i}a_{i}) > λ. Let b′ = (b′_{1}, …, b′_{n}). By the almost orthogonality, tp(b′/F) implies val (∑ x_{i}a_{i}) > λ. So some formula ϕ(x) ∈ tp(b′/F) implies the same. But F ⊧ ACV F, so there exists b = (b_{1}, …, b_{n}) ∈ F^{n} with F ⊧ ϕ(b). Thus val (∑ b_{i}a_{i}) > λ. But this contradicts the definition of λ.

For the last point, note that both K_{1} and K̂_{2} embed into the maximal immediate extension of K̂_{2}, over F. By almost orthogonality the compositum within this extension is isomorphic over F to the compositum within any other. Lemma 6.23 (domin-val) Let K_{0} ⊂ K_{1} ⊂ ⋯ ⊂ K_{n} be algebraically closed valued fields, K = K̂_{0}, L = K̂_{n}. Assume: val (K_{0}) is cofinal in val (K_{n}); and for each i, K_{i+1}/K_{i} is purely ramified, or purely inertial, or K_{i+1} ⊂ K̂_{i}. Let K̄ be an immediate extension of K. Then:

(1) K̄, L are almost orthogonal over K, and M = LK̄ is an immediate extension of L.

(2) L is dominated by L_{res} over K ∪ val (L). Proof (1) Let L^{−} = K̂_{n−1}, M^{−} = L^{−}K̄. We may assume inductively that K̄, L^{−} are almost orthogonal over K, and M^{−} is an immediate extension of L^{−}, hence of K_{n−1}. We will show that M^{−}, L are are almost orthogonal over L^{−}, and M is an immediate extension of L. From this (1) follows by transitivity.

If K_{n} ⊆ K̂_{n−1}, then L^{−} = L and there is nothing to prove. Otherewise K_{n}/K_{n−1} is purely ramified, or purely inertial. Since M^{−} is an immediate extension of K_{n−1}, it follows that L^{−}K_{n}/L^{−} is purely ramified, or purely inertial; for instance in the purely inertial case, using (K_{n−1})_{res} = (L^{−})_{res} we have

tr.deg._{L^{−}}L^{−}K_{n} ≤ tr.deg._{K_{n−1}}K_{n} = tr.deg._{(K_{n−1})_{res}} (K_{n})_{res} ≤ tr.deg._{(L^{−})_{res}} (L^{−}K_{n})_{res}

so the numbers are equal. This also shows that L^{−}K_{n} is almost orthogonal to M^{−} over L^{−}. By Lemma 6.22, L is almost orthogonal to M^{−} over L^{−}, and M^{−}L is an immediate extension of L. But M = M^{−}L.

(2) follows by [Haskell-Hrushovski-Macpherson], Theorem 6.13, taking K̄ to be a maximal immediate extension of K̂. □

Lemma 6.24 (gauss) Let k be an algebraically closed field, K = k(t)^{a}. Let v be a valuation of K/k, R the valuation ring. Let X be a quasi-projective scheme over R, with X(K) and X(k) finite. Consider the residue homomorphism R → k, and the induced map r : X(R) → X(k). If p ∈ X(k) and r^{−1}(p) has n distinct points, then p has geometric multiplicity ≥ n on the scheme X_{0} = X⊗_{R}k. More generally, this holds true if r^{−1}(p) has n points counted with multiplicities on (X⊗_{R}K).

Proof Find an affine open subscheme containing the said n points q_{1}, …, q_{n}, and thus reduce to X = Spec^{σ} A, A a finitely generated R-algebra. Let B be the image of A in A⊗_{R}K. As X is 0-dimensional, A⊗_{R}K is a finite dimensional K-space, and so the finitely generated R-submodule B is a free R-module of finite rank. It follows that dim_{k}(A⊗_{R}k) ≥ dim_{k}(B⊗_{R}k) = dim_{K}(B⊗_{R}K). □

6.6 Structure theory

Example 6.25 Let K be a transformal discrete valuation ring, with value group ℤ[σ], and residue field F (a)_{σ}, where g(a) = 0 for some difference polynomial g over F while g′(a) ≠ 0. Then there exists a 1-generated transformal discrete valuation ring over F, with the same value group and residue field.

Proof We may assume K is Henselian. Pick t such that val (t) generates the value group as a ℤ[σ]-module. Lift g to O_{K}(x), and lift a to O_{K}. Then val (g(a)) > 0, so val ((g−t)(a) > 0). By one step of Hensel’s lemma, we can perturb a so that val ((g − t)(a)) > val (t). Then val g(a) = val (t). So F (a)_{σ} has the same residue field and value group as K.

Example 6.26

1. Let F be a valued field. Form F (t)_{σ}, and let L be the completion. Let a = ∑_{n=0}^{∞} t^{σ^{n}} ∈ L, K = F (t, a)_{σ}. We have σ(a) − a − t = 0.

2. Let y^{σ} − y = t^{−1}. Then (F (t)_{σ})^{inv}(y) is an immediate extension of (F (t)_{σ})^{inv}:

y = t^{−1/σ} + t^{−1/σ^{2}} + …

F (t, y)_{σ} is a ”ramified” extension of F (t)_{σ}, in the sense that dimℚ(ℚ⊗( val (F (t, y)_{σ})/ val (F (t)_{σ}))) = 1.

Proposition 6.27 (vt4) Let K = K^{a} be a transformal valued field with value group val (K) ⊂ ℚ_{σ}. Assume: (*) for any c ∈ K, { val (c − b) : b ∈ σ(K)} has a greatest element.

Let L be a valued difference field extension of K, with tr.deg._{K}(L) < ∞, val (L) = val (K), and L_{res} = K_{res} . Then L̂ = K̂.

Proof Let a ∈ L; we will show that a ∈ K̂. We may assume val (a) > 0. Let a_{n} = σ^{n}(a), C_{n} = {max(0, val (a_{n} − b)) : b ∈ K}, C = C_{0}. Then C_{n} is closed downwards in the nonnegative elements val (K). C_{n} cannot have a maximal element, since L is an immediate extension of K.

If C is unbounded, then a ∈ K̂. Suppose for contradiction that C is bounded. We have ℚ_{σ} = ℚv ⊕ ℚv^{σ} + …; let E_{m} = ℚv ⊕ … ⊕ ℚv^{σ^{m}}. Let m be least such that C ⊂ E_{m} + c_{0} for some c_{0} ∈ C. Since val (L) = val (K), dividing a by an element of K with value c_{0}, we may assume C ⊂ E_{m}. Then σ(C) ⊂ E_{m+1}. Let H_{1} be the convex hull of σ(C). Claim E_{m} + H_{1} = H_{1}. Proof Let 0 < e ∈ E_{m}, 0 < c ∈ C. Since C ⊈ c + E_{m−1}, there exists c′ ∈ C , l ∈ ℕ with c′ − c > e/l. Then σ(c′ − c) > l(c′ − c) > e. So e + σ(c) ≤ σ(c′). Claim H_{1} = C_{1} Proof Let c_{1} < c_{2} < … be cofinal in C. Let b_{n} ∈ K, val (a − b_{n}) = c_{n}; let B_{n} = {x : val (x − b_{n}) ≥ c_{n}}; P = ∩_{n∈ℕ}B_{n}; B_{n}^{σ} = {x : val (x − σ(b_{n})) ≥ σ(c_{n})} P^{σ} = ∩_{n∈ℕ}B_{n}^{σ};

Then P (K) = ∅. So P^{σ}(K^{σ}) = ∅. By the assumption (*) on K/K^{σ}, P^{σ}(K) = ∅. Since a_{1} ∈ P^{σ}, it follows that there can be no b ∈ K with val (b − a_{1}) > σ(C). Thus C_{1} = H_{1}.

Now σ^{n} induces an isomorphism (K, σ(K)) → (σ^{n}(K), σ^{n+1}(K)). So the convex hull of σ^{n}(C) is C_{n} for all n; E_{m+n} + C_{n+1} = C_{n+1}; and C_{n} ⊂ H(E_{m+n}). Thus the hypotheses of Lemma 6.19 are valid of the C_{m}.

But now by 6.19, the elements a_{n} must be algebraically independent over K. This contradicts the assumption of finite transcendence degree. □

The condition (*) is easily seen to apply in the fundamental case: K = F (t)_{σ}^{a}. (See 6.16.)

Example 6.28 The composition of difference polynomials extends to F (t)_{σ}, and by continuity to the completion L of F (t)_{σ}. By 6.27, difference polynomials with nonzero linear term have a left compositional inverse in L.

For instance, the compositional inverse of t + t^{σ+1} is the repeated fraction:

\frac{t}{1 + _{\frac{t^{σ}}{1+ \frac{t^{σ^{2}}}{1+···}}}}

Transformally Henselian fields

Lemma 6.29 (hensel-c) Let K be a complete, algebraically Henselian transformal valued field over F, with value group val (K) ⊂ ℚ_{σ}. Let f ∈ O_{K}[X]_{σ} be a difference polynomial, f_{ν} = ∂_{ν}f. Suppose a ∈ O_{K}, v = val (f(a)) > 2 val (f_{1}(a)) = 2v′. Then there exists b ∈ K, f(b) = 0, val (a − b) ≥ v − v′. Proof Let e_{0} = f(a). We will find a_{1} ∈ O_{K}, val (a − a_{1}) ≥ v − v′, val f(a_{1}) ≥ σ(v). Since val (f_{1}(a) − f_{1}(a_{1})) ≥ v − v′ > v′ = val f_{1}(a), it follows that val f_{1}(a_{1}) = v′. Iterating this we get a sequence a_{n}, with v_{n} =_{def} val f(a_{n}) ≥ σ(v_{n−1}), and val (a_{n−1} − a_{n}) ≥ v_{n} − v′. As K is complete, there exists a unique b ∈ K such that the sequence a_{n} converges to b; by continuity, f(b) = 0.

To obtain a_{1}, write a_{1} = a + re, with e = e_{0}/f_{1}(a), and r ∈ O_{K} to be found. Then

f(a_{1}) = f(a + re) = e_{0} + ∑_{i=1}^{m} f_{i}(a)(re)^{i} + ∑_{ν} f_{ν}(a)(re)^{ν}

Here the f_{i} are transformal derivatives, cf. 2.12 ν runs over indices ≥ σ, while i ranges over the nonzero finite indices. Note that val f_{ν}(a)(re)^{ν} ≥ val (σ(e)) = σ(v). Thus it suffices to find r with e_{0} + ∑_{i=1}^{m} f_{i}(a)(re)^{i} = 0, or with r a root of

g(x) = 1 + x + f_{2}(a)(e^{2}/e_{0})x^{2} + … + f_{m}(a)(e^{m}/e_{0})x^{m}

(Divide through by e_{0}, noting f_{1}(a)(e/e_{0}) = 1.) This (ordinary) polynomial has derivative

g′(x) = 1 + 2f_{2}(a)(e^{2}/e_{0})x + …

Now for k ≥ 2, val (e^{k}/e_{0}) = k(v − v′) − v ≥ 2(v − v′) − v = v − 2v′ > 0. Thus res g′ is a nonzero constant polynomial. By the ordinary Hensel lemma, there exists r ∈ O_{K} with g(r) = 0, near any c ∈ O_{K} with val g(c) > 0. Letting c = −1 we see that there exists r ∈ O_{K} with g(r) = 0. □

We will tentatively call a transformal valued field satisfying the conclusion of 6.29 of characteristic 0 transformally henselian. In characteristic p > 0, we demand also that the field L be perfect, and that all the Frobenius twists (L, σ ∘ (ϕ_{p})^{n}) also satisfy the conclusion of 6.29. ( ϕ_{p}(x) = x^{p}).

Lifting the residue field

Corollary 6.30 (res-lift-lem-0) Let K = K^{a} ≤ L be transformal valued fields, with L transformally henselian. Let f ∈ K_{res} [X]_{σ} be a difference polynomial of order r, f′ = ∂_{1}f. Let ā ∈ L_{res} , f(ā) = 0, f′(ā) ≠ 0, tr.deg._{K_{res}} K_{res} (ā) = r. Then there exists a purely inertial extension K′ = K(b)_{σ} of K, K′ ≤ L, ā ∈ res (K′). Proof Pick any a ∈ O_{L} with res (a) = ā, and also lift f to O_{K}[X]_{σ}. Then val f(a) > 0, val f′(a) = 0, so by the transformal Hensel property (cf. 6.29) there exists b ∈ O_{K} with f(b) = 0, res (b) = ā. Clearly K′ = K(b)_{σ} is purely inertial over K. □

Let K be an ω-increasing transformal valued field over F. The set of elements of K satisfying nontrivial difference polynomials over F forms a difference subfield F′ of K; it is the union of all subfields of transformal transcendence degree 0 over F. The residue map res is injective on F′: to see this, suppose res (a) = 0 and a ≠ 0. Then val (a) > 0, then 0 < val(a) << val (σ(a)) << …, so a, σ(a), … are algebraically independent over F, so a ∉ F′.

Corollary 6.31 (res-lift-lem) Let K be an ω-increasing transformally henselian valued field over F. Let ā ∈ K_{res} , f(ā) = 0, f′(ā) ≠ 0, f ∈ F [X]_{σ} a difference polynomial, f′ = ∂_{1}f. Then F (ā)_{σ} lifts to K: there exists a subfield F (b)_{σ} of K such that the residue map restricts to an isomorphism F (b)_{σ} → F (ā)_{σ} (of difference F -algebras.)

Proof Pick any a ∈ O_{K} with res (a) = ā, and also lift f to O_{K}[X]_{σ}. Then val f(a) > 0, val f′(a) = 0, so by 6.29, there exists b ∈ O_{K} with f(b) = 0, res (b) = ā. The difference field F (b)_{σ} has finite transcendence degree m over F, and hence contained in F′; thus it is trivially valued, so the residue map is injective on F (b)_{σ}. since res (b) = ā, it carries F (b)_{σ} into and onto F (ā)_{σ}. □

Proposition 6.32 (res-lift) Let K be a perfect ω-increasing transformally henselian valued field over a difference field F.

Assume F^{inv} ⊂ F^{a}, and K_{res} has transformal transcendence degree 0 over F. Then res : F′ → K_{res} is surjective.

Proof By Lemma 5.3, (F′)^{inv} ⊂ (F′)^{a}; and clearly F′ is perfect. We may assume F′ = F, and show that res (K) = F. Suppose K_{res} ≠ F. In characteristic 0, choose a difference polynomial f ∈ F [X]_{σ} of smallest possible transformal order and degree, and ā ∈ K_{res} with f(ā) = 0. If f ∈ F [X^{σ}]_{σ}, then replacing ā by σ(ā) we may lower the degree of f; unless σ(ā) ∈ F. But in this case, ā ∈ F^{a}, so the order of f must be 0, so f ∈ F [X^{σ}] is impossible. Thus f ∉ F [X^{σ}]. It follows that f′ ≠ 0 and f′ has smaller order or degree. So the hypothesis of 6.31 is met, and we can increase F′.

In positive characteristic, this may not be the case, because of polynomials such as x^{σ}−x^{p}. However, after replacing σ by τ = σ(x^{1/p^{m}}) for large enough m, it is possible to find such an ā for τ, lowering the degree of f further. Assuming the residue field of finite total dimension, this permits lifting the residue field; we may then return from τ to σ. □

Remark 6.33

In positive characteristic, when K is a transformal discrete valuation ring over F, the Proposition applies to the perfect closure of K. But we can lift K_{res} to a subfield of the completion of a smaller extension L of K̂ within the perfect closure, such that L is still the completion of a transformal discrete valuation ring, and in particular val (L) ⊂ ℤ_{σ}; the details are left to the reader.

Example 6.34 (res-lift-ex)

The assumption in 6.32 that F^{a} is inversive is necessary: take F = ℚ(b, b^{σ}, …), K = F (t, c), σ(c) = b + t.

Characterizing the completion Here is one version of a theorem characterizing completions of algebraically closed valued fields of transformal dimension one over a difference field F.

Lemma 6.35 (char-comp) Let L = L^{a} be a transformal valued field over F = F^{inv}, with value group ⊂ ℚ_{σ}. Assume L is complete. Then the following conditions are equivalent:

1. L = L̂_{0} for some transformal valued field L_{0} of transformal dimension 1 over F.

2. Whenever L′ = (L′)^{a} ≤ L is complete with the same value group and residue field, L′ = L.

3. L ≃ K̂, where K = (L_{res} (t)_{σ})^{a}.

Proof Assume(1), and let L′ be as in (2). By 6.14, L has value group ℚ_{σ}, and L_{res} has transformal transcendence degree 0 over F. So L_{res} is inversive.

Let t ∈ L′ be such that val (t) generates val (L) as a ℚ[σ]-module. By 6.32, L_{res} lifts to a trivially valued difference subfield F′ of L′. Let K = (F′(t)_{σ})^{a}. Then K_{res} = L_{res} , and val (K) = val (L).

We have val (K) ≠ val (K^{σ}), while tr.deg._{σ(K)}K = 1. By 6.15, the hypothesis (*) of 6.27 holds for K. Thus by 6.27, L = K̂. Since K̂ ≤ L′ ≤ L, we have also L′ = L. This proves (2).

The same proof shows that K embeds into L; so assuming (2), we conclude (3), using the assumption of (2) in place of 6.27.

(3) implies (1) trivially. □

Lifting the value group Let K = K^{a} be a valued field. If K′/K, K′′/K′ are purely ramified, then clearly K′′/K′ is purely ramified. Thus K has a maximal purely ramified extension K′ within a given extension L (not necessarily unique.)

Let L be a transformally valued field. Let

H_{e} = H_{e}(L) = {v ∈ val (L) : (∀u > 0)(|v| < σ^{e}(u))}

H_{e} is a convex subgroup of val (L); in case val (L) = ℚ_{σ}, H_{e} is the e’th nonzero convex subgroup. When v > c for every c ∈ H_{e}, we write: v > H_{e}. Let h_{e} : val (L) → val (L)/H_{e} be the quotient map, and let

V AL_{e} = h_{e} ∘ val : L^{∗} → val (L)/H_{e}

We obtain auxiliary valuations, with residue map denoted RES_{e}. Moreover we obain a valuation V AL_{e}^{∗} : RES_{e} → H_{e}, with residue field L_{RES} . Neither of these is a transformal valuation.

When L = F (t)_{σ}^{a}, RES_{e} induces an isomorphism F (t, t^{σ}, …, t^{σ^{e−1}})^{a} → RES_{e}(L) naturally. When L = F̂ (t)_{σ}^{a} the same is true, since any element of L is close to an element of F (t)_{σ} to within a value > H.

Proposition 6.36 (lift-val) Let L = L^{a} be a transformal valued field of transformal dimension 1 over an inversive difference field F = L_{res}; with value group ℚ_{σ}. Let F < K = K^{a} ≤ L̂. Then there exists a difference field K′ ≤ L̂, with

1. K ≤ K′, and K′/K is purely ramified.

2. With e = rkℚ val (L)/ val (K′), σ^{e}L̂ = K̂′.

Proof

Let K′ be a maximal purely ramified difference field extension of K within L̂. Then K′ = (K′)^{a}.

By 6.35, L̂ = F̂ (t)_{σ}^{a} for some t. Let e = rkℚ val (L)/ val (K′). Then there exists s ∈ K′, val (s) > 0, val (s) ∈ H_{e+1}. Fix N ≥ e + 1 for a moment. Let t_{n} = σ^{n}(t), s_{m} = σ^{m}(s), t̄_{n} = RES_{N}(t_{n}), s̄_{n} = RES_{N}(s).

We saw that tr.deg._{F} RES_{N}(L) = N. Thus t̄_{0}, …, t̄_{e}, s̄_{0}, …, s̄_{N−e−1} are algebraically dependent over F. Since s ∈ H_{e+1}, s̄_{0}, …, s̄_{N−e−1} are algebraically independent over F, using the valuation val _{e}^{∗} . Let m be maximal such that t̄_{m}, …, t̄_{e}, s̄_{0}, …, s̄_{N−e−1} are algebraically dependent over F.

Write f(t̄_{m}, …, t̄_{e}, s̄_{0}, …, s̄_{N−e−1}) = 0, f ∈ F [X_{m}, …, X_{e}, s̄_{0}, …, s̄_{N−e−1})] of minimal X_{m}- degree.

We now argue that m = e. Lift f to f(X_{m}, …, X_{e}, s_{0}, …, s_{N−e−1}) ∈ O_{L}[X_{m}]_{σ}, viewed as a difference polynomial in X_{m} ( with coefficients involving the s_{i}, and X_{m+j} = (X_{m})^{σ^{j}}.) In characteristic 0, let f′ be the (first) derivative of f with respect to X_{m}; then f′ ≠ 0, and so by minimality of f, RES_{N}f′(t̄_{m}, …, t̄_{e}) ≠ 0, i.e. val f′(t_{m}, …, t_{e}) ∈ H_{N}. By 6.29, applied to the element t_{m}, there exists b ∈ L̂, val (t_{m} −b) > H_{N}, f(b, …, σ^{e−m}(b), s_{0}, …, s_{N−e−1}) = 0. Let L′ = K′(b)_{σ}. Clearly tr.deg._{K}L′ ≤ e − m, while rkℚ val (L′)/ val (K′) ≥ e − m (since val (σ^{i}(b)) = val (t_{m+i}), and val (t_{m}) << val (t_{m+1}) << … << val (t_{e−1}) << val (c) for any c ∈ K′, val (c) > 0.) However rkℚ val (L′)/ val (K′) ≤ tr.deg._{K′}L′, so these numbers are equal, and L′/K′ is purely ramified, of transcendence degree m − e. But K′ is a maximal purely ramified extension; so K′ = L′, and m = e.

In characteristic p, we first replace σ by σ ∘ ϕ^{−l} where ϕ(x) = x^{p}, reduce the equation if it is now a p’th power, till t_{m} occurs with an exponent that is not a power of p. We then use the above argument. So m = e in any characteristic.

Thus t̄_{e} ∈ RES_{N}(K′) (since this residue field is algebraically closed). So for some b_{N} ∈ K′, val (t_{e}−b_{N}) > H_{N}(L). Now letting N → ∞, we see that t_{e} ∈ K̂′. Let K′′ = F (t_{e})_{σ}^{a}; then σ^{e}L̂ = K̂′′ ⊂ K̂′, and it remains to show equality. Note that val (K′′) = ℚ[σ] val (t_{e}) = val (K′), and res (K′′) = F = res (K′). Thus K′/K′′ is immediate, so for any c ∈ K′, T (c) = { val (c − b) : b ∈ K′′} can have no maximum value (6.15). But by 6.16, for any c ∈ L̂ \ K̂′′, T (c) does have a maximal element. Thus K′ = K′′. □

Corollary 6.37 (char-comp-cor) If L satisfies 6.35 (1)-(3), and K is an algebraically closed, complete difference subfield of L, K_{res} = L_{res} , then either K is trivially valued or K satisfies the same conditions. Proof By 6.32, we may assume L, K are transformally valued fields over F, F = K_{res} = L_{res} . Let K′ be as in 6.36. Then K′/K is purely ramified, of finite transcendence degree r; there exist therefore K_{0} ≤ K,K′_{0} ≤ K′, tr.deg._{F} (K_{0}) finite, tr.deg._{K_{0}}(K′_{0}) = r = rkℚ val (K′_{0})/ val (K_{0}), K′ = KK′_{0}. We can choose K_{0} algebraically closed, and with value group equal to val (K). So K̂_{0} ≤ K̂′_{0} ≤ K̂′. K̂′ is isomorphic (by 6.35 (1 → 3)) to L, so it satisfies 6.35 (1)-(3); by (2), since K̂′_{0} is a complete difference subfield with the same value group and residue field, K̂′ = K̂′_{0}. Thus K̂_{0} ≤ K ≤ K̂′ = K̂′_{0}. If c ∈ K̂′ \ K̂_{0}, as K′_{0}/K_{0} is purely ramified, and as in the proof of 6.16, K_{0}(c)/K_{0} is also purely ramified. In particular if c ∈ K \ K̂_{0} then val (c) ∉ K_{0}; but we chose val (K_{0}) = val (K). Thus K = K̂_{0}. Now K̂_{0} satisfies 6.35 (1). □

When E/F is an extension of valued fields, let

rk_{val} (E/F) = dim ℚ(ℚ⊗ val (E)/ val (F)) + tr.deg._{F} res E

When E, F are subfields of a valued field L, we let rk_{val} (E/F) = rk_{val} (EF/F), EF being the compositum of E, F in L. A similar convention holds for transcendence degree.

Corollary 6.38 (vrk) Let K ≤ L = L^{a} be an ω-increasing transformal valued field of transformal dimension 1 over an inversive difference field F, with value group ℚ_{σ}. Then the following are equivalent:

1. rk_{val} (L/K) ≤ e

2. There exist difference fields K ≤ K_{r} ≤ K̂, and K_{r} ≤ K_{1} ≤ K_{2} ≤ L̂, with: (a) tr.deg._{F} K_{r} ≤ tr.deg._{F} K_{res}

(b) tr.deg._{K_{r}} K_{1} = tr.deg._{K_{res}} L_{res} = e_{res}

(c) tr.deg._{K_{1}}K_{2} = rkℚ val (K_{2})/ val (K_{1}) = e_{1}

(d) K̂_{2} = L̂^{σ^{e_{2}}}

(e) e_{res} + e_{1} + e_{2} ≤ e

3. There exists a difference field K′, K̂ ≤ K′ ≤ L̂, tr.deg._{K}K′ + e_{2} ≤ e, K̂′ = L̂^{σ^{e_{2}}} .

4. There exist difference fields K′, M, K̂ ≤ K′, K̂′ ≤ M, M̂ = L̂, with tr.deg._{K}K′ +

tr.deg._{K′}(M) ≤ e.

Thus, if rk_{val} (L/K) < ∞, there exists a chain of difference subfields K = M_{0} ≤ M_{1} ≤ ⋯ ≤ M_{5} = L̂ such that M_{i+1} = M̂_{i} for even i < 5, while for odd i < 5 tr.deg._{M_{i}}M_{i+1} = rk_{val} (M_{i+1}/M_{i}).

Proof Since F is inversive, and L_{res} is algebraically closed, by 6.14, L_{res} is also inversive. Thus for any d ≥ 1, res (L^{σ^{d}}) = res (L) By 6.6, rk_{val} (L̂/L̂^{σ^{d}}) = rkℚ(ℚ_{σ}/σ^{d}(ℚ_{σ})) = d.

To show that (1) implies (2), we may assume K = K^{a}. By 6.32, there exists a field of representatives F_{r} ≤ K̂ for the residue map of K̂. We have tr.deg_{F} F_{r} = tr.deg._{F} K_{res} . Let K_{r} = F_{r}K.

By 6.32 again, applied to L̂ over K_{r}, there exists a difference field F_{r} ≤ L_{1} such that res : L_{1} → L_{res} is an isomorphism. Let K_{1} = K_{r}L_{1}. Then tr.deg._{K_{r}} K_{1} ≤ e_{res} ; equality holds by comparing the residue fields. Thus (b).

Now over K_{1}, 6.36 applies, and gives K_{2} with (c,d). We have K ≤ L̂^{σ^{e_{2}}} , and e_{1} ≤ rkℚ val (L̂^{σ^{e_{2}}} /K); so e_{1} + e_{2} ≤ rkℚ( val (L̂)/ val K) = rkℚ( val L/ val K). Thus (e).

(2) obviously implies (3), with K′ = K_{2}K̂. To go from (3) to (4), let K′ be as in (3). By 6.35, L̂ ≃ ˆ (L_{res} (t)_{σ})^{a}, K̂′ = L̂^{σ^{e_{2}}} = ˆ (L_{res} (t^{e_{2}})_{σ})^{a}. Let M = ˆ (L_{res} (t^{e_{2}})_{σ})^{a}(t). Then M̂ = L̂

and tr.deg._{K′}(M) = e_{2}.

(4) implies (1) since rk_{val} is additive in towers, bounded by transcendence degree, and vanishes for completions (rk_{val} _{K} K̂ = 0.)

The final conclusion follows from (4) by taking e = rk_{val} (L/K). We obtain M_{0}, …, M_{5} with M_{i+1} = M̂_{i} for even i < 5, and tr.deg._{M_{1}}M_{2}+tr.deg._{M_{3}}M_{4} ≤ e. But e = rk_{val} (M_{2}/M_{1})+ rk_{val} (M_{4}/M_{3}), and rk_{val} (M_{i+1}/M_{i}) ≤ tr.deg._{M_{i+1}}M_{i}; so all inequalities must be equalities. □

Corollary 6.39 (vrk-dec) Let L be a transformal valued field of transformal dimension 1 over an inversive difference field F, with value group ℚ_{σ}. Let K be a nontrivially valued subfield of L. Let K′ be an extension of K within some ω-increasing transformal valued field containing L; assume tr.deg._{K}K′ < ∞. Then rk_{val} (L/K′) ≤ rk_{val} (L/K).

Proof Note that taking algebraic closure or completion does not change rk_{val} . We will thus assume all these fields are algebraically closed; and we will prove a more general statement, allowing L to be the completion of a transformal valued field of transformal dimension 1.

All residue fields are therefore also algebraically closed and, being extensions of K_{res} of transformal transcendence degree 0, are inversive. Note (say by 6.35) that L′ = K̂′L is also the completion of a transformal valued field of transformal dimension 1.

If K ≤ M ≤ L̂, then rk_{val} (L/K) = rk_{val} (L/M) + rk_{val} (M/K), and rk_{val} (L/K′) = rk_{val} (L/MK′) + rk_{val} (M/K′); so the lemma for M/K; K′ and for L/M; MK′, implies the statement for L/K; K′. By 6.38, we may thus assume one of the following cases holds:

1. L ≤ K̂

2. tr.deg._{K}(L) = rk_{val} (L/K)

In case (1), LK′ ≤ K̂′, so rk_{val} (L/K′) = rk_{val} (K̂′L/K̂′) = 0.

In case (2), rk_{val} (L/K′) ≤ tr.deg._{K′}(K′L) ≤ tr.deg._{K}(L) = rk_{val} (L/K).

□

Example 6.40 In 6.39, the hypothesis that the value group of L is ℚ_{σ} is necessary.

Let F be an inversive difference field, a ∈ F K the inversive hull of F (t)_{σ}. Consider also the field of generalized power series F ((t^{G})), where the coefficient group G is ∪_{n}σ^{−n}ℚ_{σ} ⊂ ℚ[σ, σ^{−1}] (ordered naturally.)

Let L = K(b), where b solves the equation:

σ(x) − tx = a

We can take b to be a generic over F ((t^{G})).

Let

c = a^{σ^{−1}} + a^{σ^{−2}}t^{σ^{−1}} + a^{σ^{−3}}t^{σ^{−1}+σ^{−2}} + a^{σ^{−4}}t^{σ^{−1}+σ^{−2}+σ^{−3}} + ⋯ ∈ F ((t^{G}))

Let K′ = K(c).

Then rk_{val} (L/K) = rk_{val} (K′/K) = 0. But rk_{val} (L/K′) = 1, since d = b−c is a nonzero solution of σ(x) − tx = 0, and hence val (d) = (σ − 1)^{−1} val (t) ∉ ℚ_{σ}.

Proposition 6.41 (domin) Let L be a transformal valued field of transformal dimension 1 over F. Let K = K^{a} ≤ L̂ be a complete subfield. Let K̄ be a maximally complete valued field containing K. Then L, K̄ (and even L̂, K̄) are almost orthogonal over K in ACVF.

Hence L is dominated by L_{res} over K ∪ val (L).

Proof By 6.38, there exists a tower K = K_{0} ≤ ⋯ ≤ K_{6} = L̂ of difference field extensions of K, with K_{i+1} purely ramified or purely inertial over K_{i}, or contained in K̂_{i}. (Namely K ≤ K_{r} ≤ K_{1} ≤ K_{2} ≤ K̂_{2} ≤ M ≤ L̂; with M as in 6.38 (4).) The proposition follows from 6.23.

Discussion The notation tp refers to the quantifier-free valued field type; so does the term ”dominated”. However this implies domination in the sense of transformal value fields too: if K′ is a ( transformal) valued field extension of K, with res (K′), res (L) algebraically independent over res (K), then the quantifier-free valued field type of K′/K ∪ val (L) implies the type of K′/L.

This implies the existence of canonical base change of L/K to extension fields of K, relative to a base change in res (L) and in val (L); the canonicity is such that the result goes through for transformal valued fields. We will make no use of this tool in the present paper.

7 Transformal valuation rings and analyzability

(transvalan)

7.1 The residue map on difference varieties

We begin with some criteria for the residue map to be total, injective and surjective. We will not use them much, but they clarify the picture.

Lemma 7.1 (res-dom) Let D ⊂ D′ = D[a] be difference domains; a = (a_{1}, …, a_{n}) a tuple of generators of D′ as a difference D-algebra. Assume σ(a) ∈ D[a]^{int} ( the integral closure of the domain D[a].). Then for large enough m, for any morphism h : D′ → K, with (K, R, v) an m-increasing transformal valuation field, if h(D) ⊂ R then h(D′) ∈ R.

Proof For each i, σ(a_{i}) is a root of a monic polynomial f_{i} = ∑_{l=0}^{d_{i}} c_{i,l}X^{l} ∈ D[a][X]. The coefficients c_{i,l} are themselves polynomials in a over D; take m bigger than the total degree of all these polynomials. Then if v is a valuation, with v(d) ≥ 0 for d ∈ D and v(a_{i}) ≥ δ for each i, (where δ < 0), then for each i, l, v(c_{i,l}) > mδ.

Assume (K, R, val ) is m-increasing, and h(D) ⊂ R. We may replace D, D′ by their images under h, and assume h is the inclusion. Say val (ha_{1}) ≤ val (ha_{2}) ≤ … ≤ val(ha_{n}). We have to show that δ = val (a_{1}) ≥ 0. Otherwise, as the (monic) leading monomial of f_{i}(σ(a_{i})) cannot have valuation less than all other monomials, we have d_{1}val(σ(a_{1})) ≥ val (c_{1,l}) + lval(σ(a_{1})) for some l < d_{1}. Thus (d_{1} − l)val(σ(a_{1})) ≥ val (c_{1,l}) > mδ. So val σ(a_{1}^{−1}) ≤ m val a_{1}^{−1}. This contradicts the assumption that val is m-increasing, and proves the lemma. □

Definition 7.2 (res-inj-not)

For the sake of the lemma 7.3, define a standard unramified map to be a scheme morphism Spec^{σ} S → Spec^{σ} R, where R is a (not necessarily Noetherian) commutative ring, S = R[a_{1}, …, a_{n}], and we have f_{i}(a_{1}, …, a_{n}) = 0 for some polynomials f_{1}, …, f_{n} over R, with invertible Jacobian matrix. If the (f_{i}) give a presentation of S, i.e. S = R[X_{1}, …, X_{n}]/(f_{1}, …, f_{n}), the map is said to be étale. Let us say that a map f : X → Y of schemes is unramified if for any two points of Y over a field, there exists an open subscheme Y′ of Y containing the two points, X′ = f^{−1}(Y′), with X′ → Y′ (isomorphic to) a standard unramified map. When Y is a difference scheme, we view the map Y → B_{1}Y as a map of schemes, and apply the same terminology.

Lemma 7.3 (res-inj) Let (K, R, K̄) be a strictly increasing transformal valued field. Let X be a finitely generated quasi-projective difference scheme over R, X_{0} = X⊗_{R}K̄. Assume the reduction sequence map X_{0} → B_{1}X_{0} is unramified (7.2). Then the residue map X(R) → X(K̄) is injective.

Proof Consider first the special case X = Spec^{σ} R[x]/f(x), f(x) = x + g(x) + h(x) ∈ R[x]_{σ}; where g has coefficients in the maximal ideal M of R, and h is a sum of monomials x^{ν}, ν ∈ ℕ[σ], nu > 1. Let a, b ∈ X(R) with res (a) = res (b); we will show a = b. We may assume b = 0. Then res (a) = 0 so val (a) > 0; but then val (h(a)) > val (a), and also val (g(a)) > val (a), so f(a) = 0 implies a = 0.

In general, we may assume X = Spec^{σ} A; X_{0} = Spec^{σ} A_{0}, B_{1}X_{0} = Spec^{σ} A_{0}′, with A_{0} = A⊗_{R}K̄, A_{0}′ = σ(A_{0}), with X_{0} → B_{1}X_{0} a standard unramified map (7.2): A_{0} has generators y_{1}, …, y_{n} over A_{0}′, admitting relations F_{1}, …, F_{n}, where (F_{i}) ∈ A_{0}′[y_{1}, …, y_{n}] has invertible Jacobian matrix det(∂_{y_{i}}F_{j}) ∈ G_{m}(A_{0}′).

Complete y_{1}, …, y_{n} to a system of generators of A_{0} over K̄; since the y_{i} already generate A over A_{0}′, the additional generators y_{j} may be chosen from A_{0}′. Each new y_{j} solves the inhomogeneous linear polynomial Y_{j} − y_{j} ∈ A_{0}′[Y_{j}]; adding these generators and relations to the system clearly leaves the Jacobian invertible.

Now the coefficients of the F_{i} lie in σ(A_{0}), so they may themselves be expressed as difference polynomials in the y_{i}, indeed in the σ(y_{i}). Replacing the coefficients by these polynomials, we may assume that F_{i} ∈ K̄[Y_{1}, …, Y_{n}] has coefficients in K̄, rather than in σ(A_{0}). If we convene that ∂σ(Y_{j})/∂Y_{k} = 0, the Jacobian matrix remains invertible.

Lift the generators y_{i} to x_{i} ∈ A. Then any element of A can be written as a difference polynomial over σ(A) in the x_{i}, up to an element of MA = ker A → A⊗_{R}K̄. (Here M is the maximal ideal of R. ) So A has generators x_{1}, …, x_{n}, x_{n+1}, …, x_{N}, where for i > n the image of x_{i} in A_{0} is y_{i} = 0. We still have A_{0} = A_{0}′[y_{1}, …, y_{N}]/(F_{1}, …, F_{N}), where F_{i} = y_{i} for i > n. Lift the F_{i} to f_{i} ∈ σ(A)[x_{1}, …, x_{N}]. Then in A we have a relation f_{i}(x_{1}, …, x_{N}) = g_{i}(x_{1}, …, x_{N}), where g_{i} has coefficients in M.

Consider two elements e = (e_{1}, …, e_{N}), e′ = (e′_{1}, …, e′_{N}) of X(R) specializing to the same point of X_{0}. Replacing x_{i} by x_{i} − e′_{i}, we may assume e′ = 0. Suppose e ≠ 0; let min_{i} val (e_{i}) = β > 0. In these coordinates, let L_{i} be the linear (monomials of order 1) part of f_{i}. The L_{i} are then the rows of an invertible matrix L over R, and L_{i}x−g_{i}(x)+h_{i}(x) = 0, where h_{i} is a sum of σ-monomials of degree > 1. Since e ∈ X(R), we have Le−g(e)+h(e) = 0. (Where g, h are the matrices of σ-polynomials whose rows are g_{i}, h_{i}.)

Now min_{i} val (σ(e_{i})) = σ(β) > β, so val h_{i}(e) > β. Also val g_{i}(e) > β. Writing e = −G^{−1}(g(e) + h(e)), we see that min_{i} val (e_{i}) > β, a contradiction. □

We differentiate difference polynomials using the rule ∂(x_{j}^{σ})/∂x_{i} = 0.

Lemma 7.4 (res-surj) Let (K, R, K̄) be a strictly increasing transformal valued field, with K maximally complete as a valued field. Let X = Spec^{σ} R[X]/(F, J^{−1}), where X = (x_{1}, …, x_{n}), F = (f_{1}, …, f_{n}), where the f_{i} are difference polynomials over R, and J = det ∂f_{i}/∂x_{j}. Then X(R) → X(K̄) is surjective.

Proof Let p ∈ X(K̄); we may assume p = (0, …, 0). Replacing F by J^{−1}(p)F, we can write: F (x) = x + (higher terms) + terms with coefficients from M . We seek e ∈ M^{n}, F (e) = 0. The usual proof of Hensel’s lemma provides such an e, by a transfinite sequence of successive approximations. Maximal completeness permits jumping over limit steps.

Remark 7.5 (res-surj-p)

In this paper, we will only use 7.4 when σ is the standard Frobenius, σ(x) = x^{q} on K, K a complete discrete valuation ring. Here the convention that ∂σ(x_{j})/∂x_{i} = 0 follows from Leibniz’s rule: ∂(x_{j}^{q})/∂x_{i} = 0 in characteristic p.

7.2 Sets of finite total dimension are residually analyzable

(fintdra)

Definition 7.6 (scatter-def) Let K be a valued field. A subset X ⊂ K is scattered if {|x − y| : x, y ∈ X} is finite. X ⊂ K^{n} is scattered if pr_{i}X ⊂ K is scattered for each i.

The definition can be made more generally an ultrametric spaces. We will use it in contexts where X is definable in some expansion of K, and one can envisage X in arbitrary elementary extensions. Then the notion appears sufficiently close to the usual topological one to permit our choice of the term ”scattered”.

A scattered set X may consist of clusters of points of distance α < 1. Each cluster can be enlarged, to reveal a new set of clusters separated by the residue map. After finitely many iterations, this process separates points of X.

Lemma 7.7 (scatter-analysis) Assume X is scattered. Then there are a finite number of equivalence relations E_{0} ⊂ E_{1} ⊂ … E_{n}, such that E_{0} = Id, and for each E_{i+1}-class Y of X, there exists a map f_{i}^{Y} embedding Y/E_{i} into the residue field. The E_{i} and f_{i}^{Y} are quantifier-free definable in the language of valued fields; f_{i}^{Y} is defined by a formula with a parameter depending on Y, while E_{i} requires no parameters beyond those needed to define X.

Proof First take X ⊂ K. Let ρ_{0} < ρ_{1} < … ρ_{n} be the possible values of |x − y| for x, y ∈ X. Say ρ_{i} = |c_{i}|. Let E_{i}(x, y) ≡ |x−y| ≤ ρ_{i}. Given Y, pick b ∈ Y, and let f_{i}(x) = res c_{i}^{−1}(x−b).

When X ⊂ K^{2}, we let E_{i}(x, y) ≡ (|pr_{0}(x) − pr_{0}(y)| ≤ ρ_{i}&pr_{1}(x) = pr_{1}(y)) for i ≤ n, E_{j}(x, y) ≡ |pr_{1}(x) − pr_{1}(y)| ≤ ρ_{j−n} for j > n; etc. □

Proposition 7.8 (scatter-1) Let K be a transformal valued field, with Γ a torsion-free ℤ[σ]-module. Let F ∈ K[x]_{σ} be a nonzero transformal polynomial. Then X = {x : F (x) = 0} is scattered. More generally, any X ⊂ K^{n} of finite total dimension is scattered.

Proof This reduces to X ⊂ K, using projections. When X ⊂ K has finite total dimension, there exists a nonzero difference polynomial F over K with X ⊂ {x : F (x) = 0}. Recall the transformal derivatives F_{ν}. We have F (x + y) = ∑_{ν} F_{ν}(x)y^{ν}. So if a, a + b ∈ X, then b is a root of ∑_{ν>0} F_{ν}(a)Y^{ν} = 0. If F_{ν}(a) = 0 for all ν > 0, then F is constant, so X = ∅. Otherwise, either b = 0 or val F_{ν}(a)b^{ν} = val F_{μ}(a)b^{μ} for some ν < μ; so (μ − ν) val (b) = val F_{μ}(a) − val F_{ν}(a). By the Claim below, { val F_{ν}(a) : a ∈ X} is finite for each ν; so { val (b) : b ≠ 0, a, a + b ∈ X} is also finite. Claim Let G ∈ K[x]_{σ}. Then { val G(a) : F (a) = 0} is finite.

It suffices to see that { val G(a) : F (a) = 0} remains bounded in any elementary extension L of K. In fact if F (a) = 0 then λ val (a) ∈ val (K) for some 0 ≠ λ ∈ ℕ[σ]. Let a ∈ L, F (a) = 0, c = G(a). Then K(a)_{σ} has transcendence degree ≤ deg_{σ} F over K; hence so does the subfield K(c)_{σ}. So H(c) = 0 for some nonzero H ∈ K[x]_{σ}, H = ∑_{ν} d_{ν}x^{ν}. Arguing as above, we see that (μ − ν) val (c) = val d_{ν} − val d_{μ} for some ν < μ. □

Remarks on the liaison groups In most run of the mill cases, if S is the set of zeroes of a difference polynomial F within some ball B, some transformal derivative F′ of F will have a unique root in B, and applying F′ will map S into a ball of known radius around 0; so a direct coordinatization will be possible (internalization without additional parameters.) This process will fail in those some cases where a transformal derivative is constant, but non zero. The polynomial F (X) = X^{σ} + X − a is an example. It seems likely that the examples can all be shown to have an Artin-Schreier aspect, and to become generalized Artin-Schreier extensions upon application of M_{q}.

Another approach, that does not explicitly look at the form of the polynomial, is in [Hrushovski02]. The associated groups of automorphisms of clusters over the residue field are all subgroups of the additive group; this has to do with ”higher ramification groups” (cf. [Serre]).

We will not go in these directions here. However, in the next subsection, we will explain how to assign a dimension to scattered sets, given a notion of dimension over the residue field. (We will apply this to the total dimension over the residue field, obtaining something finer than total dimension of the closure over the valued field.) It will be convenient to explain the way that this dimension is induced in a more abstract setting. Our only application however will be the above mentioned one, and the reader is welcome to use 7.9,7.19 as a dictionary.

7.3 Analyzability and inertial dimension

(anresdim)

Let L be a language with a distinguished sort V. x, y, v will denote tuples of variables; and we will write a ∈ M to mean: a is a tuple of elements of M.

Variables v = (v_{1}, …, v_{m}) will be reserved for elements of V. We use the notation ϕ(x; y) when we have in mind the formulas ϕ(x; b) with b ∈ M ⊧ T. ( This corresponds to the use of relative language, for schemes over a given scheme, in algebraic geometry.)

Data

A theory T (not necessarily complete). A set Φ of quantifier-free L-formulas and a set Φ_{fn} of basic functions of L. We assume Φ is closed under conjunctions and substitutions of functions from Φ_{fn}. The set of formulas of Φ in the variables x is denoted Φ(x); similarly Φ_{fn}(x, t) refers to domain variables x and range variable t. We allow partial functions, whose domain is given by some P ∈ Φ(x). Φ(x; y) denotes formulas in Φ(xy), together with a partition of the variables, as indicated.

A map d_{V} : Φ(v; y) → ℕ.

Example 7.9 (val-case)

L = the language of transformal valued rings, over M_{F} (t)_{σ}^{a}, with a distinguished sort V = V_{res} for the residue field. Here F is a trivially valued inversive difference field, and val (t) > 0.

T = the theory of ω-increasing transformal valued fields.

Φ(x, v) = all quantifier-free formulas ϕ(x, v) implying an ACFA quantifier-free formula of transformal dimension 0 over M, in both x and v variables.

Φ_{fn} includes polynomials, and maps of the form x → res ((x−y_{1})/(y_{1} −y_{2})); with domain {(x, y_{1}, y_{2}) : y_{1} ≠ y_{2}, val (x − y_{1}) = val (y_{1} − y_{2})}.

d_{V} (ϕ(v; y)) ≤ n iff for any M ≤ M′ ⊧ T, and b ∈ M′, ϕ(v; b) implies v ∈ X for some difference scheme X over M′_{res} of total dimension ≤ n.

Later, we will use a rank Rk(L/K) on substructures; it corresponds to rk_{val} (L/K). In this case, the V -dimension dim_{V} defined below will be referred to as the inertial

dimension.

We define V -co-analyzability (relative to T, Φ) and the V -dimension of P ∈ Φ (relative also to d_{V}), using recursion on h ∈ (1/2)ℕ. (Compare [Herwig-Hrushovski-Macpherson].) The second half-step between any two integers does not directly relate to x, but serves to provide additional parameters for future steps. Recall that the variables v refer to n-tuples from V. Since V is fixed, we will simply refer to co-analyzability (but will continue to talk of V -dimension.)

Definition 7.10 (co-an)

1. P (x; u) is 0-step coanalyzable over V (with V -dimension 0, i.e. ≤ n for all n) if T ⊧ P (x, u) ∧ P (x′, u). ⇒ x = x′.

2. P (x; u) is co-analyzable in h + 1/2 steps (with V -dimension ≤ n) if there exist Q ∈ Φ(x; u, v), co-analyzable in h steps (with V -dimension ≤ n_{1} ), R ∈ Φ(v; u) (with d_{V} (R) ≤ n_{2}), and g ∈ Φ_{fn}(x, u; v), such that

T ⊧ P (x; u) ⇒ Q(x, u, g(x, u)) ∧ R(g(x, u); u))

(and n_{1} + n_{2} ≤ n ).

3. P (x; u) is co-analyzable in h+1 steps (with V -dimension ≤ n) if there are finitely many Q_{j} ∈ Φ(y; u) such that T ⊧ P (x; u) ⇒ ⋁_{j}(∃y)Q_{j}(y, u), and for each j,

(P (x; u) ∧ Q_{j}(y, u)) ∈ Φ(x; uy)

is coanalyzable in h + 1/2 steps (with V -dimension ≤ n).

We say P is (V -)co-analyzable if it is so in some finite number of steps. We will write

dim_{V} (P) ≤ r

for: “ P is (V -)co-analyzable, of V -dimension ≤ r. ”. Clearly if T ⊧ P ⇒ P′ and dim_{V} (P′) ≤ r then dim_{V} (P) ≤ r.

In applications, it is convenient to apply this terminology to ∞-definable P = ∧_{i∈I}P_{i}. An ∞-definable P (x, u) is given, by definition, by a collection {P_{i}(x; u_{i}) : i ∈ I}, P_{i} ∈ Φ(x; u_{i}). u may be an infinite list, containing all the finite lists u_{i}. . We have in mind P (M) =_{def} ∩_{i∈I}P_{i}(M). We simply define: dim_{V} (P) = inf{dim_{V} (∧_{i∈I′}P_{i}) : I′ ⊂ I, I′ finite.}.

When K ≤ L ≤ M ⊧ T, and a ∈ L, we will write dim_{V} (a/K) ≤ r if there exists P ∈ Φ(x; y), dim_{V} (P) ≤ r, and b ∈ K, such that M ⊧ P (a, b).

Example 7.11 Assume P (x; y) is V -co-analyzable in 1 step. In this case,P (x; y) is V - internal, i.e. if b ∈ M ⊧ T, there exists an M-definable injective map f : P (x, b) → V^{eq}. All such injective maps have the same image, up to an M-definable bijection; if V is stably embedded, and the given dimension on V is an invariant of definable bijections, then the V -dimension of P equals the dimension of the image of any of these maps f.

See [Hrushovski02] (appendix B) for a general treatment of internality, and associated definable groups.

Example 7.12 Let P (x) ∈ Φ(x), E_{0} ⊂ … ⊂ E_{h} ∈ Φ(x, x′) equivalence relations on P (x), such that for each i, the set of E_{i+1}-classes is internal relative to the E_{i}-classes. (I.e. for every class Y of E_{i}, there exists a function f_{Y} = f(y, b) : Y → V^{eq}, f ∈ Φ_{fn}, parameters b depending on Y, such that f_{Y} is injective modulo E_{i−1}.) In this situation P is said to be V -analyzable. This is a stronger notion than co-analyzability. If the V -dimension of Y/E_{i+1} is n_{i} for each E_{i}-class Y, then ∑ n_{i} is an obvious upper bound for the V -dimension of P.

Remark 7.13 In the case of valued fields, 7.9, if P is scattered, then it is in fact analyzable.

Consider structures K ⊧ T_{∀}; (more precisely, subsets K of models M of T, closed under Φ_{fn}, with the formulas in Φ interpreted according to M.) We will consider pairs K ≤ L with L generated over K by a tuple c; write L = K(c) in this case. Let

tp_{Φ}(c/K) = {ϕ(x; b) : ϕ(x; y) ∈ Φ, b ∈ K, L ⊧ ϕ(c, b)}

Assume given an ℕ ∪ ∞-valued function Rk on such Φ-types. We will assume that the Rk does not depend on the choice of a generator c of L/K, and write Rk(L/K) for Rk(c/K) when L = K(c).

Lemma 7.14 (str-form) Assume:

1. If Rk(L/K) ≤ n, a ∈ V (L), then for some ϕ(v; u) ∈ Φ(v; u) with d_{V} (ϕ) ≤ n, and b ∈ K, M ⊧ ϕ(a, b).

2. If K ≤ K′ ≤ L ≤ M, Rk(L/K′) + Rk(K′/K) = Rk(L/K).

3. If K ≤ K′ ≤ M, c ∈ M, Rk(K(c)/K) < ∞, then Rk(K′(c)/K′) ≤ Rk(K(c)/K).

Let K ≤ L ⊧ T_{∀}, with Rk(L/K) ≤ n. Let a ∈ L, a/K co-analyzable. Then dim_{V} (a/K) ≤ n.

Proof We use induction on the number of steps of co-analyzability (the case of 0 steps being clear.) Suppose a/K is co-analyzable in h + 1/2 steps, i.e. L ⊧ P (a, b), b ∈ K, P co-analyzable in h + 1/2 steps. Then there exists Q ∈ Φ(x; u, v), co-analyzable in h steps, g ∈ Φ_{fn}(x, u; v), and d = g(a, b), such that M ⊧ Q(a; b, d). By induction, there exists Q′ ∈ Φ(x; y′, v) and b′ ∈ K with M ⊧ Q(a, b′, d), dim_{V} (Q′) ≤ Rk(L/K(d)). (Any parameters from K(d) can be written as terms in elements b′ of K, and d.)

By (1), there exists R′(v; u) ∈ Φ(v; u), d_{V} (R′) ≤ Rk(K(d)/K), and b′′ ∈ K, with R′(d, b′′).

Let P′(x; y, y′, u) = Q′(x, y′, g(x, y)) ∧ R(g(x, y); y′′)). Then P′(a; b, b′, b′′) shows that the V -dimension of a/K is ≤ Rk(L/K(d)) + Rk(K(d)/K) = Rk(L/K).

Now suppose a/K is co-analyzable in h + 1 steps. Then L ⊧ P (a, b), b ∈ K, T ⊧ P (x; u) ⇒ ⋁_{j}(∃y)Q_{j}(y, u), and for each j, (*) P (x; u) ∧ Q_{j}(y, u) is coanalyzable in h + 1/2 steps. Let r = Rk(L/K),

Ξ = {S ∈ Φ(x; u, u′, y) : dim_{V} (S) ≤ r}

and let Ξ′ = {¬S(x, b, b′, y) : S ∈ Ξ, b′ ∈ K}.

Claim T_{∀} ∪ tp_{Φ}(a/K) ∪ Ξ′ ∪ {Q_{j}(y, b)} is inconsistent. Proof Suppose otherwise. Then in some M ⊧ T_{∀}, K ≤ M, we can find a′ ⊧ tp_{Φ}(a/K), and d with Q_{j}(d, b), such that (**) for any S ∈ Ξ, M ⊧ ¬S(a, b, b′, d). Let K′ = K(d), L′ = K(a′, d). by (3), Rk(L′/K′) ≤ r. By (*), a′/K′ is coanalyzable in h + 1/2 steps. Thus by induction, dim_{V} (a′/K′) ≤ r. So there exists S ∈ tp_{Φ}(a′/K′) with dim_{V} (S) ≤ r. We can take S = S(x, b, b′, d), S ∈ Φ(x, u, u′, y). So S ∈ Ξ. But this contradicts (**). □

By compactness, for some finite disjunction ⋁_{j′} S_{jj′}(x, u, u′, y) (with S_{jj′} ∈ Ξ) and some P′(x, b, b′) ∈ tp_{Φ}(a/K) implying P (x, b),

T_{∀} ⊧ P′(x, u, u′) ∧ Q_{j}(y, u). ⇒ ⋁_{j′} S_{jj′}(x, u, u′, y)

so dim_{V} (P′(x, u, u′) ∧ Q_{j}(y, u)) ≤ r. By 7.10 (3), dim_{V} (P′) ≤ r. □

Dually, we have:

Proposition 7.15 (form-str) Let Rk satisfy the hypotheses of 7.14. Let P ∈ Φ(x; y) be V -co-analyzable. (Or more generally, P = ∧_{i∈I}P_{i}, P_{i} ∈ Φ(x; y).) Assume: whenever K ≤ M ⊧ T, a ∈ M, b ∈ K, if P (a, b) then Rk(K(a)/K) ≤ r. Then dim_{V} (P) ≤ r

Proof For any M ⊧ T, if a, b ∈ M, M ⊧ P (a, b), letting K be the substructure of M generated by b, we see that there exists Q ∈ Φ(x; y), M ⊧ Q(a, b), dim_{V} (Q) ≤ r. By compactness, T ⊧ P ⇒ ⋁_{j} Q_{j}, with dim_{V} (Q_{j}) ≤ r. It follows that dim_{V} (P) ≤ r. □

Specializing this to our example 7.9, we have:

Proposition 7.16 (vrk-vdim) Let M be an algebraically closed ω-increasing transformal valued fields of transformal dimension 1 over an inversive difference field, with value group ℚ_{σ}. Let ϕ(x) ∈ Φ(x) be a quantifier-free formula in the language of transformal valued fields over M, cf. 7.9. Assume ϕ- is V_{res} -analyzable, and: for any ω-increasing transformal valued field extension L = M(c) of M, with ϕ(c), rk_{val} (L/M) ≤ n. Then ϕ has inertial dimension ≤ n.

Proof This is a special case of 7.15. Note that the proof of 7.14 uses Rk(K′/K′′) only for finitely generated extensions K′, K′′ of K of transformal dimension 0 over K. For such extensions, take Rk(K′/K′′) = rk_{val} (K′/K′′). Then hypothesis (1) of 7.14 is clear since if rk_{val} (K′/K′′) ≤ n then tr.deg.K′_{res} /K′′_{res} ≤ n. (2) follows from additivity of transcendence degree and vector space dimension. And the truth of (3) is the content of 6.39 □

Remark 7.17 There is a canonical function Rk satisfying 7.14 (1-3). To define it, let • Rk_{0}(L/K) = sup{d_{V} (a/K) : a ∈ V (L)}

• Rk_{n+1/2}(L/K) = sup{Rk_{n}(L/K′) + Rk_{n}(K′/K) : K ≤ K′ ≤ L}

• Rk_{n+1}(L/K) = sup{Rk_{n}(LK′/K′) : K ≤ K′, L ≤ M}

• Rk_{∞}(L/K) = sup_{n} Rk_{n}(L/K)

Remark 7.18 (R-descent) Let P (x; y) ∈ Φ(x; y) be V co-analyzable in h steps, with V -dimension ≤ n, with respect to T, Φ.

1. For some finite T_{0} ⊂ T, Φ_{0} ⊂ Φ, Φ_{fn}_{0} ⊂ Φ_{fn}, P has V -dimension ≤ N with respect to T_{0}, Φ_{0}, Φ_{fn}_{0}.

2. Let M_{q} ⊧ T_{0} be a family of models of T_{0}, indexed by an infinite set of integers {q}, and suppose that for any P (v, y) ∈ Φ_{0} , for some β, for all q and all b ∈ M_{q}, |P (M, b)| ≤ βq^{d(P)}. Then for any P (x, y) ∈ Φ_{0} of V -dimension ≤ n, for some β, for all q and all b ∈ M_{q}, |P (M, b)| ≤ βq^{n}.

Proof (1) is obvious from the definition; (2) also follows immediately from the definition, by induction on the number of steps. (For each n, the half-step from n to n + 1/2 increases the V -dimension and the exponent; the half-step from n + 1/2 to n + 1 increases only the constant β.)

Example 7.19 (R-descent-ex) (continuing 7.9). Here T_{0} will be, for some k, the theory of k-increasing transformal valued fields. The family of models M_{q} can be taken to be the fields K_{q}(t)^{alg}, endowed with a nontrivial valuation over K_{q}, and with the q-Frobenius automorphism. The validity of the assumption will be seen in 9.8.

7.4 Direct generation

(dirgen)

Let F be an inversive difference field, K a transformal discrete valuation ring over F. Then K is generated as a difference field by a subfield K(0) with tr.deg._{F} K(0) < ∞, and (letting K(n) be the field generated by K(0) ∪ … ∪ σ^{n}K(0)), tr.deg._{K(0)}K(1) = 1. We wish to show that this automatically implies: K_{res} ⊂ (K(0)_{res} )^{a}. We phrase it a little more generally.

Lemma 7.20 (val2.1) Let F be an inversive difference field, and let (K, R, M, K̄) be a weakly transformal valued field extending (F (t)_{σ}, F [t̆]_{σ}, tF [t̆]_{σ}, F). Let K(0) ⊂ K(1) ⊂ … be subfields of K with K = ∪_{n}K(n), t ∈ K(1); let R(n) = K(n) ∩ R, K̄(0) = res(R(0)).

Assume tr.deg._{F} K(0) < ∞, tr.deg._{K(n)}K(n + 1) ≤ 1, and σ(R(n)) ⊂ R(n + 1).

Then K̄ ⊂ K̄(0)^{a}.

Proof Let us first reduce to the case that K is a transformal valued field, and the val(t_{n}) are cofinal in Γ. Let Γ_{∞} be the convex subgroup of Γ generated by the elements val(t_{n}). (If r ∈ R = ∪_{n}R(n), val(r) ≤ val(t_{n}) then σ(r) ≠ 0, so Γ′ of 6.4(3) contains Γ_{∞}.) Let K̂, R̂, M̂, π, R_{Γ_{∞}} be as in 6.4(6), and let K̂(n) = π(K(n) ∩ R̂).

Note that σ extends to R̂: an element of R̂ has the form ba^{−1}, with a, b ∈ R, 0 ≤ val(a) ≤ val(t_{n}) for some n. Then val(σ(a)) ≤ val(t_{n+1}) (in particular, σ(a) ≠ 0.) Let σ(ba^{−1}) = σ(b)σ(a)^{−1}.

It follows that K̂ is a transformal valued field. Since all elements of F [t̆]_{σ} have valuation in Γ_{∞}, (K̂, R̂, M̂, K̄) extend (F (t)_{σ}, F [t̆]_{σ}, tF [t̆]_{σ}, F). This effects the reduction.

Thus σ extends to an endomorphism of K; and we have σ(K(n)) ⊂ K(n + 1).

Let Γ_{n} denote the divisible hull within Γ(= Γ_{∞}) of {val(a) : a ∈ K(n)}. So Γ_{n} is a group of finite rank.

Claim 1 Let Γ_{I} = {u ∈ Γ : u << σ(u)}. If one of u, σ(u) ∈ Γ_{I} and 0 < u ≤ v ≤ σ(u) then v ∈ Γ_{I}. Each t_{m} ∈ Γ_{I}. Proof If σ(u) ≤ mu, then σ^{2}(u) ≤ mσ(u), so neither u nor σ(u) are in Γ_{I}. Thus u << σ(u). Let 0 < u ≤ v ≤ σ(u). If mu ≤ v for all m, applying σ we obtain mv < mσ(u) ≤ σ(v) for all m, so v << σ(v). If v < mu for some m, then v ≤ mu << σ(u) ≤ σ(v).

Claim 2 If u ∈ Γ_{n} ∩ Γ_{I} , then Γ_{n} ≠ Γ_{n+1}, and in fact Γ_{n+1} has an element greater than any element of Γ_{n}. Proof In an ordered Abelian group, if 0 < u_{1} << u_{2} << u_{3} << … then u_{1}, u_{2}, u_{3}, … are linearly independent. Thus there is no infinite <<-chain of elements of Γ_{n}. So we may take u to be <<-maximal in Γ_{n} ∩ Γ_{I}. But then u << σ(u), σ(u) ∈ Γ_{I} so we cannot have σ(u) ∈ Γ_{n}. At the same time σ(u) ∈ Γ_{n+1}. So Γ_{n} ≠ Γ_{n+1}.

Claim 3 For all n ≥ 0, Γ_{n} ≠ Γ_{n+1}. Moreover, Γ_{n+1} has an element greater than any element of Γ_{n}. Proof As val(t) ∈ Γ_{1} ∩ Γ_{I}, the previous claim applies for n ≥ 1. It applies to n = 0 too if Γ_{0} has an element v ≥ val(t). (Since v ≤ val(t^{σ^{m}}) for some m, so v ∈ Γ_{I}.) Otherwise, Γ_{0} < val (t) ∈ Γ_{1}. This covers all the cases.

Claim 4 K(n + 1)_{res} ⊂ (K(n)_{res} )^{a}. Proof This follows from valuation theory, the previous claim, and the assumption tr.deg._{K(n)}K(n+ 1) ≤ 1.

This finishes the proof of the lemma. □

As in definition 6.2,let

Δ = {v ∈ val(K) : −val(t) < nv < val(t), n = 1, 2, …}

Lemma 7.21 (val2.2) In 7.20, we can also conclude: Δ ⊂ Γ_{0}.

Proof Note Δ ⊂ Γ_{∞} = ∪_{n}Γ_{n}. By Claim 3 of 7.20, Γ_{n+1} has an element greater than any element of Γ_{n}. But tr.deg._{K(n)}K(n + 1) ≤ 1 implies rkℚ(Γ_{n+1}/Γ_{n}) ≤ 1, and it it follows that Δ ∩ Γ_{n} = Δ ∩ Γ_{n+1}. By induction, Δ ∩ Γ_{n} ⊂ Γ_{0}, and the lemma follows. □

Corollary 7.22 (val2.4) In 7.20, let d = tr.deg._{F} K(0). Then R/tR has a unique minimal prime ideal P. The total dimension of R/tR is ≤ d; if equality holds, then tR ∩ R(0) = P ∩ R(0) = (0). Ditto the total valuative dimension rk_{val} (K).

If K is ω-increasing, then ^{σ}\sqrt{P} = M.

Proof Let Δ be as in 7.21, and let P = {a ∈ R : val(a) ∉ Δ}. Then P/tR is the ideal of nilpotents of R/tR, and P is prime.

Let R′ = {a/b : a, b ∈ R, val(b) ∈ Δ}. R′ is a valuation ring, with maximal ideal P R′, and value group Γ/Δ. R′ is not in general invariant under σ, but R′ ∩ K_{n} has value group Γ_{n}/Δ, and by 7.20, 7.21 the rank of this value group grows with n. Thus as in 7.20, if K′ is the field of fractions K′ of R/P, then K′ ⊂ K′(0)^{a}, where K′(0) = resR′(0). So tr.deg._{F} K′ = tr.deg._{F} K′(0) ≤ tr.deg.K(0); the second inequality is strict unless the R′- valuation is trivial on K(0), and in this case R(0) ∩ P = (0).

The total dimension of R/tR equals that of R/P R since P/tR = \sqrt{(}0) in R/tR, and so is bounded by the transcendence degree of the field of fractions of this domain. The total dimension of R/tR equals the reduced total dimension, r.dim(R/tR) = t.dim(R/M) = tr.deg._{F} K̄, plus the transformal multiplicity, or total dimension of Spec R/t over Spec K. As there is a chain of prime difference ideals of length rk_{ram}(K) between M and P, this total dimension is ≥ rk_{ram}(K). Thus

t.dim(R/tR) ≥ tr.deg._{F} K̄ + rk_{ram}(K) = rk_{val} (K)

So rk_{val} (K) ≤ d, and if equality holds then P ∩ R(0) = (0).

The last statement is clear from the definition of P: if K is ω-increasing, and a ∈ M, then val (a) > 0; we have val (a) << val (σ(a)) << …, so they are ℚ- linearly independent, and hence cannot all be in Δ; thus σ^{m}(a) ∈ P for some m ≤ rkℚΔ < ∞. □

8 Transformal specialization

(transspec)

8.1 Flatness

Intersection theory leads us to study difference schemes ”moving over a line”; the behavior of a difference scheme X_{t} depending on a parameter t, as t → 0. In his Foundations of Algebraic Geometry, Weil could say that (a, t) specializes to (a′, 0) (written (a, t) → (a′, 0) ) if (a′, 0) lies in every Zariski closed set that (a, t) lies in; i.e. the point of Spec^{σ} X corresponding to (a′, 0) lies in the closure of the (generic) point (a, t). By Chevalley’s lemma, this automatically implies the existence of a valuation ring, and the theory that comes with that. We think of that as indicating the existence of a ”path” from (a, t) to (a′, 0). In difference schemes, closure and pathwise closure do not coincide. We will use transformal valuations to define the latter.

We can view the valuations (or open blowing up) as revealing new components, that cannot be separated with difference polynomials, but can be separated using functions involving for instance x^{σ−1}. (This can also be given responsibility for the failure of the dimension theorem in its original formulation, cf. [Cohn].)

The above phenomenon can lead to a special fiber with higher total dimension than the generic fiber, or even with infinite total dimension; in this case it cannot be seen as a smooth movement of a single object. To prevent this, we need to blow up the base. It is convenient to replace the affine line once and for all with the spectrum of a transformal valuation scheme (a posteriori, a finite blowing-up will suffice to separate off occult components in any particular instance. In general, we will use the language of valuation theory in the present treatment, so that blowing ups occur only in the implicit background.) Over a valuative base, we will show 8.2 that total dimension does not increase.

The lack of jumps in total dimension is analogous to the preservation of dimension (”flatness”) of the classical dynamic theory; but it is not the Frobenius transpose of the classical statement. The latter corresponds rather to the preservation of transformal dimension, a fact that holds true already over the usual transformal affine line, without removing hidden components. The good behavior of total dimension is related under Frobenius to another principle of classical algebraic geometry; what Weil called the ”preservation of number”.

We actually require something more than the flatness of total dimension when measured globally over a fiber. Consider a difference subscheme X of an algebraic variety V, or more generally a morphism j : X → [σ]_{k}V. Say X is evenly spread (along V, via j) if for any proper subvariety U of V, j^{−1}([σ]_{k}U) has total dimension < dim (V). We need to know that if the the generic fiber X_{t} is evenly spread, then the same is true of the special fiber. To achieve this, we need to replace the naive closure X_{0} of with a pathwise closure X_{→0} =′′ lim_{t→0}X′′_{t}, and the total dimension by a valuative dimension. (cf. 8.9).

As a matter of convenience, since we are interested in the generic point and in one special point at a time, we localize away from the others.

Ă_{k} will be used as a base for moving difference varieties, analogously to the affine line in rational equivalence theory of algebraic varieties. Let X be a difference scheme over Ă_{k}; we will write X_{t} for the generic fiber X ×Ă_{k} Spec^{σ} k(t)_{σ}, and X_{0} for the special fiber X ×Ă_{k} Spec^{σ} k (referring respectively to the inclusion k[t̆]_{σ} → k(t)_{σ} and the map k[t̆]_{σ} → k, t ↦ 0.)

Let X be a difference subscheme of \underline{V} = [σ]_{k}V ×_{k} Ă_{k}, V an algebraic variety over k. We denote by X[n] ⊂ V × V^{σ} × … × V^{σ^{n}} the n’th weak Zariski closure of X, and similarly X_{t}[n], X_{0}[n].

Definition 8.1 (flat1.5) (cf. 8.4.) We will say that a difference scheme X of finite type over Ă_{k} is flat over Ă_{k} if in every local ring, y ↦ t_{n}y is injective.

When X is algebraically reduced, this is equivalent to: X has no component contained in the special fiber X_{0}.

Lemma 8.2 (flat2) Let X be flat over Ă_{k}. If X_{t} has total dimension d over k(t)_{σ}, then X_{0} has total dimension ≤ d over k. Proof View X[n] as a scheme over Spec k[t̆]_{σ}. Since X_{0}[n] ⊂ X[n]_{0}, it suffices to show that dim X[n]_{0} ≤ d (where X[n]_{0} is the fiber of X[n] above t_{0} = t_{1} = … = 0). X[n] arises by base extension from a scheme Y of finite type over Spec S′, where S′ is a finitely generated k-subalgebra of S = k[t̆]_{σ}. Let K = k(t)_{σ}, K[m] = k(t_{0}, …, t_{m}), S[m] = k[t̆]_{σ}[m] = k[t̆]_{σ} ∩ K[m]; then for some m we can take S′ = k[t̆]_{σ}[m]; and dim_{K[m]}(Y ⊗_{S[m]}K[m]) = d. Claim Let A be a finitely generated S[m]-algebra, such that x ↦ t_{m}x is injective on A. Suppose dim_{K[m]}(A⊗_{S[m]}K[m]) = d. (This refers to Krull dimension.) Let A′ = A/JA, where J is the ideal generated by (t_{m}, t_{m}t_{m−1}^{−1}, t_{m}t_{m−1}^{−2}, …). Then A′ is an S[m]/J = S[m−1]- algebra, x ↦ t_{m−1}x is injective on A′ , and dim_{K[m−1]}(A′⊗_{S[m−1]}K[m − 1]) ≤ d Proof Suppose t_{m−1}c ∈ JA; so t_{m−1}c = t_{m}t_{m−1}^{−r}a for some r ∈ ℕ and a ∈ A. As t_{m−1}|t_{m}, x → t_{m−1}x is injective on A. So c = t_{m}t_{m−1}^{−r−1}a. But then c ∈ JA. This shows that x ↦ t_{m−1}x is injective on A′.

To see that the dimension remains ≤ d, let B = A⊗_{K[m]}K[m − 1][t_{m}]′, where K[t_{m}]′ is the localization of K[m − 1][t_{m}] at t_{m} = 0. It is easy to see, by looking at the numerator, that t_{m} is not a 0-divisor in B. We are given that B⊗_{K[m−1][t_{m}]′}K[m − 1](t_{m}) has Krull dimension ≤ d. Thus Spec B → Spec K[m − 1][t_{m}] has generic fiber of dimension ≤ d, and has no component sitting over t_{m} = 0, so it has special fiber of dimension ≤ d as well (over t_{m} = 0); thus B/t_{m}B has Krull dimension ≤ d. But B/t_{m}B = A′⊗_{S[m−1]}K[m − 1], proving the claim.

The lemma follows upon m + 1 successive applications of the Claim. □

Remark 8.3 (move1)

Using the main theorems of this paper concerning Frobenius reduction, and the translation this affords, one can conclude:

1. A statement similar to 8.2 holds for transformal dimension; this corresponds to [Hartshorne] III 9.6 (or, using the dimension theorem in ℙ^{m} ×ℙ^{1}, where X ⊂ ℙ^{m}, note that no component of the special fiber t = 0 can have smaller codimension than a component of a generic fiber of an irreducible variety projecting dominantly to ℙ^{1}.)

2. When X is a closed subscheme of V ×_{k} Ă_{k}, V a proper algebraic variety over k, one can conclude that X_{0} has total dimension equal to d (though not necessarily reduced total dimension d). (The Frobenius specializations give systems of curves over the affine line, having about q^{d} points over a generic point of the affine line, hence the fiber over 0 cannot be of size O(q^{d−1}).)

3. One cannot expect every component of X_{0} to have total dimension d (even with completeness and irreducibility assumptions). Nor can one expect the reduced total dimension to equal d. It suffices to note that the diagonal Δ of ℙ^{1} can be moved to ({pt} × ℙ^{1}) ∪ (ℙ^{1} × {pt}) (and pull back with the graph of Σ.)

The conclusion of 8.2 below for ordinary schemes is usually obtained from flatness hypotheses; cf. e.g. [Hartshorne] III 9.6. This suggested the terminology, as well as the following remark. We have not been able to use it, though, since it is not obvious how to reduce to schemes of finite type while retaining flatness. The lemma refers to the k-algebra structure of k[t̆]_{σ}, disregarding the difference structure.

Remark 8.4 (flat1) Let k be a difference field, M be a k[t̆]_{σ} module, such that t_{m}y = 0 implies y = 0. Then M is a flat k[t̆]_{σ}- module.

Proof By [Hartshorne] III 9.1A, it suffices to show that for any finitely generated ideal I of A_{k}, the map I⊗_{R}M → M is injective. Now k[t̆]_{σ} is a valuation ring, so any finitely generated ideal is principal, I = k[t̆]_{σ}c. The map in question is injective if cy = 0 implies y = 0. But for some m ∈ ℕ, c|t_{m} in k[t̆]_{σ}, so t_{m}y = 0, and thus y = 0.

8.2 Boolean-valued valued difference fields

We wish to consider finite products of transformal valuation domains (corresponding to a finite disjoint unions of difference schemes.) As we wish to include them in an axiomatizable class (for purposes of compactness and decidability), we will permit arbitrary products, and so will discuss briefly Boolean-valued transformal valuation domains. Compare [Lipshitz-Saracino73], [Macintyre1973]; but here we will need them only at a definitional level.

Definition 8.5 (cf. [shoenfield].) Let T be a theory in a language L. The language L_{boolean} has the same constant and function symbols as L; and for each l-place formula ϕ of L, a new l-place function symbol [ϕ], taking values in a new sort B; also, functions ∪, ∩, ¬, 0, 1 on B. T_{boolean} is the theory of all pairs (M, B), where (B, ∪, ∩, ¬, 0, 1) is a Boolean algebra, and M is a Boolean-valued model of T. Thus [R](a_{1}, …, a_{l}) is viewed as the truth value of R(a_{1}, …, a_{l}).

Axioms: universal closures of [ϕ&ψ] = [ϕ] ∩ [ψ], [¬ϕ] = ¬[ϕ]; if ϕ = (∃x)ψ, [ϕ] ≥ [ψ], and (∃x)([ϕ] = [ψ]).

And: [ϕ] = 1, where T ⊧ ϕ.

When B = [2] is the 2-element Boolean algebra, we obtain an ordinary model of T.

If (M, B) ⊧ T_{boolean}, and h : B → B′ is a homomorphism of Boolean algebras, we obtain another model (M, B′) of T_{boolean}, by letting [ϕ]′ = h([ϕ]).

Thus when (M, B) ⊧ T_{boolean}, X = Hom(B, 2), for each x ∈ X we obtain a model M_{x} of T.

When T is a theory of fields, it is not necessary to have B as a separate sort. A model M of T_{boolean} is a ring, and B can be identified with the idempotents of M, via the map [x = 1] : M → B. (The sentence (∀y)(∃x)(y = 0 → x = 0 & y ≠ 0 → x = 1), true in T, must have value 1 in T_{boolean}, and shows that for any m ∈ M there exists an idempotent b with [m = 0] = [b = 0].)

When L has an additional unary function symbol σ and T states that σ(0) = 0, σ(1) = 1, T_{boolean} implies that σ fixes the idempotents.

We will just need the case when T is the theory of transformal valued fields (K, R, M). A model of T_{boolean} is then a difference ring, called a multiple transformal valued field. {r : [val(r) ≥ 0] = 1} is a subring S, called a boolean-valued transformal valuation domain. Thus a boolean-valued transformal valuation domain is a certain kind of difference ring R, without nilpotents, such that for any ultrafilter U on the Boolean algebra B of idempotents of R, R/U is a transformal valuation domain. This class of rings is closed under Cartesian products.

When the Boolean algebra is finite, we say T is finitely valued. A finitely valued transformal valuation domain is just a finite product of transformal valuation domains.

8.3 Pathwise specialization

We will formulate a notion of a ”specialization” lim_{t→0} X_{t} of a difference variety X_{t} over F_{t} = F (t)_{σ}, as t → 0. This will be a difference subscheme of X_{0}; we will denote it more briefly as X_{→0}. In addition, there will be certain data over each point of X_{→0} determining the ”multiplicity” of that point as a point of specialization.

A valuative difference scheme over Ă_{k} is a difference scheme T = Spec^{σ} R, R a transformal valuation domain extending k[t̆]_{σ}, the valuation ring of k(t)_{σ}.

Note that any family of closed difference subschemes of a difference scheme X has a ”union”, a smallest closed subscheme containing each element of the family. This correspond to the fact that the intersection of any family of well-mixed difference ideals is again one. This makes possible the following definition.

Definition 8.6 (vs) Let X be a difference scheme of finite type over Ă_{k}. We define the pathwise specialization X_{→0} to be the smallest well-mixed difference subscheme Y of X_{0} such that for any valuative difference scheme T over Ă_{k} , and any morphism f : T → X over X → Ă_{k}, f(T_{0}) ⊂ Y (i.e. the restriction of f to the fiber over 0 factors through Y.)

Intuitively, T is a kind of transformal smooth curve, and f(T) marks a path from points on X_{t} to points on X_{0}.

Remark 8.7 (vs-r)

(i) Since T is perfectly reduced, X_{→0} depends only on the perfectly reduced difference scheme underlying X. (Though of course X_{→0} need not itself be perfectly or even transformally reduced.)

(ii) One could think of allowing more singular paths, using weakly transformal valuation rings, with nilpotent elements permitted. Presumably, when X_{t} is reduced, and irreducible even in the valuative sense, the two definitions yield the same object.

(iii) A simple point of X_{t} may specialize to a singular point of X_{0}; or several simple points may specialize to one. This multiplicity of specialization is not faithfully reflected in geometric multiplicity on X_{→0}. (For instance, when X is the union of of several ”transformal curves” C_{i} meeting at a point p of the special fiber, each C_{i} will map into X, but separately; so that p is registered just once on X_{→0}.) Thus the local algebra of points on X_{→0} is insufficient to capture the data needed over each point.

One can modify the definition of X_{→0} so that the local algebra will capture the multiplicities; cf. X_{→0}′ in §15.4. We will deal with the issue differently, replacing the local algebra information by scattered sets (definable in the language of transformal valued fields, and residually analyzable.) This approach appears to us much more transparent.

Lemma 8.8 (vs-lim)

Let Z be an algebraic component of X_{→0}. Then there exists a valuative scheme T over Ă_{k}, and a morphism T → X over Ă_{k}, such that Z is contained in the image of T_{0} in X_{0}.

Proof Moving to an open affine difference subscheme of X, we can assume X = Spec^{σ} R; Z corresponds to an algebraically prime difference ideal p; there are homomorphisms h_{j} : R → A_{j} ⊂ L_{j}, L_{j} a transformally valued field extending k[t̆]_{σ}, with corresponding transformal valuation domain A_{j}, such that p contains ∩_{j∈J}h_{j}^{−1}(tA_{j}). Let h : R → A_{∗} = Π_{j}A_{j} ⊂ L_{∗} = Π_{j}L_{j} be the product homomorphism; then h^{−1}(tA_{∗}) ⊂ p, and A_{∗} is a Boolean-valued transformal valuation domain, with Boolean algebra E of idempotents.

Let U_{0} be the filter in E generated by all e ∈ E such that for some r ∈ R \ p, h(r) ∈ tA_{∗} + eA_{∗}. If r_{1}, …, r_{n} ∈ R \ p, and h(r_{i}) ∈ tA_{∗} + e_{i}A_{∗}, then (since p is prime) r = r_{1} · . . . · r_{n} ∈ R \ p; and with e = e_{1} ∩ … ∩ e_{n}, h(r) ∈ tA_{∗} + eA_{∗}; so e ≠ 0. Thus I_{0} is a proper filter, and so extends to an ultrafilter U, with complementary maximal ideal I. Composing h with the map A_{∗} → A_{∗}/I =: Ā ⊂ L_{∗}/I =: L̄, we obtain h^{∗} : R → Ā ⊂ L̄, Ā a transformal valuation domain extending Ă_{k}, such that for r ∈ R \ p, h(r) ∉ tĀ. Thus p contains h^{−1}(tĀ). □

We come to the main estimate for the ”equivalence of the infinite components”. Given a difference scheme X, we have the residue map X_{t} → X_{0} (on the points in a transformal valued field.) If X has finite total dimension over Ă_{k}, the pullback of a difference subvariety W ⊂ X_{0} is a certain scattered set ^{∗}W ⊂ X_{t}. Under certain conditions of direct presentation, we find an upper bound on the inertial dimension of ^{∗}W; it can be smaller than the total dimension of the closure of ^{∗}W within X_{t}.

Let V be an algebraic variety over an inversive difference field k, dim (V) = d. Let VĂ_{k} = [σ]_{k}V ×_{k} Ă_{k}, V_{t} = V ⊗_{k}k(t).

Proposition 8.9 (key) Let X be a closed difference subscheme of VĂ_{k}. Then (1) implies (3), and (1 & 2) implies (3 & 4).

1. Consider transformally valued fields L generated over k(t)_{σ} by c ∈ X_{t}(L) ⊂ V (L). Whenever tr.deg._{k(t)_{σ}} L ≥ d, equality holds, and σ(c) ∈ k(t, c)^{a}.

2. Every weakly Zariski dense (in V) algebraically integral subscheme of X_{0} is transformally reduced.

3. X_{→0} is evenly spread out along V.

4. for any subvariety W of V with dim (W) < d = dim (V),

∗W = {x ∈ X_{t}(R) : res_{X}(x) ∈ W }

has inertial dimension < dim (V).

Explanations:

(1e) A point c of X_{t}(L) corresponds to a morphism from a local difference ring of X_{t}, into L; let k(c)_{σ} denote the field of fractions of the image. The inclusion X_{t} ⊂ [σ]_{k(t)_{σ}} V_{t} allows us to restrict c to a point of V (L), with a morphism from a local ring of V into L; let k(c) denote the field of fractions of the image. Then (1) states that if tr.deg._{k(t)_{σ}} k(c)_{σ} = d, then k(c)^{a} = (k(c)_{σ})^{a}. It follows also that k(c)^{a} = k(V).

(2) implies that a weakly Zariski dense algebraic component is actually a component of X_{0}. Note (2e) that any weakly Zariski dense (in V) difference scheme has total dimension ≥ dim (V).

(3) means: every algebraic component Y of X_{→0} of total dimension ≥ d is weakly Zariski dense in V. Note that (1) is a weak form of the statement that X_{t} is evenly spread out along V.

(4): Here R denotes the valuation ring, a quantifier-free formula in the language of transformal valued rings. res_{X} denotes the map X_{t}(R^{L}) → X_{0}(L_{res} ) induced by res on a transformally valued field L. Proof (1) ⇒ (3) :

We may assume X = Spec^{σ} A. Let Y be an algebraic component of X_{→0}, and let p be the corresponding algebraically prime difference ideal of A. By 8.8, there exists a transformally valued field (K, R) extending (k(t)_{σ}, Ă_{k}) and a difference ring morphism h : A → R over k[t̆]_{σ}, such that p contains h^{−1}(tR); so A/p embeds into a quotient of R/tR.

Note that h^{−1}(tR) may be bigger than (kerh, t), and h(A) cannot be assumed to generate R (cf. §15.4). But we may assume h(A) generates K as a field over k(t)_{σ}. By (1), tr.deg._{k}K ≤ d.

Now t is not a 0-divisor in A/ ker(h). By 8.2, A/(ker(h), t) has total dimension ≤ d, hence so does A/h^{−1}(tR). Moreover if tr.deg._{k}K < d, or if p is not a minimal prime of A/h^{−1}(tR), then then Y = Spec^{σ} A/p has total dimension < d. So we may assume tr.deg._{k}K = d (hence by (1), k(V) embeds into K via h), and p is a minimal prime of A/h^{−1}(tR).

Let K[0] = k(V)^{a} ∩ K. Let K[n] = K[0](t_{0}, …, t_{n−1})^{a} ∩ K. Then σ(K[0]) ⊂ K[1]^{a}; so σ(K[n]) ⊂ K[n + 1]. Now 7.20-7.22 apply to ∪_{n}K[n]. If R/tR has total dimension < d, then so does A/p, hence Y. In this case (3) holds trivially. Otherwise by 7.22, the nilradical of R/tR lifts to an algebraically prime difference ideal P of R, with P ∩ K[0] = (0). If h(x) ∈ P then h(x^{n}) ∈ tR for some n, so x^{n} ∈ p, and thus x ∈ p. So h^{−1}(P) ⊂ p. But p is a minimal prime, so h^{−1}(P) = p. Thus p ∩ k(V) = (0), i.e. Y is weakly Zariski dense in V.

(1&2) ⇒ (4) : We will use 7.16. Let M be an ω-increasing transformal valued field extending K = k(t)_{σ}, with valuation ring R = R^{M}; let c ∈ X_{t}(R^{M}), res (c) ∈ W. Let K = k(t, c)_{σ}. We must show that rk_{val} K/k(t)_{σ} < dim (V).

If tr.deg._{k(t)_{σ}} K < dim (V), this is clear. Otherwise, let K(0) = k(c), K(n) = k(c, …, σ^{n}(c), t, …, σ^{n−1}(t)). By (1), σ(c) ∈ k(c, t)^{a} so σ^{n+1}(c) ∈ k(σ^{n}(c), σ^{n}(t))^{a}. So the hypotheses of 7.20 hold. By 7.22, R/tR has a unique minimal prime ideal P; and rk_{val} (K) < d, unless rk_{val} (K) = t.dim(R/tR) = d, and P ∩ R ∩ K(0) = (0).

In the latter case, we will obtain a contradiction. Let A be the image in R of the local ring of X, corresponding to the point c. Let A(0) be the image of the corresponding restriction to a point of V, as in (1e). So A(0) ⊂ R^{M} ∩ K(0).

Thus A(0) ∩ P = (0). So Z = Spec^{σ} A/(P ∩ A) is a closed difference subscheme of X_{0}, weakly Zariski dense in V. By 8.2, Z has total dimension ≤ d; being weakly Zariski dense in V, it must have dimension d. By (2), Z is a component of X_{0}. So P ∩ A is transformally prime in A.

Let K_{A} (resp. K′ ) be the field of fractions of A/P (resp. R/P). Then tr.deg._{k}K_{A} = d = tr.deg._{k}K′. So K′ ⊂ (K_{A})^{a}. By lemma 2.1, P is transformally prime. But if a ∈ R, val (a) > 0, then σ^{k}( val (a)) > val (t) for some k; so σ^{k}(a) ∈ P; and thus a ∈ P. Hence P is the maximal ideal of R. Since P ∩ K(0) = 0, the residue map is an isomorphism on K(0). But c ∉ W, res (c) ∈ W; a contradiction □

9 Frobenius reduction

(frobred)

9.1 The functors M_{q}

These functors may be viewed as difference-theoretic analogs of “reduction mod p”; we reduce mod p, and also “reduce” σ to a Frobenius map. In the case of subrings of number fields, the situation is what one ordinarily describes using the Artin symbol.

Let R be a difference ring. Let q be a power of a prime number p. We let J_{q}(R) be the ideal generated by p = p · 1_{R} together with all elements r^{q} − σ(r). Let M_{q}(R) = (R/J_{q}(R)). This is a difference ring, on which σ coincides with the Frobenius map x ↦ x^{q}. In particular, J_{q} is a difference ideal.

Lemma 9.1 (localizations) Let T be a multiplicative subset of R with σ(T) ⊂ T, T̄ the image of T under the quotient map in M_{q}(R). Let R[T^{−1}] be the localized ring. Then M_{q}(R[T^{−1}]) = M_{q}(R)[T̄^{−1}].

Proof In whatever order it is obtained, the ring is characterized by the universal property for difference rings W with maps R → W, p = 0 in W, such that the image of T is invertible, and σ(x) = x^{q}. □

Lemma 9.2 Let f : S_{i} → R be surjective maps of difference rings. Then M_{q}(S_{1}) ×_{M_{q}(R)} M_{q}(S_{2}) = M_{q}(S_{1} ×_{R} S_{2})

Proof Similar to that of 9.1

Lemma 9.3 (kernels) Let h : R → S be a surjective difference ring homomorphism, with kernel I. Then the kernel of M_{q}(h) : M_{q}(R) → M_{q}(S) is (I + J_{q}(R))/J_{q}(R).

Proof The kernel is h^{−1}(J_{q}(S))/J_{q}(R). If h(r) ∈ J_{q}(S), h(r) = ∑ t_{j}(s_{j}^{q} − σ(s_{j})) + ps′ for some t_{j}, s_{j}, s′ ∈ S. Writing s_{j} = h(s̄_{j}), t_{j} = h(t̄_{j}), s′ = h(s̄′), r̄ = ∑_{j} t̄_{j}(s̄_{j}^{q} − σ(s̄_{j})) + ps̄′, we obtain r − r̄ ∈ I, r̄ ∈ J_{q}(R), so r ∈ I + J_{q}(R). □

Definition 9.4 Let X be a difference scheme. Let J_{q} be the difference-ideal sheaf on X generated by the presheaf: J_{q}(U) = J_{q}(O_{X}(U)). M_{q}(X) is the closed subscheme corresponding to J_{q}.

The above lemmas show that for a difference ring R without zero - divisors, M_{q}(Spec^{σ} R) = Spec^{σ} M_{q}(R).

Let us say that a q-Frobenius difference scheme is a difference scheme in which σ coincides with x ↦ x^{q} on every local ring. Any ordinary scheme together with a map into Spec 𝔽_{q} admits a canonical q-Frobenius difference scheme structure.

M_{q} yields a functor on difference -schemes into q-Frobenius difference schemes.

Remark 9.5

Let K be a number field, σ an automorphism, R a finitely generated difference subring of K. Then R is also finitely generated as a ring. Spec M_{q}(R) may be empty, and may be reducible. It is always a reduced scheme: if d = [K : ℚ], then σ^{d}(x) = x on R, so x^{q^{d}} = x on M_{q}(R), hence M_{q}(R) has no nilpotents.

Remark 9.6

Let R be as above, S a finitely generated domain over R of positive but finite transcendence degree d, and let σ be the identity automorphism. Then for all but finitely many q, M_{q}(S) is reducible. Indeed every homomorphism of S into K_{q} factors through M_{q}(S). We will see later that the number of homomorphism of M_{q}(S) into K_{q} is O(1)q^{d}; these all fall into the field GF (q). The relevant Galois group has order d, so the number of prime ideals of M_{q}(S) is at least O(1)q^{d}/d, and in particular > 1. But M_{q}(S) is finite, hence it is reducible.

On the other hand, if S = ℤ[X], σX = X + 1, then M_{p}(S) is a domain for all primes p. Even when the fraction field of S is a “regular” extension of K, in the sense that it is

linearly disjoint over K from any difference field of finite transcendence degree over K, and M_{q}(R) is a field, M_{q}(S) can be reducible. Example: R = ℤ[X, Y : σ(X) = Y X].

It would be interesting to study this question systematically.

Notation 9.7 (Xqy) Let X be a difference scheme over Y, q a prime power, y a point of M_{q}(Y) valued in a difference field L. We denote X_{q,y} = (M_{q}(X))_{y} (cf. 3.13). Thus if X = Spec^{σ} R, Y = Spec^{σ} D, R a D-algebra, then X_{q,y} = Spec M_{q}(R)⊗_{M_{q}(D)}L.

9.2 Dimensions and Frobenius reduction

We will see that the functors M_{q} take transformal dimension to ordinary algebraic dimension. When the transformal dimension of X is zero, M_{q}(X) will be a finite scheme; in this case the total dimension is a (logarithmic) measure of the rate of growth of M_{q}(X) with q. Transformal multiplicity becomes, in a similar sense, geometric multiplicity, while transformal degree is related to the logarithm of the projective degree.

By the size |Y | of a 0-dimensional scheme Y over a field, we will mean the number of points, weighted by their geometric multiplicity. Thus for a k-algebra R, the size of Spec R over Spec k is just dim_{k}(R).

Lemma 9.8 (si-dim) Let X be a difference scheme of finite type over a Noetherian difference scheme Y.

1. Assume X has transformal dimension ≤ d over Y. Then for all prime powers q, and all points y of M_{q}(Y), X_{q,y} has dimension at most d as a scheme over L_{y}.

2. If X has reduced total dimension ≤ e over Y, then there exists b ∈ ℕ such that for all large enough prime powers q, and all points y ∈ M_{q}(Y)(L), L a difference field, the zero-dimensional scheme X_{q,y} over L_{y} has at most bq^{e} points.

Remark The statement of 9.8(2) refers to the number of points, multiplicities ignored. We will see later that it remains true if multiplicities are taken into account. The proof of this refinement is more delicate in that when passing to a proper difference subscheme X′ of X one cannot forget X; even if a point is known to lie on X′ one must take into account the multiplicity of X , not of X′, at that point. Proof The lemma reduces to the case X = Spec^{σ} (R), Y = Spec^{σ} (D), R a finitely - generated D-algebra. We may assume Y is irreducible and perfectly reduced, i.e. D is a difference domain. We wish to reduce the lemma further to the one-generated case, R = D[a, σ(a), …]. We use Noetherian induction on Spec^{σ} D and, with D fixed, on Spec^{σ} S, and thus assume the lemma is true for proper closed subsets. Claim If D ⊂ S ⊂ R and the lemma holds for Spec^{σ} R_{U′} over U′ and for Spec^{σ} S_{U} over U, whenever U′, U are Zariski open in Spec^{σ} S, Spec^{σ} D respectively, then it holds for Spec^{σ} R over Spec^{σ} D. Proof We prove this for (1); the proof for (2) is entirely similar. Let p ∈ Spec^{σ} R, p∩D = p_{D}, p ∩ S = p_{S}. Let d′ the relative dimension of S_{p_{S}} over D_{p_{D}}, d′′ the relative dimension of R_{p} over S_{p_{S}}. Then d′ + d′′ ≤ d. There exists an nonempty open affine neighborhood U′ of p_{S} in Spec^{σ} S such that Spec^{σ} R has relative dimension ≤ d′′ over any point of U′. There exists an nonempty open affine neighborhood U of p in Spec^{σ} D such that Spec^{σ} S has relative dimension ≤ d′ over any point of U. Let q be a large prime, y ∈ M_{q}(Spec^{σ} D). Then as the lemma is assumed to hold in those cases, (Spec^{σ} S′_{U})_{q,y} has dimension at most d′′ over U′, while (Spec^{σ} R_{U})_{q,y} has dimension at most d′. It follows that for any p ∈ Spec^{σ} R with p ∩ R ∈ U′ and p ∩ D ∈ U, the lemma holds. By additivity of transcendence degree in extensions, the lemma holds for Spec^{σ} R over Spec^{σ} D at a neighborhood of any such p. Outside of U, or of U′, the lemma still holds by Noetherian induction. □

We continue to use Noetherian induction on Spec^{σ} D. Let k be the field of fractions of D. We may assume R is perfect, since factoring out the perfect ideal generated by 0 changes neither the transformal or reduced total dimension nor the physical size of the zerodimensional schemes X_{q,y}. In this case O is the intersection of finitely many transformally prime ideals p_{i} of R, and it suffices to prove the lemma separately for each R/p_{i}. Thus we may assume R is a difference domain.

By the Claim, we may assume R is generated a single element a as a difference D-algebra. If R is isomorphic to the difference polynomial ring over D , the transformal dimension

of Spec^{σ} R over Spec^{σ} D equals 1, and we must show the dimension of specM_{q}(S) over specM_{q}(D) is everywhere at most 1. This is clear since M_{q}(R) is generated over M_{q}(D) by a single element.

Otherwise, there exists a relation G(a, σ(a), …, σ^{m}(a)) = 0 in F, G over D. Moreover in case (2), we can choose m ≤ e. We may write G(a) = ∑ c_{i}a^{ν_{i}(σ)}, where c_{i} ∈ S, c_{i} ≠ 0, the ν_{i} are finitely many integral polynomials, of degree at most the total dimension r of Spec^{σ} R over Spec^{σ} S.

Note that for all sufficiently large q, the values ν_{i}(q) are distinct, indeed ν_{i}(q) < ν_{j}(q) if i < j. The set Y′ of transformally prime ideals of D containing one of the nonzero coefficients of G is a proper closed subset of Spec^{σ} S; by induction the lemma is true over Y′. On the other hand if y ∈ Spec^{σ} S \ Y′, then the monomials of M_{q}(G) are distinct and their coefficients are non-zero in L_{y}; it follows that M_{q}(R)⊗L_{y} is 0-dimensional over L_{y}, and moreover has at most max_{i} ν_{i}(q) ≤ O(q^{e}) points. This finishes the proof of the lemma. □

9.3 Multiplicities and Frobenius reduction

The order of magnitude of multiplicity upon Frobenius reduction will be shown to be bounded by the transformal multiplicity. We will begin with two elementary and purely algebraic lemmas regarding multiplicities.

9.3.1 An explicit example

We begin with an example in one variable; it shows explicitly the distribution of multiplicities of the points of M_{q}(X), where X is a difference scheme defined by F (X, X^{σ}, …, X^{σ^{n}}) = 0. If F is irreducible, there will be at most about q^{n−l−1} points with multiplicity of order q^{l}. The general picture is similar (cf. 5.8), but we have not succeeded in reducing it to the example (because of tricky additivity behavior of transformal multiplicity), and will give an independent proof.

The Hasse derivatives D^{ν}f of a polynomial f are defined by the Taylor series expansion:

f(X + U) = ∑_{ν}(D^{ν}f)(X)U^{ν}

Here X = (X_{0}, …, X_{n}), U = (U_{0}, …, U_{n}), X + U = (X_{0} + U_{0}, …, X_{n} + U_{n}), ν is a multiindex (ν_{0}, …, ν_{k}), U^{ν} = Π_{i}U_{i}^{ν_{i}}. Write sup supp ν for the highest i such that ν_{i} > 0.

Example 9.9 (mult-1) Let K be a field, f ∈ K[X_{0}, …, X_{n}] an irreducible polynomial, f ∉ K[X_{1}, …, X_{n}]. For k ≥ 1, let V_{k} be the Zariski closed subset of 𝔸^{n+1} defined by the vanishing of all D^{ν}f with sup supp ν < k.

1. V_{k} = 𝔸^{k} × U, U ⊂ 𝔸^{n+1−k}, dim (U) < n − k.

2. Assume K has characteristic p > 0, and let q be a power of p with q > deg_{X_{i}}(f) for i < k. Let F (X) = f(X, X^{q}, …, X^{q^{n}}). Let S be a subscheme of 𝔸^{1} defined by an ideal I with F ∈ I, and let a ∈ S, (a, …, a^{q^{n}}) ∉ V_{k}. Then the multiplicity of S at a is bounded by

Mult_{a}(S) ≤ ∑_{i<k} deg_{X_{i}} fq^{k−1} ≤ (deg f)q^{k−1}

Proof (1) If (a_{0}, …, a_{n}) ∈ V_{k}, then using the Taylor series expansion, we see that all polynomials D^{ν}f are constant on 𝔸^{k} × {(a_{k}, …, a_{n})}; so 𝔸^{k} × {(a_{k}, …, a_{n})} ⊂ V_{k}. Thus V_{k} = 𝔸^{k} × U for some U. Since f is irreducible, and involves X_{0}, V_{k} is a proper subset of the zero set V (f). Thus dim V_{k} < dim V (f) = n. So dim U < n − k.

(2) We may assume a = 0. Let g be the sum of monomials of f involving only the variables X_{0}, …, X_{k−1}. Then D^{ν}(f − g)(0) = 0 if sup supp ν < k. As 0 ∉ V_{k}, D^{ν}f(0) ≠ 0 for some such ν. Thus g ≠ 0. Let G(X) = g(X, …, X^{q^{(k−1)}}). A monomial Π_{i}X_{i}^{a_{i}} of g turns into the monomial X^{∑ a_{i}q^{i}} of G; as by assumption a_{i} < q for each i, no cancellation occurs; so G ≠ 0. Write G(X) = X^{m}H(X), H(0) ≠ 0. Then for some I,

F (X) = G(X) + X^{q^{k}}I(X) = X^{m}(H(X) + X^{q^{k}−m}I(X))

Note m < q^{k}. So F (X)/X^{m} is a polynomial not vanishing at 0, thus m bounds the multiplicity of F at 0. □

9.3.2 Algebraic lemmas

Lemma 9.10 (gen-M) Let k be a field of characteristic p, [k : k^{p}] = p^{e} (in characteristic 0, let e = 0.) Let S be a k-algebra, generated by m elements, M a maximal ideal of S with dim_{k}(S/M) < ∞. Then M is generated by at most m + e + 1 elements.

Proof Let h : S → K be a surjective homomorphism with kernel M, K a finite extension field of k. Let s_{1}, …, s_{m} be generators for S as a k-algebra.

Consider first the case e = 0, i.e. K/k separable. Using the primitive element theorem, K is generated over k by one element a = h(s_{0}); and one relation, the minimal monic polynomial P (a) = 0, P ∈ k[X]. We have h(s_{i}) = Q_{i}(a) = h(Q_{i}(s_{0})) (some Q_{i} ∈ k[X]), so s_{i} − Q_{i}(s_{0}) ∈ M; and clearly M is generated by the m + 1 elements P (s_{0}), s_{i} − Q_{i}(s_{0}).

In general, K/k is generated by e + 1 elements, and e + 1 relations. (Let K ≤ K_{s} ≤ K, where K_{s}/k separable, K/K_{s} purely inseparable. Then K_{s}/k can be presented by one generator a_{0} and one relation, as above. K/K_{s} is generated by ≤ e generators a_{1}, …, a_{e} and ≤ e relations; these can be viewed as a relation over k between a_{0} and the a_{i}.) As above, the maximal ideal M is generated by the preimages of the above relations. □

Lemma 9.11 (mult-2-0) Let k be a field, R a finite dimensional local k - algebra, with maximal ideal M. Let S be a finitely generated R-module. Then dim_{k}S ≤ (dim_{k}R)(dim_{k}(S/MS))

Proof Let A be a k-subspace of S with dim_{k}(A) = dim_{k}(S/MS) and A + MS = S. Let T be the k-span of RA. So T is an R-submodule of S, and dim_{k}(T) ≤ (dim_{k}R)(dim_{k}(S/MS)). Also T + MS = S. By Nakayama (applied to the R-module S/T), T = S.

Lemma 9.12 (mult-2) Let f : X → Y be a morphism of finite schemes over Spec (k). Let p ∈ Spec Y, X_{p} the fiber of X above p, and q ∈ X_{p}. Then Mult_{q}X ≤ (Mult_{q}X_{p})(Mult_{p}Y)

Proof Let R, S be the local rings of Y, X at p, q respectively. Then the local ring of X_{p} at q is S/MS, M being the maximal ideal of R. We must show that (dim_{k}S) ≤ (dim_{k}R)(dim_{k}(S/MS)). This follows from 9.11.

Lemma 9.13 (mult-2-3) Let S a local ring, with finitely generated maximal ideal M. Suppose S/M^{r} has length n < r. Then M^{n} = 0.

Proof We have S ⊃ M ⊃ M^{2} ⊃ … ⊃ M^{r}. M^{i}/M^{i+1} is an S/M-space of some finite dimension. As the length of S/M^{r} is n < r, we must have M^{i} = M^{i+1} for some i ≤ n. By Nakayama, M^{i} = 0. So M^{n} = 0.

N.B. Let I be an ideal of a ring R with p · 1_{R} = 0, and let q be a power of p. Let ϕ_{q}(x) = x^{q}. Then I^{q} might mean one of three things: the ideal RI^{q} generated by q-fold products of elements of I; the ideal ϕ_{q}(I) of ϕ_{q}(R); or possibly the ideal Rϕ_{q}(I) of R. We will refrain from writing I^{q} altogether, and use the above notations.

Remark 9.14 (q-bq) Let S be a ring, with ideal M generated by s_{1}, …, s_{b}. Assume p = 0 in S, and let q be a power of p. Then SM^{bq} ⊂ Sϕ_{q}(M) ⊂ SM^{q}

Proof The ideal SM^{bq} has generators of the form s_{1}^{m_{1}} · . . . · s_{b}^{m_{b}} with m_{1} + … + m_{b} = bq. Necessarily m_{i} ≥ q for some i, so s_{i}^{m_{i}} and hence the product are in ϕ_{q}(M). □

Lemma 9.15 (mult-2-4) Let S be a ring of characteristic p > 0, with maximal ideal M, and let q be a power of p. Assume M is finitely generated. Then ϕ^{q}(S) has a unique maximal ideal ϕ_{q}(M), with residue field k = ϕ_{q}(S)/ϕ_{q}(M). Note S is an S^{q}-module, so S/Sϕ_{q}(M) is a k-space. Assume dim_{k}(S/Sϕ_{q}(M)) < q, or even just

(#) dim_{k}(S/SM^{q}) = n < q

Then S has length ≤ n (as an S-module.)

Proof Clearly S/SM^{q} has length ≤ n as an S-module. By Lemma 9.13, SM^{n} = 0, so SM^{q} = 0. Thus S = S/SM^{q} has length ≤ n. □

Let S be a local ring of characteristic p > 0, with maximal ideal M, generated by b elements. For r a power of p, write S_{r} = ϕ_{r}(S), M_{r} = ϕ_{r}(M); so S_{r} is local, with maximal ideal M_{r} (the elements of S \ M are units; hence ϕ_{r}(S \ M) consists of units of S_{r}, while ϕ_{r}(M) is a proper ideal of S_{r}. These sets are thus disjoint, while their union equals S_{r}.)

Lemma 9.16 (mult-2-5) Let S be a local ring of characteristic p > 0, with maximal ideal M, generated by b elements. Let r, r′ be powers of p. Assume ϕ_{r}(S/SM_{r′}) has length n as an S_{r}-module, and (nb + 1)r < r′. then SM^{rb^{n+1}} = 0. In particular (S_{r}M_{r})^{b^{n+1}} = 0 , so S_{r} has length < b^{b^{n+1}}.

Proof The assumption that ϕ_{r}(S/SM_{r′}) has length n refers to the ring S/SM_{r′}, and the function ϕ_{r}(x) = x^{r} in that ring. Being a quotient of S, S/SM_{r′} is an S-module, and ϕ_{r}(S/SM_{r′}) is an S_{r}-module. Note

ϕ_{r}(S/SM_{r′}) = (ϕ_{r}(S) + SM_{r′})/(SM_{r′}) ≃_{S_{r}} ϕ_{r}(S)/(SM_{r′} ∩ ϕ_{r}(S))

Thus S_{r}/(SM_{r′} ∩S_{r}) has length n as an S_{r}-module. (It is an S_{r}/(M_{r})^{r′}-module; this ring has finite length as a module over itself, so all nontrivial finitely generated modules below have nonzero finite length.) Consider the S_{r}- modules N_{i} = (SM^{r′} ∩S_{r})+(S_{r}M_{r})^{i}. They lie between S_{r} and SM_{r′} ∩ S_{r}, and form a descending chain. Consider i = 1, 1 + b, 1 + b + b^{2}, . . .. As the length of S_{r}/(SM_{r′} ∩ S_{r}) is n, we can find 1 ≤ i, bi < j ≤ b^{n+1}, with N_{i} = N_{j}.

By Remark 9.14, SM^{rb} ⊂ SM_{r}, so SM^{rbi} ⊂ S(M_{r})^{i} ⊂ SN_{i}. On the other hand, SN_{j} ⊂ SM^{rj}, provided rj < r′. But rb^{n+1} < r′, and N_{i} = N_{j}; so SM^{rbi} ⊂ SM^{rj} ⊂ SM^{rbi+1}. By Nakayama, SM^{rbi} = 0. So SM^{rb^{n+1}} = 0.

In particular (S_{r}M_{r})^{b^{n+1}} = 0. As M_{r} is also generated by b elements, (S_{r}M_{r})^{l} can be generated by ≤ b^{l} elements, so S_{r} has length ≤ length(S_{r}/M_{r})+length(M_{r}/(S_{r}M_{r})^{2})+… ≤ 1 + b + b^{2} + … + b^{b^{n+1}−1} < b^{b^{n+1}}. □

9.3.3 Relative reduction multiplicity

We first observe a natural relationship between transformally radicial extensions and relative reduction multiplicity. The proof is less straightforward than it ought to be; one reason is that the notion of multiplicity of a point on a scheme is somewhat delicate, and does not easily permit devissage. Recall that when q is a power of the prime p, K_{q} denotes a (large) algebraically closed field of characteristic p, endowed with the map x ↦ x^{q}. For any scheme Y, Y (K_{q}) denotes the set of K_{q}-valued points of Y, i.e. difference scheme maps y : Spec^{σ} K_{q} → Y. Recall also that X_{q,y} denotes M_{q}(X) ×_{f,y} Spec^{σ} (K_{q}).

Definition 9.17 Let f : X → Y be a morphism of difference schemes. X has reduction multiplicity ≤ k over Y if for some integer B, for all large prime powers q, and all y ∈ Y (K_{q}), and z ∈ X_{q,y}, Mult_{z}X_{q,y} ≤ Bq^{k}

Definition 9.18 Let X be a difference scheme of finite type over a finitely generated difference field K. We will say that X is generically of k-bounded reduction multiplicity over K if there exists a finitely generated difference domain D ⊂ K, and a difference scheme X_{0} of finite type over D, with X ≃ X_{0} ×_{Spec^{σ} D} Spec^{σ} K, such that X_{0} is of reduction multiplicity ≤ k over Spec^{σ} D.

Lemma 9.19 (si-dim-3.0) Let X be a difference scheme of finite type over a Noetherian difference scheme Y. Assume X is transformally radicial over Y, and of total dimension ≤ e over Y. Then X has reduction multiplicity ≤ e over Y

Proof

We assume inductively the lemma holds for X_{Y′} over Y′ for any proper closed Y′ ⊂ Y. Note that if the lemma holds for X_{Y′} as well as for X_{Y \Y′}, then it holds for X. Thus it suffices to prove the lemma for any open subset of Y. We may thus assume Y is irreducible; and indeed that Y = Spec^{σ} R, R a difference domain; and it suffices to prove the lemma for some difference localization R′ of R in place of R.

The case of X is more delicate; we may still assume the lemma is true for any proper closed subset; but if the lemma holds for a closed subset and its complement, it is not clear that it holds for X. Still if X is a union of open subsets, it suffices to prove the lemma for each separately. Thus we may take X = Spec^{σ} S, S a finitely generated transformally radicial R-algebra. We may assume S is well-mixed. Claim 1 Let R′ be a difference R-subalgebra of S, finitely generated as an R-module. If the conclusion of the lemma holds for X over Spec^{σ} R′, then it holds for X over Spec^{σ} R. Proof For any y ∈ M_{q}(Spec^{σ} R)(L) and y_{1} ∈ (Spec^{σ} R′)_{q,y}, there exists d ∈ ℕ with

Mult_{y_{1}}(Spec^{σ} R′)_{q,y} ≤ d

By assumption, for some b_{1},

Mult_{z}X_{q,y_{1}} ≤ b_{1}q^{e}

So by 9.12, with b = db_{1}, Mult_{z}X_{q,y} ≤ bq^{e}. □

Find a sequence of subrings R = S_{0} ⊂ S_{1} ⊂ … ⊂ S_{n} = S such that S_{k+1} = S_{k}[a_{k}] and σ(a_{k}) ∈ S_{k}. We will also use induction also on the length n of this chain.

Let L be the field of fractions of R,S_{L} = L⊗_{R}S. Effecting a finite localization of R, we may assume S embeds into S_{L}. (The ideal of polynomials over L vanishing at (a_{1}, …, a_{k}) can be taken to be generated by polynomials over R.) We will further use induction on the Noetherian rank of S_{L}.

Consider the ideal I_{L} of S_{L} = L⊗_{R}S generated by elements a with σ(a) = 0, a^{2} = 0. I_{L} is generated by finitely many elements b_{1}, …, b_{m}. After a finite localization of R, we may assume that the b_{i} lie in S. Let R′ = R[b_{1}, …, b_{m}]. R′ is finitely generated over R as an R-module. By the claim, it suffices to prove the lemma for S over R′. Every prime of R′ must contain the b_{i}, so this amounts to proving the lemma for S′ = S/I_{L}. If I_{L} ≠ 0, then S′_{L} = S′⊗_{R}L has smaller Noetherian rank than S_{L}; so using induction on this ordinal, we have the lemma for S′ and hence for S. Thus we may assume I_{L} = 0.

So S_{L} has no nonzero elements a with σ(a) = a^{2} = 0. (Since S embeds in S_{L}, S has no such elements either.)

Next, suppose S_{L} has a zero-divisor c with σ(c) ∈ L. If σ(c) ≠ 0, then passing to a finite localization we may assume σ(c) is a unit in R; thus if cd = 0 then σ(d) = 0; so replacing c by d if necessary we may assume σ(c) = 0. Let J = {s ∈ S_{L} : cs = 0}. Then S_{L}c,J are nontrivial difference ideals of S_{L}, and after further localization of R, J ∩ S is nontrivial. The lemma holds for S/(J ∩ S) and for S/Sc by Noetherian induction. Moreover if e ∈ S_{L}c ∩ J then e^{2} = 0 and σ(e) = 0, so e = 0 by the previous paragraph. So S_{L}c ∩ J = 0. The lemma follows for S.

Thus we may assume there are no zero divisors c ∈ S_{L} with σ(c) ∈ L.

Consider first the case that there exists a polynomial F ∈ R[X] with nonzero leading coefficient r, such that F (a_{1}) = 0. By the remark preceding the claim, it suffices to prove the lemma for S[r^{−1}] over R[r^{−1}]; so we may assume F is monic. In this case S_{1} is a finitely generated R-module. By the induction hypothesis, the lemma holds true for S over S_{1}. But now by the Claim it holds for S over R.

Assume now there is no such F; so S_{1} as an R-algebra is the polynomial ring in one variable. Moreover for f ∈ R[X], f(a_{1}) is not a zero-divisor in S. And S has Krull dimension e over R. It follows algebraically that - after replacing R by a finite localization - every fiber of the map of algebraic schemes Spec S → Spec R[a_{1}] has dimension ≤ e − 1.

Let Y′ = Spec^{σ} S_{1}. Clearly for any q and any y ∈ M_{q}Y, and y′ ∈ Y′_{q,y}, Mult_{y′}Y′_{q,y} ≤ q

Using the case e − 1 assumed inductively, there exists b such that for any large q, and any y′ ∈ Y′(L), and z ∈ X_{q,y′}, Mult_{z}X_{q,y′} ≤ bq^{e−1}

Let y ∈ M_{q}Y (L), z ∈ X_{q,y}; let y′ ∈ Y′_{q,y} be the intermediate. We then have Mult_{z}X_{q,y′} ≤ bq^{e−1} and Mult_{y′}Y′_{q,y} ≤ q; by 9.12, Mult_{z}X_{q,y} ≤ bq^{e} the lemma follows. □

9.3.4 Transformal multiplicity and reduction multiplicity

Lemma 9.20 (trans-mult-red) Let X be a difference scheme of finite type over a finitely generated difference domain D. Assume X is of generic transformal multiplicity ≤ n. Then there exists a nonempty open Y ⊂ Spec^{σ} D and a bound b_{0} such that for all inversive difference fields L and L -valued points y of Y and x of B_{n+1}(X_{y}), for every local ring R of (X_{y})_{x}, σ^{n}(R) is a finite-dimensional L-space, of dimension ≤ b_{0}.

Proof Let K be the inversive closure of the field of fractions of D. We may take X = Spec^{σ} S. Let S_{K} = S⊗_{D}K.

We first consider the case of generic y ∈ Spec^{σ} D, i.e. of difference field extensions L of K. Let S_{L} = S⊗_{D}L (so Spec^{σ} S_{L} = X_{y}). Let x be an L-valued point of B_{n+1}(X_{y}), i.e. x : σ^{n+1}(S_{L}) → L a difference ring homomorphism. Let x_{k} = x|σ^{n+1}(S_{K}).

We have a natural homomorphism

S_{K}⊗_{x_{k}}L → S_{L}⊗_{x}L

and it is easily seen to be surjective. By 5.11, σ^{n}(S_{K}⊗_{x_{k}}L) has total dimension 0. Thus the homomorphic image σ^{n}(S_{L}⊗_{x}L) also has total dimension 0. By Lemma 4.31, S_{L}⊗_{x}L is a finitely generated L-algebra, hence so is σ^{n}(S_{L}⊗_{x}L). By 4.6, dim_{L}(σ^{n}(S_{L}⊗_{x}L)) < ∞.

It follows by compactness that for some d_{0} ∈ D and b_{0}, for all inversive difference fields L and difference ring homomorphisms y : D → L with y(d_{0)} ≠ 0, and all L-difference algebra homomorphisms x : σ^{n+1}(S_{y}) → L, dim_{L}(σ^{n}(S_{y}⊗_{x}L)) < b_{0}. To see this, let F_{0} be a finite set of generators of S as a difference D -algebra, F = F_{0} ∪ … ∪ σ^{n}(F_{0}). Then the image of F generates S_{y}⊗_{x}L as an L-algebra, for any x, y. Let F_{1} = F, …, F_{m+1} = F F_{m}. Then dim_{L}(σ^{n}(S_{y}⊗_{x}L)) < ∞ iff for some m,

(*) for any c, d ∈ σ^{n}F_{m}, the image of cd in S_{y}⊗_{x}L is in the L-span of the image of σ^{n}F_{m}.

Let y_{i}, x_{i}, L_{i} be a sequence of such triples, with y_{i} approaching the generic point of Spec^{σ} D; let S_{i} = S_{y_{i}}⊗_{x_{i}}L_{i}; we have to show that dim_{L_{i}}(σ^{n}(S_{i})) is bounded. Let (y_{∞}, x_{∞}, L_{∞}, S_{∞}) be an nonprincipal ultraproduct of the (x_{i}, y_{i}, L_{i}). Then dim_{L}(σ^{n}(S_{y_{∞}}⊗_{x_{∞}}L_{∞})) =< ∞, so a fortiori the image of this ring in S_{∞} has finite dimension over L_{∞}. Thus for some m, (*) above holds for the image of F_{m} in S_{∞}, hence also for the image of F_{m} in S_{i}, for almost all i. This proves that dim_{L_{i}}(σ^{n}(S_{i})) is bounded, as required.

The existence of the open difference subscheme Y and the bound b_{0} now follow by a standard compactness argument.

□

Corollary 9.21 (trans-mult-red-q) Let f : X → Y be a a morphism of difference schemes of finite type, of relative transformal multiplicity ≤ n. Then there exists b ∈ ℕ such that for all prime powers q > b, for any perfect field L and any y ∈ (M_{q}Y)(L), every closed point x ∈ M_{q}(X_{y}) has multiplicity ≤ bq^{n} on M_{q}(X_{y}).

Proof We may assume X = Spec^{σ} R, Y = Spec^{σ} D, R a finitely generated D-algebra. By lemma 9.10, the local rings of M_{q}(X_{y}), y : M_{q}(D) → L, L a perfect field, have maximal ideals generated by a bounded number of elements. (D itself is generated by a finite number e of elements, so that quotient domains of D have fields of fractions k with [k^{q} : k] ≤ q^{e}, but we just consider perfect L, so actually 9.10 applies with e = 0.) Thus Lemma 9.16 applies, and translates Lemma 9.20 to say that x′ = x|σ^{n+1}(R⊗_{y}L) has bounded multiplicity on M_{q}(B_{n+1}(X_{y})). By Lemma 9.19, X/(Y × B_{n+1}(X)) has reduction multiplicity ≤ n. So Mult_{x}M_{q}(X)_{y,x′} ≤ O(q^{n}). By Lemma 9.12, x has multiplicity O(q^{n}) on M_{q}(X_{y}). □

Corollary 9.22 (si-dim+) Let X be a difference scheme of finite type over a Noetherian difference scheme Y, of total dimension d over Y. Then there exists b such that for all large enough prime powers q, and all y ∈ M_{q}(Y), the zero-dimensional scheme X_{q,y} over L_{y} has size at most bq^{d}.

Proof By the usual Noetherian induction on Y we may assume Y = Spec^{σ} D, D a difference domain with field of fractions K; and we may pass to a localization of D by a finite set. Moreover base change will not change the total relative dimension or the size, so we may replace D by a finitely generated extension within the inversive hull K′ of K^{alg}. Let X′ = X⊗_{Y} Spec^{σ} K′. We enlarge D within K so that the Mlt_{k}X′ and their components X_{k,j} are defined over D. The components X_{k,j} have reduced total dimension ≤ d − k; so by 9.8, for all y ∈ Y and all sufficiently large q, (X_{k,j})_{y,q} has O(q^{d−k}) points, and by 5.8, away from Mlt_{k+1}(X), these points have multiplicity ≤ O(q^{k}) on M_{q}X. Thus X_{q,y} is divided into d groups (Z_{k}X)_{q,y}, the k’th having at most O(q^{d−k}) of multiplicity (on M_{q}X) at most O(q^{k}); so altogether there are O(q^{d}) points, multiplicity counted.

The corollary 9.22 can also be obtained using Lemmas 4.36 and Bezout theorem methods (see 10.25). One could take things up from that point with 9.19. However, the methods of this section give information about the distribution of multiplicities (9.23), and in particular allow estimating the multiplicity on X of points on smaller-dimensional X′ ⊂ X, (9.24). This does not seem apparent from the Bezout approach (though one can use Bezout methods to estimate the multplicities of these points on X′.)

Corollary 9.23 (si-dim++) Let X_{0} be a difference scheme of finite type over a difference field K. Assume X_{0} has total dimension d, and that no algebraic component of X_{0} of total dimension d has transformal multiplicity > 0. Then there exist a finitely generated subdomain D of K, a difference scheme X over D with X_{0} = X ×_{Spec^{σ} D} Spec^{σ} K, and an integer b, so that for all large enough prime powers q, and all y ∈ M_{q}(Spec^{σ} D), the zero-dimensional scheme X_{q,y} over L_{y} has size at most bq^{d}; moreover all but bq^{d−1} points of X_{q,y} (counting multiplicities) have multiplicity < b.

Proof As in the proof of 9.22, we consider reduced subschemes Y of Z_{k}X = Mlt_{k}(X) \ Mlt_{k+1}(X). Here however, for k ≥ 1, we use 5.8(3) to conclude that the reduced total dimension of Y is ≤ d − k − 1. This gives an O(q^{d−k−1})-bound on multiplicities, by 5.8. As we are considering Y away from Mlt_{k+1}, we obtain O(q^{d−1}) points counted with their multiplicities on X. □

Corollary 9.24 (si-dim+++) Let X_{0} be as in Corollary 9.23. For any proper subscheme X′ of X, of total dimension d′ < d, the number of points of X′, counted with their multiplicities on X, is O(q^{d−1}).

Proof By Corollary 9.23, only O(q^{d−1}) points have high multiplicity on X_{q,y}. These can therefore be ignored. The rest have multiplicity O(1) on X_{q,y}, and by Corollary 9.22, their number is O(q^{d′}). Thus even with multiplicity, the number is O(q^{d−1}).

Part II

Intersections with Frobenius

10 Geometric Preliminaries

(geopre)

We show here that the main theorem is invariant under birational changes (10.27) and in the appropriate sense under taking finite covers (10.29). We use this together with de Jong’s version of resolution of singularities to reduce to the smooth case, where intersection theory applies. We deal with the possible inseparability, and note another proof of a crude initial bound (10.25) on the set whose size we are trying to estimate.

Our basic reference, here and in later sections, is [Fulton]. We will begin with some basic lemmas on proper and improper intersections, degrees and correspondences.

10.1 Proper intersections and moving lemmas

(moving)

Let X be a nonsingular variety over an algebraically closed field. We write codim_{X}(U) for dim (X) − dim (U). A variety (or cycle) is said to have pure (co)dimension k if each irreducible component has (co)dimension k. Two subvarieties U,V of X of pure codimension k,l respectively are said to meet properly if either U ∩ V = ∅, or codim_{X}(U ∩ V) ≥ k + l. By the dimension theorem for smooth varieties, each component of U ∩V must have codimension at most k + l; thus the intersection is proper iff each (nonempty) component of U ∩ V has codimension k + l in X. The properness of the intersection is equivalent to the properness of intersection of all irreducible components of U with those of V.

Lemma 10.1 (proper1) Let U, V, W be pure-dimensional varieties. Assume U, V meet properly, and U ∩ V, W meet properly. Then U meets V ∩ W properly.

Proof Let U, V, W have codimensions k, l, m. Then U ∩ V = ∅ or codim_{X}(U ∩ V) = k + l; so U ∩ V ∩ W = ∅ or codim_{X}(U ∩ V ∩ W) = k + l + m; this last condition is equivalent to the assumptions, and is symmetric in U, V, W.

Lemma 10.2 (proper2) Assume X, Y are smooth varieties. Let T be an irreducible subvariety of X × Y, and let π[k]T be the subvariety of X, whose points are {p ∈ X : dim (T ∩ ({p} × Y)) ≥ k}. If U is a subvariety of X meeting each component of each π[k]T properly, then U ×Y meets T properly. If U ×Y meets T properly, then U meets πT properly.

Proof Note that codim_{X×Y} (U × Y) = codim_{X}(U); let c be the common value. Let W be an irreducible component of (U × Y) ∩ T. Say dim (W) = dim (πW) + k. Then πW ⊂ π[k]T. So πW ⊂ π[k]T ∩ U,and dim (πW) ≤ dim (π[k]T ∩ U) ≤ dim (π[k]T) − c ≤ dim (T) − k − c Thus dim (W) ≤ dim (T) − c. This shows that U × Y meets T properly.

For the remaining statement, let k = dim (T)−dim (πT). Then πT = π[k]T. So dim (T)− codim_{X}(U) ≥ dim ((U × Y) ∩ T) ≥ k + dim (U ∩ πT); hence dim (πT) − codim_{X}(U) ≥ dim (U ∩ πT).

Lemma 10.3 (proper3) Let U ⊂ (X × Y), V ⊂ (Y × Z) be complete varieties, all of pure dimension d. Let W = pr_{XZ}((U × Z) ∩ (X × V)). Let T be a subvariety of X × Z. Assume

U meets each component of X × pr_{Y} [l]V and of pr_{X}[l]T × Y properly for each l; V meets each component of pr_{Y Z}[l]((T × Y) ∩ (U × Z)) properly; Then U × Z meets X × V properly, and W meets T properly.

Proof 1. By the first assumption and 10.2, U × Z meets X × V properly.

2. Similarly, T × Y meets U × Z properly.

3. By the second assumption, X × V meets ((T × Y) ∩ (U × Z)) properly.

4. By 10.1, and (2),(3), T × Y meets (U × Z) ∩ (X × V) properly.

5. By the last statement of 10.2, T meets W properly.

Let X be a smooth algebraic variety over an algebraically closed field. A cycle is a formal sum, with integer coefficients, of irreducible subvarieties of X.

To each subscheme U of X, one associates a cycle [U]; it is the sum of the irreducible components of U, with certain nonnegative integer coefficients ([Fulton], 1.5).

Notation 10.4 For any cycle I, write I = Σ_{k}{I}_{k}, where {I}_{k} is a k-dimensional cycle.

B^{i}(X) denotes the group of cycles whose components have codimension i, up to algebraic equivalence ([Fulton] 10.3). An operation · can be defined on B^{∗}(X) = ⊕_{i}B^{i}(X), making it into a commutative graded ring with unit. This operation is determined by the fact that [U] · [V] = [U ∩ V] when U, V are irreducible subvarieties of X meeting properly.

Most of the time, we will be able to use the finer notion of rational equivalence; but since we are primarily interested in intersection theory, the difference between two algebraically equivalent cycles will not concern us. In particular, a 0-cycle ∑ n_{i}p_{i} is determined up to algebraic equivalence by ∑ n_{i} ∈ ℤ; we will identify the group of 0-cycles with ℤ.

Lemma 10.5 (moveall) Let Y be a smooth projective variety over an algebraically closed field k, and let U be an irreducible subvariety of Y, and f : Y → V a flat morphism. There exists a purely transcendental extension K of k, and a cycle U′ on Y defined over K, such that U, U′ are rationally equivalent, and for any subvariety W of Y defined over k, each component C of U′ meets W properly, and f(C) = V.

Proof This is a version of the moving lemma described in [Hoyt] and [Fulton] 11.4.1, and the classical proof works. Say Y ⊂ ℙ^{N}; let G be the Grassmanian of linear subspaces of ℙ^{N} of codimension one more than the codimension of U in Y. Pick a point p of G generic over k, representing a linear subspace L, and also g ∈ P GL_{N}, generic over k(p). Given any k-subvariety U of Y, form the cone C(L, U), and consider also gC(L, U). Write C(L, U) ∩ Y = U + ∑ m_{i}U′′_{i}, gC(L, U) ∩ Y = ∑ m′_{j}U′_{j}, and let U[1] = ∑ m′_{j}U′_{j} − ∑ m_{i}U′′_{i}. If U is not a variety but a cycle, a formal sum ∑ U = n_{i}U_{i} of varieties of the same dimension, define U = ∑ n_{i}U_{i}[1]. So U[1] is rationally equivalent to U, is defined over a purely transcendental extension of k, and for any subvariety W of Y defined over k, each component C of U[1] either meets W properly, or satisfies dim (C ∩ W) < dim (C′ ∩ W) for some component C′ of U. Let U[2] = U[1][1], U[3] = U[2][1], etc. Then U[m] is rationally equivalent to U, is defined over a purely transcendental extension of k, and for any subvariety W of Y defined over k, each component C of U[m] either meets W properly, or satisfies dim (C ∩ W) ≤ dim (C′ ∩ W) − m for some component C′ of U. It follows that U′ = U[m + 1] satisfies our requirements. The argument that f maps each component onto V is given separately, in 10.6; to apply it, embed ℙ^{N} into some larger ℙ^{N′} first by the d-uple embedding, so that any four distinct points of ℙ^{N} are linearly independent in ℙ^{N′}. □

Lemma 10.6 (moveall-tl) Let U ⊂ Y ⊂ ℙ^{N} be projective varieties, with no three or four points of Y linearly dependent in ℙ^{N}. Let f : Y → V a flat morphism to an irreducible variety. Let dim (Y) = n, dim(U) = d = dim (V), n + 1 < N.

1. Let G be the Grassmanian of linear subspaces of ℙ^{N} of codimension n+1. For L ∈ G, let C(L, U) be the cone on U with center L. Then for generic L ∈ G, and any component U′ of C(L, U) ∩ Y other than U, f(U′) = V.

2. Let W be any subvariety of ℙ^{N} of codimension n − d, and let g ∈ P GL_{N} be generic. Then for any component U′ of gW ∩ Y, f(U′) = V.

Proof (1) Let

M_{2} = {(L, y, y′, p, p′) ∈ G × (Y \ U)^{2} × (ℙ^{N})^{2} : p ≠ p′ ∈ L, y ∈ C(p, U), y′ ∈ C(p′, U), fy = fy′}

Then M_{2} projects to G × Y^{2}, and the image of the projection contains

M = {(L, y, y′) ∈ G × (Y \ U)^{2} : y ≠ y′ ∈ C(L, U), fy = fy′}

Indeed if (L, y, y′) ∈ M, then there exist p, p′ ∈ L and u, u′ ∈ U with y, u, p and y′, u′, p′ colinear. Since no three or four distinct points of Y are linearly dependent in ℙ^{N}, we must have p ≠ p′.

So dim (M) ≤ dim (M_{2}).

Let G_{p,p′} = {L ∈ G : p, p′ ∈ L}; we have dim G_{p,p′} + 2N ≤ dim (G) + 2(N − (n + 1)). We will also use: if y ∈ C(p, U)\U, then p ∈ C(y, U), and dim C(y, U) = dim (U) + 1. Now compute, using the maps (L, y, y′, p, p′) ↦ (y, y′, p, p) ↦ (y, y′) ↦ fy:

dim (M_{2}) ≤ dim (V) + 2(dim (Y) − dim (V)) + 2(d + 1) + dim G_{p,p′} =

= d + 2(n − d) + 2(d + 1) + (dim (G) − 2(n + 1)) = d + dim (G).

Thus we see that for generic L ∈ G, dim {(y, y′) : (L, y, y′) ∈ M} ≤ d.

Now let U′ be a component of C(L, U) not containing U, and let Ũ = U′ \ U. If dim f(U′) < d, then dim {(y, y′) ∈ Ũ^{2} : fu = fu′} ≥ (dim f(U′))+2(dim (U′)−dim f(U′)) > d, a contradiction. So dim f(U′) = d. Since V is irreducible and U′ is projective, f(U′) = V.

For (2), a similar computation shows that dim {(g, y, y′) : y ≠ y′ ∈ Y ∩gW, fy = fy′} ≤ d, and we conclude as above.

□

If f : S → V is a morphism of varieties, and U a subvariety of S, recall the notation: f[1](U) := {v ∈ V : dim f^{−1}(v) ≥ 1}.

Lemma 10.7 (move2) Let V be a smooth projective variety over a difference field k, and let S be a subvariety of Y = (V × V^{σ}), dim (S) = dim (V) = d.

There exists a cycle S′ = ∑_{i} m_{i}[U_{i}] on Y defined over k, such that S, S′ are rationally equivalent, and

[σ_{k}]pr_{V} [1](U_{i}) ∩ (U_{i} ⋆ Σ) = ∅

Moreover, each U_{i} as well as the varieties involved in the rational equivalence of S with S′ other than S itself all have dominant projections to V and to V^{σ}.

Proof Using 10.5, find a purely transcendental extension k(s) of k and a cycle S′ on Y, rationally equivalent to S, such that any subvariety of Y defined over k^{alg} meets S′ properly, and such that each component of S′ maps dominantly to V. We will show that S′ has the required properties. It then follows easily, by specializing s into k (avoiding finitely many proper Zariski closed sets), that such a cycle and such a rational equivalence exist over k too.

Let s_{0} = s, and K = k(s_{0}, s_{1}, s_{2}, …), σ^{n}(s_{0}) = s_{n} (with s_{0}, s_{1}, … algebraically independent over k.) Let V_{n} = σ^{n}(V); note V_{n} is defined over k.

Let S_{0} be any component of S′, S_{n} = σ^{n}(S_{0}). By 10.5 applied to the inversive hull of k^{alg}, any subvariety of σ^{n}(Y) defined over k^{alg} meets S_{n} properly . Since σ^{n}(Y) and S_{n} are defined over k(s_{n}), and k(s_{0}, …, s_{n−1}, s_{n+1}, s_{n+2}, …)^{alg} is linearly free from k^{alg}(s_{n}) over k^{alg}, in fact any subvariety of σ^{n}(Y) defined over k(s_{0}, …, s_{n−1}, s_{n+1}, s_{n+2}, …)^{alg} meets S_{n} properly.

Claim Let W_{1} be any proper subvariety of S_{0}, defined over k(s_{0}). Then W_{1} ⋆ Σ = ∅. To prove the claim, let c = (d − dim (W_{1}))/2 > 0. We will inductively define W_{n} satisfying:

1. W_{n} ⊂ V_{0} × … × V_{2^{n}−1}

2. dim (W_{n}) ≤ d − 2^{n}c (or W_{n} = ∅).

3. W_{n} is defined over k(s_{0}, …, s_{2^{n}−2})

4. (W_{1} ⋆ Σ)[2^{n} − 1] ⊂ W_{n}

For n = 1, (1-4) hold by assumption. Assume they hold for n. Then (W_{n})^{σ^{2^{n}}} ⊂ V_{2^{n}} × … × V_{2^{n+1}−1}, has dimension ≤ d − 2^{n}c, and is defined over k(s_{2^{n}}, …, s_{2^{n+1}−2}).

Thus W^{∗} = W_{n}×(W_{n})^{σ^{2^{n}}} has dimension ≤ 2d−2^{n+1}c, and is defined over k(s_{0}, …, s_{2^{n}−2}, s_{2^{n}}, …, s_{2^{n+1}−2}). Let π : (V_{0} × … × V_{2^{n+1}−1}) → (V_{2^{n}} × V_{2^{n}+1}) be the projection. Then π[k]W is defined over k(s_{0}, s_{2^{n}−2}, s_{2^{n}}, s_{2^{n+1}−2}), hence meets S_{2^{n}−1} properly; by 10.2, π^{−1}S_{2^{n}−1} meets W^{∗} properly. Let W_{n+1} = π^{−1}S_{2^{n}−1} ∩ W^{∗}. Then dim W_{n+1} ≤ dim (W^{∗}) − d = d − 2^{n+1}c, and (1)-(4) hold. This proves the claim.

When 2^{n}c > d, (2) forces W_{n} = ∅, so by (4), W_{1} ⋆ Σ = ∅.

In particular, as pr_{0} : S_{0} → V dominantly, and dim (S_{0}) = dim (V), pr_{0}[1](S_{0}) ≠ V; so letting W_{1} = S_{0} ∩ pr_{0}^{−1}(pr_{0}[1](S_{0})), we have dim (W_{1}) < d, and we can apply the claim.

□

Remark 10.8 (move-rem)

In particular, in 10.7 one can conclude: [σ]_{K}U_{i} ⋆ Σ has transformal dimension 0. (See 4.44).

An alternative treatment of a moving lemma for difference schemes can be given by transposing the proof of [Fulton],11.4.1 to difference algebra, almost verbatim. The methods used there - projective cones, moving via the group of automorphisms of projective space, ”counting constants” - work very well for difference varieties and transformal dimension in place of varieties and dimension. One can also use the dimension growth sequence to get finer results.

10.2 Degrees of cycles

In general, the intersection product gives no direct information about improper intersections. In products of projective spaces, we can obtain such information, as in [Fulton], Example 8.4.6.

Notation 10.9 Let H, H′ be hyperplane divisors on ℙ^{n}, ℙ^{m}, respectively, s = pr_{1}^{∗}H, t = pr_{2}^{∗}H′. For a subvariety U of ℙ^{n} × ℙ^{m}, U = Σa_{ij}s^{i}t^{j} as a cycle up to rational equivalence; the a_{ij} are called the bidegrees. ([Fulton], Example 8.4.4). If U is of pure dimension k and i + j = k, we have a_{ij} = (U · s^{n−i}t^{m−j}). By taking a representative in general position, it follows that the bidegrees a_{ij} are non-negative integers.

In a product of more than two projective spaces, multi-degrees are defined analogously.

A divisor H on a variety X is said to be very ample if it is the pullback of a hyperplane divisor of projective space, under some projective embedding of X. The projective degree of U ∈ B^{i}(X) under this embedding can then be expressed as U ·_{X} H^{i}, and written deg_{H}(U).

We will denote projection from a product X × X′ × … to some of its factors X × X′ by pr_{X,X′}.

Notation 10.10 (nota3) If A, B are cycles on P, a multi-projective spaces, write A ≤ B if B − A is effective. Equivalently, if {V }^{ijk} = V · h_{1}^{i}h_{2}^{j}h_{3}^{k} are the multi-degrees of a cycle V on P,

{A}^{ijk} ≤ {B}^{ijk}

for each i, j, k. Note that this partial order is preserved by the intersection product.

Let H_{i} be the hyperplane divisor on ℙ^{n}, H = pr_{1}^{∗}H_{1} + pr_{2}^{∗}H_{2}. Then H is very ample. (It corresponds to the Segre embedding of ℙ^{n} × ℙ^{n} in ℙ^{N}.)

The following lemmas will be used in §11.

Lemma 10.11 (proj0) Let D_{i} be a very ample divisor on a projective variety X_{i}, Y = X_{1} × X_{2}, dim (X_{i}) = d_{i}, D = pr_{1}^{∗}D_{1} + pr_{2}^{∗}D_{2}. Let π = pr_{1} : Y → X_{1} be the projection.

1. Let U be a k-dimensional subvariety of X_{1}. Then deg_{D}(U×X_{2}) ≤ \binom{k+d_{2}}{k} deg_{D_{2}}(X_{2}) deg_{D_{1}}(U)

2. Let W be an l-dimensional subvariety of Y. Then deg_{D_{1}}(π_{∗}W) ≤ deg_{D}(W)

Proof

1. dim (U × X_{2}) = k + d_{2}, and deg_{D}(U × X_{2}) = (pr_{1}^{∗}D_{1} + pr_{2}^{∗}D_{2})^{k+d_{2}}(U × X_{2}) == ∑_{i+j=k+d_{2}} \binom{i+j}{i}(D_{1}^{i}D_{2}^{j} · U × X_{2}) . The only nonzero factor is i = k, j = d_{2}.

2. deg_{D}(W) = ∑_{i+j=l} \binom{l}{i}(pr_{1}^{∗}D_{1}^{i}pr_{2}^{∗}D_{2}^{j}) · W ≥ pr_{1}^{∗}D_{1}^{l} · W = D_{1}^{l} · pr_{∗}W, using the

projection formula for divisors ([Fulton] 2.3 ). The last quantity equals deg_{D_{1}}(π_{∗}W). □

We can obtain some information about a proper intersection in a smooth variety Y by viewing it as an improper intersection in projective (or multiprojective) space. Note that two rationally equivalent cycles on Y are a fortiori rationally equivalent on the ambient projective space, so have the same degrees.

Lemma 10.12 (trans1) Let Y be a smooth subvariety of projective space ℙ^{m}. Let U, V be properly intersecting subvarieties of Y, W = U ·_{Y} V. Then deg(W) ≤ deg(U) deg(V)

Proof We may represent W by an effective cycle ∑ m_{i}W_{i}, where W_{i} are the components of U ∩ V and m_{i} = i(W_{i}, U · V; Y) are the intersection multiplicities.

Let L be a generic linear subspace of ℙ^{m} of dimension m − dim (Y) − 1, and let C be the cone over U with vertex L, cf. [Fulton] Example 11.4.1. C is a subvariety of ℙ^{m}, “union of all lines meeting U and L”.

Note that deg(C) = deg(U): take a generic linear space J of dimension complementary to C in ℙ^{m}, so that J meets C transversally in deg(C) points. The cone E on J with center L is (over the original base field) a generic linear space, and so meets U transversally, in deg(U) points. But both numbers equal the number of pairs (p, q) ∈ J × L such that p, q, r are colinear for some r ∈ U.

By [Fulton] Example 11.4.3, each W_{i} is a proper component of C ∩W, and m_{i} = i(W_{i}, C · V; ℙ^{m}).

By the refined Bezout theorem [Fulton] Example 12.3.1, ∑_{i} m_{i} deg(W_{i}) ≤ deg(C) deg(V). Thus deg(W) ≤ deg(U) deg(V). Notation 10.13 (not2.0) Let D be a very ample divisor on Y, and let U be a cycle, or a rational equivalence class of cycles. Define

|U|_{D} = sup_{S} inf_{U′,U′′} deg_{D}(U′) + deg_{D}(U′′)

where S ranges over all cycles of Y , and U′, U′′ range over all pairs of effective cycles such that U is rationally equivalent to U′ − U′′, and U′, U′′ meet S properly. Corollary 10.14 (mult1) Let Y be a smooth variety, D a very ample divisor on Y. Let |X| = |X|_{D}. Let U,V be cycles on Y. Then |U · V | ≤ |U||V | Proof Let W be an arbitrary cycle on Y, of dimension dim (Y) − dim (U) − dim (V). Write V = V′ − V′′, with V, V′ effective and meeting W properly, and |V | = deg(V′) + deg(V′′). Write U = U′ − U′′ similarly, with U′, U′′ meeting V′ ∩ W, V′′ ∩ W properly. Let X_{1} be an effective cycle representing U′ · V′, supported on the components of U′ ∩ V′; and similarly X_{2}, X_{3}, X_{4} for U′ ∩ V′′,...,U′′ ∩ V′′. Then by 10.1, the X_{i} meet W properly. By 10.12, deg(X_{1}) ≤ deg(U′) deg(V′), etc. Thus ∑_{i} | deg(X_{i})| ≤ |U||V |. Since W was arbitrary, |U · V | ≤ |U||V |. Lemma 10.15 (cyfin) For any cycle R on Y, and very ample H on Y, |R|_{H} is finite. Proof Say D, Y, U are defined over an algebraically closed field k. Since k is an elementary submodel of an algebraically closed field K with tr.deg._{k}K infinite, in the definition 10.13 of |R|_{H}, one can restrict the sup_{S} to range over cycles S defined over k, while allowing the U′, U′′ to be defined over K. The finiteness is now immediate from Lemma 10.5. □

10.3 Correspondences

(correspondences)

Let us recall the language of correspondences, (cf. [Fulton], 16.1). Let X,X′ be smooth, complete varieties of dimension d over an algebraically closed field. A correspondence R on X × X′ is a d-cycle on X × X′; we will write somewhat incorrectly R ⊂ (X × X′). The transpose of R is the corresponding cycle on X′×X; it is denoted R^{t}. If T is a correspondence of X′ × X′′, then one defines the composition by: T ∘ R = pr_{X,X′′ ∗}(pr_{X,X′}^{∗}R · pr_{X′,X′′}^{∗}T)

When the context does not make clear which product is in question, we will use X ∘ Y and X^{∘n} for the composition product and power, and X · Y, X^{·n} for the intersection product and power. Notation 10.16 (not2) Suppose X and X′ are smooth complete varieties of dimension d, given together with very ample divisors H,H′ on X,X′. Let R ⊂ (X × X′) be a correspondence. Write deg_{cor}(R) = [p × X′] · R where p is a point of X

deg_{cor}(R) is intended to denote the degree of R as a correspondence. If R is an irreducible subvariety R and pr_{X} restricts to a dominant morphism π : R → X, then deg_{cor}(R) is the degree of π. In particular that if R is the graph of a function, then deg_{cor}(Φ) = 1. By contrast, we will write deg(R) or deg_{cy}(R) for deg_{pr_{1}^{∗}H+pr_{2}^{∗}H′} R. If R = R_{1} − R_{2} with R_{1}, R_{2} effective (irredundantly), we let deg_{|cy|}(R) = deg_{cy}(R_{1}) − deg_{cy}(R_{2}).

Note that deg_{cor}(Φ) = 1, deg_{cor}(Φ^{t}) = q^{d}, deg_{cor}(S) = δ.

Lemma 10.17 (cy) Let R ⊂ X × Y and T ⊂ Y × Z be correspondences. Computing their degrees with respect to very ample divisors H_{X},H_{Y},H_{Z} on X,Y,Z respectively:

1. deg_{cor}(T ∘ R) = deg_{cor}(T) deg_{cor}(R).

2. deg_{cor}(R) ≤ deg_{|cy|}(R)

3. |T ∘ R| ≤ \binom{2d}{d}^{2}|T ||R|

4. Suppose Z = X and H_{X} = H_{Z}. Then

((T ∘ R) ·_{X^{2}} Δ_{X}) = (R · T^{t}) ≤ |T^{t}||R|

Proof

1. Let P_{XY} be the correspondence p × Y, where p is a point of X. Then deg_{cor}(R) = (P_{XY} · R) = Δ_{X} · (P_{XY}^{t} ∘ R). (See (4) below for the last equality.) For R the cycle of an irreducible variety, and hence in general, P_{XY}^{t} ∘ R is a multiple of P_{XX}^{t}. So

P_{XY}^{t} ∘ R = deg_{cor}(R)P_{XX}^{t}

Similarly P_{Y Z}^{t} ∘ T = deg_{cor}(T)P_{Y Y}^{t}, so P_{XZ}^{t} ∘ T = P_{XY}^{t} ∘ P_{Y Z}^{t} ∘ T = deg_{cor}(T)P_{XY}^{t}. Thus:

(P_{XZ}^{t} ∘ T ∘ R) = deg_{cor}(T)(P_{XY}^{t} ∘ R)

The desired formula follows. upon intersecting this with Δ_{X}.

2. Here we may assume R ≥ 0, so that deg_{|cy|}(R) = deg_{cy}(R). On X, H^{·n} = c[pt] for some c ≥ 1; so

deg_{cor}(T) = (P_{XY} · R) = c^{−1}(pr_{X}^{∗}H)^{n} · R ≤ deg_{cy}(R)

3. Both R ∘ T and |R|,|T | depend on R,T only up to rational equivalence, so we may change R and T within their rational equivalence class. By 10.3, R,T may be replaced so that deg_{|cy|}(R) ≤ |R|, deg_{|cy|}(T) ≤ |T |, (R × Z) ∩ (X × T) is a proper intersection, and pr_{X,Z}((R × Z) ∩ (X × T)) meets properly a given cycle S. Then R ∘ T = pr_{X,Z}_{∗}((R × Z) ∩ (X × T)), and by 10.11(1),10.12, and 10.11(2), deg_{|cy|}(W) ≤ (\binom{2d}{d} deg_{|cy|}(R)\binom{2d}{d} deg_{|cy|}(T)| ≤ \binom{2d}{d}^{2}|R||T |; and W meets S properly. Taking supremum over S, the statement follows.

4. The equality (T ∘ R) · Δ_{X} = (R · T^{t}) follows from the projection formula, and the definition of composition of correspondences; both are equal to the triple intersection product (R × X) · (X × T) · Δ_{13}. The inequality is clear from 10.12. □

10.4 Geometric statement of the uniformity

(geostat)

To formulate the theorem geometrically, we will need to consider V, q, and S ⊂ V × V^{ϕ_{q}} all varying separately. Below, B will be the base over which V varies. B′ will be the base for S; it need not be a subscheme of B × B, though one loses little or nothing by thinking of that case. A scheme over B is a scheme S together with a morphism α : S → B. We will not always have a notation for α; Instead, if L is a field and b ∈ B(L), we will write S_{b} for the fiber of S over the point b. If U → V is a dominant map of irreducible varieties over k, then the (purely inseparable) degree of U/V is the (purely inseparable) degree of [k(U) : k(V)].

Notation 10.18 (not)

1. B and B′ are reduced, irreducible, separated schemes over ℤ[1/m] or 𝔽_{p}. (Not necessarily absolutely irreducible.) A (base change) map β : B′ → B^{2} is also given. We will refer to the cases as “characteristic 0” and “characteristic p”, respectively. Let V be a scheme over B. View V^{2} as a scheme over B^{2}. Let S be a (B′−) subscheme of V^{2} ×_{B^{2}} B′. For L a field and b ∈ B′(L), denote β(b) = (b_{1}, b_{2}) ∈ V^{2}(L). We will assume that V_{b_{1}},V_{b_{2}} and S_{b} are varieties, and view S_{b} as a subvariety of V_{b_{1}} × V_{b_{2}}

2. We further assume that for b ∈ B′, S_{b},V_{b_{1}},V_{b_{2}} are absolutely irreducible; the projection map S_{b} → V_{b_{2}} is a quasi-finite map; dim (V_{b_{1}}) = dim (V_{b_{2}}) = dim (S_{b}) = d, deg(S_{b}/V_{b_{1}}) = δ < ∞, and if B is over 𝔽_{p}, the purely inseparable degree of S_{b} over V_{b_{2}} is δ′_{p}.

3. Let q be a prime power (power of p), a ∈ B(K_{q}), b ∈ B′(K_{q}) such that β(b) = (a, ϕ_{q}(a)). In this situation, we denote:

V_{b}(S, q) = {c ∈ V_{a}(K_{q}) : (c, ϕ_{q}(c)) ∈ S_{b}(K_{q})}

In this language, the asymptotic version of ACFA 1 is the following: Theorem 1B’ Let B be a reduced, separated scheme of finite type over ℤ[1/m] or 𝔽_{p}. Let B′, V, S, β be as in 10.18. For any sufficiently large q, if b ∈ B′(K_{q}), and b_{2} = ϕ_{q}(b_{1}), then there exists c ∈ V_{b_{1}}(K_{q}) with (c, ϕ_{q}(c)) ∈ S_{b}(K_{q}).

While we need a mere existence statement, we see no way to prove it without going through a quantitative estimate. We formulate this estimate as follows. Note the similarity to the Lang-Weil estimates; these are the special case when the S_{b} are the diagonals. Theorem 1B

Let assumptions be as in Theorem 1B’. Then there exists a nonempty open subscheme B′′ of B′ and constants ρ and δ^{∗} > 0 such that if b ∈ B′′(K_{q}), β(b) = (a, ϕ_{q}(a)), then V_{b}(S, q) is finite, of cardinality

#(V_{b}(S, q)) = δ^{∗}q^{d} + e with |e| ≤ ρq^{d− \frac{1}{2}}

with δ/δ′_{p}

Theorem 1B’ follows from Theorem 1B by restriction to an open subset of V, and using Noetherian induction on B′.

The statement 1B refers to number of points, without multiplicities. In fact each point (excepting O(q^{d−1})) occurs with multiplicity δ′_{p}, so that the number of points counted with multiplicity is δq^{d} + O(q^{d− \frac{1}{2}} ).

10.5 Separability

Lemma 10.19 (sep) In Theorem 1B, we may assume that for b ∈ B′, S_{b}/V_{b_{2}} is separable.

Proof If B⊗Spec ℚ ≠ ∅, this may simply be achieved by replacing B′ by an open subscheme B′′ so that for b ∈ B′′, deg(S_{b}/V_{b_{2}}) is a constant δ′, and then further by the open subscheme B′′′ = B′′⊗_{ℤ[1/m]}ℤ[1/(mδ′)]. Othewise we may assume B is over 𝔽_{p}, and δ′_{p} = p^{l}. We will modify B′ and S so as to obtain a similar situation with δ′_{p} = 1; the modified objects will be denoted by a but B and V are left the same.

Let F_{B} : B^{2} → B^{2} be the map (Id, ϕ_{p^{l}}). Let B̃′ = B′ ×_{B^{2}} B^{2}, where the implicit map B^{2} → B^{2} is F_{B}. (Thus if B′ is a subscheme of B^{2}, B̃′ = F_{B}^{∗}(B′)_{red}) is the reduced scheme underlying the pullback of B′ by F_{B}^{∗}). Now F_{B} induces a map F_{2} : B̃′ → B′. Also let F_{1} denote the map (Id, ϕ_{p^{l}}) : V^{2} → V^{2}, and let

F = (F_{1} ×_{F_{B}} F_{2}) : V^{2} ×_{B^{2}} B̃′ → V^{2} ×_{B^{2}} B′

Finally let S̃ = F^{∗}(S)_{red} be the reduced pullback of S via F. Also if q is a power of p, let q̃ = q/p^{l}. We now claim that δ̃′_{p} = 1, δ̃ = δp^{ld}/δ′_{p}, and that if b̃ ∈ B̃′(K_{q}), b = F (b̃), then V_{b}(S, q) and V_{b}(S̃, q̃) coincide as sets. This is easily seen by going to fibers: if β̃(b̃) = (b_{1}, b̃_{2}) then β(b) = (b_{1}, b_{2}) with b_{2} = ϕ_{p^{l}}(b̃_{2}). If further (a_{1}, ã_{2}) is a generic point of S̃_{b}, and a_{2} = ϕ_{p^{l}}(ã_{2}), then (a_{1}, a_{2}) is a generic point of S_{b}. We have δ = (K_{q}(a_{1}, a_{2}) : K_{q}(a_{1})), δ̃ = (K_{q}(a_{1}, ã_{2}) : K_{q}(a_{1})), and the degree computations are elementary. The truth of Theorem 1B for S now follows from the same for S̃.

10.6 Rough bound (Bezout methods)

We give here an alternative proof of the rough upper bound on the number of points on the intersection of a variety with Frobenius, 9.22, using Bezout’s theorem. We require Fulton’s ”refined” version since the intersection can have components of various dimensions (if only at infinity.) We need to reduce to an intersection of divisors, partly in order to arrive at a situation where intersection multiplicities agree with geometric multiplicities.

Lemma 10.20 (bezout-1) Let l, k ∈ ℕ. Let U_{i} be a d_{i}-dimensional irreducible subvariety of ℙ^{n} (i = 1, …, r). Let V be a subscheme of ℙ^{n} cut out by hypersurfaces of degree b. Then there exists a subscheme V′ ⊇ V such that for any i, any d_{i} − k-dimensional irreducible component of U_{i} ∩ V is also an irreducible component of U_{i} ∩ V′; and V′ is the intersection of k hypersurfaces of degree b.

Proof Say V is cut out by P_{j} = 0 (j = 1, …, l), with P_{j} a homogeneous polynomial of degree b. Let M be a k × l matrix, with generic coefficients, i.e. avoiding a finite number of proper Zariski closed sets of matrices, to be specifed below. Let (Q_{1}, …, Q_{k}) = M(P_{1}, …, P_{l}). Let V′[m] be the zero set of {Q_{1}, …, Q_{m}}, V′ = V′[k]. Evidently V′ ⊇ V. Fix U = U_{i}, d = d_{i}, and a d − k-dimensional irreducible component C of U ∩ V. Then C ⊆ (U ∩ V′), and we have to show that C is an irreducible component of U ∩ V′, i.e. that C is contained in no larger irreducible subvariety of U ∩ V′. Let U′ be a Zariski open neighborhood in U of the generic point of C, not meeting any of the other irreducible components of U ∩V. Let C′ = (C ∩U′). Then dim (C′) = d − k. Claim For m ≤ k, dim (U′ ∩ V′[m]) ≤ d − m.

For m = 0 this reads dim (U′) = d, which is clear. Assume m < k and the inequality is true for m; to show that it holds for m + 1, let E be an irreducible component of U′ ∩ V′[m]. Then dim (E) ≤ d − m, and it suffices to show that dim (E ∩ V′[m + 1]) < d − m. If dim (E) < d − m there is nothing to show. Otherwise, dim (E) = d − m > d − k; as dim (U′ ∩ V) = dim (C′) = d − k, we have E ⊈ V. Thus not all P_{j} vanish on E. By genericity of the m + 1’st row of M over the previous rows, Q_{m+1} does not vanish on E. So E ⊈ V′[m + 1]. Thus dim (E ∩ V′[m + 1]) < dim(E) ≤ d − m, proving the claim.

In particular, dim (U′ ∩ V′) ≤ d − k; so C′ is an irreducible component of U′ ∩ V′. It follows that C is an irreducible component of U ∩ V′. □

If U is a variety and C is an irreducible subvariety of U, let dim_{C}(U) be the dimension of U at a neighborhood of the generic point of C; i.e. dim_{C}(U) = min dim (U ∩W) where W ranges over Zariski open subsets of ℙ^{n} containing a dense subset of C. Equivalently, dim_{C}(U) is the maximal dimension of a component of U containing C. Let codim_{C}(U) = dim_{C}(U)−dim (C).

Let V be a subscheme of ℙ^{n} cut out by hypersurfaces of degree b, and let V′[k] be defined as in the proof of Lemma 10.20 above. Corollary 10.21 (bezout-1c) Let U be a variety, and let C be an irreducible component of U ∩ V. Then C is a component of U ∩ V′[k], with k = codim_{C}(U).

The geometric multiplicity of U ∩ V′[k] along C is at least that of U ∩ V along C. Proof Let {U_{i}} be the irreducible components of U containing C, d_{i} = dim (U_{i}). Then C is an irreducible component of each U_{i}. By Lemma 10.20, C is an irreducible component of U_{i} ∩ V′[k_{i}] with k_{i} = d_{i} − dim (C). Now k_{i} ≤ k, so V ⊆ V′[k] ⊆ V′[k_{i}]; hence C is also an irreducible component of U_{i} ∩ V′[k]. Since this is true for each U_{i}, C is an irreducible component of U ∩ V′[k].

The statement regarding geometric multiplicity, is immediate since V ⊆ V′[k] as schemes. Here are a few special cases, with U = ℙ^{n}. Let C be an irreducible subvariety of a variety

P, F a set of hypersurfaces of P. We say C is weakly cut out by F if it is a component of the intersection scheme ∩F.

More generally, if the cycle [∩F] = ∑ n_{i}C_{i} with C_{i} the cycle of an irreducible variety, and 0 ≤ m_{i} ≤ n_{i}, we say that the cycle ∑ m_{i}C_{i} is weakly cut out by F.

Let S be a n − k-dimensional subvariety of ℙ^{n}, weakly cut out by hypersurfaces of projective degree b. Then S is weakly cut out by k hypersurfaces of degree b.

More generally, if C = ∑ m_{i}C_{i} is weakly cut out by l hypersurfaces of degree b, and dim (C_{i}) ≥ n − k for each i, then C is weakly cut out by k hypersurfaces D_{i} of degree b.

The proof of the next lemma will use positivity properties of intersection theory in multiprojective space. Let ℙ = ℙ^{n_{1}} × ℙ^{n_{2}} × ℙ^{n_{3}} be a product of three projective spaces. Let T_{i} be divisors on ℙ, with multi-degrees (a_{i}, b_{i}, c_{i}) with respect to the standard divisors H_{i}, pullbacks of the hyperplane divisors on ℙ^{n_{i}}. Then

{[T_{1} ∩ … ∩ T_{m}]}_{dim (ℙ)−m} ≤ T_{1} · . . . · T_{m}

Here on the left side we have the proper part of the cycle corresponding to the schemetheoretic intersection, and on the right-hand side the intersection cycle in the sense of intersection theory. The inequality ≤ has the sense of 10.10. More explicitly, one considers the intersection of T_{1} × … × T_{m} with the diagonal embedding of ℙ in ℙ^{m}. This intersection has proper components W_{1}, …, W_{l}, and other distinguished varieties Z_{j}. By [Fulton] 6.1, we have the canonical decomposition

T_{1} · . . . · T_{m} = ∑_{i} m_{i}[W_{i}] + ∑_{j} m_{j}α_{j}

where α_{j} are certain cycles on the W_{ν}. We push forward to ℙ to obtain the same equality for cycles on ℙ. Now the tangent bundle to multi-projective space is generated by global sections ([Fulton] 12.2.1(a,c)), hence ( [Fulton] 12.2(a)) each α_{j} is represented by a nonnegative cycle. So

T_{1} · . . . · T_{m} ≥ ∑_{i} m_{i}[W_{i}]

On the other hand, by [Fulton] 7.1.10, m_{i} is precisely the geometric multiplicity of W_{i} in the intersection scheme T_{1} ∩ … ∩ T_{m}; so {[T_{1} ∩ … ∩ T_{m}]}_{dim (ℙ)−m} = ∑ m_{i}[W_{i}]. The claimed Bezout inequality follows.

We now consider an analogue of Lemma 10.20 when two degrees b, B are involved. Assume U is a subscheme of ℙ^{n} cut out by hypersurfaces of degree ≤ b, while V is a subscheme of ℙ^{n} cut out by hypersurfaces of degree ≤ B.

Let Ξ(U, V, l, k) be the set of components C of U ∩ V with dim (C) = l, dim_{C}(U) = k; write deg(Ξ(U, V, l, k)) = ∑_{C∈Ξ(U,V,l,k)} m_{C} deg(C) where m_{C} is the geometric multiplicity of C in U ∩ V, and deg(C) is the projective degree.

Lemma 10.22 (bezout-2) deg(Ξ(U, V, l, k)) ≤ b^{n−k}B^{k−l}

Proof Say U is cut out by homogeneous polynomials P_{i}^{U} of degree b, while V is cut out by homogeneous polynomials P_{j}^{V} of degree B. As in Lemma 10.20, let U′[m] (respectively V′[m]) be the scheme cut out by m generic linear combinations of the P_{i}^{U} (respectively P_{j}^{U}.)

Let C ∈ Ξ(U, V, l, k). By the dimension theorem, dim (C) ≥ dim_{C}V + dim_{C}U − n, so dim_{C}V ≤ n − k + l. It follows using the genericity in the definition of V′[j] that for j ≤ k − l we have dim_{C}V′[j] = n − j. (The inequality ≥ is true by the dimension theorem, since V′[j] is the intersection of j hypersurfaces.) In particular, dim_{C}V′[k − l] = n − (k − l).

By Corollary 10.21, C is an irreducible component of U ∩V′[k−l]. We have codim_{C}V′[k− l] = n − (k − l) − l = n − k. Hence again by 10.21, C is an irreducible component of U′[n − k] ∩ V′[k − l]; with geometric multiplicity at least m_{C}. Now this is an intersection of n − k hypersurfaces of degree b and k − l hypersurfaces of degree B, so by the refined Bezout theorem as discussed above, ∑ m_{C} deg(C) ≤ b^{n−k}B^{k−l}. □

Corollary 10.23 (bezout-3) Let L be an algebraically closed field. f_{1}, …, f_{r}, g_{1}, …, g_{s} be polynomials in n-variables, with deg(f_{i}) ≤ b, deg(g_{j}) ≤ B, where b ≤ B ∈ ℕ. Let J be the set of dimensions of components of the zero set Z(f_{1}, …, f_{r}), and assume W = Z(f_{1}, …, f_{r}, g_{1}, …, g_{s}) is finite. Then |W | ≤ ∑_{j∈J} b^{n−j}B^{j}.

Lemma 10.24 (fin-proj) Let H be the hyperplane divisor on ℙ^{n}, H_{12} = pr_{1}^{∗}H + pr_{2}^{∗}H on ℙ^{n} × ℙ^{n}. Let U be a purely k-dimensional cycle of ℙ^{n} × ℙ^{n}, cut out by hypersurfaces with H_{12}-degrees ≤ b. Then

deg_{H_{12}}[U ∩ Φ_{q}]_{l} ≤ b^{2n−k}(q + 1)^{k−l}

Proof The subvariety Φ_{q} ⊂ ℙ^{n} × ℙ^{n} is cut out by \binom{n}{2} hypersurfaces F_{ij} of bidegree (q, 1) (namely, those defined by X_{i}^{q}Y_{j} = X_{j}^{q}Y_{i}). The statement thus follows from Lemma 10.22.

Lemma 10.25 (fin) Let assumptions be as in 10.18 (1),(3). Assume further that for any b ∈ B′, the projection maps S_{b} → V_{b_{2}} has finite fibers. Then for all large enough q, and any b ∈ B′, V_{b}(S, q) is finite. Moreover, for some constant ρ, #V_{b}(S, q) ≤ ρq^{d}.

Proof Note that the assumptions remain true if V is replaced by a proper subscheme U and S by any component of S ∩ U^{2} ×_{B^{2}} B′. Hence using Noetherian induction, we may assume the lemma holds for proper closed subschemes of V.

As in 10.19, we may assume the projection S_{b} → V_{b_{2}} is generically separable. Let b ∈ B′(K_{q}); note that V_{b}(S, q) is the set of points of a scheme over K_{q}, and let C be an irreducible component over K_{q}. Let a_{1} be a generic point of C, a_{2} = ϕ_{q}(a_{1}). Then (a_{1}, a_{2}) ∈ S_{b} so the field extension [K_{q}(a_{1}, a_{2}) : K_{q}(a_{2})] is finite and separable. Since it is also purely inseparable, K_{q}(a_{1}) = K_{q}(a_{1}^{q}), so K_{q}(a_{1}) is a perfect, finitely generated field extension of K_{q}. It follows that K_{q}(a_{1}) = K_{q}, so that C is finite. Since C was an arbitrary component, V_{b}(S, q) must be finite.

For the quantitative bound, we may (again using Noetherian induction and stratifying if needed) embed V_{a} in a projective space ℙ_{n}; the dimension and degree of the embedding are bounded irrespective of a ∈ B, as well as the bidegrees (f_{1}, f_{2}) of the resulting embedding of S_{b} in ℙ_{n} × ℙ_{n}. Now V_{b}(S, q) is the projection to V_{a_{1}} of the intersection of S_{b} with the graph Φ_{q} of the Frobenius map ϕ_{q} : ℙ_{n} → ℙ_{n}. By 9.8 or 10.24 the intersection has O(q^{d}) points, and the lemma follows. □

Remark 10.26 (transverse)

(1) If S ⊂ V × V^{σ} are varieties over k, and S → V^{σ} is separable in the sense that k(a) ∈ k(b)^{sep} for any L ⊃ k and (a, b) ∈ S(L), then S, Φ_{q} meet properly. Indeed if (a, b) ∈ (S ∩ Φ_{q})(L) then k(a) ⊂ k(a^{1/q})^{sep}, so k(a)^{sep} is perfect, and it follows that a ∈ k^{sep}. See 5.4.

(2) ([Lang59], [Pink92]) if S → V^{σ} is étale, then S and Φ_{q} intersect transversally. (Look at the tangent spaces: T Φ_{q} is horizontal at each point, while T S → T V^{σ} is injective.)

10.7 Finite covers

Lemma 10.27 (birational) (Birational invariance). Let B,V,B′,S satisfy the hypotheses of Theorem 1B. Suppose Ṽ is an open subscheme of V, so that for a the generic point of B, Ṽ_{a} is an open subvariety of V_{a}. Let S̃ = S ∩ Ṽ^{2} ×_{B^{2}} B′. Let B̃′ be an open subscheme of B′ such that the hypotheses of 10.18 hold. Then Theorem 1B is true for B,V,B′,S iff it holds for B,Ṽ,B̃′,S̃.

Proof The invariants δ,δ′_{p} are the same for the two families. The sets V_{b}(S, q) and Ṽ_{b}(S̃, q) are also the same, except for points in F_{b}(S^{∗}, q), where F is a closed subscheme of V complementary to Ṽ at the generic fiber, and S^{∗} is the restriction of S to F. By 10.25, the size of F_{b}(S^{∗}, q) is of the order of the error term in the expression for V_{b}(S, q) in 1B. □

The first item of the next lemma will be used to reduce to smooth varieties. The second will not be used, but originally gave some indication of the correctness of the form of Theorem 1B, since it shows that the desired lower bound in this theorem follows from the upper bound. (Once once one knows the theorem for the easy case of single difference equations),

10.28 (2qin)

#(V_{b}(S, q)) ≤ δ^{∗}q^{d} + e with e ≤ ρq^{d− \frac{1}{2}}

We will write such statements as

#(V_{b}(S, q)) ≤ δ^{∗}q^{d} + O(q^{d− \frac{1}{2}} )

The point is that the implicit coefficient depends only on the system (B′, V′, S, β) and not on the choice of b or of q.

Lemma 10.29 (birat2) Let B,V,B′,S satisfy the hypotheses of 1B. Let h_{B} : B̃ → B and h_{V} : Ṽ → V be compatible quasi-finite maps (base extension for B, and finite cover of V). Let B̃′ = B′ ×_{B^{2}} B̃^{2}. We get an induced map h : (Ṽ)^{2} ×_{B^{2}} B̃′ → V^{2} ×_{B}^{2} B′. Let S̃(j) be the various irreducible components of h^{−1}S, that are nonempty in the generic fiber.

(a) Suppose the conclusion of 1B holds for each B̃,Ṽ,B̃′,S̃(j). Then it holds also for the original system.

(b) Conversely, suppose the conclusion of 1B hold for (B, V, B′, S), and the conclusion of 10.28 holds for each B̃,Ṽ,B̃′,S̃(j). Then the conclusion of 1B holds for each B̃,Ṽ,B̃′,S̃(j).

Proof We can first apply the base change from B to B̃ without changing V (i.e. replacing V by V ×_{B} B̃); this clearly makes no difference to 1B. Thus we may assume B̃ = B, and B̃′ = B′. Let a be the generic point of B. Let h_{a} : Ṽ_{a} → V_{a} be the map induced by h on the fibers. Outside of a proper subvariety of V_{a}, all fibers of h_{a} have the same size s. This proper subvariety may be removed using 10.27. Similarly we may assume the various S̃(j) are disjoint. Let b be a generic point of B′. Let

H = h_{b_{1}} × h_{b_{2}} : Ṽ_{b_{1}} × Ṽ_{b_{2}} → V_{b_{1}} × V_{b_{2}}

We have H^{−1}S_{b} = ∪_{j}S(j)_{b}. Thus

H^{−1}V_{b}(S, q) = ∪_{j}V_{b}(S̃(j), q)

By the constant size of the fibers of h_{a},

s#V_{b}(S, q) = Σ_{j}#V_{b}(S̃(j), q)

A simple exercise in degrees of field extensions shows:

δ^{∗}s = ∑_{j} δ_{j}^{∗}

Where δ_{j}^{∗} refers to S̃(j), cf. 1B. (Because of 10.19, only the case δ′_{p} = 1 need be considered in the applications).

Now (a) if 1B holds S̃(j), we have

#V_{b}(S̃(j), q) = δ_{j}^{∗}q^{d} + O(q^{d− \frac{1}{2}} )

Summing over j , using the previous two displayed equalities, we obtain

s#V_{b}(S, q) = δ^{∗}sq^{d} + O(q^{d− \frac{1}{2}} )

and division by s yields (a). (b) follows similarly, using the principle that if ∑_{i} a_{i} = ∑_{i} a′_{i} and each a_{i} ≤ a′_{i}, then each a_{i} = a′_{i} (here all up to O(q^{d− \frac{1}{2}} ).) □

10.8 Reduction to smooth varieties

Lemma 10.30 (smooth) In Theorem 1B, we may assume that V is an open subscheme of V̄, S = S̄ ∩ V^{2}, with V̄ smooth and projective over B.

Proof In characteristic zero, we can use Hironaka’s resolution of singularities; the generic fiber V_{b} of V is birational to an open subvariety of a smooth, proper variety, and by 10.27 the theorem is invariant under birational changes of this type. Similarly, in general, we can use 10.29 and the following theorem of de Jong ([deJong]):

For any variety V over a field k there exists a finite extension k̃ of k, and a smooth projective variety Ṽ over k̃, and a finite, dominant morphism from an open subvariety of Ṽ to V. □

Remark 10.31

For theorem 1B’ the reduction to the smooth case is not needed on this (geometrical and numerical) level; a similar but easier reduction shows directly (using [deJong]) that the instances of the axiom scheme ACFA referring to open subvarieties of smooth complete varieties imply the rest of the axioms.

11 The virtual intersection number

(virtual)

This section is devoted to a proof of theorem 11.2. It is formally similar to Theorem 1B; but it refers to the virtual intersection number, in the sense of intersection theory, and not to the actual intersection. By virtue of 10.30,10.19 we can now assume that the varieties V_{a} are smooth and complete, and S_{b}/V_{b_{2}} separable. But as a result of the completion process, the intersection may have infinite components. The effect of these components on the intersection number will have to be estimated in the next section. Lemma-Notation 11.1 (sm) Let μ, ν be bounds for the absolute degree and the sum of Betti numbers of V_{b}, so that:

• For b ∈ B, there exists a very ample divisor H_{b} on V_{b}, such that |S_{b}| = |S_{b}|_{H_{b}} ≤ μ • For b ∈ B, letting X be the variety V̄_{b} base-extended to an algebraically closed field, we

have Σ_{i≤2dim (X)}dim H^{i}(X, ℚ_{l}) ≤ ν. Proof To show the existence of either μ or ν, it suffices to show that the numbers in question are bounded on a Zariski open subset of V; then one may use Noetherian induction. Let b be a generic point of V, and pick a very ample H_{b} on V_{b}. The construction of 10.15 can be described by a first-order formula concerning b. This formula must remain valid for a Zariski neighborhood of b. Thus |S_{b′}| is uniformly bounded on this neighborhood. The same argument, using the constructibility of the higher derived images of constant sheaves, shows that Σ_{i≤2dim (X)}dim_{Z/lZ}H^{i}(X, Z/lZ) is bounded on a Zariski neighborhood; this number bounds also the sum of the Betti numbers. Theorem 11.2 (2qc) Let notation be as in 1B and 10.30. Let b ∈ B′(K_{q}) , β(b) = (a, ϕ_{q}(a)), X = V̄_{a}, X′ = V̄_{ϕ_{q}(a)}. Let Φ_{b} ⊂ (X × X′) be the graph of Frobenius. Then

S_{b} ·_{X×X′} Φ_{b} = δq^{d} + e with |e| ≤ \binom{2d}{d}^{2}μνq^{d− \frac{1}{2}}

Example 11.3 (cycle-t) In the case of projective space, i.e. V̄_{b_{1}} = ℙ_{d}, we can immediately prove Theorem 11.2 cycle-theoretically (without transcendental cohomology.)

Let S̄_{b} ⊂ ℙ_{d}^{2} be the closure of S_{b}. Consider the intersection of S̄_{b} with Φ_{q}, the graph of Frobenius on ℙ^{d}. In the intersection theory sense, it can be computed in terms of the bidegrees:

[Φ_{q}] = ∑_{i} q^{i}h_{1}^{i}h_{2}^{d−i}

[S] = ∑_{i} a_{i}h_{1}^{i}h_{2}^{d−i}

with a_{0} = δ. So

[Φ_{q} · S] = ∑_{i} a_{i}q^{d−i} = δq^{d} + O(q^{d−1})

□

For curves, Weil’s “positivity” proof of the Riemann Hypothesis ([Weil48]) works here too. ( The notation follows §10.3. ) Example 11.4 (weil) Let C be a smooth, complete curve of genus g over a field K of characteristic p > 0. Let q = p^{m}. Let ϕ_{q} be the q-Frobenius, and let C′ = C^{ϕ_{q}}. Let S ≤ C × C′ be an irreducible subvariety. Then

S · Φ_{q} = q deg_{cor}(S) + deg_{cor}(S^{t}) + e

with

|e| ≤ ((2g)(2 deg_{cor}(S) deg_{cor}(S^{t}) − |S · S^{t}|))^{1/2}q^{1/2}(≤ O(1)q^{1/2})

Proof Let A_{0} be the group of divisors on C × C′, up to rational equivalence, A = ℚ⊗A_{0}. Define a symmetric bilinear form:

β(X, Y) = deg_{cor}(X) deg_{cor}(Y^{t}) + deg_{cor}(Y)deg_{cor}(X^{t}) − X · Y

If X,Y are divisors on C × C′, let X^{t}Y denote the intersection-theoretic composition X^{t} ∘ Y. It is a correspondence on C × C; and we have:

Δ.(X^{t}Y) = X.Y, deg_{cor}(X^{t}Y) = deg_{cor}(Y)deg_{cor}(X^{t})

Thus β(X, Y) = β_{C}(X^{t}Y, Δ) where β_{C} is defined like β, but on C × C.

Let Z = Φ_{q}. We have Z^{t}Z = ZZ^{t} = qΔ (where Δ denotes the diagonal of C, respectively C′.) By associativity of composition, for any X and Y,

(Z^{t}X)^{t}(Z^{t}Y) = X^{t}(ZZ^{t})Y = qX^{t}Y

Hence β(X, Y) = β_{C}(X^{t}Y, Δ) = (1/q)β_{C}((Z^{t}X)^{t}, (Z^{t}Y)). Now according to Weil, β_{C}(W^{t}, W) ≥ 0; hence β(X, X) ≥ 0. This non-negativity extends to A⊗ℝ.

By Cauchy- Schwartz, we have β(X, S) ≤ β(X, X)^{1/2}β(S, S)^{1/2}. Now

β(Z, Z) = β_{C}(qΔ, Δ) = qβ_{C}(Δ, Δ) = q(2 − Δ · Δ) = q(2 + (g − 2)) = 2gq
(Z · S) = qdeg(S) + deg(S^{t}) − β(S, Φ_{q})

So e^{2} ≤ β(S, S)(2gq), and the estimate follows. □

But in general, we will have to decompose [S] and Φ_{q} cohomologically and not cycletheoretically.

It will be convenient to renormalize the norm, and write ||S|| = \binom{2d}{d}^{2} |S|. Then (by 10.17) ||S ∘ T || ≤ \binom{2d}{d}^{4} |S| |T | = ||S|| ||T ||

When dealing with the étale cohomology groups, we will always work over an algebraically closed field k, fix a prime l ≠ char(k), and fix an isomorphism of ℤ_{l}(1) with ℤ_{l}. See the section of [Deligne 74] on “orientations”, and [Kleiman68]. (For the groundwork, see [Deligne 77] and the references there, or [Freitag-Kiehl] or [Milne1980]).

Notation 11.5 (not3) If R is a correspondence on X × X′, we let η_{i}(R) : H^{i}(X, ℚ_{l}) → H^{i}(X′, ℚ_{l}) be the endomorphism of H^{i}(X, ℚ_{l}) induced by R ; cf [Kleiman68], 1.3. When X = X′, we let τ_{i}(R) the trace of this endomorphism.

When R is a correspondence on X as in 11.5, the Lefschetz trace formula ([Kleiman68], 1.3.6) relates the intersection number of R with the diagonal to the endomorphisms η_{i}(R) as follows: 11.6 (lef)

R · Δ_{X} = Σ_{i=1,...,2n}(−1)^{i}τ_{i}(R)

For the top cohomology, we have 11.7 (top) Let n = dim (X). Then dim H^{2n}(X) = 1, and τ_{2n}(R) = deg_{cor}(R) is the degree of S as a correspondence (10.16).

(The first statement is in [Kleiman68] 1.2; hence τ_{2n} acts as multiplication by a scalar, and this scalar can be seen to be deg_{cor}(R) by computing the effect of η_{2n}(R) on the image of a point under the cycle map.)

We will make weak use of Deligne’s theorem ([Deligne 74]).

11.8 (deligne) Let X be a smooth projective variety over k, the algebraic closure of a finite field. Suppose X descends to 𝔽_{q}. Let Φ be the graph of the Frobenius correspondence x ↦ x^{q} on X × X. Then all the eigenvalues of η_{i}(Φ^{t}) are algebraic numbers, of complex absolute value q^{i/2}.

We wish to study eigenvalues of compositions of two correspondences S,T. This can probably be done using the remark 11.14 below. However we will take another route, starting with a slight generalization of [Lang59], V.3 lemma 2.

Lemma 11.9 (cp0) Let b, t > 0 and ω_{j}, c_{j} be complex numbers (j = 1, …, s). Suppose

lim sup t^{−n} |Σ_{j}c_{j}ω_{j}^{n}| ≤ b

Then one may partition {1, …, s} into disjoint sets J_{1},J_{2} such that

Σ_{j∈J_{2}}c_{j}ω_{j}^{n} = 0

for all n, and

|ω_{j}| ≤ t

for each j ∈ J_{1}.

Proof Let M be the maximal norm of an ω_{j}. Dividing the ω_{j} and t by M, we may assume M = 1.

Consider first the ω_{j} lying on T, the group of complex numbers of norm one. Say they are ω_{1}, …, ω_{s′}; note that summing only over j ≤ s′ does not effect the lim sup in the hypothesis. Let S be the closed subgroup of T^{s′} generated by the point α = (ω_{1}, …, ω_{s′}). Now S is a compact group. In every neighborhood of S, there are infinitely many powers α^{m}; for these we have (c_{1}, …, c_{s′}) · α^{m} arbitrarily small; hence there exists a point γ in the closure of this neighborhood, with (c_{1}, …, c_{s′}) · γ = 0. So such points are dense in S, and thus (c_{1}, …, c_{s′}) · γ = 0 for all γ ∈ S. We may thus put all the ω_{i}, i ≤ s′, into J_{2}. Now removing them from the sum has no effect, so the hypothesis applies to ω_{s′+1}, …, ω_{s}, and we may proceed by induction. □

Lemma 11.10 (deg1) Let R and T be commuting correspondences on a smooth projective variety X of dimension d over an algebraically closed field K, T ∘ R = R ∘ T. Fix an embedding of ℚ_{l} into ℂ. Let c_{ij}, e_{ij} be the corresponding eigenvalues of the endomorphisms η_{i}(R), η_{i}(T), considered as complex numbers. Let J_{1} be the set of (i, j) such that e_{ij} has complex norm at most ||T ||, and J_{2} the rest. Then for any m, k Σ_{(i,j)∈J_{2}}(−1)^{i}c_{ij}^{k}e_{ij}^{m} = 0 Proof The following claim is immediate from 10.17 (3) and (4): Claim For all m, (R ∘ T^{∘m} · Δ_{X}) ≤ |R| ||T ||^{m}

Let γ = ∑_{(i,j)∈J_{1}} |c_{ij}| + |R|. By definition of J_{1}, |Σ_{(i,j)∈J_{1}}(−1)^{i}c_{ij}e_{ij}^{m}| ≤ (γ − |R|)||T ||^{m} By 11.6, (R ∘ T^{∘m}) · Δ_{X} = Σ_{i}(−1)^{i}τ_{i}(RT^{m}) = Σ_{ij}(−1)^{i}c_{ij}e_{ij}^{m} Thus

|Σ_{(i,j)∈J_{2}}(−1)^{i}c_{ij}e_{ij}^{m}| ≤ γ||T ||^{m}

By 11.9, one can partition J_{2} into disjoint sets J_{3},J_{4} such that Σ_{(i,j)∈J_{3}}(−1)^{i}c_{ij}e_{ij}^{m} = 0 for all positive integers m, and |e_{ij}| ≤ ||T || for each (i, j) ∈ J_{4}. But then J_{4} ⊆ J_{1} by definition of J_{1}, so J_{4} = ∅. Thus J_{2} = J_{3}, and we have shown what we wanted for k = 1. Applying this to R^{k} in place of R yields the lemma. □

The following Proposition does not mention Frobenius, and could be stated over arbitrary fields, but the proof we give uses [Deligne 74].

Proposition 11.11 (bound) Let R be a correspondence on a smooth projective variety X over k, the algebraic closure of a finite field. Then every eigenvalue of η_{i}(R) is an algebraic number, of complex absolute value at most ||R||.

It suffices to show that in every complex embedding, every eigenvalue has absolute value at most ||r||. Fix therefore a complex embedding, and view the eigenvalues of η_{i}(R) as complex numbers e_{ij}. Let T = Φ_{q}^{t} be a Frobenius correspondence on X, with q large enough so that T ∘ R = R ∘ T as correspondences. Let J_{2} index the set of e_{ij} of absolute value > |R|. Let c_{ij} be the eigenvalues of T, with corresponding indices. By 11.10, for any m and k,

Σ_{(i,j)∈J_{2}}(−1)^{i}c_{ij}^{m}e_{ij}^{k} = 0

Suppose for contradiction J_{2} is nonempty. Let i_{max} be the highest index represented in J_{2}, and let J_{3} = {j : (i_{max}, j) ∈ J_{2}}. Let d_{ij} = (−1)^{i}c_{ij}q^{−i_{max}/2}, and let d_{j} = d_{i_{max},j}, e_{j} = e_{i_{max},j}. Then |d_{j}| = 1, |d_{ij}| < 1 for i < i_{max}, and Σ_{(i,j)∈J_{2}}d_{ij}^{m}e_{ij}^{k} = 0 for any k,m. For fixed m,k we have Σ_{(i,j)∈J_{2}}d_{ij}^{m+m′}e_{ij}^{k} = 0 Let m′ approach infinity, becoming highly divisible; then d_{ij}^{m′} approaches 0 for i < i_{max}, while d_{j}^{m′} approaches 1. We obtain Σ_{j∈J_{3}}d_{j}^{m}e_{j}^{k} = 0 for any k,m. Dually, let J_{4} = {j ∈ J_{3} : |e_{j}| = L}, where L is the highest value attained by an e_{j}. Then as above we get Σ_{j∈J_{4}}d_{j}^{m}(e_{j}/L)^{k} = 0 for any k,m. But now letting both m and k approach infinity, each d_{j}^{m} and each (e_{j}/L)^{k} approach 1, and we obtain |J_{4}| = 0, a contradiction. Thus J_{2} = ∅ and we are done.

Corollary 11.12 (bound2) Let X be a smooth complete variety over the algebraic closure k of a finite field. Let X′ be the image of X under the q-Frobenius automorphism of K, and let T = Φ_{q}^{t} ⊂ X′ × X be the transposed Frobenius correspondence. Let R ⊂ X × X′ be a correspondence. Then every eigenvalue of η_{i}(T ∘ R) is an algebraic number, of complex absolute value at most q^{i/2} ||R||.

Proof For some m, X and S are defined over 𝔽_{q^{m}}. Taking m’th powers, it suffices to prove that every eigenvalue of η_{i}((T ∘ R)^{m}) is an algebraic number, of absolute value at most q^{im/2}||R||^{m}. Let X_{i} be the image of X under ϕ_{q}^{i}, and R_{i} ⊂ X_{i} × X_{i+1} the image of R. Then (T ∘ R)^{m} = Φ_{q^{m}}^{t} ∘ R_{m−1} ∘ … ∘ R Let R^{∗} = R^{m−1} ∘ … ∘ R. Then (10.17) ||R^{∗}|| ≤ Π_{i=0,...,m−1} ||R_{i}|| = ||R||^{m}. R^{∗} is a correspondence on X. By 11.11, every eigenvalue of η_{i}(R^{∗}) is algebraic of absolute value at most |R|^{m}. Now note that Φ_{q^{m}}^{t} is a correspondence on X, commuting with R^{∗}, and (11.8) with algebraic eigenvalues of absolute value q^{im/2}. Thus η_{i}((T ∘ R)^{m}) = η_{i}(Φ_{q^{m}}^{t} ∘ R^{∗}) has algebraic eigenvalues of absolute value (||R||q^{i/2})^{m}. The conclusion follows.

Lemma 11.13 (2qcr) In Theorem 11.2 we may assume K_{q} is the algebraic closure of the prime field; in other words it suffices to prove the result for b chosen from a finite field.

Proof We may assume V (and V ×_{B} V) are flat over B. Let T be an irreducible component of B′ ∩ β^{−1}Φ_{B}, where Φ_{B} ⊂ B^{2} is the graph of Frobenius on B. By [Fulton] (10.2), the number

S̄_{b} ·_{V̄_{b_{1}} ×V̄_{b_{2}}} Φ_{b}

is constant for (b_{1}, b_{2}) ∈ T. Thus one can replace X, X′ by V_{b_{1}}, V_{b_{2}}, where b is a point of T rational over a finite field. □

proof of Theorem 11.2

By 11.13, we may assume K is the algebraic closure of a finite field. By 11.6, S_{b} · Φ_{b} = ((Φ_{b})^{t} ∘ S_{b}) · Δ_{X} = Σ_{i≤2d}τ_{i}((Φ_{b})^{t} ∘ S_{b}) By 11.12,

|Σ_{i<2d}τ_{i}((Φ_{b})^{t} ∘ S_{b})| ≤ \binom{2d}{d}^{2}μνq^{d− \frac{1}{2}}

By 11.7, τ_{2d}((Φ_{b})^{t} ∘ S_{b}) = deg_{cor}((Φ_{b})^{t} ∘ S_{b}) = deg_{cor}(S_{b})q^{d}. □

We conclude the section with the rationality lemma mentioned but not used above.

Lemma 11.14 (2vars) Let X be a smooth projective variety over a field K, and let S, T be correspondences on X. Write a power series in two variables:

F (s, t) = Σ_{m,n≥0}S^{m} · T^{n}s^{m}t^{n}

Then F is a rational function, with denominator of the form f(s)g(t).

Proof Using Grothendieck’s cohomological representation, it suffices to prove more generally that if V is a complex vector space, and S, U two linear transformations of V, then Σ_{m,n≥0}tr(S^{m}U^{n})s^{m}t^{n} is a rational function of two variables.

We will actually show that

Σ_{m,n}S^{m}U^{n}s^{m}t^{n} ∈ End(V)[[s, t]]

lies in End(V)⊗ℂC(s, t), so that all the matrix coefficients are rational functions. (With denominators as specified.) Now

Σ_{m,n≥0}S^{m}U^{n}s^{m}t^{n} = (Σ_{m}S^{m}s^{m})(Σ_{n}U^{n}t^{n})

So it suffices to consider Σ_{m}S^{m}s^{m}. We can sum the geometric series:

Σ_{m≥0}S^{m}s^{m} = (1 − Ss)^{−1} = (det(1 − Ss)^{−1})S′

For a certain matrix S′ with coefficients polynomial in s. □

12 Equivalence of the infinite components: asymp-

totic estimate

(equivalence)

While Theorem 1.1 promises actual points of intersection, Theorem 11.2 refers rather to an intersection number, i.e. to the intersection with a better placed S′ rationally equivalent to S. We will now bridge the gap by giving an asymptotic estimate of the difference between the number of isolated points of these intersections. Here we do so numerically and roughly (up to lower order of magnitude); this requires only tying together some previous threads. This suffices for the proof of Theorem 1.1. However our methods are essentially precise and motivic, and later we will describe the language needed to bring this out.

We will actually bound at once both the equivalence of the infinite components, and the contribution of isolated points of high multiplicity.

We work with a single fiber of the data 10.18: for some b ∈ B′(K_{q}) with ϕ_{q}(b_{1}) = b_{2}, we let X̃_{1} = V_{b_{1}}, X̃_{2} = V_{b_{2}}, S̃ = S_{b}. Thus we have a d-dimensional variety X̃_{1} over K_{q}, X̃_{2} = X̃_{1}^{ϕ_{q}}, and an irreducible subvariety S̃ ⊂ X̃_{1} × X̃_{2}. We will bound a certain quantity associated with the intersection of S̃ with the graph of Frobenius. We will write O(q^{l}) for a quantity bounded by Cq^{l}, where C is a constant independent of q, and dependent on X̃_{1}, X̃_{2}, S̃ in a way that remains bounded when the varieties are obtained as above from the data 10.18, with b varying.

We assume (cf. 10.19,10.30) that the projection from S̃ to X̃_{2} is étale, and that X̃_{1} is an open subvariety of a smooth, complete variety X_{1}. (And thus X̃_{2} of X_{2} = X_{1}^{ϕ_{q}}). Let Φ = Φ_{q} ⊂ Y = X_{1} × X_{2} denote the graph of ϕ_{q} : X_{1} → X_{2}, and let S be the Zariski closure of S̃ in X_{1} × X_{2}. Proposition 12.1 (.1) Let e = (Φ_{q} · S) − |{Φ_{q} ∩ S̃}|. Then |e| ≤ O(q^{d−1})

Observe that the points of Φ_{q} ∩ S̃ are simple, by by 10.26; so it makes no difference whether |{Φ_{q} ∩ S̃}| is counted with multiplicities.

Almost all Frobenius specializations Recall that a Frobenius field K_{q} is an algebraically closed difference field of characteristic p > 0 made into a difference field via x ↦ x^{q} (q = p^{m}, m ≥ 1). |Y | denotes the size of a 0-dimensional scheme, i.e. the number of points with multiplicities.

We will consider difference schemes Y of finite type over a difference field k. Y arises by base extension from a scheme Y_{D} over a finitely generated difference domain D with fraction field K. We will say: ”for almost all (q, h), ⋯ ” to mean: for some such D and Y_{D}, for all sufficiently large prime powers q, and all h : M_{q}(D) → K_{q} . . ..

When K is infinite, by enlarging D, one sees that a simpler formulation is equivalent: ”for some D, Y_{D}, for all prime powers q and all h : h : M_{q}(D) → K_{q} . . ..” When concerned with only almost all (q, h), we will not mind increasing D within K, and

so can assume that some such Y_{D} has been fixed, and write Y_{q,h} for M_{q}(Y_{D})⊗_{h}K_{q}. Different choices of Y_{D} will give the same Y_{q,h} for almost all (q, h).

Let D ⊂ k be a finitely generated difference domain, and let D[t]′ be a finitely generated sub-D-difference algebra of k[t̆]_{σ}. By enlarging D[t]′, and D, we may obtain the form D[t]′ = D[t^{σ−n}, f^{−1}]_{σ}, where f ∈ D[t]_{σ} and f(0) is invertible in D. Consider morphisms h : M_{q}(D) → K_{q} into Frobenius fields K_{q} with q ≥ n. For any such h, let h_{t}^{∗} : D[t]′ → K_{q}(t) be the unique difference ring morphism extending h and with h_{t}^{∗}(t^{σ−n}) = t^{q−n}. Sometimes we will consider it as a map D[t]′ → K_{q}[t, f^{−1}(t)] or D[t]′ → K_{q}[t̆] := K_{q}[t, g^{−1} : g(0) ≠ 0]; in this case we will write h̆_{t}.

When Y is a difference scheme of finite type over k[t̆]_{σ}, we fix D[t]′ as above; given q and h : M_{q}(D) → K_{q}, we define Y_{q,h} = Y_{q,h_{t}^{∗}}.

Let k be a perfect, inversive difference field, V an absolutely irreducible, projective (or complete) algebraic variety over k, dim (V) = d.

Let C(V) be the set of subvarieties S of V × V^{σ}, such that any component C of S (over k^{alg}) has dimension d, and the projections pr _{0}^{S} : S → V, pr _{1}^{S} : S → V^{σ} are dominant. Let ℤC(V) denote the free Abelian group generated by C(V). For any subscheme W of V ×V^{σ}, let [W] be the formal sum of the components of W of dimension d, weighted by their geometric multiplicities. (cf. [Fulton].)

Write S ∈ C(V)_{sep} if in addition, k(S) is a separable finite extension of k(V^{σ}). In this case, the projection pr _{1}^{S} is étale above a nonempty Zariski open subset of V^{σ}. Let V_{et}(S) be the largest smooth open subvariety U of V such that pr _{1}^{S} is étale above U^{σ}.

In general, for S ∈ C(V), let pr_{1}[1](S) = {a ∈ V : dim pr _{1}^{S}^{−1}(a) > 0}. Let V_{fin}(S) = (V \ pr_{1}[1](S)^{σ^{−1}}, and S̃ = S ∩ (V_{fin}(S) × V_{fin}(S)^{σ}). Thus V_{fin}(S) is the largest open subvariety of V such that, with this definition, pr_{1} : S̃ → Ṽ^{σ} is quasi-finite.

Let X(S) = (S̃ ⋆ Σ). So X(S) has total dimension d.

Let C_{f} (V) = {S ∈ C(V) : pr_{0}[1](S) ∩ (S ⋆ Σ) = ∅} = {S ∈ C(V) : X(S) = S ⋆ Σ}. (cf. 4.44 .) ℤC_{f} (V) is the free Abelian group generated by C_{f} (V).

The X(S) with S ∈ C_{f} (V) are directly presented, and of total dimension d; and moreover of transformal multiplicity 0.

Asymptotic equivalence Given S ∈ C(V), defined over a finitely generated difference domain D, let X = X(S), let η_{0}(S; q, h) be the number of points in X_{q,h}(K_{q}), and let η(S; q, h) = δ′_{p}η_{0}(S; q, h) (cf. 10.18).

We say S, S′ ∈ C(V) are asymptotically equinumerous (S ∼_{a} S′) if if there exists a finitely generated difference domain D ⊂ k (containing any given finite set), such that for some b, for almost all q, for all h : M_{q}(D) → K_{q}, |η(S; q, h) − η(S′; q, h)| ≤ bq^{d−1}.

We extend the terminology to ℤC(V) by additivity.

(We could instead count the points of X(S)_{q,h} with their geometric multiplicities, or with appropriate intersection multiplicities; asymptotically this makes no difference, and so we chose the approach involving least irrelevant complications.)

Let ℤC(V)_{rat} be the group of cycles in ℤC(V) that are rationally equivalent to 0, by a rational equivalence involving cycles in ℤC(V); i.e. ℤC(V)_{rat} is generated by cycles \underline{S}_{0} − \underline{S}_{∞} where \underline{S} ⊂ (V × V^{σ}) × ℙ^{1} is a d + 1-dimensional variety, with \underline{S}_{0}, \underline{S}_{∞} ∈ C(V). Similarly define ℤC(V)_{rat^{sep}} within ℤC(V)_{sep}. Recall 10.7 :

Lemma 12.2 (ae0) For any S ∈ C(V) there exists S′ ∈ ℤC_{f} (V) with S − S′ ∈ ℤC(V)_{rat}.

Lemma 12.3 (ae1) Let S ∈ C(V), and let Z be a proper Zariski closed subset of V. Then |X_{q,h} ∩ Z| = O(q^{d−1}).

Proof By 4.44 , 9.24 . □

Lemma 12.4 (ae2) Let \underline{S} be an irreducible d + 1-dimensional variety of (V × V^{σ}) × ℙ^{1}. Let t denote a generic point of ℙ^{1} over k. Assume S_{t} = \underline{S}_{t} and S_{0} = \underline{S}_{0} are in C(V)_{sep}. Then S_{t} ∼_{a} S_{0}.

(Here we identify S_{0}, a variety over k, with the same variety viewed by base change as a variety over k(t). The separability assumption could be dispensed with, as in 12.5 , 10.19 .) Proof Let VĂ_{k} = V ×_{Spec^{σ} k} Ă_{k}. V_{t} = V ×_{k} k(t)_{σ}, V_{0} = V denote the fibers of this map over the generic point of Ă_{k} and the point t = 0 respectively.

Let X_{t} be the closure of X(S_{t}) in V_{t} (i.e. the smallest k(t)_{σ}- definable closed difference subscheme containing X(S_{t}).) Let X be the closure in VĂ_{k} of X_{t}; it is a k- closed difference subscheme of \underline{V} = [σ]_{k}V ×_{k} Ă_{k}, flat over Ă_{k}, whose fiber over k(t)_{σ} is X_{t}.

Let W = V_{0} \ V_{et}(S_{0}); and let ^{∗}W be as in Proposition 8.9 ( a formula in the language of transformal valued fields over k(t)_{σ}.) By definition of X(S_{t}), and 4.44 , condition (1) of 8.9 holds; hence it holds also for the closure X_{t}. (Cf. 4.7 ). Since S_{0} ∈ C(V), condition (2) of 8.9 holds too; see 4.44 (2). Thus (4) holds, i.e. ^{∗}W has inertial dimension < dim (V).

Consider a (large) prime power q. Fix a discrete valuation of L = L_{q} = K_{q}(t)^{a} over K_{q} with val (t) = 1; let R_{q} denote the valuation ring. We will show that |X(S_{t})_{q,h}(L)| = |X(S_{0})(L)| mod O(q^{d−1}). Since q is large enough, X(S_{0})_{q,h} is a finite scheme; so X(S_{0})_{q,h}(L) = X(S_{0})_{q,h}(K_{q}).

Since V_{t} is projective, V_{t}(R_{q}) = V_{t}(L), and the residue homomorphism induces a map r_{V} : V_{t}(L) → V_{0}(L); restricting to r_{0} : (X_{t})_{q,h}(L) → (X_{0})_{q,h}(K_{q}). Let Y = (X_{t})_{q,h} \^{∗} W. Let r_{1} = r_{0}|Y. By definition of ^{∗}W, r_{1} : Y → V_{et}(S_{0}).

Injectivity of r_{1} follows from 7.3. An alternative argument: by 10.26 , X(S_{0})_{q,h} has only simple points on V_{et}(S_{0}); by 6.24 , r_{1} must be injective.

Surjectivity. We can use a Hensel lemma approach (analogous to 7.4 ; here only algebraic and not transformal valuation fields are in question, 7.5 .)

But we prefer to vary the visualization. Let ϕ denote the q-Frobenius, Φ ⊂ V × V^{ϕ} the graph of ϕ, \underline{Φ} = Φ × ℙ^{1}. Then \underline{S} ∩ \underline{Φ} is an algebraic set of relative dimension 0 over ℙ^{1}. By the dimension theorem, it has no isolated points. So it consists of horizontal curves, together with other curves and higher-dimensional components above the non-quasi-finite locus Let p ∈ (X_{0})_{q,h}(K_{q}), p ∉ W; then (p, 0) must lie on one of the horizontal curves C of \underline{S} ∩ \underline{Φ}. Let (p′, t) ∈ C with t a generic point of ℙ^{1}. Then p′ ∈ X(S_{t}). (If p′ ∈ pr_{1}[1](S_{t}) then p ∈ pr_{1}[1](S_{0}) ⊆ W.) Thus there exists a place of K_{q}(t)^{a} into K_{q} with p′ ↦ p. For the valuation corresponding to this place, p lies in the image of X_{t}(L) under the residue map; but all such valuations are isomorphic. This proves the surjectivity of r_{1}. □

We thus exhibited a bijection between X(S_{0})_{q,h}(K_{q}) \ W = X(S_{0})_{q,h}(K_{q}) ∩ V_{et}(S) and X(S_{t})_{q,h}(L)\ ^{∗}W (L). Now by 12.3 , at most O(q^{d−1}) of X(S_{0})_{q,h}(K_{q}) lies on W (even when counted with multiplicities on X(S_{0})_{q,h}(K_{q}).) By 7.18 (cf. 7.19 ), ^{∗}W (L) has O(q^{d−1}) points. This proves the lemma. □

Lemma 12.5 (ae3) Let S ∈ ℤC(V)_{rat}. Then S ∼_{a} 0.

Proof Since 12.4 was stated for C(V)_{sep}, we wish to reduce to S ∈ ℤC(V)_{rat^{sep}}. To this end, let ϕ(x) = x^{p^{l}}, where p^{l} = max_{T} [k[T] : k[V^{σ}]]_{insep}. Here T runs over all components of S, and any additional components needed to witness the rational equivalence of S to 0. Define τ by σ = τϕ; let V′ = V^{ϕ}; consider the map g : (V × V^{σ}) → (V′ × V^{σ}), g(v, w) = (ϕ(v), w). Then g_{∗}S ∈ ℤC(V)_{sep}. By [Fulton] §1.4, rational equivalence is preserved under proper push-forwards; (straightforwardly), so that g_{∗}S ∈ ℤC(V)_{rat^{sep}}. Thus as in 10.19, the statement for g_{∗}S implies the same for S, and we may assume S ∈ ℤC(V)_{sep}. (An alternative here, if we wished to use multiplicities, would be to use the projection formula for intersection multiplicities, |T ∩ Φ_{q}| = |g_{∗}T ∩ Φ_{q/p^{l}}|, and a bound for the geometric multiplicities showing them to be equal.)

We may take S = [\underline{S}_{0}]−[\underline{S}_{∞}], \underline{S} an irreducible d+1-dimensional variety of (V ×V^{σ})×ℙ^{1}. Let t denote a generic point of ℙ^{1} over k. Then by 12.4 , [\underline{S}_{t}] ∼_{a} [\underline{S}_{0}] and [\underline{S}_{t}] ∼_{a} [\underline{S}_{∞}]. So [\underline{S}_{0}] − [\underline{S}_{∞}] ∼_{a} 0 , as required. □

We repeat the statement of 12.1 in this language, and prove it. Corollary 12.6 Let S ∈ C(V)_{sep}. For almost all (q, h), the numbers |X(S)_{q,h}|, (S_{q,h} · Φ_{q}), Φ_{q} ∩ S̃ all differ by at most O(q^{d−1}). Proof We will use [Fulton] Prop. 7.1 and the notions of multiplicity there. By 12.2, there exists S′ ∈ ℤC_{f} (V) with S − S′ ∈ ℤC(V)_{rat}. By 12.5, S ∼_{a} S′. Since rational equivalence preserves intersection numbers, S′_{q,h} · Φ_{q} = S_{q,h} · Φ_{q}. Thus we may assume S = S′, i.e. we may assume S ∈ C_{f} (V). As in 12.5, 10.19, we may assume in addition that S ∈ CV sep. By 10.26, the intersection S_{q,h} ∩ Φ_{q} on (V × V^{σ})_{q,h} is proper; so (S_{q,h} · Φ_{q}) is the number of points of S_{q,h} ∩ Φ_{q}, counted with intersection multiplicities. On V_{et}(S)_{q,h} the geometric multiplicity of the intersection is 1; while on Z = V \ V_{et}(S), by 12.3, the total number of points of intersection (with geometric multiplicities) is O(q^{d−1}). Thus counting the points of X(S)_{q,h} with intersection or with geometric multiplicities, or without multiplicities agree up to O(q^{d−1}). Since the intersection multiplicities are sandwiched in between, and have value 1 whenever the geometric multiplicity is 1, they yield the same result. □

13 Proofs and applications

(proofs)

13.1 Uniformity and Decidability

Throughout this paper we have been dealing with a variable Frobenius on an essentially fixed variety. When the variety is not itself defined over the fixed field of the Frobenius, this is slightly strange; given the algebraic variety Y = X × X^{ϕ} there is (at most) one Frobenius on it. We have explained the uniformity in three different ways:

1. By fixing, not Y and S, but only the degrees of their projective completions. (As in Theorem 1.1)

2. Using the formalism of 10.18 (As in 1B)

3. Using difference schemes. In this language, one could state Theorem 1A thus:

Theorem 1.1C Let D be a finitely generated difference domain, X an absolutely transformally integral difference scheme of total dimension d over Spec^{σ} D. Then there exist b ∈ ℕ and c ∈ D such that for prime power q > b,and any y ∈ M_{q}(D[c^{−1}]), X_{q,y} is nonempty, indeed has δ^{∗}q^{d} + O(q^{d−1/2}) distinct points.

Here if K is the field of fractions of D and L is the transformal function field of X, then δ^{∗} = deg_{lim}(L/K)/ι′(L/K), where deg_{lim} is the limit degree, and ι′ is the purely inseparable dual degree, cf. 5.4.

Uniformity statements such as Theorem 1.1 amount to the same in any of the formulations. To demonstrate this, we show (2) implies (3) implies (1), and (2). proof that Theorem 1.1B is equivalent to Theorem 1.1C

In effect Theorems 1.1B and 1.1C make the same statement about points of a difference scheme X; but Theorem 1.1B assumes the generic fiber X of X is directly presented, while Theorem 1.1C assumes instead that X is irreducible. One may pass from the directly presented to the general case using 4.36; and from the irreducible case to the general case using 4.43. proof of Theorem 1B

We make the assumption stated in 10.30, and use the notation there. (We also accept the conclusion of 10.19.) By Theorem 11.2, S̄ · Φ_{q} has degree as stated in 1B. Theorem 1B now follows from Proposition 12.1.

□

proof that Theorem 1.1B implies Theorem 1

Suppose given a family X_{i} of affine varieties over algebraically closed fields k_{i} of characteristic p_{i}, and S_{i} ⊂ (X×X^{ϕ_{q_{i}}}), with the assumptions of Theorem 1.1 holding, and the degrees of the projective completions of the X_{i} and S_{i} bounded, but with (q^{−(d− \frac{1}{2} )}(|S(k)∩Φ_{q}(k)|)−aq^{d} unbounded. Let (L, σ) be an ultraproduct of the (k_{i}, ϕ_{q_{i}}). Since the degrees and dimensions are bounded, the corresponding ultraproducts of the X_{i} and S_{i} are ordinary subvarieties X,S ⊂ X × X^{σ} of finite dimensional affine space over L. Let D be a finitely generated difference sub-domain of L, with X,S defined over L; replacing D by a localization, we may assume X,S remain absolutely irreducible varieties of dimension d when reduced modulo any prime ideal of D. This puts us in the situation of Theorem 1.1B. For almost every i, one has a difference ring homomorphism D → k_{i}, mapping X to X_{i}. By Theorem 1.1B, X_{i} ∩ Φ_{q} has about aq^{d} points. A contradiction. □

Corollary 13.1 Let k be an algebraically closed difference ring, X a difference variety over K of transformal dimension l. Then there exists a finitely generated difference domain D ⊂ l, such that X descends to D, and for almost all q, for all h : D → K_{q}, dim (X_{q,h}) = l.

Proof X admits a definable map to [σ]_{K}𝔸^{l}, whose image contains the complement of a proper difference subvariety of [σ]_{K}𝔸^{l}. Thus for almost all q, h,X_{q,h} admits a dominant map to 𝔸^{l}, and so has dimension ≥ l. The converse is 9.8. □

We note one more form of Theorem 1.1; where in effect quasi-finiteness is replaced by the weaker assumption of finite total dimension.

Theorem 13.2 (1D) Let V be a quasi-projective variety over K_{q}, and let S ⊂ (V ×V^{ϕ_{q}}) be an irreducible subvariety. Assume dim (S) = dim (V) = d, and pr_{V}|S is a generically quasifinite map of degree a. If the scheme S∩Φ_{q} is 0-dimensional, then it has size ≤ aq^{d}+O(q^{d− \frac{1}{2}} )

Proof By taking ultraproducts, we find (V, S) over a difference domain D, with X = [σ]S∩Σ of finite total dimension; and we must show that |X_{q,h}| ≤ aq^{d} + O(q^{d− \frac{1}{2}} ). By Theorem 1.1, it suffices to show that X has total dimension d, and that every algebraic component of X of total dimension d is Zariski dense in V. This follows from 15.7. □

ACFA Proof of Theorem 1.4 That T_{∞} contains ACFA follows from Theorem 1.1. Now it is shown in [Chatzidakis-Hrushovski] that ACFA is nearly complete: the full elementary theory of a model (K, σ) of ACFA is determined by the isomorphism type of (K_{0}, σ|K_{0}), where K_{0} is the subfield of K consisting of points algebraic over the prime field. By the Cebotarev density theorem, every isomorphism type of (K_{0}, σ|K_{0}) can occur with σ_{0} an ultrapower of Frobenius maps. So every completion of ACFA is consistent with T_{∞} = ACF A. Thus every sentence of T_{∞} is a consequence of ACFA.

Decidability The decidability referred to in this paper, in particular in Theorem 1.6, is in the sense of Gödel; it corresponds to the dichotomy: finite/infinite, and not to: finite/bounded, or to distinctions between different degrees of boundedness. The proof of Proposition 12.1 can routinely be seen to be effective in this sense. We mention two instances. Suppose one has a smooth variety Y over a finite field, and two subvarieties V,X on Y, meeting improperly. One knows there exists Y′ rationally equivalent to Y, intersecting X transversally. Then without further work, this is effective: it is merely necessary to search (over some finite field extension) for a Y′ and for data demonstrating the rational equivalence. To take another example, suppose a sentence is shown to be true in every (boolean-valued) ω-increasing transformal valuation field, by whatever methods. Since the class of such transformal valuation fields is an elementary class, one can search for an elementary formal proof, with assured success; the proof will only use that the valuation is m-increasing, for some m. Thus the statement will be true in every valuation field with the automorphism σ(x) = x^{q}, as soon as q ≥ m; and we have found m effectively.

Theorems I or 1B follow from Propositions 11.2 and 12.1. To use Theorem 11.2, we require an effective bound on the Betti numbers dim H^{i}(X, ℚ_{l}) of a smooth projective variety X over an algebraically closed field. In characteristic zero, by Artin’s comparison theorem, one can compute the Betti numbers using singular cohomology. For this it suffices to search and find a triangulation. In positive characteristic, the situation is somewhat less clear a priori, but the proof in SGA4 or in [Freitag-Kiehl] of the finiteness of these numbers is everywhere effective. An explicit bound is given in [Katz01] (thanks to Emmanuel Kowalski for the reference.):

Fact 13.3 ([Katz01]) Let b(n, m) be the maximum possible Betti number of a smooth subvariety of ℙ^{n} of degree m. Then b(n, m) is bounded by a recursive function.

Proof of Theorem 1.6. Let T be the theory of all K_{p}. The completions of T are:

• The theories T_{p} of the individual K_{p}; these are just the theory of algebraically closed fields of characteristic p, with an axiom stating (∀x)(σ(x) = x^{p}). Call this axiom α_{p}.

• The completions of T^{∞}_{0}, the extension of ACFA stating that in addition, the field has characteristic zero.

Let β_{n} be the disjunction of α_{p} for p ≤ n.

Thus T is axiomatized by sentences of the form:

β_{n} ∨ ϕ

with ϕ an axiom of ACFA; the problem is only to know, given ϕ, a value of n for which this disjunction is true. Now the proof of 11.2 estimates an intersection number S̄ · Φ_{q} as aq^{d}, with an error of ≤ b(q^{d− \frac{1}{2}} ), with b bounded effectively and independently of q. 12.1 gives effectively another constant c, such that |S ∩Φ_{q}|−(S̄·Φ_{q}) ≤ cq^{d−1}. Thus it suffices to choose n such that cq^{d−1} + b(q^{d− \frac{1}{2}} ) < aq^{d} for q > n.

This shows that the axioms above are recursively enumerable. On the other hand, every sentence is equivalent, modulo these axioms, to a bounded Boolean combination of existential sentences describing a finite extension of the prime field. The decidability problem for the theory is thus reduced to the same problem for such sentences, and this is settled by Cebotarev. □

13.2 Finite simple groups

In the case of algebraic simple groups G of dimension n, over an algebraically closed field, the proof is classical (cf. e.g. Humphreys, Linear Algebraic Groups): any conjugacy class X of G generates G in at most dim (G) + 1 steps. One considers X_{n}, the set of n-fold products of elements of X. Then dim (X_{n}) is nondecreasing, and must eventually stabilize: dim (X_{n−1}) = dim (X_{n}), n ≤ dim (G). At this point, one considers the stabilizer of the Zariski closure of X_{n}, and concludes it is a group of dimension equal to dim (X_{n}), and containing a conjugacy class. Simplicity of G implies that the stabilizer equals G.

Boris Zilber realized that the proof generalizes to groups of finite Morley rank; the key to his proof was a different definition of the stabilizer, using the dimension theory directly without reference to closed sets.

Difference equations of finite total dimension do not have finite Morley rank, but they do have ”finite S1-rank”, cf [Chatzidakis-Hrushovski]. More precisely, such a dimension theory applies to their solution sets in a universal domain for difference fields; not necessarily to smaller fields.

In the presence of finite S1-rank, the stabilizer must be defined in a significantly different way than in finite Morley rank; it can however be defined, and enjoys similar properties, sufficient to make the proof go through. See [Hrushovski-Pillay].

The ultraproducts of the groups G_{n}(q) are a priori a group defined in the same way over the ultraproduct U of the difference fields GF (p)^{alg}, ϕ_{q}). The proof presented in [Hrushovski-Pillay] would then go through, provided one knows that U has an appropriate dimension theory. The present paper completes the proof by showing that U is a universal domain for difference fields.

13.3 Nonstandard powers and a suggestion of Voloch’s

We first restate and prove 1.5. Let F = GF (p)^{alg} be the algebraic closure of a finite field, and let G be the automorphism group. G is isomorphic to the profinite completion of Z, and we consider it with this topology.

Let F = GF (p)^{alg} . Let Υ be the set of automorphisms σ of F such that (F, σ) has ACFA. Then Υ is co-meager in the sense of Baire.

Proof Fix a variety V over F. Then V is defined over GF (q) for some power q = p^{l} of p. There are l possibilities for σ|GF (q); fix one of them, say ϕ_{p^{j}}; let V′ = V^{ϕ_{p^{j}}}; and then fix S ⊂ V × V′ as in the statement of ACFA. There are countably many possible V, j, V′, S altogether. It suffices therefore to show that for each such set of data, the set of σ meeting the particular instance of ACFA relative to V, V′, S is comeager among all σ with σ|GF (q) = ϕ_{p^{j}}. This set is an open set of automorphisms, so it suffices to show that it is dense. For this we must show that for any l′ and j′ with l|l′ and j′ = j(modl), there exists σ ∈ Υ with σ|GF (p^{l′}) = ϕ_{p^{j′}}, and such that the relevant instance of ACFA holds. However, by Theorem 1.1, any given instance of ACFA holds for any sufficiently large (standard) power of the Frobenius automorphism; in particular one can pick a Frobenius power p^{j′′} large enough, and such that j′′ = j′ (mod l′). Thus the open set in question is dense, and Υ is comeager. □

The examples of Cherlin-Jarden

Let l be a positive rational. The equation E_{l} : σ(x) = lx, x ≠ 0 implies σ^{n}(x) = l^{n}x. Thus E_{l} has no solutions fixed by σ^{n}, hence no algebraic solutions. So (ℚ̄, σ) is not a model of ACFA.

It can also be shown that if a_{l} ∈ E_{l}, with l varying over the positive rational primes, then the a_{l} are algebraically independent over ℚ. This justifies the assertion made after the statement of 1.5: any model of ACFA of characteristic 0 must have infinite transcendence degree over ℚ.

However,as a rule, an axiom of ACFA is true in (ℚ̄, σ) for a generic σ. For instance, among order-one difference varieties, this rule admits only a few concrete families of exceptions, of the form σ(x) = f(x) with f a fractional linear transformation, and σ(x) = ax + b, with x ranging over an elliptic curve.

Voloch’s question on two primes

Let p be a rational prime, and let Ω_{p} be the roots of unity of order prime to p. L_{p} = ℚ(Ω_{p}) be the maximal prime-to-p cyclotomic extension of ℚ. Let p_{1}, p_{2} be two distinct primes on the integers of L_{p}, lying above the rational prime p. Let V be a variety, say given by linear equations L over ℤ. Voloch suggested investigating the solutions to L modulo both primes, with coordinates on Ω_{p}:

(1) V (v_{1}, v_{2}) = {x = (x_{1}, …, x_{k}) ∈ Ω_{p}^{k} : L(x) ∈ p_{1} ∩ p_{2}}

He noted that x_{2} is conjugate to x_{1} by an automorphism θ of L_{p}, and that on Ω_{p}, one may write: θ(x) = x^{n}, with n ∈ (Π_{l≠p}ℤ_{l})^{∗}. The question is therefore equivalent to solving, in GF (p)^{alg}, the equations:

(2) (x_{1}, …, x_{k}) ∈ V, (x_{1}^{n}, …, x_{k}^{n}) ∈ V

Under what conditions is the number of solutions finite?

In a very special case, the present results are relevant. Assume n = p^{m} + 1, where m ∈ (Π_{l≠p}ℤ_{l}); or more generally, that n = f(p^{m}), f is a fixed nonconstant polynomial over ℤ. Choose m ∈ (Π_{l≠p}ℤ_{l}) generically. Then the structure (GF (p)^{alg}, x ↦ x^{n}) is interpretable in (GF (p)^{alg}, x ↦ x^{p^{m}}). The latter is a model of ACFA by 1.5, hence we have a precise criterion for the answer to (2). In particular, it is easy to see that that if V is defined by a single linear form with at least 3 variables, then (1) has infinitely many solutions.

13.4 Consequences of the trichotomy

The Lefschetz principle suggests transferring the results of [Chatzidakis-Hrushovski] to statements about the Frobenius. The propositions below were found in this way; though in retrospect they can be proved using only a very small part of the results of the present paper. Nonetheless the general Lefschetz principle gives the propositions a general context and makes the proofs immediate.

The first and third propositions are direct translations of Theorem 1.10 in [Hrushovski01], using Theorem 1.4. Note that the numerical bounds obtained there can be construed as firstorder statements.

Proposition 13.4 Let f ∈ ℤ[T] be an integral polynomial in one variable, with no cyclotomic factors. Let L(X_{1}, …, X_{k}) be a linear form over ℚ, k ≥ 3. Let X(q) be the roots of unity of order f(q), in 𝔽̃_{q}. Let Y (q) be the set of k-tuples a = (a_{1}, …, a_{k}) from X(q) satisfying L(a) = 0, but such that no proper sub-sum is zero. Then there exists an absolute bound B(f, L) and p_{0} such that for all primes p ≥ p_{0} and all powers q of p, |Y (q)| ≤ B(f, L). The bounds p_{0} and B(f, L) are effectively computable from f and L.

Proof If σ is an automorphism of a field K, let X(σ) = {a ∈ K : a^{f(σ)} = 1}, and let Y (σ) be be defined analogously to Y (q). By Theorem 1.10 in [Hrushovski01], the axioms of ACFA together with the axiom scheme of fields of characteristic 0, imply that X(σ) is finite; say |X(σ)| ≤ B where B = B(f, L) is finite. Hence a finite set of axioms of ACFA imply that if the field characteristic is larger than some p_{0}, then Y (σ) is finite. By Theorem 1.4, the said axioms hold in all algebraically closed fields of large enough positive characteristic, when σ(x) = x^{q}, q any power of the characteristic. □

Remark 13.5

1. An upper bound for B(f, L) can be given explicitly; it is doubly exponential in k and in the degree of f, with coefficients using the absolute values of the coefficients. This follows from [Hrushovski01], Proposition 1.11, by plugging in the parameters.

2. Using a positive - characteristic version of [Hrushovski01], Theorem 1.10, one can determine p_{0}; the proposition holds for all primes p such that f(p^{a/b}) ≠ 0 for all rational a/b.

3. One could take L over a number field instead of over ℚ, still with a uniform bound, independent of L.

4. One can also compute the set of numbers that actually occur as |Y (q)|, or and describe the set of q for which a particular number occurs.

By taking f(T) = T − 2, we obtain the following corollary; in hindsight it is easy to find a direct proof, but we keep it as an example. For a prime p, let E_{2}(p) be the set of natural numbers e such that 2 is in the multiplicative subgroup of (Z/eZ)^{∗} generated by p. (This includes the e such that p is a primitive root mod e.) Let

R_{2}(p) = {x ∈ GF (p)^{alg} : (∃e ∈ E_{2}(p))(x^{e} = 1)}

Corollary 13.6 Let p be an odd prime. Let L = ∑ c_{i}X_{i} be a linear form in k ≥ 3 variables. Let Y be the set k - tuples (a_{1}, …, a_{k}) ∈ R_{2}(p)^{k} with ∑ c_{i}X_{i} = 0 , but no proper sub-sum equals 0. Then |Y | ≤ 2^{k2^{(k−1)}}

Proof This follows from the proposition (and the remarks), since if x ∈ R_{2}(p) then x^{q−2} = 1 for some power q of p.

Similar remarks will apply to the propositions below. First, a non-linear analog. For simplicity we consider one-variable q-nomials over ℤ, but the results of [Chatzidakis-Hrushovski] allow an analysis of the behavior of any system of q-polynomials, in several variables, with respect to the question raised below; and the bounds obtained are independent of parameters, if the polynomial is over a larger field.

Let h(X, Y) ∈ ℤ[X, Y]. Let us say that h is special if there exists a curve C, either C = G_{m} or C an elliptic curve, a coset S of a subgroup of C ×C, correspondences U, V ⊂ (C ×ℙ^{1}) (U,V are irreducible curves, projecting dominantly to C and to ℙ^{1}), and points (a, b, c, d) such that (a, b),(a, c),(b, d), are generic points of S, U,V respectively, and h(c, d) = 0.

(If h is special, then the roots of the q-nomial h(X^{q}, X) are in uniform algebraic correspondence with either roots of unity as in the previous proposition, or an elliptic analog, or the points of 𝔽_{q}.)

Let us say that an affine variety U ⊂ 𝔸^{n} is special if it is explicitly a product of curves and points; i.e. there exists a partition of the coordinates of 𝔸^{n}, corresponding to an isomorphism j : 𝔸^{n} → 𝔸^{k_{1}} × … 𝔸^{k_{l}}, such that j(U) = C_{1} × … × C_{l}, with C_{i} a point or a curve. Theorem 13.7 (tricho1) Let h_{i}(X, Y) ∈ ℤ[X, Y] be non-special polynomials, and let U ⊂ 𝔸^{n} be an affine variety over ℤ. Then either h is special, or there exists a finite union W of special subvarieties of U (with W defined over ℚ) such that for all sufficiently large primes p, and all powers q of p, if h_{i}(a_{i}, (a_{i})^{q}) = 0 and (a_{1}, …, a_{n}) ∈ U, then (a_{1}, …, a_{n}) ∈ W. Proof Assume h is not special. By the trichotomy theorem of [Chatzidakis-Hrushovski], in any model ℚ ≤ M ⊧ ACV F,

X = {x : h(x, σ(x))}

is a stable, stably embedded definable set, whose induced structure is superstable of rank one, and distintegrated: the algebraic closure relation on X(M) \ ℚ^{a} is an equivalence relation.

Let U ⊂ 𝔸^{n} be an affine variety. If (a_{1}, …, a_{n}) ∈ U and a_{i} ∈ X, let w_{o}(a) = {i ≤ n : a_{i} ∈ ℚ^{a}}, w_{+}(a) = {i ≤ n : a_{i} ∉ ℚ^{a}} and let {w_{k}(a) : k = 1, …, m} be the classes of the equivalence relation on w_{+}(a) defined by: i ∼ j iff ℚ(a_{i})^{a} = ℚ(a_{j})^{a}. Let a(k) = (a_{i} : i ∈ w(k)), so that we can write a = (a(0), a(1), …, a(m)). Let C(k) be the locus over ℚ^{a} of a(k); so C(0) is finite, and C(k) is a curve for k ≥ 1. Let S(a) = {a(0)} × C(1) × … × C(m)}; it is a special variety, defined over ℚ^{a}. By disintegration of X, the fields ℚ(a(0)), …, ℚ(a(m)) are linearly free. Thus a is a generic point of S(a). Since a ∈ U, we have: S_{a} ⊂ U. Now a was an arbitrary point of U(M) ∩ X(M)^{n}, in a model M, and so by compactness, there are finitely many special S_{i}, defined over ℚ^{a}, such that U(M) ⊂ ∪_{i}S_{i}. Further taking the union of all ℚ-conjugates, we find a ℚ- definable finite union S of special varieties, with U ∩ X^{n} ⊂ S (in any difference field extension of ℚ.) The statement on the large primes follows immediately by compactness. □

Similar results exist for q-nomials of higher order, and for commutative algebraic groups. In particular, for semi-Abelian varieties, we have: Proposition 13.8 Let A be a commutative algebraic group scheme over a scheme Y of of finite type over ℤ, with generic fiber A_{Y} a semi-Abelian variety. Let f be an integral polynomial in one variable, with no cyclotomic factors. Let V be a subscheme of A. For any closed point y ∈ Y, let A_{y} be the fiber of A over y. Let ϕ denote the Frobenius endomorphism of A_{y}, relative to the residue field k_{y}, and let ψ be any finite power of ϕ. Let X(y, ψ, f) be the kernel of the endomorphism f(ψ) on A(k̃_{y}).

Then there exists a Zariski open Y_{0} ⊂ Y and finitely many group subschemes B_{i} of A over Y_{0}, with B_{i} ⊂ V, and an effectively computable integer B, such that for any closed point y ∈ Y, X(y, ψ, f) ∩ V is contained in at most B translates of some of the B_{i}.

The proof is analogous to that of 13.7. □

14 Jacobi’s bound for difference equations

Consider a system u of n difference equations u_{1}, …, u_{n} in n variables x_{1}, …, x_{n}; assume u_{i} has order h_{k}^{i} with respect to the variable x_{i}. Assume that the system defines a difference scheme of finite total dimension d. What bound can one impose on d, based on the data (h_{k}^{i})?

Jacobi considered this problem for differential equations, and gave as a bound:

j = max_{θ∈Sym(n)} Σ_{k=1}^{n}h_{k}^{θ(k)}

Jacobi’s proof was criticized by Ritt [Ritt], and the problem is considered open. See [Ritt], [KMP], [Cohn79] for partial results.

The transposition to difference equations was made by Cohn, with analogous partial results; cf. [Lando], [KMP]. Jacobi’s original idea was reduction to the linear case (the ”equations of variation”), and then the the case of constant coefficients; and present results still follow this line. However when the equations have hidden singularities, the reduction to the linear case requires additional assumptions, or new ideas.

We show here that Jacobi’s bound is correct for difference equations by a quite different method, Frobenius reduction. We also get an “explanation” for the curious determinant-like formula.

Cohn refined the problem by setting h_{k}^{i} = −∞ if x_{i} does not occur in u_{k}; we accept this refinement. (It not only gives a better bound in many cases, but allows for greater flexibility in manipulating the equations.)

Consider first the analogous problem for algebraic equations. Let U_{1}, …, U_{n} be polynomials in n variables x_{1}, …, x_{n}, over a field k; suppose U_{i} has degree H_{i}^{j} in the variable x_{i}. Let Z(U) be the scheme cut out by {U_{i}}, and let Z_{0} be the 0-dimensional part of Z(U).

Lemma 14.1 (jacobi-alg) |Z_{0}| ≤ J = ∑_{θ∈Sym(n)} ∏_{k=1}^{n} H_{k}^{θ(k)}

Proof Let V = (ℙ^{1})^{n} be the n’th Cartesian powers of the projective line. The tangent bundle of V is generated by its global sections, since this is true for each factor ℙ^{1}. The normal bundle to V embedded diagonally in V^{n} is hence also generated by its global sections. Thus by [Fulton] Theorem 12.2, each contribution to the intersection of V with a subvariety of V^{n} of codimension n is represented by a non-negative 0-cycle. The number of components of intersection, in particular the number isolated points, is therefore bounded by the intersection number (just as in [Fulton], Example 12.3.)

Each equation U_{i} determines a hypersurface [U_{i}] in V (the closure in V of the subscheme of (𝔸^{1})^{n} cut out by U_{i}.) Z_{0} is the 0-dimensional part of [U_{1}]∩⋯∩[U_{n}] ≃ ([U_{1}]×⋯×[U_{n}])∩V. By the above discussion, |Z_{0}| ≤ [U_{1}] · . . . , ·[U_{n}]. It remains to compute this intersection number.

Now [U_{i}] is rationally equivalent to ∑_{j=1}^{n} h_{i}^{j}D_{j}, where D_{j} is the divisor defined by the equation x_{j} = 0, the pullback of a point on the j’th copy of ℙ^{1}. We have D_{j}^{2} = 0, since {pt}^{2} = 0 in ℙ^{1}. Thus D_{θ}(1) · . . . · D_{θ}(n) = 0 if θ is not injective, while D_{θ}(1) · . . . · D_{θ}(n) = D_{1} · . . . · D_{n} = {pt} if θ is a permutation. Thus

[U_{1}] · . . . , ·[U_{n}] = ∏_{i=1}^{n} ∑_{j=1}^{n} h_{i}^{j}D_{j} = J

□

Theorem 14.2 Let k be a difference field u_{i} ∈ k[x_{1}, …, x_{n}]_{σ} a differential polynomial of order h_{i}^{j} in x_{j}. Let W be a component of the difference scheme Y cut out by h_{1} = ⋯ = h_{n} = 0. Then the total dimension of W, if finite, is no larger than Jacobi’s bound j.

Proof By assumption, W is transformally reduced, of total dimension w. Let v be a differential polynomial that does not vanish on W, but vanishes on every other component of Y. Adding a variable y and the equation vy = 1 does not change the Jacobi bound (using Cohn’s convention, the new equation has order 0 in the variable y, and all other equations have order −∞ in this variable.) Thus we may localize away from the other components; so we may assume that Y itself has finite reduced total dimension w. We must show that w ≤ j.

Let D ⊂ k be a finitely generated difference domain, with u_{i} ∈ D[x_{1}, …, x_{n}]_{σ}. Given a homomorphisms ϕ : D → K_{q}, let U_{i} be the polynomial obtained by replacing σ(x) by x^{q} and d by ϕ(d) in u_{i}.

Let H_{i}^{j} = deg_{x_{j}} H_{i}. Writing u_{j} as a sum of σ-monomials, they each involve u_{j}^{ν} for some ν ∈ ℕ[σ], and we have H_{i}^{j} = ν_{j}(q) for the highest such ν. This ν has the form dσ^{h_{i}^{j}} + ⋯ (lower terms), so H_{i}^{j} = dq^{h_{i}^{j}} + O(q^{h_{i}^{j} −1}). Thus lim_{q→∞} log_{q} H_{i}^{j} = h_{i}^{j}.

By 9.8, for some D, for almost all q and ϕ, Y_{q} = {x : U_{1}(x) = ⋯ = U_{n}(x) = 0} is finite; and by Theorem 1.1C, for some a, b ∈ ℚ with a > 0, |Y_{q}| = aq^{w} + e_{q}, |e_{q}| ≤ bq^{w−1/2}. On the other hand , by 14.1, |Y_{q}| ≤ J. So

aq^{w} + e_{q} ≤ ∑_{θ∈Sym(n)} ∏_{k=1}^{n} H_{k}^{θ(k)}

Taking log_{q} and letting q → ∞ we obtain

w ≤ max_{θ∈Sym(n)} ∑_{k=1}^{n} h_{k}^{θ(k)}

□

Remark 14.3 Ritt’s conjecture for difference equations.

As Cohn observed ([Cohn87] for differential equations, personal communication for difference equations), the validity of the refined Jacobi bound implies the Ritt dimension conjecture: given a system of m difference equations in n variables, if m < n then every component of the solution set has transformal dimension ≥ n − m. This can also be deduced directly from Theorem 1.1, as in 15.1.

15 Complements

(complements)

15.1 Transformal dimension and degree of directly presented difference schemes

(transdimdeg)

Here is a variant of 4.43, valid for all components. The proof assumes Theorem 1.4, and illustrates the way the main result of this paper may be used in difference algebra.

Proposition 15.1 (illustrate-dirdim) Assume theorem 1.4. Let X be a smooth algebraic variety over a difference field K. Let S ⊂ X × X^{σ} be an absolutely irreducible subvariety, dim (X) = d, and assume dim S = e + d. Let Z = {x ∈ X : (x, σ(x)) ∈ S} Then any component of Z has transformal dimension ≥ e.

Proof Let Z(1), …, Z(m) be the components. Suppose for contradiction that trans. dim. (Z(1)) < e. We may assume K is the field of fractions of a finitely generated difference domain D. By the main theorem, for almost all q, and almost y ∈ Spec M_{q}(D), M_{q}(Z)_{y} = ∪_{j}M_{q}(Z(j)_{y}), M_{q}(Z(1))_{y} ⊄ ∪_{j>1}M_{q}(Z(j))_{y}, and dim (M_{q}(Z(j))_{y}) ≤ trans. dim. (Z(1)) < e. It follows that M_{q}(Z)_{y} has a component of dimension < e. However M_{q}(Z)_{y} = S_{y} ⋆ Φ_{q}. Since (for almost all y) S_{y} ⊂ X_{y} × (X^{σ})_{y} is irreducible of dimension d + e, and X_{y} × (X^{σ})_{y} is smooth, the dimension theorem implies that every component of this intersection has dimension ≥ (d + e) + d − 2d = e; a contradiction. □

Remark 15.2 (ww+) Moreover, in 15.1, if h : Z → W is a morphism of difference schemes, and W has transformal dimension 0, then each component of each fiber of h has transformal dimension ≥ e.

The proof is similar, using the fact that M_{q}(W)_{y} is finite, and thus the components of the fibers of M_{q}(h)_{y} are also components of M_{q}(Z)_{y}.

Note that 15.1 is equivalent to to following purely geometric statement:

Lemma 15.3 (ww) Let U, V, S be quasi-projective varieties over an algebraically closed difference field k, with V ⊂ U^{σ}, and S ⊂ (U × V). Let T be the set of absolutely irreducible varieties T ⊂ S such that (pr_{1}T)^{σ} = pr_{2}T. (Where pr_{i}T is the Zariski closure of the i’th projection.) Let e ≥ 1. Assume dim (S) ≥ dim (U) + e. Then any maximal T ∈ T satisfies dim (T) ≥ dim (pr_{1}T) + e. Problem 15.4 Is there a simple direct proof of 15.3? Is the smoothness assumption necessary, even if one just wants one component of transformal dimension e?

The difficulty here is associated with singularities. If all varieties encountered were smooth, it would be easy to prove 15.3: Let T_{0} ∈ T . Let U′ be a component of pr_{1}[1](S) (cf. 4.44) containing pr_{1}(T_{0}). Let S′ = (U′ × U) ∩ S; then dim (S′) ≥ dim (S) − (dim (U) − dim (U′)) ≥ dim (U′) + e. Thus one can proceed by induction.

In attempting to find a direct proof of 15.3, I considered projecting at the right moment to ℙ^{m} by a finite map, and using the dimension theorem there. To do this, one needs to know Lemma 15.6 below; and in order to reduce to the case of purely positive transformal dimension, one finds oneself requiring a stronger statement than 15.3, namely 15.2 above. I did not carry through this proof to the end, but the lemmas that came up seem sufficiently suggestive in themselves to be stated here. Definition 15.5 (pure-pos) An irreducible difference variety U over K is purely positive transformal dimensional if every differential rational morphism on X into a difference variety of transformal dimension 0 is constant. Proposition 15.6 (pos-stat) Let K be a difference field, U, V algebraic varieties over K, f : U → V a quasi-finite map. Let X be an irreducible difference subvariety of V, X(K) Zariski dense in V. Assume X is purely positive transformal dimensional. Then for some Zariski open Ũ ⊂ U, f^{−1}(X) ∩ U is an irreducible difference variety (and also purely positive transformal dimensional.) Proof [Chatzidakis-Hrushovski]. Corollary 15.7 (www) Let U, S be affine varieties over an algebraically closed difference field k, with S ⊂ (U × U^{σ}). Let X = S ⋆ Σ ⊂ [σ]_{k}U. Then either X has positive transformal dimension, or else it has total dimension ≤ dim (U).

In the latter case, if U is absolutely irreducible, every algebraic component of X of total dimension equal to dim (U) is Zariski dense in U. Proof Let (a_{0}, a_{1}, …) be a generic point of an algebraic component C of U. (I.e (a_{0}, a_{1}, …, a_{n}) → (a_{1}, …, a_{n+1}) is a specialization over k, but not necessarily an isomorphism; and (a_{0}, a_{1}) ∈ S.)

Let d_{n} = tr.deg._{k}k(a_{0}, …, a_{n}). If, for some n, d_{n} < d_{n+1}, let U_{n}, V_{n}, S_{n} be the k-loci of (a_{0}, …, a_{n}), (a_{1}, …, a_{n+1}), (a_{0}, …, a_{n+1}) respectively. Then the hypothesis of 15.3 holds. So there exists a point c in a difference field extension of k, with c_{n} = σ^{n}(c) satisfying: (c_{0}, …, c_{n+1}) ∈ S_{n}, and of positive transformal dimension over k. In particular, (c_{0}, c_{1}) ∈ S, so c ∈ X and X has positive transformal dimension.

Otherwise, d_{n+1} ≤ d_{n} for each n. So d_{n} ≤ d_{0} ≤ dim (U) for each n. This shows that C has total dimension ≤ dim (U). Moreover if c_{0} is not a generic point of U, then C has total dimension < dim (U). □

15.2 Projective difference schemes

(projective)

ℕ[σ]- Graded rings Let ℕ[σ] denote the set of polynomials over ℕ, with indeterminate labeled σ. Consider difference rings R graded by the ring ℕ[σ]. In other words, we are provided with a decomposition of R as an Abelian group:

R = ⊕_{n∈ℕ[σ]}H_{n}

Multiplication in R induces maps H_{n} × H_{m} → H_{m+n}. The action of σ is assumed to carry H_{n} to H_{nσ}. Such a structure will be called a ℕ[σ]-graded ring.

We will assume R is generated as a difference ring by H_{0} ∪ H_{1}. A difference ideal P is called homogeneous if it is generated by the union of the sets P ∩ H_{n}.

Projective difference schemes Let R be a ℕ[σ]-graded ring, with homogeneous components H_{n}. We define an associated difference scheme, Proj^{σ} (R), as follows. The underlying space is the set of homogeneous transformally prime ideals, not containing H_{1}. The topology is generated by open sets of the form: W_{a} = {p : a ∉ p}, with a ∈ H_{k} for some k ∈ ℕ[σ]. Such sets are called affine open. One assigns to W_{a} the difference ring:

R_{a} = \{R[a^{−n} : n ∈ ℕ[σ]]\}_{0} = \{ \frac{b}{a^{n}} : n ∈ ℕ[σ], b ∈ H_{nk}\}

and glues.

Explanation: the localized difference ring R[a^{−n} : n ∈ ℕ[σ]] has a natural grading, with an element of the form b/a^{n} (b ∈ R, n ∈ ℕ[σ]) in the homogeneous component of degree deg(b) − n deg(a). {R[a^{−1}]}_{0} is the degree-zero homogeneous component of this ℕ[σ]- graded ring. To verify that this yields, uniquely, a difference scheme structure, observe that {R[(a_{1}a_{2})^{−n} : n ∈ ℕ[σ]]}_{0}, the affine ring corresponding to W_{a_{1}a_{2}}, is the difference ring localization of the ring corresponding to W_{a_{1}} by the element \frac{a_{2}^{deg a_{1}}}{a_{1}^{^{deg} ^{a_{2}}}} . Note also that the prime ideals of R_{a} can be identified with the elements of W_{a} (a homogeneous prime ideal of R is the kernel of a homomorphism h : R → L, L a difference field, with h(a) ≠ 0. Then h restricts and extends to R_{a}. Conversely, let g : R_{a} → L′ be a homomorphism. Then g extends to a graded homomorphism ḡ : R → L′[t^{ℕ[σ]}], with ḡ(a) = t.)

We could alternatively directly describe the structure sheaf in terms of section, as in the definition of Spec^{σ} ; the local ring at p is defined as

\{ \frac{b}{a} : (∃k) a, b ∈ H_{k}, b ∉ p\}

ℕ[σ]-graded ring associated to a graded ring Let D be a difference domain, and

let R be a graded D-algebra in the usual sense, R = ⊕_{i∈ℕ}R_{i}. Assume the homogeneous component of degree 0 is a domain.

Lemma 15.8 [σ]_{D}R has a unique ℕ[σ]-grading, compatible with the grading of R. {[σ]_{D}R}_{0} = [σ]_{D}(R_{0}).

Proof For n ∈ ℕ[σ], n = ∑_{i} k_{i}σ^{i}, let H_{n} be the subgroup of [σ]_{D}R generated by the products Π_{i}σ^{i}(a_{i}) with a_{i} ∈ R_{k_{i}}. (We write here a_{i} also for the image of a_{i} in [σ]_{D}R.) Clearly the H_{n} generate [σ]_{D}R between them, and any ℕ[σ]-grading must make H_{n} the homogeneous component of degree n. Thus it is only a question of showing that the H_{n} are in direct sum. Let F be a free R_{0}-algebra, graded so that the free generators span F_{1} as a R_{0}-module, and let h : F → R be a surjective homomorphism of graded R_{0}-algebras. Let F̄ = [σ]F; it is clear that the lemma holds for F̄. Let J be the ℕ[σ]-ideal generated by ker(h). Also J is generated by J ∩ F̄_{1}, so J is homogeneous, and J ∩ F̄_{0} = (0). The ℤ-graded part of (F̄)/J is F/ker(h) ≅ R. Thus we obtain a map R → (F̄)/J. This map extends to a difference ring map [σ]_{D}R → (F̄)/J. We also have a map F̄ → [σ]_{D}R. By the universal properties, these are isomorphisms, and [σ]_{D}R is ℕ[σ]-graded.

Difference structure on projective schemes Let D be a difference domain.

Lemma 15.9 (5.13) Let R be a graded D-algebra. Then Proj^{σ} [σ]_{D}R is the difference scheme obtained by gluing together Spec^{σ} [σ]_{D}R_{a}, where R_{a} runs over the various localized rings R_{a} = {R[a^{−1}]}_{0}, a ∈ R_{1}. Proof First, for any a ∈ R_{n}, n ∈ ℕ, [σ]_{D}(R[a^{−1}]) = ([σ]_{D}R)[a^{−1}], the second localization being taken in the sense of difference rings. This is clear, since both are the universal answers to the following problem: a difference ring R̄, a homomorphism h : R → R̄, with h(a) invertible in R̄. Next, one verifies that when R is graded, the two induced gradings on these rings are the same; in particular, [σ]_{D}(R_{a}) = ([σ]_{D}R)_{a}, using the notation of the definition of Proj^{σ} , and the previous lemma. □

Lemma 15.10 There are canonical isomorphisms:

• M_{q}([σ]_{D}R) = R⊗_{D}M_{q}(D)

• M_{q}(Spec^{σ} [σ]_{D}R) = Spec (R) ×_{Spec D} M_{q}(Spec D)

• When R is a graded D-algebra, M_{q}(Proj^{σ} ([σ]_{D}R)) = Proj (R⊗_{D}M_{q}(D)) Proof

• By the universal properties.

• Apply Spec to the previous.

• By 15.9 and the previous item.

Multi-projective varieties Suppose R is graded by ℕ[σ]^{k}, in place of ℕ[σ]. We can define Proj^{ℕ[σ]^{k}}R analogously to the case k = 1. Sometimes if the intended structure of R is clear we will just write Proj^{σ} R or Proj R. The underlying space is the set of homogeneous difference ideals, not containing a homogeneous component R_{j} (for any j ∈ ℕ[σ]^{k}). The basic affine open sets are the sets of primes not containing b = a_{1}a_{2} … a_{k}, where a_{i} has degree (0, …, 0, 1, 0, …, 0). The localization R[b^{−1}, σ(b^{−1}), …] has a graded structure, and the associated ring is defined as the graded component of degree 0.

Note 15.11 (htd)

Let p ∈ Proj R. Then there exists a_{i} of degree (0, …, 0, 1, 0, …, 0) with a_{i} ∉ p. Since p is homogeneous, and prime, the a_{i} are algebraically independent over R_{0}/(p ∩ R_{0}). Thus if R is a domain, and a D-algebra, the transformal transcendence degree of R over D is k more than the transformal dimension of Proj R over D.

The ideals J_{q} used in defining M_{q} are not homogeneous. However, a ring graded by ℕ[σ]^{k} can be (forgetfully) viewed as graded by ℕ^{k}, in many ways; any ring homomorphism from ℕ[σ] to ℕ will give such a way. Given q, let h_{q} : ℕ[σ] → ℕ be the homomorphism with σ ↦ q, and also let h_{q} : ℕ[σ]^{k} → ℕ^{k} denote the product homomorphism. Use h_{q} to view R as ℕ^{k} graded: the graded component of degree n is by definition the sum of the graded components of degree ν, over all ν with h_{q}(ν) = n. Then J_{q} is homogeneous for this grading; the generators can be taken to be σ(r) − r^{q} with r in some homogeneous component, and these elements are homogeneous in the h_{q}-induced grading. Therefore M_{q}(R) = R/J_{q}(R) has a natural ℕ^{k}-graded structure. We can thus view M_{q} as a functor on ℕ[σ]^{k}-graded difference rings into ℕ^{k}-graded rings.

Notation 15.12 The difference polynomial ring in one variable over a difference ring R is denoted R[t^{ℕ[σ]}]. The localization of this ring by t is denoted R[t^{ℤ[σ, σ^{−1}]}], and is called the ℤ[σ, σ^{−1}]-polynomial ring over R. The k-variable version R[tℤ[σ, σ^{−1}]_{1}, …, tℤ[σ, σ^{−1}]_{k}] is called the ℤ[σ, σ^{−1}]^{k}-polynomial ring over R.

Note: Lemma 15.13 (22) Let R = ⊕{R_{a} : a ∈ ℕ[σ]^{k}} be an ℕ[σ]^{k}-graded ring. Let a_{i} ∈ R have degree (0, …, 0, 1_{(i)}, 0, …, 0). Let S = R[a_{i}^{σ^{n}} : i = 1, …, k, n ∈ ℤ] be the localization by a_{1}, …, a_{k}, and let S_{0} be the degree-0 component. Then S is the ℤ[σ, σ^{−1}]^{k}- polynomial ring over S_{0}:

S = S_{0}[a_{1}^{ℤ[σ, σ^{−1}]}, …, a_{k}^{ℤ[σ, σ^{−1}]}]

Proof An element c of degree (n_{1}, …, n_{k}) of this ring can be written as ba_{1}^{n_{1}} … a_{k}^{n_{k}}, where b = ca_{1}^{−n_{1}} … a_{k}^{−n_{k}} ∈ S_{0}. By homogeneity, any difference - algebraic relation among the a_{i} over S_{0} implies a monomial relation among them. But ba_{1}^{n_{1}} … a_{k}^{n_{k}} = 0 implies b = 0 ∈ S_{0}, since the a_{i} are invertible.

Lemma 15.14 (Mq-proj) Let R be a ℕ[σ]^{k}-graded ring. Then M_{q}(Proj^{σ} (R)) ≅ Proj (M_{q}(R))

Proof To simplify notation we treat the case k = 1. Let a ∈ R_{1}. By 15.13, R[a^{−1}, a^{−σ}, …] = R_{a}[a^{ℤ[σ]}] is a ℤ-polynomial ring over R_{a}. Thus M_{q}(R_{a}) = {M_{q}(R_{a}[a^{ℤ[σ]}])}_{0}.

Write ā for the image of a modulo J_{q}. Then

M_{q}(R[a^{−1}, a^{−σ}, …]) = M_{q}(R_{a})[ā^{−1}]

Taking degree-0 components,

M_{q}(R_{a}) = {M_{q}(R[a^{−1}, a^{−σ}, …])}_{0} = {M_{q}(R)[ā^{−1}]}_{0} = (M_{q}(R))ā

Thus

M_{q}(Spec^{σ} (R_{a})) = Spec (M_{q}(R)ā)

Now the difference schemes Spec^{σ} (R_{a}) glue together to give Proj^{ℕ[σ]^{k}}R, while the schemes Spec (M_{q}(R)_{a}) glue together to give Proj (M_{q}(R)). (As a runs through R_{1}, ā runs through a generating set for M_{q}(R)_{1}.) The lemma follows after verifying that the gluing maps agree.

Remark 15.15 (biprojring)

If D is a difference domain, D[X, Y] = D[X_{1}^{ℕ[σ]}, …, X_{n}^{ℕ[σ]}, Y_{1}^{ℕ[σ]}, …, Y_{m}^{ℕ[σ]}], bi-graded so that X_{1}^{a_{1}} … X_{n}^{a_{n}}Y_{1}^{b_{1}}, …, Y_{m}^{b_{m}} is homogeneous of degree (∑ a_{i}, ∑ b_{i}), then Proj D[X, Y] is the product over D of the σ- projective spaces of dimensions m,n. Conversely, Proj of any bi- or multi-graded ring can be viewed as a fiber product of simple Proj^{σ} of some component rings.

Morphisms between graded rings Let R be a ℕ[σ]^{k}-graded ring, and S a ℕ[σ]^{l}-

graded ring. Let h : R → S be a surjective σ - ring homomorphism. Assume h(R_{c}) ⊂ S_{λ(c)}, where λ : ℕ[σ]^{k} → ℕ[σ]^{l} is an injective ℕ[σ]-linear map. Then one obtains a map h^{∗} : Proj S → Proj R, as follows. If q is a homogeneous transformally prime ideal on S, not containing a homogeneous component, then so is h^{−1}(q) on R. Let h^{∗}(q) = h^{−1}(q). h^{∗} is continuous: the open set W_{a}^{R} defined by an element a ∈ R pulls back to the set W_{h(a)}^{S}. Finally h induces a difference ring homomorphism on the graded rings, R[a^{−1}, a^{−σ}, …] → S[h(a)^{−1}, h(a)^{−σ}, …], respecting the grading in the same sense as h; and in particular induces a map R_{a} → S_{h(a)}.

15.3 Blowing up

(blowup)

We give two constructions of blowing up. We show that the transformal blowing-up reduces under Frobenius to a scheme containing the usual blowing-up, and of the same dimension, without resolving the interesting questions regarding their exact relation. One construction can be viewed as the result of a deformation along the affine t line (of transformal dimension 1) while the other is a limit of a deformation along t = σ(t) (total dimension 1.) The latter seems to have no analog in the Frobenius picture. Nevertheless they are shown to give (almost) the same result.

Blowing up difference ideals Let R be a difference ring, X = Spec^{σ} (R), and J a finitely generated difference ideal. Consider the ordinary polynomial ring R[t] over R, and set σ(t) = t. Let R[Jt] be the subring of R[t]:

R[Jt] = ∑_{n∈ℕ}(Jt)^{n}

Observe that R[Jt] is finitely generated as a difference ring, if R is. Indeed if V is a set of generators for J as a difference ideal, and Y a set of generators for R as a difference ring, then Y ∪ V t is a set of generators for R[Jt].

Moreover, the degree 0 difference subring of the difference ring localization R[Jt][(at)^{−1}] is finitely generated as a difference ring. If a ∈ V, it is generated by Y ∪ \frac{V t}{at} ∪ \{\frac{σ(a)t}{at} , \frac{at}{σ(a)t}\}.

View R[Jt] as a (non-Noetherian) graded ring with an endomorphism; form the ordinary scheme-theoretic Proj , [Hartshorne] II 2; and consider the difference subscheme described in 3.10 above. Let

X̃_{J} = Fix^{σ} Proj (R[Jt])

A basic open affine of Proj (R[Jt]) has the form W_{a} = Spec (R[ \frac{J}{a}]) , with a ∈ J. The intersection of W_{a} with the set of difference ideals is contained in Spec (R[ \frac{J}{σ^{n}(a)} ]) for each n. Thus one sees that X̃_{J} has an open covering by open subschemes of the form Spec^{σ} R[ \frac{J}{a}]_{σ}, a ∈ J, where the square brackets [ ]_{σ} here refer to localization of difference rings.

There is a natural map π : X̃_{J} → X, considered as part of the structure.

Compatibility with localization There are two (closely related) points here.

First, suppose π : X̃_{J} → X is the blowing up at J, X′ = Spec^{σ} R′ is an open subscheme of X, R′ = R[a^{−1}, a^{−σ}, …], with inclusion map i : X′ → X, and J′ = R′J. Then X̃′_{J′} = π^{−1}(X′). This is immediately verified.

Secondly, suppose J′ is another finitely generated difference ideal, J ⊂ J′, and for every p ∈ Spec^{σ} R, if R_{p} is the local ring, then JR_{p} = J′R_{p}. Then X̃_{J} = X̃_{J′}. Indeed, every prime ideal containing J must contain J′, so X̃_{J′} is covered by the open affines Spec^{σ} R[ \frac{J′}{a}] with a ∈ J. Next, one may find an open covering of Spec^{σ} R, such that the restrictions of J and J′ to each open set agree. It follows that Spec^{σ} R[ \frac{J′}{a}] = Spec^{σ} R[ \frac{J}{a}] since they are equal locally (on R).

Now define X̃_{J} and

π : X̃_{J} → X

for a general difference scheme X and quasi-coherent σ - ideal presheaf J on X, by gluing.

If i : W → X is a closed subscheme of a scheme X, with corresponding difference ideal sheaf J = ker(O_{X} → i_{∗}O_{W}), we write

X̃_{W} = X̃_{J}

We define the exceptional divisor E_{W} to be the difference scheme inverse image of W under this map.

A second construction: the closed blowing up of ideals In order to obtain

an embedding of the blowing-up in σ-projective space, we use a different construction. For this construction, we will blow up an ideal (or quasi-coherent ideal presheaf) I instead of a difference ideal (or difference ideal sheaf) J. The two constructions coincide when I = J is both a finitely generated ideal and a difference ideal, so there is no confusion in denoting both by R̃_{I} or R̃_{J}. However, to emphasize the difference we will temporarily use a superscript ^{c} for the closed blowing-up of ideals.

Let R be a difference ring, and let I be a finitely generated ideal. We define the closed blowing-up ring R̃_{I}^{c} as a subring of R[t^{ℕ[σ]}]:

R̃_{I}^{c} = ∑_{n∈ℕ[σ]} I^{n}t^{n}

where if n = ∑_{i} σ^{k(i)}, I^{n} is the subgroup of R generated by elements of the form Π_{i}f_{i}, with f_{i} ∈ σ^{k(i)}(I).

R̃_{I} inherits a ℕ[σ]-grading from R[t^{ℕ[σ]}], with R the homogeneous component of degree 0.

If X = Spec^{σ} R, let

X̃_{I}^{c} = Proj^{σ} (R̃_{I}^{c})

and let π : X̃_{I}^{c} → X be the natural map, corresponding to the inclusion R → R̃_{I}^{c}.

If X is any difference scheme, and I a quasi-coherent ideal presheaf of O_{X} (considered forgetfully as a sheaf of rings), one can check as above that the blowing ups of Spec^{σ} U at the ideals I(U), and their canonical maps to U, glue together to give a difference scheme over X, denoted X̃_{I}^{c}.

Definition 15.16 Let X be a difference scheme, I a quasi-coherent ideal presheaf. Let Y = X̃_{I}^{c}, π : Y → X the structure map. Then we obtain an ideal presheaf π^{∗}I on Y. We define the exceptional divisor to be the subscheme of Y defined locally by π^{∗}I (or equivalently by the difference ideal sheaf generated by π^{∗}I.)

Comparing the blowing-ups We include a result comparing the two blowing-ups as the finitely generated ideal I approaches J.

When J is already generated as a difference ideal by I ∩ σ^{−1}(I), the difference between the two blowing-ups is a proper closed subscheme Z of the exceptional divisor. It can be interpreted as follows. In either version of the blowing up, a point of the exceptional divisor corresponds to a “direction of approach” to the difference subscheme defined by J. However in the closed blowing up, directions in which σ(y)/y approaches infinity as y → 0 are allowed; in the open blowing up they are not.

If one blows up after applying M_{q}, I and J become identified, and one obtains only “directions of approach” in which σ(y)/y approaches 0; thus these points are bounded away from Z.

There is more to be said here:

1. As I → J, the closed blowing up X̃_{I}^{c} appears to change fairly gently. Perhaps the different X̃_{I}^{c} can be compared by transformally birational radicial morphisms. (cf. Definition 4.30.)

2. Outside Z, the closed blowing ups contains points representing directions in which σ(y)/y is finite but nonzero; upon applying M_{q}, these points form a detachable divisor. (Is it possible to get rid of them before?)

3. On the other hand it may be interesting precisely to study blowing up one-generated difference ideals. Note that after M_{q}, such ideals become principal, so blowing them up has no effect on smooth varieties. For instance, it appears possible that one obtains a better definition of ”irreducible difference variety” by demanding that the irreducibility persist to the strict transform under such principal blowing ups.

Proposition 15.17 Let R be a difference ring, J a finitely generated difference ideal,

V a set of generators of J, and I the ideal generated by V ∪ σ(V). Let X = Spec^{σ} R. Then X̃_{J} is isomorphic to an open subscheme of X̃_{I}^{c}. Specifically, let

Z(V) = {p ∈ Proj^{σ} (R̃_{I}) : V t ⊂ p}

Then

X̃_{J} = X̃_{I}^{c} \ Z(V)

Proof Both difference schemes are obtained by gluing together affine schemes Spec^{σ} S, where for some a ∈ V, S is a subring of R[1/a, 1/σ(a), …]. Namely:

S = S_{1} = \{ \frac{c}{a^{n}} : n ∈ ℕ[σ], c ∈ I^{n}\}

according to the closed construction; for the other, one checks that

S = S_{2} = \{ \frac{c}{a^{n}} : n ∈ ℕ[σ], c ∈ J^{n}\}

Note that Spec^{σ} (S_{1}) (respectively Spec^{σ} (S_{2})) embeds naturally as an open subscheme of X̃_{I}^{c} (resp. X̃_{J}). In the first case, by definition of the closed set Z(V), the union of these open subschemes over a ∈ V is precisely X̃_{I}^{c} \ Z(V). in the second case, it is X̃_{J}, since V generates J as a difference ideal. We will now fix a ∈ V and show that S_{1} = S_{2}. The resulting isomorphisms of open subschemes are natural and glue together to give the required isomorphism.

Clearly S_{1} ⊂ S_{2}. For the other direction, it suffices to check that \frac{c}{a} ∈ S_{1} when c ∈ J. \{c : \frac{c}{a} ∈ S_{1} forms an ideal of R. J is generated as an ideal by ∪_{k}I^{σ^{k}}, so we may take c ∈ I^{σ^{k}} for some k, so that \frac{c}{a^{σ^{k}}} ∈ S_{1}. But also a^{σ} ∈ σ(V) ⊂ I, so \frac{a^{σ}}{a} ∈ S_{1}, hence by applying σ, \frac{a^{σ^{i+1}}}{a^{σ^{i}}} ∈ S_{1} for i < k. Multiplying these k + 1 elements, we obtain \frac{c}{a} ∈ S_{1}.

Blowing up and M_{q}-reduction

Lemma 15.18 (5.25) Let R be a difference ring, I an ideal. Let R̃_{I}^{c} be the closed blowing up ring. Let S = M_{q}(R), and Ī = M_{q}(I) = (I + J_{q}(R))/J_{q}(R). Let S̃c ; Ī = ∑_{n∈ℕ}(Īt)^{n} ⊂ S[t].

There exists a natural surjective homomorphism of graded rings

j_{q} : M_{q}(R̃_{I}^{c}) → S̃c ; Ī

Proof Clearly M_{q}(R[t^{ℕ[σ]}]) ≅ S[t]; by restriction, we get an isomorphism

R̃_{I}^{c} /(J_{q}(R[t^{ℕ[σ]}]) ∩ R̃_{I}^{c}) ≅ S̃c ; Ī

Now clearly J_{q}(R̃_{I}^{c}) ⊂ (J_{q}(R[t^{ℕ[σ]}]) ∩ R̃_{I}^{c}), yielding the surjective homomorphism

M_{q}(R̃_{I}^{c}) = R̃_{I}^{c}/J_{q}(R̃_{I}^{c}) → S̃c ; Ī

The naturality can be seen via the universal property of M_{q}. □

Lemma 15.19 (5.26) Let X = Spec^{σ} R be an affine difference scheme, I an ideal of R, Ī ⊂ M_{q}(R) the image of I under M_{q}. There exists a natural embedding of M̃_{q}(X)Ī as a closed subscheme of M_{q}(X̃_{I}^{c}).

Moreover, if V is a sub-ideal of I, generating the same difference ideal, and Z(V) = {p ∈ Proj^{σ} (R̃_{I}^{c}) : V t ⊂ p}, then the image of M̃_{q}(X)Ī is disjoint from M_{q}(Z(V)).

Proof From 15.18 we obtain a map

j_{q}^{∗} : Proj M̃_{q}(R)c ; Ī → Proj M_{q}(R̃_{I}^{c})

This can be composed with 15.14. The ”moreover” is clear, since if V t ⊂ j_{q}^{−1}(p) then j_{q}(V t) ⊂ p. However V and I generate the same difference ideal, so if V̄ denotes the image of V in M_{q}(R), then V̄ and Ī generate the same ideal. Thus V̄ t ⊈ p for a homogeneous ideal

p ∈ M̃_{q}(R)c ; Ī. □

Corollary 15.20 (5.27) Let X be a difference scheme, I a quasi-coherent ideal presheaf. Let Ī be the M_{q}-image of I. There exists a natural embedding of M̃_{q}(X)Ī as a closed subscheme of M_{q}(X̃_{I}).

Proof This reduces to the local case, 15.19, by gluing. □

Note 15.21

(5.28) In 15.20, assume I is a sub-presheaf of a difference ideal sheaf J, and I ∩ σ(I) generates J. Then the image of M̃_{q}(X)Ī in M_{q}(X̃_{I}) is contained in the M_{q}-image of the open blowing up of X at J.

A geometric view of blowing up difference schemes When X = Proj (R) is

itself a projective or multi-projective difference variety, blowing-ups of X have a natural multi-projective structure. Suppose R is graded by ℕ[σ]^{k}. Then R[t^{ℕ[σ]}] is naturally graded by ℕ[σ]^{k+1}; this induces a grading of R̃_{I}^{c}.

Let I be a homogeneous ideal of the ℕ[σ]^{k}-graded difference ring R. Let I be the corresponding ideal sheaf on X = Proj^{ℕ[σ]^{k}}R. Then

X̃_{I} ≅ Proj^{ℕ[σ]^{k+1}} R̃_{I}^{c}

naturally.

Let R be a difference ring; view it as a scheme X = Spec R, endowed with an endomorphism. Let I = I(0) be a finitely generated ideal of R. Let B_{1}(R, I) be the affine coordinate ring of the blow-up of Spec R at I, in the sense of algebraic geometry ([Fulton]). Thus B_{1}(R, I) may be viewed as a subring of R[t]:

B_{1}(R, I) = R + ∑_{i≥1} I^{n}t^{n} ⊂ R[t]

We can also view B_{1}(R) as an extension of R. Let I(1) be the ideal of B_{1}(R, I) generated by σ(I) ⊂ R ⊂ B_{1}(R, I). Proceeding inductively, let

B_{n+1}(R, I) = B_{1}(B_{n}(R, I), B_{n}(R, I)σ^{n+1}I)

B_{1}(R, I) is naturally ℤ-graded, with R the homogeneous component of degree 0. This grading inductively builds up to a ℤ^{n}-grading on B_{n}(R, I). We let

X_{n} = P roj^{ℤ^{n}}(B_{n}(R, I))

On X_{n} we have the exceptional divisor E_{n} corresponding to the ideal σ^{n}(I)B_{n}(R, I). Let B_{∞}(R, I) be the direct limit of the rings B_{n}(R, I). Each of these rings is naturally embedded in R[t, t^{σ}, …], and B_{∞}(R, I) can be taken to be the union of the rings B_{n}(R, I) within R[t, t^{σ}, …]. While the individual B_{n}(R, I) do not in general admit a difference ring structure, B_{∞}(R, I) is clearly a difference subring of R[t, t^{σ}, …], and we thus view it as a difference ring.

Lemma 15.22 R̃_{I}^{c} ≃ B_{∞}(R, I) as difference ring extensions of R.

Proof They actually coincide as subrings of R[t, t^{σ}, …]. □

X̃_{I}^{c} can thus be viewed as a limit of the projective system of schemes X ← X_{1} ← X_{2} ← . . .. The exceptional divisor E on X̃_{I}^{c} corresponds to the intersection of the pullbacks of all the E_{n}. Observe that the image of E on X_{n} will usually have unbounded codimension.

Example 15.23

Let R = [σ]k[X_{1}, …, X_{n}] be the affine difference ring of affine n-space, with k a difference field. Then Spec^{σ} R is affine n-space over k. However Spec R is an infinite product of schemes Y_{i}, each isomorphic to affine n-space over k. Suppose I is an ideal of R generated by an ideal I_{0} of k[X_{1}, …, X_{n}], corresponding to a subscheme S_{0} of affine space, and let Z_{i} be the result of blowing up Y_{i} at I_{0}, EZ_{n} the exceptional divisor. Then X_{n} can be identified with

Z_{1} × Z_{2} × … × Z_{n} × Y_{n+1} × Y_{n+2} × …
□

In the above example, we used the following observation concerning ordinary blowing-up of varieties: when S, T are schemes, and C a subscheme of S, ˜ (S × T)_{C×T} can be identified with S̃_{C} × T.

15.4 Semi-valuative difference rings

(pinched-sec)

Valuative schemes of finite total dimension over Ă_{k} can be viewed as transformal analogs of smooth curves. But moving a finite scheme over ℙ^{1}, we encounter singular curves as well. Their ultraproducts lead to considering semi-valuative schemes. We take a look at these in the present subsection; it will not be required in the applications. Lemma 15.31 explains their potential role in a purely schematic treatment of transformal specializations.

An m-semi-valuative A_{0}-algebra A is by definition a local k[t̆]_{σ}-algebra contained in a Boolean -valued valuation ring R over A_{0}, such that σ^{m}(t)R ⊂ A. We will be interested in this when A_{0} = k[t̆]_{σ} or when A_{0} = k[T] in characteristic p. An ultraproduct of the latter will be an instance of the former. When R is a transformal valuation domain, we say A is an m-pinched transformal valuation ring.

Definition 15.24 (vmult) Let R be a transformal valuation domain, A a difference subring of R, t ∈ R. If A ∩ t^{m}R ⊂ A, we will say that A is an m-pinched valuative domain (with respect to (R, t).)

Let X_{0} = Spec^{σ} R/tR; we have a map f : X_{0} → Spec^{σ} A/tA. If Y is a component of Spec^{σ} A/tA, write r.dim(X_{0}/Y) for the relative reduced total dimension of f^{−1}(Y) ⊂ X_{0} over Y. The valuative multiplicity of Y is v.mult(Y) = rk_{ram}(K) + r.dim(X_{0}/Y). Finally, the valuative dimension of Y (viewed as a part of the specialization of A) is vt.dim(Y) = r.dim(f^{−1}(Y)) + rk_{ram}(K).

Example 15.25 A plane curve with a point pinched to order m, inside formal ring of normalization.

More precisely, the local ring of the curve has the required property, inside the local ring of the normalization, over k[t] rather than k[t̆]_{σ}; an ultraproduct of such rings leads to k[t̆]_{σ}. We consider this class of examples in more detail.

Consider curves C ⊂ ℙ^{n} over an algebraically closed field k, P ∈ C. C may be reducible and singular. (For simplicity, assume C is reduced.) The normalization C̃ of C is defined to be the disjoint sum of the normalizations of the irreducible components of C. Let A A be the local ring of C at P, B the product of the local rings of C̃ at the points above P.

Let T ∈ A be a non-zero-divisor, corresponding to the restriction of a linear birational map ℙ^{n} → ℙ^{1}.

Example 15.26 (pinch) Let C ⊂ ℙ^{n} be a curve of degree \underline{d}. Let P ∈ C, A, B, C̃, T as above. Then T ^{^{\frac{3}{4}} \underline{d}^{2}}B ⊂ A.

(For our purposes here, we could equally well replace A, B by their completions Â, B̂ for the T -adic topology; i.e. the conclusion T ^{^{\frac{3}{4}} \underline{d}^{2}} B̂ ⊂ Â is what we really need.) Proof We may assume P = 0 ∈ 𝔸^{n} ⊂ ℙ^{n}, T a linear map on 𝔸^{n}. Let Y be a generic linear map on 𝔸^{n}. Projecting C via (T, S) to 𝔸^{2} we obtain a plane curve of degree d, whose local ring k[S, T] ∩ A is if anything smaller that A, while the local ring of C̃ does not change since the map (T, S) is birational on each component of C. So we can assume C ⊂ Spec k[S, T] is a plane curve.

Assume first that C is irreducible. By [Hartshorne] I ex. 7.2, the arithmetic genus p_{a}(C) satisfies p_{a}(C) = \binom{\underline{d}}{2} ≤ \underline{d}^{2}/2. By [Hartshorne] IV Ex. 1.8 (a), since the genus of the normalization C̃ is non-negative, the A-module B/A has length l ≤ p_{a}(C). Thus A + T^{i}B = A + T^{i+1}B for some i < l. In B/A , we have T (T^{i}B/A) = (T^{i}B/A), so by Nakayama, T^{i}B/A = 0, and hence T^{l}B ⊂ A.

If C is reducible (at 0), it is a union of irreducible curves C_{i}, of degrees \underline{d}_{i}, with ∑ \underline{d}_{i} = \underline{d}. More precisely, C is cut out by Π_{i}f_{i}, f_{i} ∈ k[S, T] relatively prime polynomials of degrees \underline{d}_{i}; C_{i} is the curve cut out by f_{i}.

Let B_{i} be the local ring of the normalization C̃_{i} of C_{i}. Then B = Π_{i}B_{i}. Let A_{i} be the local ring of C_{i} itself. We can view A as a subring of Π_{i}A_{i}. We will show that (T^{\underline{d}^{2}/4}Π_{i}A_{i}) ⊂ A. By the irreducible case, T^{\underline{d}^{2}/2}Π_{i}B_{i} ⊂ Π_{i}A_{i}. So T^{3\underline{d}^{2}/4}B ⊂ A. It suffices therefore to show that T^{\underline{d}^{2}/4}A_{i} ⊂ A for each i; i.e. that there exists a_{i} ∈ A whose image in A_{j} is T^{\underline{d}^{2}/4}, and whose image in every other A_{j} is 0. Take for instance i = 1.

Let f_{1}′ = f_{2}f_{3} · . . . · f_{n}, \underline{d}_{1}′ = deg(f_{1}′). Note that \underline{d}_{1} + \underline{d}_{1}′ = \underline{d}, so \underline{d}_{1}\underline{d}_{1}′ ≤ \frac{1}{4} \underline{d}^{2}. Now the curves (f_{1}), (f_{1}′) intersect in a subscheme of size ≤ \underline{d}_{1}\underline{d}_{1}′, by Bezout’s theorem. So T^{\underline{d}^{2}/4} = rf_{1} + r′f′_{1}, r, r′ ∈ k[S, T]. The element r′f′_{1} of A is the one we looked for: it equals T^{\underline{d}^{2}/4} module f_{1}, and equals 0 modulo f_{j} for j ≠ 1. □

Here is a view of some of the combinatorics of the above example. Lemma 15.27 (semigroup) Let E be a sub-semi-group of ℕ, generated by a finite X with greatest element b. Then for some Y ⊂ ℤ/bℤ, for all n ≥ b(b − 1) + 1, n ∈ E iff nmod b ∈ Y. Proof Let Y (i) be the image in ℤ/bℤ of E ∩[ib+1, (i+1)b]. Clearly ∅ ⊂ Y (0) ⊆ Y (1) ⊆ …, so for some i < b we have Y (k) = Y (k + 1). Now X + Y (k) ⊂ Y (k + 1)(mod b), so X + Y (k) = Y (k)(mod b), and it follows that E + Y (k) = Y (k)(mod b). The claim follows, with Y = Y (k). □

Remark 15.28 (val2.5) In 7.22, let A be an m-pinched valuative subring of R, with respect to t, t ∈ A. Then A/tA has total dimension ≤ d over F; if equality holds, then tA ∩ K[0] = (0) (in fact tR ∩ K[0] = (0). ) Proof The inequality is immediate from 8.2; so we will concentrate on the case: tA∩K[0] ≠ (0), and show that A/tA has total dimension < d. (But the argument also serves in general to show the weak inequality.)

Assume tR ∩ K[0] ≠ (0). The proof of 7.20-7.22 shows that R/t_{m}R has total dimension < m + d. (In general, it has total dimension ≤ m + d.)

Since t_{m}R ⊂ A, we have (t_{m}R ∩ A)^{2} ⊂ t_{m}A: if t_{m}r_{i} ∈ A, then (t_{m}r_{1})(t_{m}r_{2}) = t_{m}(t_{m}r_{1}r_{2}) ∈ t_{m}(t_{m}R ∩ A). So the difference rings A/t_{m}A, A/(t_{m}R ∩ A) differ only by nilpotents, and thus have the same total dimension. Now A/(t_{m}R ∩ A) embeds into R/t_{m}R, and so is well-mixed, and has total dimension < m + d (resp. ≤ m + d.)

Let J_{n} be the smallest well-mixed algebraically radical ideal of A containing t_{n}. So J_{m} = t_{m}R ∩ A, by the above argument; thus ^{A}\sqrt{J}_{m} = (\sqrt{J}_{m}) ∩ A is an algebraically prime ideal.

Let A′′ be a finitely generated F -subalgebra of A/tA. Lift the generators to A/t_{m}A, and let A′ be the F [t̆]_{σ}[m − 1]-subalgebra of A/t_{m}A generated by them. We have seen that A/t_{m}A has total dimension < m + d, hence so does A′, and thus also A′/\sqrt{0}_{A′} ≤ A/J_{m}. In this ring, t_{m} is not a 0-divisor. Thus the proof of 8.2 takes over, and shows that A′/tA′ has total dimension < d. Hence so does A′′. □

The usefulness of 15.28 is limited by the fact that it does not apply to an arbitrary algebraic component of A/tA. We can use valuation-theoretic rather than schematic multiplicities:

Lemma 15.29 (val2.6) In 7.22, let A be an m-pinched valuative subring of R, with respect to t, t ∈ A. Let Y be a component of Spec^{σ} A/t. Then

v.mult(Y) + r.dim(Y) ≤ d

If equality holds, then P = 0. If every weakly Zariski dense algebraic component of R/tR is Zariski dense, then Y is Zariski dense in Spec R[0]. Proof By the argument in 15.28, A/t_{m}A, A/(t_{m}R ∩ A) differ only by nilpotents; they thus have the same reduced total dimension. Similarly, the maps A/t_{m}A → A/tA, A/(t_{m}R∩A) → A/(tR ∩ A) are isomorphisms on points, and also respect the reduced total dimension. Thus r.dim(A/tA) = r.dim(A/(tR ∩ A)) ≤ r.dim(R/tR); r.dim(Y) + r.dim(X_{0}/Y) ≤ r.dim(f^{−1}(Y)). So vt.dim(Y) ≤ vt.dim(f^{−1}(Y)) ≤ vt.dim(R/tR). The lemma follows from 7.22. □

By an m- semi-valuative difference scheme (over Ă_{k}), we will mean one one of the form Spec^{σ} A, where A is an m-semi-valuative k[t̆]_{σ}-algebra. We will also say that the singularity of Spec^{σ} A is of order ≤ m.

(We think of Spec^{σ} A as a disjoint union of generalized curves over Ă_{k}, with a pinching of order m over t = 0.)

X_{→0}′ is the variant of X_{→0} defined using semi-valuative difference schemes, rather than valuative ones. (2d+1 -semi-valuative difference schemes will do; any larger number will lead to substantially the same scheme. cf. 15.26.) X_{→0}′ has the same points as X_{→0}, but may be thicker at the edges. X_{→0}′ can probably also be defined as the intersection of the closed projections of of (X̂)_{0} as X̂ ranges over all open blow-ups of X, with vertical components removed. Lemma 15.30 (vs-lim-pinch) Let Z be an algebraic component of X_{→0}′. Then there exists a T = Spec^{σ} Ā, Ā a pinched valuative domain over k[t̆]_{σ}, and a morphism T → X over Ă_{k}, such that Z is contained in the image of T_{0} in X_{0}. Proof Same as the proof of 8.8, except that we consider h_{j} : R → B_{j} ⊂ A_{j} ⊂ L_{j}, with T^{σ^{2d+1}}A_{j} ⊂ B_{j}. The condition is that p contains ∩_{j∈J}h_{j}^{−1}(tB_{j}), and in the conclusion p contains h^{−1}(tB̄). □

Lemma 15.31 (semival)

Let D ⊂ k be a difference domain such that X descends to D (X′ ×_{D} k = X, X′ a difference scheme over D.) Let Y^{0} (resp. Y) be a finitely presented well-mixed difference scheme with X_{→0} ⊂ Y ×_{D} k. Then for some a_{0} ∈ K, with F = D[1/a_{0}],

(i) For all difference fields L and all homomorphisms h : F → L, if X_{h} = X′⊗_{F [t̆]_{σ}} L̆_{t}, (X_{h})_{→0}′ ⊂ Y_{h} (as schemes.)

(ii) For all sufficiently large q, all h : F → L = K_{q}, if X̄ = Xh̆_{t}, for any reduced (but

possibly reducible) curve C over L and map C → X̄, C ∩ X̄_{0} ⊂ Y_{h}^{0} (as varieties.) (iii) For all sufficiently large q, all h : F → L = K_{q}, if X̄ = Xh̆_{t}, for any reduced (but

possibly reducible) curve C over L and map C → X̄, C ∩ X̄_{0} ⊂ Y_{h} (as schemes.)

Proof The assumption that Y^{0} is finitely presented over D means that locally, on Spec^{σ} A ⊂ X, Y^{0} is defined by a finitely generated ideal I (among well-mixed ideals.) Let r_{1}, …, r_{j} be generators for I. Then (i) and (ii) amount to showing that each r_{i} lies in a certain ideal. In (i), the condition is that r_{i} should vanish on the image f(T_{0}) for any valuative T and f : T → (Y^{0})_{h}. In other words, given a map g : A → R with R an 2d + 1-semi-valuative ring over L̆_{t}, where d is the total dimension of X_{t}, that g(r_{i}) ∈ tR. If (i) fails, there are F_{ν} approaching K and h_{ν} : F → L_{ν}, and g_{ν} : A → R_{ν}, with g(r_{i}) ∉ tR_{ν}. Taking an ultraproduct, we obtain g_{∗} : S → R_{∗} with g(r_{i}) ∉ tR_{∗}. But this contradicts the definition of X_{→0}′, and the assumption X_{→0}′ ⊂ Y^{0}.

As for (ii) and (iii), let A be the product of the local rings at 0 of C, and let R be the corresponding product of the local rings of the the normalizations of the components of C. Each such local ring is a discrete valuation domain, and can be viewed as a transformal valuation domain using the Frobenius map x ↦ x^{q}. By Lemma 15.26, if \underline{d} is a projective degree of C, then t^{^{\frac{3}{4}} \underline{d}^{2}}R ⊂ A. Now by 10.24, \underline{d} ≤ O(1)q^{d}. So \underline{d}^{2} ≤ O(1)^{2}q^{2d} ≤ q^{2d+1} (for large enough q.) so t^{σ^{2d+1}}R ⊂ A. Continue as in (i).

□

References

[Ax] J. Ax., The elementary theory of finite fields, Annals of Math. 88 (1968), pp. 239-271

[Bombieri] Enrico Bombieri, Thompson’s Problem (σ^{2} = 3), Invent. Math. 58 (1980), no. 1., pp. 77-100

[Chatzidakis-Hrushovski] Z. Chatzidakis, E. Hrushovski, Model theory of difference fields,

Trans. Amer. Math. Soc. 351 (1999), pp. 2997-3071

[Chatzidakis-Hrushovski-Peterzil] , Model Theory of difference fields II: the virtual structure and the trichotomy in all characteristics, Proc. London. Math. Soc. (3) 85 (2002) pp. 257-311

[Cohn] Richard M. Cohn, Difference Algebra, Interscience tracts in pure and applied mathematics 17, Wiley and Sons New York - London - Sydney 1965

[Cohn79] Cohn, R.M., The Greenspan bound, Proc. AMS 79 (1980), pp. 523-526

[Cohn87] Cohn, R.M., Order and Dimension, Proc. AMS 87 (1983), pp. 1-6.

[deJong] A.J. de Jong, Smoothness, semi-stability and alterations, preprint obtained from

                       ftp://ftp.math.harvard.edu/pub/AJdeJong/alterations.dvi

[Deligne 74] P. Deligne, La conjecture de Weil, I., Publ. Math. Inst. Hautes Etude. Sci. Publ. Math. 43 (1974) pp. 273-307 [Deligne 81] P. Deligne, La conjecture de Weil, II., Publ. Math. Inst. Hautes

Etude. Sci. Publ. Math. 52 (1981) pp. 313-428

[Deligne 77] P. Deligne, SGA 4 \frac{1}{2} , Springer-Verlag Lecture notes in mathemat-

ics 569, Berlin-Heidelberg 1977

[Fujiwara97] Fujiwara, Kazuhiro, Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture. Invent. Math. 127 (1997), no. 3, 489–533

[Fulton] Fulton, W., Intersection Theory, Springer-Verlag Berlin-Tokyo 1984

[Freitag-Kiehl] E. Freitag and R. Kiehl, Étale Cohomology and the Weil Conjec-

ture, Springer-Verlag, Berlin-Heidelberg 1988

[Fried-Jarden86] Fried, M and Jarden, M, Field Arithmetic, Springer-Verlag, Berlin

1986

[Gromov] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 109–197

[Hartshorne] Robin Hartshorne, Algebraic Geometry, Springer-Verlag New

York - Berlin 1977

[Haskell-Hrushovski-Macpherson] D. Haskell, E. Hrushovski, D. Macpherson, Definable sets

in algebraically closed valued fields, Part I (preprint.)

[Haskell-Hrushovski-Macpherson] Definable sets in algebraically closed valued fields, Part II,

preprint

[Herwig-Hrushovski-Macpherson] Bernhard Herwig, Ehud Hrushovski, Dugald Macpherson, Interpretable groups, stably embedded sets, and Vaughtian pairs J. London Math. Soc. (2) 68 (2003) pp. 1-11

[Hoyt] William L. Hoyt, On the moving lemma for rational equivalence, J. of the Indian Math. Soc. 35 (1971) p. 47-66

[Hrushovski01] Hrushovski, E., The Manin-Mumford Conjecture and the Model Theory of Difference Fields, Annals of Pure and Applied Logic 112 (2001) 43-115

[Hrushovski02] Hrushovski, E., Computing the Galois group of a linear differential equation, in Differential Galois Theory, Banach Center Publications 58, Institute of Mathematics, Polish Academy of Sciences, Warszawa 2002

[Hrushovski-Pillay] Hrushovski, E. and Pillay, A., Definable subgroups of algebraic groups over finite fields, Crelle’s journal 462 (1995), 6991; Groups definable in local fields and pseudo-finite fields, Israel J. of Math., 85 (1994) pp. 203-262

[Jacobi] Jacobi, C.G.J De investigando ordine systematis aequationum differentialium vulgarium cujuscunque, (Ex. Ill. C. G. J. Jocbi maunscriptis posthumis in medium protulit C. W. Borchardt), Journal für die riene und angewandte Mathematik, Bd. 64, p. 297-320

[Katz01] Nicholas M. Katz, Sums of Betti numbers in arbitrary characteristic, Finite Fields Appl. 7 (2001), 29–44

[Kleiman68] S.L. Kleiman, Algebraic cycles and the Weil conjectures, in Dix exposes sur la cohomologie des schemas, A. Grothendieck and N. H. Kuiper (eds.), Mason et Cie., Paris, North Holland, Amsterdam 1968

[KMP] Kondrat’eva, M. V.; Mikhalv, A. V.; Pankrat’ev, E. V. Jacobi’s bound for systems of differential polynomials. (Russian) Algebra, 79–85, Moskov. Gos. Univ., Moscow, 1982.

[Lando] Lando, B.A., Jacobi’s bound for first order difference equations, Proc. AMS 52 (1972), pp. 8-12

[Lang59] Serge Lang, Abelian varieties, Springer -Verlag New-York Tokyo 1983

[Lang-Weil] Lang, S. and A. Weil, Number of points of varieties in finite fields,

Amer. Jour. Math., 76 (1954), 17-20

[Lipshitz-Saracino73] Lipshitz L., Saracino D., The model companion of the theory of commutative rings without nilpotent elements, Proc. Amer. Math. Soc. 38, 1973, 381-387.

[Macintyre95] A. Macintyre, Generic automorphisms of fields, Annals of Pure

and Applied Logic 88 Nr 2-3 (1997), 165 – 180

[Macintyre1973] Macintyre A.J., Model-completeness for sheaves of structures,

Fund. Math. 81, 1973/74, no. 1, 73-89.

[Milne1980] J.S. Milne, Étale Cohomology, Princeton University Press, Prince-

ton, NJ 1980

[Pink92] Richard Pink, On the calculation of local terms in the Lefschetz-Verdier trace formula and its applications to a conjecture of Deligne, Annals of Mathematics 135 (1992) pp. 483-525

[Ritt] Ritt, J. F., Jacobi’s problem on the order of a system of differential equations, Ann. of Math., (2) 36 (1935), 303-312

[Scanlon] Quantifier elimination for the relative Frobenius, in Valuation Theory and Its Applications Volume II , conference proceedings of the International Conference on Valuation Theory (Saskatoon, 1999), Franz-Viktor Kuhlmann, Salma Kuhlmann, and Murray Marshall, eds., Fields Institute Communications Series, (AMS, Providence), 2003, 323 - 352

[Seidenberg1980] A. Seidenberg, editor, Studies in Algebraic Geometry, Studies in Mathematics, vol. 20, Mathematical Association of America 1980 [Serre] J. -P. Serre, Local fields, Springer Verlag 1979. [shoenfield] Shoenfield, J. R., Mathematical Logic. Reprint of the 1973 second

printing. Association for Symbolic Logic, Urbana, IL; A K Peters, Ltd., Natick, MA, 2001 ; Shoenfield, J. R., Unramified forcing, in Axiomatic Set Theory, Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967, pp. 357–381 ; Amer. Math. Soc., Providence, R.I 1971

[Weil48] André Weil, Variétés abeliennes et courbes algébriques, Hermann, Paris, 1948