Introduction to Berkovich analytic spaces, Math 80738
Announcements:
There will be no classes on the week
of January 4 ---11.
General
Info:
- The lectures will be on Mondays 10-11:45 at Sprintzak 202 and Wednesdays 16-17:45 at Sprintzak 102. I will try to give new material on
Mondays, keeping Wednesday's meetings more close to recitation style.
Also, we will discuss on Wednesdays relevant background from the theory of
valued fields. Here are my
notes on that topic (from the course I gave in the spring term).
- The course will be based on my expository notes that can be
found here. The notes
cover relatively large amount of material and contain various exercises
but not proofs. During the course I will try to cover most of the notes
and provide most of the proofs. I do not suggest to read them, but some
sources of proofs are as follows: (1) The book "Spectral theory and analytic
geometry over non-Archimedean fields" by Berkovich, (2) a large article of Berkovich
in Publ. IHES 93, (3) the book "A systematic approach to
rigid analytic geometry" of Bosch, Guntzer and Remmert.
- The evaluation will be based on your participation on
Wednesdays. For example, an hour long lecture given by students in the end
(in this case, I will assign a topic for the lecture and provide a
relevant source). We’ll see how it goes – might depend on the number of
students that want the grade.
Lectures:
- October 28. Introduction, section 2.1, half of section 2.2
from the notes. Including: semi-normed and normed rings and modules,
bounded homomorphisms and equivalent norms,
separated completion, Banach rings and modules.
- October 30. Ostrowski’s theorem – classification of semivaluations on Z (in fact, the description of
M(Z)), the dichotomy – archimedean/non-archimedean.
- November 4. Completed tensor
products. Berkovich spectrum M(A) of a Banach ring (A) and its compactness.
- November 6. Extension of
valuation for algebraic extensions of analytic fields: existence due to
non-emptyness of M(A), and uniqueness due to
equivalence of norms on finite dimensional vector spaces.
- November 11. The maximum
modulus principle for |f| on Berkovich spectrum,
Berkovich spectrum of Z, affine line over
analytic fields.
- November 13. Hensel’s lemma.
- November 18. Affinoid algebras over analytic fields. Weierstrass preparation and division theorems.
- November 20. Continuity of
roots and completed algebraic closure of real-valued fields.
- November 27. Corollaries of Weierstrass theory: Noether
normalization and Hilbert Nullstehlensatz.
Reduction of claims about affinoid algebras to
the strictly affinoid case.
- November 29. Invariants of
algebraic and transcendental extensions of real-valued fields. The
fundamental inequality and Abhyankar’s inequality.
- December 2. Finite banach modules over Noetherian Banach
rings.
- December 4. Arbitrary
valuations and basic theory of valuation rings.
- December 9. Reduction of affinoid rings and the theorem about reduction of
finite homomorphisms.
- December 11. Banach open map theorem.
- December 16. End of proof of
the finiteness theorem. Formulations of Gerritzen-Grauert
theorem, Tate’s acyclicity and Kiehl’s theorems.
- December 18. Valuation rings
and their characterization as maximal local domain with respect to
domination. A valuative description of integral
closure. Extension of valuations in algebraic extensions.
- December 23. Proofs of the
main theorems. Here are the notes
with a detailed proof of Gerritzen-Grauert.
Rational and Laurent covers and refinements, and main ideas of the proofs
of Tate’s and Kiehl’s theorems.
- December 30. The category of
k-analytic spaces (the definition with charts). Closed immersions, finite
morphisms, separated morphisms.
- January 1. Formal schemes and
some applications (Zariski’s connectedness theorem).