Algebraic geometry II, Math 80990

Announcements:

The schedule was changed! The lecture on Wednesdays was moved to Sundays 9-11. The new room is Shprintzak 116.

General Info:

Instructor:

· Michael Temkin

· E-mail: temkin $http://www.math.huji.ac.il/~temkin/images/at.gif$ math.huji.ac.il

· Office: Manchester House 308 (tel. 02-6584575)

· Office Hours: by appointment

· Lectures: Sundays 9-11 at Shprintzak 116, Mondays 10-12 at Shprintzak 25.

Texts:

· We will mainly use Hartshorne's "Algebraic geometry", sections II.8, III.6-12 and chapter IV.

Grading:

· Based on a lecture students will give in the end of the course.

Homework:

· The exercises will be assigned on Wednesdays or Thursdays on this webpage. They are not for submission, but I urge you to work on them. I will be glad to discuss the homework problems with you.

Homework Assignments:

More difficult problems are marked with *.

HW 1. The first homework is here.
HW 2. The second homework is here.
HW 3. The third homework is here.
HW 4. The fourth homework is here.
HW 5. The fifth homework is here.
HW 6. The sixth homework is here.
HW 7. The seventh homework is here.
HW 8. The eighth homework is here.
HW 9. The ninth homework is here.
HW 10. The tenth homework is here.

Lectures:

February 17. Complete non-singular curves and the Riemann-Roch theorem, IV.1 (the basic version concerning the Euler characteristic of invertible sheaves on such curves). Started Kahler differentials, II.8 (a reference is Matsumura's book).
February 19. Basics on Kahler differentials (first and second fundamental sequences and computations in basic cases), modules of differentials on schemes (beginning of II.8 in Hartshrone's text).
February 24. Smooth varieties over a field and equivalence of two definitions of smoothness: geometric regularity (i.e. regularity after any base field extension) and the definition in terms of the module of differentials. Serre's duality on smooth curves: the formulation.
March 2. Reminds on Weil and Cartier divisors (some material of section 6 in chapter II), degree of divisors on complete curves, pullbacks of Weil divisors on curves. Corollaries of Serre's duality: degree of the canonical class, curves of genus 0 and 1 (section 1 in chapter IV).
March 3. Proof of Serre's duality theorem (except the two theorems on residues: existence and the residue formula), a good reference is Serre's "Algebraic groups and class fields", chapter 2.
March 9. Differentials of complete rings, existence of residues.
March 10. The residue formula. Hurwitz theorem (section 2 of chapter IV in Hartshorne's text).
March 23. Inseparable maps of curves, Luroth theorem, a brief review on Jacobians of curves.
March 24. A brief review on moduli problems and moduli space of curves of genus g. Flat morphisms (section 9 in chapter III of the text).
March 30. Flatness of a family of closed subschemes of P^n_T and Hilbert polynomial.
March 31. Flatness and dimension.
April 28. Smooth morphisms.
May 11. Smooth rmophisms (ended section 10 in chapter III), category theory.
May 12. Category theory: limits and colimits, adjoint functors, additive and abelian categories.
May 18. Main theorem on existence and uniqueness of derived functors.
May 19. Examples: Ext, Tor, inner Ext, group cohomology, cohomology of sheaves, higher direct images. Balancing of the derived bifunctors Ext and Tor.
May 25. Support and associated primes of a module. Beginning of the dimension theory: Hilbert series of a graded ring and Hilbert-Samuel function of modules over local rings.
May 26. Dimension theory of modules: the equality d(M)=delta(M)=dim(M) for finitely generated modules over noetherian local rings (Matsumura's "Commutative ring theory", 13.4) and corollaries.
June 1. Depth of rings and modules, Cohen-Maucaulay rings, conditions S_n and R_n, Red=S_1+R_0 and Nor=S_2+R_1 theorems.
June 2. Kozhul complexes and regular sequences.
June 8. The theory of regular rings (Serre's theorems about finite global dimension and localization) of a regular ring).
June 9. Ext and inner Ext in the category of O_X-modules.