Fundamental concepts in commutative algebra, Math 80599

Announces:

January 23. Some details about the exam: it contains 5 problems for 25 points, and you should choose 4. In addition there is a less typical bonus problem for 10 points. In large, the theoretic part of the exam covers the material of [AM] that was studied in class (more or less chapters 1—9) and practical problems are related to this material. Due to your requests I provide here a concrete exam’s syllabus.
November 28. I recommend to look at the book on commutative algebra of Altman and Kleiman, which is available on the web. It also includes nice sections on categories.
November 7. We got new rooms for lectures. I updated the general info in accordance.
October 31. Due to your request, the deadline for submitting homework will be on Thursdays midnight, starting with HW1 (which should now be submitted on November 2).
October 22. The lecture on October 24 will be given by Uri Brezner.

General Info:

Instructor:

· Michael Temkin

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

· Office: Ros 78A (tel. 02-6584575)

· Office Hours: by appointment

· Lectures: Mondays 10-11:45 at Ros 70, Tuesdays 16-17:45 at Mathematics 110.

Teaching Assistants:

· Uri Brezner

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

· Office: Ros 26 (tel. 02-6584846)

· Office Hours: by appointment

· Recitations: Tuesdays 12-13:45 at Sprinzak 116.

Texts:

· The commutative algebra part will be due to "Introduction to Commutative Algebra" by Atiyah and Macdonald.

Grading:

· The final grade will be obtained as follows: 50% of the exam grade plus 50% of the homework grade. The grade for homework will be computed by averaging 10 best weekly homework grades.

The exam will be only on the commutative algebra part of the course, and an exam syllabus will be posted in the end of the course.

Homework:

· The exercises are assigned on Tuesdays on this webpage, and should be submitted to Uri's mailbox until the end of the next Thursday (any time before 24:00 is ok). Late (or non-submitted) homework will not be graded. Homework may include a non-mandatory part marked by *, which is not for submission but can be helpful for deeper understanding of the material. You are welcome to discuss it with me or Uri.

Homework Assignments:

More difficult problems are marked with *.

October 24. The first homework is here. Due date is October 31. (Changed to November 2.)
October 31. The second homework is here. Due date is November 9.
November 7. The third homework is here. Due date is November 16.
November 14. The fourth homework is here. Due date is November 23.
November 21. The fifth homework is here. Due date is November 30.
November 28. The sixth homework is here. Due date is December 7.
December 5. The seventh homework is here. Due date is December 14.
December 12. Eighth homework: solve problems 18,19,20 after chapter 1 and problem 21 after chapter 3 in Atyah-Macdonald. Due date is December 21.
December 19. Ninth homework is here. Due date is December 28.
December 27. Tenth homework is here. Due date is January 4.
January 2. Eleventh homework is here. Due date is January 11.
January 9. Twelfth homework is here. Due date is January 18.
January 16. Thirteenth homework: solve problems 10, 11, 20, 21, 22, 35 after chapter 5. Due date is January 25.

Lectures:

October 23. Some basic definitions from chapter 1, including commutative rings with units, their homomorphism, R-algebras, polynomial rings, maximal/prime/principal ideals, quotient rings, nilpotent elements and zero divisors.
October 24. End of chapter 1 and some exercises after it: extension and contraction of ideals with respect to a homomorphism, Chinese remainder theorem, some properties of the spectra.
October 30. Beginning of chapter 3: localizations of rings, its universal property, and behavior of ideals under localization.
October 31. Categories, limits and colimits (including products and coproducts, fibred products, equalizers and coequalizers), complete and cocomplete categories. Criteria of completeness and cocompleteness. Here are my written notes about categories. (Two precautions: (1) I'll polish it and maybe add some details, (2) The order of exposition is different. I introduce representable functors first and deduce the criteria from Yoneda lemma. We will study (or at least formulate) this next week.)
November 6. Adjoint functors, continuous and cocontinuous functors.
November 7. Started R-modules (beginning of chapter 2). Definitions, submodules, quotients, limits and colimits, kernels and cokernels.
November 13. Different versions of Nakayama lemma, additive invariants and K_0 of subcategories of R-Mod, exact sequences, the snake lemma.
November 14. Bilinear maps, tensor products, adjunction between tensor products and Hom's, exactness properties of tensor products and Hom's, further properties of tensor products.
November 20. Extension of scalars, localization of modules.
November 21. Classification of adjoint pairs of functors between categories of modules, tensor products of algebras and basic examples, including quotients, localizations, products of field extensions. Definition of flat modules.
November 27. Faithfully flat modules and simple instances of faithfully flat descent.
November 28. Affine algebraic varieties over an algebraically closed field. Hilbert Nullstellensatz and the correspondence between varieties and radical ideals.
December 4. Morphisms between affine varieties. Function (or coordinate) ring k[V] of an affine variety V, and the equivalence of categories it induces: k-Aff^op ==Red.f.g.k-Alg. The opposite functor Max(A).
December 5. Examples, products, fiber products, modification needed to construct varieties over an arbitrary field.
December 11. Affine space over an arbitrary perfect field. Projective spaces P^n, graded rings, and homogeneous ideals.
December 12. Cones, projective varieties, projective Nullstellensatz. Started a new topic: general varieties and schemes. So far, gave a local definition of regular functions on an affine variety V and proved the theorem that it can be identified with an element of k[V] (i.e. agrees with the old definition of polynomial functions).
December 18. Preasheaves and sheaves. Arbitrary algebraic varieties (V,O_V) over k, where V is a topological space and O_V is the structure sheaf of regular functions from V to k. Morphisms of varieties and the theorem that the category of affine varieties embeds fully faithfully into the category of arbitrary varieties. Analogs for affine schemes – regular functions and the structure sheaf.
December 19. Noether and Artin modules and rings. Modules of finite length, composition series and Jordan-Holder theorem.
December 26. Noetherian rings and Hilbert basis theorem.
January 1. Primary ideals and their properties in general rings and in Noetherian rings (chapter 4 and the end chapter 7).
January 2. Primary decomposition and the two uniqueness theorems.
January 8. Artin rings (chapter 8).
January 9. Integral dependence (started chapter 5): basic properties of integral and finite homomorphisms.
January 15. Going down theorem, integrally closed domains.
January 16. Going up theorem, Noether normalization theorem.
January 22. Dimension theory of finitely generated algebras over a field.
January 23. Unique factorization domains, valuation rings, Dedekind rings (chapter 9 without invertible ideals).