742

Math 742

Commutative Algebra and Galois Theory

Prof: Andrei Caldararu

Grader: Evan Dummit

Caldararu's office hours: Monday 1:30pm.

Grader's office hours: Wednesday 2:15pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.

This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using Emil Artin's notes which are available here, but I may change my mind before we start on it.

SYLLABUS

In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course.

WEEKS 0.5-2:

Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]

WEEK 3:

Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.

WEEK 4:

An A-B-bimodule M induces a functor Mod-A -> Mod-B given by -- \otimes_A M. The equivalence of categories Mod-k and Mod-M_n(k). Restriction and extension of scalars. Algebras. Tensor product of algebras, the case of C \otimes_R C. Exactness of tensor products, flatness.

WEEK 5

Simple rings. Structure of finite dimensional simple rings over a field. Brauer group, computation of Br(R) = Z/2Z. Semisimple modules and rings. Wedderburn and Artin-Wedderburn theorems. Maschke's theorem on semisimplicity of k[G] for a finite ring. Tensor product of algebras and interpretation as fiber product of affine schemes. Localization (definition and basic properties).

WEEK 6

Rings and modules of fractions. Definitions and universal properties. Examples A_f, A_p. Modules of fractions. The operation S^{-1} is exact, corresponds to tensoring with S^{-1}A, so S^{-1}A is flat. Local properties, examples. Ideals in rings of fractions, in particular primes in A_p are primes in A which are contained in p.

WEEK 7

Integral dependence and valuations. Various characterizations of integral dependence, integral closure forms a ring, which is integrally closed. Relations to rings of fractions. Going up and going down. Valuation rings definition and basic properties. Existence. The integral closure of A is the intersection of all valuation rings which contain A. Nullstellensatz.

Below you will find a repository of homework problems.

HOMEWORK 1 (due Feb 4)

Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21

HOMEWORK 2 (due Feb 15)

Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28; page 31: 2, 3, 4, 8, 9

HOMEWORK 3 (due Feb 22)

Atiyah-Macdonald, page 31: 10, 11 (second part is hard!), 12, 13, 14, 20, 24

HOMEWORK 4 (due Mar 1)

Some non-commutative algebra exercises. You may find some references to read here. Or, even better, you can go and read Chapter I of the excellent book "Noncommutative Algebra" by Benson Farb and R. Keith Dennis. (On reserve in the library.) Most of these exercises are from this book.

1) Let A, A' be k-algebras, with subalgebras B, B', respectively. If the centralizers of B, B' are C, C' (in A, A', resp.), then the centralizer of B\otimes_k B' in A\otimes_k A' is C\otimes C'.

2) Find all the two-sided ideals of M_n(R), where R is a ring. Conclude that if D is a division algebra, M_n(D) is simple. (Hint: all two-sided ideals are of the form M_n(I) for some two-sided ideal I of R.)

3) Let N be a submodule of the R-module M. If N and M/N are semisimple, does it follow that M is semisimple?

4) Let M be a module such that every submodule is a direct summand. Show that M is semisimple as follows:

(a) Show that every submodule of M inherits the property that every submodule is a direct summand.

(b) Show that M contains a simple submodule: choose any finitely generated non-zero submodule M' of M. Let M" be a maximal submodule of M' such that M" is not equal to M' (why does it exist?). Then M'/M" is simple.

(c) Let M_1 be the submodule of M generated by all simple submodules. Show that M_1 = M.

5) Let A be a simple k-algebra with center k such that [A:k] = p^2 for a prime number p. Prove that A is either a division algebra or A is M_p(k).

HOMEWORK 5 (due Mar 8)

Atiyah-Macdonald page 31: 5, 6, 10; page 43: 1, 3, 5, 12, 13, 14

HOMEWORK 6 (due Mar 15)

Atiyah-Macdonald page 43: 15, 18, 19, 20, 21, 22, 25; page 67: 3, 9

HOMEWORK 7 (due Mar 22)

Atiyah-Macdonald page 67: 1, 2, 17, 28, 30, 31.

Also prove that the integral closure of R = k[x,y]/(y^2-x^3) is isomorphic to k[t] (we discussed this in class), as follows. Consider the map R -> k[t] given by x|->t^2, y|->t^3. Show that it is injective, so we can consider R as a subring of k[t]. Moreover, they have the same field of fractions. Now on one hand, k[t] is integrally closed, so the closure of R must be included in it. On the other hand, t is integral over R, so k[t] is contained in the integral closure of R.

HOMEWORK 8 (due Apr 12)

Atiyah-Macdonald page 84: 2, 4, 5, 7, 14, 26, 27 Milne page 24: 1.1--1.4