## Math 221      Commutative Algebra

Harvard Fall 2007
Instructor: Thomas Lam
Course Assistant: Dustin Clausen (dclausen[at]fas)
Lectures: Monday, Wednesday, Friday 12-1      Science Center 310
Office Hours: Monday 2-3pm, Tuesday 4-5pm (Science Center 435)

Announcements:

Announcements will be posted here.

Handouts: First Day

Problem sets: There will be problem sets roughly once a week. Collaboration on homework is permitted, but you are not allowed to just copy someone else's work. You have to mention on your problem set who you worked with.

Solutions below are written by Dustin.

Pset 1 (Due 09/28): [AM, Ch1] Exercise in text: 1.13; Exercises at end: 6, 15, 16, 17, 18, 19. Solution.

Pset 2 (Due 10/01):
(1) Prove that a unique factorization domain is normal (i.e. equal to its normalization).
(2) What is the normalization of A = k[x,y]/(y^2-x^3)? Here k is a field.
(3) (David and Soroosh ask) Let I be an ideal in a ring A. Is Ann(I/I^2) = I + Ann(I)?
[AM, Ch2] Exercises at end: 2, 3 (there is nearly nothing to do here except follow instructions).
[AM, Ch5] Exercises at end: 8, 9. Solution.

Pset 3 (Due 10/10):
(1) ([Reid, Ex. 5.2]) Describe the irreducible components of V(J) for J = (y^2-x^4, x^2-2x^3 -x^2y+2xy+y^2-y) in k[x,y,z]. Here k is algebraically closed.
[AM, Ch 5] Exercises at end: 16, 18. For 16, you may use Exercise 2.
[AM, Ch 6] Exercises at end: 1, 5, 6, 7, 8, 9. Solution

Pset 4 (Due 10/15): [AM, Ch 3] Exercises at end: 6, 9, 20, 21, 22, 23. Solution

Pset 5 (Due 10/22):
(1) If M is the (not necessarily direct) sum of submodules N and L is it true that Ass(M) = Ass(N) \cup Ass(L)?
(2) Let I = (x,y) \subset A = k[x,y,z]/(xy-z^2). Find Ass(A/I^2). (Optional: find a primary decomposition of I^2.)
(3) Let A be Noetherian and M, N finite modules over A. Show that Ass Hom(M,N) = Supp M \cap Ass N. (Hint: you may find it useful to prove that (Hom(M,N))_P = Hom_(A_P)(M_P,N_P))).
[AM, Ch 3] Exercises at end: 16 (you may use exercises quoted in the hints even if we have not proved them)
[AM, Ch 4] Exercises at end: 2. Solution

Pset 6 (Due 10/29): Hand in 5 problems from Homological algebra problem set. Later on you'll be asked to hand in more problems, so you can decide whether to do all the easy problems now or leave them for later.really detailed Solution

Pset 7 (Due 11/05): Hand in 2 more problems from the homological algebra pset.
[AM, Ch 10] Exercises at end: 2, 3, 9. As usual you are allowed to use quoted problems even if we have not done them.
Explain how Exercise 9 implies the version of Hensel's Lemma in class, at least when f'(a) is a unit. It is enough to prove the Hensel's Lemma in class for a maximal ideal. (Added 11/04: its also okay to assume f(x) is monic).

Pset 8 (Due 11/14): Hand in 2 more problems from the homological algebra pset.
[AM, Ch 8] Exercises at end: 2, 3, 4.
[AM, Ch 9] Exercises at end: 5, 7.

Pset 9 (Due 11/28):
[AM, Ch 11] Exercises at end: 1, 4, 6. ([AM] uses "height" for what I call "codim").
[Eis, Ch 10] (1) Let A be Noetherian and P \subset A a prime. Let Q \subset A[x] be prime such that Q intersects A in P. Show that Q has one of two forms (similar to the case A = Z we did a long time ago in class):
(a) Q = P[x] (and codim Q = codim P)
(b) There is some f(x) \in A[x] with leading coefficient not in P such thatQ is the set of g(x) \in A[x] such that ag(x) \in P[x] + (f(x)) for some a \in A-P. (and codim Q = codim P + 1).
Thus dim A[x] = dim A + 1 (cf. [AM, Ch 11, Ex. 7].)
(2) Let a, b be a regular sequence in a domain A. Show that ax-b is a prime in A[x].

The last two psets cover two topics which are (mostly) independent of what I cover in class in the last three weeks.

Pset 10 (Due 12/03):
(1) Read Theorems 5.11, 5.16 in [AM].
(2) [AM, Ch 3] Exercises at end: 18
(3) [AM, Ch 5] Exercises at end: 10, 11
(4) [AM, Ch 7] Exercises at end: 24, 25 (you'll need to read 20-23)
Going up and down together with Noether Normalization is enough to prove that transcendence degree of a geometric ring is equal to Krull dimension: if k[x_1,..,x_d] \subset A is integral then going up/down gives the first equality in dim(A) = dim(k[x_1,...,x_d]) = d = tr.deg_k(k[x_1,...,x_d]) = tr.deg_k(A). The last two pages of [AM, Ch 11] prove something stronger: the equality holds locally at every maximal ideal.

Pset 11 (Due 12/15): Derivations and differentials

Prerequisites: Mathematics 123 or equivalent.
Grading: Based solely on the problem sets.
Textbook: Introduction to Commutative Algebra, by Atiyah and Macdonald. You are not required to buy this book, but I intend to follow it for a large part of the course. Most problem sets will also be taken from this book.
A cheaper alternative to [AM] is Undergraduate Commutative Algebra, by Miles Reid, which is a fun read.
Other reference books:
Commutative Ring Theory, Matsumura
Commutative algebra with a View Toward Algebraic Geometry, Eisenbud