# Commutative Algebra

This fall semester (2021) I am teaching our graduate course on commutative algebra GR6261.

Tuesday and Thursday 11:40 -- 12:55 in room 507 math.

If you are an undergraduate and want to register for this course contact me.

The TA is Noah Olander. He is in the help room on Monday 2-4 and Wednesday 2-3.

Grades will be based on the weekly problem sets and a final exam.

My office hours will be 9 - 11 am on Wednesday.

The course will roughly cover Part I of what people sometimes call "old Matsumura". This is the book "Commutative Algebra" by Hideyuki Matsumura (I will specifically use the 2nd edition published in 1980, but I think the material I will cover is also contained in the first edition).

Lectures. It is strongly encouraged to attend the lectures. Below is a rough list of the material we discussed.

1. September 9. Talked about rings, ideals, prime ideals, Spec(A) as a topological space, Spec as a functor, examples of spectra, Chinese remainder theorem, existence of maximal and minimal ideals. See (1.A), (1.C), and (1.D).
2. September 14. Talked about multiplicative subsets, localization at multiplicative subsets, localization at f in A, localization at a prime of A, Spec of a localization, Spec of a quotient A/I, local rings, local homomorphisms of local rings, Jacobson radical, Nakayama's lemma. See (1.E), (1.F), (1.H), (1.K).
3. September 16. Talked about Noetherian rings, Artinian rings, finite modules over Noetherian rings, finite type ring maps, length of modules (see Jordan-Holder discussion here for example), characterize Artinian rings as A with length_A(A) finite. See (2.A) and (2.C).
4. September 21. Talked about Artinian and Noetherian rings, dimension of a ring (defn + examples), connected spaces, irreducible spaces, examples, irreducible closed subsets of Spec(A) are V(p) for prime ideal p of A, characterization of Spec(A) irreducible, connected components of topological space, irreducible components of topological space, irreducible components of Spec(A) correspond to minimal prime ideals of A, Noetherian topological space, Spec of a Noetherian ring is a Noetherian topological space, a Noetherian topolgical space has a finite number of irreducible components. Homological algebra: complexes, cohomology, maps of complexes, action on cohomology, homotopies between maps of complexes, homotopy category of complexes, quasi-isomorphisms, (left) resolutions.
5. September 23. Homological algebra, continued. We talked about (left) resolutions, free resultions, projective resolutions, finite free resolutions, existence of free resolutions, existence of finite free resolutions, examples, uniqueness of free resolutions in terms of maps between them and homotopies between those maps, example, horseshoe lemma for resolutions of modules which form a short exact sequence, construction of left derived functors of an additive functor, example of Tors, example of Exts, right exactness of - ⊗A N and of HomA(-, N), long exact sequences of Tor and Ext in first variable.
6. September 28. Snake lemma Tag 07JV, long exact sequence associated to short exact sequence complexes Tag 0117, well definedness of left derived functors (independence of choices), left derived functors L_iF transform a short exact sequence into a long exact sequence, examples of Tors and Exts from nonzerodivisor f in a ring A by taking M = A/fA, functoriality of Ext and Tor in second variable, Ext and Tor transform short exact sequences in second variable into long exact sequences, Fact: Tor_i(M, N) is isomorphic to Tor_i(N, M) bifunctorially in M and N.
7. September 30. Defined flatness of modules and ring maps. Some lemmas about Tor and flat modules. Completely proved Thm 1 on page 17 of Matsumura's book. Examples of flat modules and flat ring maps. Localization is flat.
8. October 5. Why k[x^2, xy, y^2] → k[x, y] is not flat. List of properties of flatness: transitivity (3.B), base change (3.C), localization (3.D), base change for Tor and Ext (3.E), nonzerodivisors and flatness (3.F), flat modules are sometimes free, for example finite flat modules over local rings (3.G) Tag 00NZ, flatness and ideals (3.H). Why k[x, y] → k[x, y/x] is not flat. Why A → A[x]/(f) is flat when f is a monic polynomial in x over A. Flatness of a ring map can be checked at the primes (3.J). Theorem 2 (4.A) on page 25 of Matsumura.
9. October 7. Faithfully flat ring maps (4.C), (4.D), going down (5.A) but we skipped going up, (5.D) going down for flat ring maps, discussion of Chevalley's theorem for polynomial maps C^n → C^m saying that the image of a Zariski closed subset Z of C^n is Zariski constructible in C^m, general stuff on constructible sets (6.A), (6.B), and Proposition (6.C) characterizing constructible sets
10. October 12. Definition of generic points of irreducible topological spaces and definition of sober topological spaces, see Tag 004X. Lemma 1: Spec of a ring is sober. Lemma 2: given φ: A → B TFAE: (i) Ker(φ) contained in Nil(A), (ii) the image of Spec(φ) is dense, (iii) every min prime A in image Spec(φ), (iv) every min prime p of A of the form φ^{-1}(q) for some min prime q of B. Lemma 3: Let A be a Noetherian ring. Any constructible subset of Spec(A) is the image of Spec(B) for some finite type A-algebra B. Theorem (Chevalley): For a finite type ring map A → B of Noetherian rings the corresponding map Spec(B) → Spec(A) sends constructible subsets to constructible subsets.
11. October 14. Defined integral extensions. Proved (**)/(5.E) from Matsumura but not parts (v) and (vi). Defined the notions of specialization/generalizations. Proved (6.I) Theorem 8.
12. October 19.
13. October 21.
14. October 26.
15. October 28.
16. November 2. Election day no classes.
17. November 4.
18. November 9.
19. November 11.
20. November 16.
21. November 18.
22. November 23.
23. November 25. Thanksgiving day no classes.
24. November 30.
25. December 2.
26. December 7.
27. December 9.

Problem sets. If you email your problem sets, please email them directly to Noah! Some of the exercises will be impossible, so it should not be your goal to do each and every one of them. Moreover, these exercises are not always doable purely with the material discussed in the course -- sometimes you'll have to look up things online or in books and use what you find.

1. Due 9-16 in class:
1. For a ring A prove that the standard open subsets D(f) form a basis for the topology on Spec(A).
2. Give an example of a ring A such that Spec(A) is not Hausdorff.
3. Let k be your favorite field. Describe the spectrum of A = k[x, y]/(xy) by listing all the primes in some manner and describing the topology in words.
4. Let E be a subset of a ring A. Show that V(E) is the same as V(I) where I is the radical of the ideal generated by E.
5. Let I, J be ideals of a ring A. What is the condition on I and J for V(I) and V(J) to be disjoint?
6. Let A be a ring and let M be an A-module. Define the support Supp(M) of M as the set of primes p such that the localization M_p of M at p is nonzero. Show that Supp(M) is a closed subset of Spec(A) when M is a finite A-module.
7. Give an example of a ring A with exactly 6 prime ideals p_1, p_2, p_3, m_1, m_2, m_3 with p_i minimal primes, m_i maximal primes, and p_i contained in m_j for i not equal to j.
2. Due 9-23 in class:
1. Give an example of a non-Noetherian ring whose spectrum is a Noetherian topological space. Give an example of a non-Artinian ring whose spectrum is a singleton.
2. Do exercise Tag 076G.
3. Do exercise Tag 02DL.
4. Let A be a ring. Show that the following are equivalent: (a) A is Noetherian, (b) the category of finite A-modules is an abelian category.
5. Give an example of a countable ring with uncountably many prime ideals.
6. Let A be a ring and let a, b be elements of A. Set R = A[x, y]/(ax + by). Show that R is flat over A if a and b generate the unit ideal of A. Show by an example that R is in general not flat over A.
3. Due 9-30 in class:
1. Let A = k[x] be the polynomial ring over a field k. Show that all the left derived functors of the additive functor F which sends an A-module M to F(M) = {m in M with xm = 0} are zero. (This includes the zeroth left derived functor of F.)
2. Let A = k[x]/(x^2) where k is a field. Denote ε the image of x in A. Often A is called the ring of dual numbers. Let F be the additive functor which sends an A-module M to F(M) = {m in M with εm = 0}. Show that none of the left derived functors of F are zero.
3. Do exercise Tag 0CYH.
4. Do exercise Tag 0FWR.
5. Do exercise Tag 0CRC.
6. Let A = k[x, y] and M = A/(x, y). Compute Ext^i_A(M, A) for all integers i.
4. Due 10-7 in class.
1. Do exercise Tag 02CR
2. Do exercise Tag 02CU
3. Let A = k[ε] be the ring of dual numbers over a field k. Show that an A-module M is flat over A if and only if it is a free A-module.
4. Let A = k[ε] be the ring of dual numbers over a field k. Let a, b, c be elements of k. Let B = A[x, y]/I where I is the ideal generated by x^2 - aε, xy - bε, y^2 - cε. Show that B is flat over A if and only if a = b = c = 0.
5. Let A be a ring such that every A-module is flat (such a ring is called absolutely flat). Show that every prime ideal of A is a maximal ideal. (Much more is true. This is just a puzzle. Try it yourself before googling this notion.)
6. Theoretical question that I suggest you skip (maybe just do some parts of it if you are interested).
1. Read the section on cohomological delta-functors Tag 010P.
2. Define what is a homological delta-functor {L_n, delta_{A → B → C}} by reversing the arrows in Tag 010P.
3. Define what is the dual notion of a universal delta-functor in the setting of homological delta-functors (beware of direction of arrows).
4. Let L_iF be the left derived functors (as defined in the lectures) of some additive functor F on Mod_A. Show L_iF form a universal homological delta-functor.
5. Show that for a fixed N the functors L_i(M) = Tor_i(M, N) form a universal homological delta-functor. (You don't have to answer this as this is trivial from the previous part and the definition of Tor_i(M, N) in the lectures as the ith left derived functor of M ↦ M ⊗ N. But I wanted to add it here to contrast with the next part.)
6. Show that for a fixed M the functors L_i(N) = Tor_i(M, N) also form a universal homological delta-functor. (Hints: use the long exact sequences constructed in the lecture to see that these L_i form a delta-functor; use a suitably formulated dual of Lemma Tag 010T and show that if N is free then L_i(N) = 0 for i > 0.)
7. Conclude that Tor_i(M, N) and Tor_i(N, M) are isomorphic as bi-functors. (Hint: use uniqueness of universal homological delta-functors.)
5. Due 10-14 in class.
1. Let k be a field and A = k[x, y]. Let m = (x, y) be the maximal corresponding to the "origin" of the Spec(A). Let U = Spec(A) ∖ {m}. Construct a finite type ring map φ: A → B with Spec(φ) equal to U. Can you do it so that B is a domain? Can you do it so that B is a domain with the same fraction field as A? (This last question is very difficult to answer.)
2. Do exercise 02DR
3. Do exercise 0FKE
4. Let k be a field. Let φ: k[x, y] → k[z] be a k-algebra homomorphism. Show that φ has a nonzero kernel. Compute an element of the kernel if k = Q and x maps to 3z^2 + 1 and y maps to 2z^2 + 5z + 7. (Suggest using computer algebra.)
5. Do exercise 02D0 (This exercise gives a "nonstandard" proof of the Hilbert Nullstellensatz for the complex numbers. Do this if you are interested only.)
6. Due 10-21 in class:
1. Do exercise Tag 02CJ
2. Do exercise Tag 02CL
3. Do exercise Tag 0CR8
4. Do exercise Tag 0CRA
5. Let p be a prime number. Let F_p be the field with p elements. Denote F_p[x, y]≤d the space of polynomials of total degree at most d. Show that "most" elements of F_p[x, y]≤d are irreducible as d tends to ∞.

Stacks project: The chapter on commutative algebra.