Commutative Algebra

Professor A.J. de Jong, Columbia university, Department of Mathematics.

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.

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).
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).
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).
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.
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 Hom_A(-, N), long exact sequences of Tor and Ext in first variable.
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.
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.
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.
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
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.
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.
October 19. Recapped material on integral extensions last time. Gave examples of integral extensions. Recalled the definition of finite ring maps. Lemma: finite ring maps are integral. Proved lemma in the Noetherian case. Lemma: If B is generated by b_1, ..., b_n over A and if each b_i is integral over A, i.e., b_i satisfies a monic polynomial equation with coefficients in A, then B is finite over A. Corollary: given a ring map A → B the set of elements B' ⊂ B integral over A form a sub A-algebra of B. Remark: If φ: A → B is a finite ring map then the fibres of Spec(φ) are finite, discrete topological spaces. We proved this via the observation that the fibres of Spec(φ) are spectra of finite dimensional algebras over fields and such an algebra is Artinian. We defined associated primes of a module M over a Noetherian ring. We proved (7.B) Proposition from the book. We discussed corollaries 1 and 2 of the Proposition on page 50 of the book. We did some examples of associated primes of modules.
October 21. We discussed (7.D) Theorem 9 about the relationship between Ass(M) and Supp(M). We discussed (7.C) Lemma about the associated primes and support of a localization. We discussed (7.E) Theorem 10 about filtering modules. Next, we stated (7.F) Lemma about the set of associated primes of an extension of modules. Finally, we proved that the set of associated primes of a finite module over a Noetherian ring is finite (7.G) Proposition. GRADED stuff: we defined graded rings and graded modules over graded rings. Examples are graded polynomial rings. We stated and proved a lemma about when graded rings A are Noetherian: namely, A_0 should be Noetherian and A should be finitely generated over A_0. In this case A_n is a finite A_0 module for all n. An example is to start with a Noetherian local ring (B, m, k) and consider A = ⨁ m^n/m^{n + 1}. We stated the theorem about Hilbert polynomials of graded modules.
October 26. We stated and proved a slightly stronger version of (10.F) Theorem 14: Namely, let A be a graded ring and M a graded A-module. Assume A_0 is Artinian, A is generated by A_1 over A_0, A_1 is a finite A_0-module, and M is a finite A-module. Then the function which sends n to length_{A_0}(M_n) is eventually polynomial. We also stated a variant of this theorem using Euler-Poincare functions. Next we discussed and proved Artin-Rees (11.C) Theorem 15 and its corollaries (namely, the Remark on page 69 following the theorem, (11.D) Theorem 16, and Corollary 1, 2, 3.
October 28. Dimension theory. We defined the height ht(p) of a prime ideal p in a Noetherian ring A as the supremum of the lengths of decreasing chains of primes starting with p. We define the dimension of A as the supremum of the heights of its prime ideals. Thus dim(A) is also the supremum of the lengths of chains of prime ideals in A. Of course dim(A) is equal to dim(Spec(A)) where the dimension of a topological space is defined as the supremum of the lengths of chains of irreducible closed subsets, see Tag 0054. If (A, m) is local then dim(A) = ht(m). Then ht(p) = dim(A_p). We always have dim(A) ≥ dim(A/p) + ht(p). We discussed a few examples of dimensions of rings. Given a Noetherian local ring (A, m, k) and a finite A-module M we define three invariants: dim(M) = dim(Supp(M)) = dim(A/Ann(M)), d(M) defined as the degree of the numerical polynomial which sends n to length_A(M/m^nM), and r(M) which is the minimum integer r such that there exist x_1,...,x_r in m with length_A(M/(x_1, ..., x_r)M)) < ∞. Main theorem of the lecture: dim(M) = d(M) = r(M).
November 2. Election day no classes.
November 4. We discussed some consequences of the theorem proved last time:
1. (12.I) Thm 18, see Tag 00KV and Tag 0BBZ
2. for a Noetherian local ring (A, m, k) we have dim(A) ≤ dim m/m^2, see page 78
3. we defined system of parameters, regular local rings, and regular systems of parameters, see 78 and Tag 00KU
4. we proved (12.K) Proposition, see Tag 02IE
5. (*) dim(B) ≤ dim(A) + dim(B/mB) for a local homomorphism A → B of local Noetherian rings, see (13.B) Theorem 19 part (1) or Tag 00OM
6. if GD holds then equality in (*) see Tag 00ON
7. if A → B is a ring map which is surjective on spec and we have GD or GU then dim(A) ≤ dim(B), see Tag 00OH
8. we proved dim(A[x]) = dim(A) + 1 which is (14.A) Theorem 22
November 9. We defined catenary rings and universally catenary (u.c.) rings. We discussed the key property characterizing a catenary ring A namely that if p ⊂ q ⊂ r are primes of A then ht(r/p) = ht(r/q) + ht(q/p). Lemma: if A is catenary then any quotient or localization of A is catenary. Lemma: if A is u.c. then any finite type A-algebra is u.c. (14.C) Theorem 23: A ⊂ B is a finite type extension of Noetherian domains. q a prime of B lying over p in A. Then (*) ht(q) ≤ ht(p) + trdeg_A(B) - trdeg_{k(p)}(k(q)). We have equality in (*) when either A is u.c., or B is a polynomial ring over A. Definition: given a finite type extension A ⊂ B of Noetherian domains we say the dimension formula holds between A and B if (*) is an equality for all q in B lying over p in A. Corollary: for a Noetherian ring A the following are equivalent:
1. A is u.c.
2. A is catenary and for all primes p and all finitely generated extensions A/p ⊂ B the dimension formula holds between A/p and B.
This gives us a criterion we can try to use to show that a given ring A (for example a field) is u.c.!
November 11. We discussed Noether normalization (N.N.). The first version says that given a finite type algebra A over a field k there exists a finite injective map k[y_1, ..., y_r] → A. We prove this and we proved that r = dim(A) and r = trdeg_k(A) if A is a domain. Cor 1: dim(A/p) = trdeg_k(kappa(p)) when p is a prime of A. Cor 2: if m is a maximal ideal in A then kappa(m) is a finite extension of k. Cor 3: if k is algebraically closed, then any maximal ideal of the polynomial ring k[x_1, ..., x_n] is of the form (x_1 - a_1, ..., x_n - a_n) for some a_1, ..., a_n in k. This (part of) the Hilbert Nullstellensatz. The second version of N.N. says that given a prime p of k[x_1, ..., x_n] there exists a finite injective map φ: k[y_1, ..., y_n] → k[x_1, ..., x_n] such that φ^{-1}(p) = (y_{r + 1}, ..., y_n) where r = trdeg_k(kappa(p)). The third version of N.N. says that given a prime p of a domain A of finite type over k, there exists a finite injective map φ: k[y_1, ..., y_n] → A such that φ^{-1}(p) = (y_{r + 1}, ..., y_n) and in this case r = trdeg_k(kappa(p)) and n = trdeg_k(A). We discussed that in order to prove that a field is u.c. it now suffices to prove "Going down for integral over normal".
November 16. We proved "GD for integral over normal" by the method discussed in Tag 037E ; this result is also in Matsumura, see (5.E) Theorem 5 part (v), but his proof is different. Combined with the discussions in the previous lectures we get the following corollaries:
1. every field k is u.c.,
2. the dimension formula holds for any homomorphism of finite type k-algebras which are domains,
3. more generally, if A → B is a homomorphism of finite type k-algebras and q' ⊂ q ⊂ B are primes lying over p' ⊂ p ⊂ A then we have ht(q/q') = ht(p/p') + trdeg_k(p') k(q') - trdeg_k(p)k(p'),
4. if A is a finite type k-algebra and p' ⊂ p are two (distinct) primes with no prime strictly in between, then trdeg_k k(p') = trdeg_kk(p) + 1,
5. Fix a finite type k-algebra A and set X = Spec(A).
  1. the topological space space X is sober and Noetherian,
  2. δ : Spec(A) → Z defined by p ↦ trdeg_kk(p) is a dimension function (see Tag 02I8),
  3. δ ≥ 0 and δ(p) = 0 if and only if p is closed,
  4. every point specializes to a closed point, and
  5. the closed points of X are dense in every locally closed subset of X.
  Lament: what are all the properties that X and δ possess?
6. We discussed some questions about the topological space X:
  1. Given x, y ∈ X do x and y have homeomorphic small open nbhds?
  2. Given Y, Z ⊂ X such that Y ∩ Z is nonempty, does dim(Y ∩ Z) ≥ dim(Y) + dim(Z) - dim(X) hold?
November 18. We discussed some left-over material from the last few lectures. We proved that a finite type algebra A over a field is Jacobson, i.e., that the radical of an ideal I of A is equal to the intersection of the maximal ideals containing I. This is part of (14.L) Theorem 25. We discussed how this is equivalent to saying that X = Spec(A) is a Jacobson topological space ( Tag 005T). Let φ: A → B be a k-algebra homomorphism between finite type k-algebras. Then Spec(φ) sends closed points to closed points and more generally points get mapped to points having "smaller or equal dimension". We discussed (14.K) Corollary 4 in some detail.
November 23. We talked about topological rings and modules. Some of this is discussed in Tag 0B1Y, Tag 07E7, and Tag 0AMQ (but don't read all of this of course). We defined linearly topologized rings and modules. We discussed the completion of a topological module which has a linear topology. See second paragraph of Tag 07E7. Given a ring A and an ideal I we discussed the I-adic topology on A and on any module M over A. The I-adic completion M^ of M is the completion of M with respect to the I-adic topology. In a formula: M^ = lim M/I^nM. We say M is I-adically complete if the canonical map M → M^ is an isomorphism. We discussed an example of A, I, M such that M^ is not I-adically complete.
November 25. Thanksgiving day no classes.
November 30. We continued our discussion of completion. An inverse system of modules over a ring R is a sequence of homomorphisms ... → M_3 → M_2 → M_1 of R-modules. The inverse limit lim M_n is the set of sequences (x_n) of elements x_n ∈ M_n such that x_{n + 1} always maps to x_n. An inverse system of short exact sequences 0 → A_n → B_n → C_n → 0 means that we have short exact sequences as indicated and maps ... → A_3 → A_2 → A_1, ... → B_3 → B_2 → B_1, and ... → C_3 → C_2 → C_1 compatible with the maps in the short exact sequences. Then we always get an exact sequence 0 → lim A_n → lim B_n → lim C_n. Lemma: If the maps A_{n + 1} → A_n are surjective, then also lim B_n → lim C_n is surjective and we get a short exact sequence of limits. Let R be a ring and I an ideal. All topologies are I-adic and all completions are I-adic completions. Using the lemma above we proved that if φ: M → N is surjective (it is enough if it is surjective modulo I), then φ^: M^ → N^ is surjective too. Then we discussed and proved Lemma 05GG which is about the case where I is finitely generated; it says the the I-adic completion M^ is always I-adically complete. We also proved that if R is Noetherian, then we additionally have
1. completion is an exact functor on the category of finitely generated R-modules,
2. M^ = M ⊗ R^ for finite R-modules M,
3. R → R^ is flat,
4. if (R, m, k) is local and I ⊂ m, then R → R^ is faithfully flat, R^ is local too with maximal ideal m^ = mR^ and residue field R^/m^ = k.
The proof used Artin-Rees in a key step! Completion for Noetherian rings is discussed in Section 0BNH. The completion of k[x_1, ..., x_n]_{(x_1, ..., x_n)} is equal to the power series ring k[[x_1, ..., x_n]].
December 2. We discussed the Cohen structure theorem, whose precise statement you can find in Section 0323. I sketched how this statement can be used to prove every complete Noetherian local ring is universally catenary, starting with facts about power series rings (over fields or over Cohen rings). Then we talked about injective modules. Here are some references (but please don't read all of this)
1. Tag 02D* discusses basic stuff about injective modules. It also proves the statement that any module can be mapped injectively into an injective module (last lemma on the page),
2. Tag 08XI discusses essential extensions in arbitrary abelian categories, bvut see the last lemma for why this is the same as our definition in the lecture
3. Tag 08XN discusses injective modules and essential extensions
4. Tag 08YI discusses injective hulls and existence of injective hulls.
December 7. We discussed Proposition 08YA and what it means (but we didn't talk about the proof). Then we talked about Matlis duality for Artinian local rings which is discussed in Tag 08YW. Finally, we proved the first few lemmas of Tag 08Z1
December 9. We will discuss the rest of Tag 08Z1 and then we'll say something about Grothendieck's local duality theorem.

Problem sets. If you email your problem sets, please email them to Noah with a cc to me. 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.

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.
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.
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.
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.)
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.)
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 ∞.
Due 10-28 in class.
1. Please read the definition of Euler-Poincare functions and Hilbert functions and Hilbert polynomials in Def 027Y.
2. Do exercise Tag 02E2.
3. Do exercise Tag 02E3.
4. Do exercise Tag 02E6.
5. Do exercise Tag 02E8.
6. Do exercise Tag 0AAP. Please use that a curve C as in the exercise is always equal to V(f) for some irreducible f in C[x, y].
Due 11-4 in class.
1. Do exercise Tag 0D5M where φ_R is defined in Section Tag 0D5F.
2. Do exercise Tag 0D5N with the same definition of φ_R.
3. Find a Noetherian local ring R of positive dimension such that φ_R(100) > φ_R(101).
4. Find a Noetherian local ring R with minimal primes p and q such that dim(R/p) = 1 and dim(R/q) = 2.
Due 11-11 in class.
1. Do exercise Tag 0EEM.
2. Do exercise Tag 0FWQ.
3. Do exercise Tag 0D1T (this is a nontrivial but worthwhile theoretical exercise which asks about the relationship between smoothness and regularity of local rings).
4. Do exercise Tag 0CVS (this one is hard because to answer it, you'll have to use a bunch of stuff on the dimension of finite type algebras over fields which I won't have discussed in the lectures yet).
Due 11-18 in class.
1. Let k be your favorite field. Show that A = k[x, y, z]/(xy^2 - z^3) is a domain.
2. Let k be your favorite field. Show that the domain A = k[x, y, z]/(xy^2 - z^3) is not normal.
3. Let k be your favorite field. Show that A = k[x, y, z, w]/(xy - zw) is a domain.
4. Let A be a domain. Let x, y in A be nonzero such that y is a nonzerodivisor on A/xA. Show in order that
  1. if xa = yb for some a, b in A then there exists a unique c in A such that a = yc and b = xc
  2. show that x is a nonzerodivisor on A/yA
  3. show that y^n is a nonzerodivisor on A/x^mA for all n, m > 0.
5. With same assumptions as in the previous exercise show that the intersection of A[1/x] and A[1/y] inside the fraction field equals A.
6. Let k be your favorite field. Consider the domain A = k[x, y, z, w]/(xy - zw).
  1. show that A[1/x] is normal,
  2. show that A[1/y] is normal,
  3. show that A is normal.
  Hints: a localization of a normal domain is normal, a polynomial ring over a field is normal, and use the previous exercise.
Due 12-2 in class.
1. Given a field k and a finite type k-algebra A set X = Spec(A). Let x, y ∈ X be closed points. Find k, A, X, x, y such that there exist open neighbourhoods U and V of x and y such that for any x ∈ U' ⊂ U and any y ∈ V' ⊂ V with U' and V' open we have that U' is not homeomorphic to V'.
2. Given a field k and a finite type k-algebra A set X = Spec(A). Let Y, Z ⊂ X be closed subsets. Find k, A, X, Y, Z such that X, Y, Z are irreducible and such that Y ∩ Z is nonempty of dimension < dim(Y) + dim(Z) - dim(X).
3. Do exercise Tag 0CS1 (this exercise is about depth which I didn't lecture about yet -- feel free to replace it by exercise 6 below about completion)
4. Do exercise Tag 0CS2 (this exercise is about depth which I didn't lecture about yet -- feel free to replace it by exercise 7 below about completion)
5. Do exercise Tag 0CT1 (this exercise is about depth which I didn't lecture about yet -- feel free to replace it by exercise 8 below about completion)
6. (this exercise is an alternative to exercise 3) Let A → B be a ring map. Let J ⊂ B be an ideal such that B is J-adically complete. Let b ∈ J. Show that there exists an A-algebra map A[[x]] → B mapping x to b. In what sense is the map you constructed unique?
7. (this exercise is an alternative to exercise 4) Let k be a field. Let R = k[[t]] endowed with the (t)-adic topology. Given an example of a topological R-module M whose topology is linear but not (t)-adic.
8. (this exercise is an alternative to exercise 5) Give an example of a local Noetherian domain (A, m) such that the completion of A with respect to its maximal ideal m is not a domain. (Hint: use the result from Exercise Tag 02DM.) Please explain carefully what you have to show in order to show that the completion is not a domain; maybe don't do all the calculations.
Due 12-9 in class.
1. Let A be a domain with fraction field K. Show that K is injective as an A-module.
2. Let k be a field. Let A be a k-algebra which is finite dimensional as a k-vector space. (Of course A is then an Artinian ring.) For an A-module M denote D(M) = Hom_k(M, k) viewed as an A-module by precomposition: for λ ∈ D(M) and a ∈ A the element aλ ∈ D(M) is the map x ↦ λ(ax).
  1. Show that D(-) is a contravariant functor on the category Mod_A of A-modules.
  2. Show that D(-) induces an anti-self-equivalence on the full subcategory Mod^f_A of finite A-modules.
  3. Why does D(-) not give an anti-self-equivalence of all of Mod_A?
  4. Deduce that ω_A/k = D(A) is an injective A-module.
3. Let k be a field. Let A = k[x]/(x^5). Show that A is an injective A-module.
4. Let k be a field. Let A = k[x, y]/(x^3, y^7). Show that A is an injective A-module. (Hint: use exercise 2 part 4.)
5. Let k be a field. Give an example of a finite dimensional k-algebra A such that A is not an injective A-module.

Stacks project: The chapter on commutative algebra.