Commutative Algebra

This Fall semester (2019) I am teaching our graduate course on commutative algebra and algebraic geometry.

Tuesday and Thursday, 10:10 -- 11:25 AM in room 507 math.

The TA is Carl Lian.

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

Final Exam Thursday Dec 19 from 9AM - 12PM in Room 507. The questions will be about commutative algebra only (and not on the material about sheaves etc from the last 3 lectures).

Review of lectures and suggested readings

1. Lecture I: In this lecture we discussed Chevalley's theorem for constructible sets in Cn. We used the following definitions
1. An algebraic set is a subset of Cn which is the common zero set of a collection of polynomials.
2. A constructible set is a subset of Cn which is a finite union of sets of the form Z ∩ {f ≠ 0} where Z is an algebraic set.
With this terminology Chevalley's theorem says that the image of a constructible set under a polynomial map CnCr is constructible. I sketched how to prove this theorem, but we'll redo the proof in the setting of rings and spectra of rings later.
2. Lecture II: We discussed some notational conventions. We discussed the spectrum of a ring, its topology, and its functoriality. We gave a couple of examples. We discussed localization of a ring at a multiplicative subset. We discussed the equalities Spec(A/I) = V(I) and Spec(A_f) = D(f) as topological spaces.
3. Lecture III: We discussed Noetherian rings. We poved A[[x]] is Noetherian if A is Noetherian. We showed how Spec(A) is a Noetherian topological space if A is a Noetherian ring. We discussed decomposition into irreducible components. We discussed how Spec(A) is a sober topological space in the sense that any irreducible closed subset has a unique generic point. We proved a Noetherian topological space has a finite number of irreducible components.
4. Lecture IV: We defined a constructible subset of a Noetherian topological space. We stated and proved Chevalley's theorem in the setting of finite type homomorphisms of Noetherian rings. We stated some facts in field theory. Finally, we proved Hilbert Nullstellensatz (first version).
5. Lecture V: We first talked about applications of the Hilbert Nullstellensatz: closed points map to closed points for maps Spec(B) → Spec(A) if A → B is a map of finite type k-algebras and we discussed that closed points are dense in constructible subsets of Spec(A). This then implied the radical of an ideal I of A is the intersection of all maximal ideals of A containing I. We discussed trascendence degree of field extensions. We discussed Krull dimension of topological spaces. We gave some examples of weird behaviour of dimension. We discussed dimension functions. We stated our goal theorem: the transcendence degree over k of the residue field of a prime of A gives a dimension function on Spec(A).
6. Lecture VI: We restated the goal. We first proved some lemmas on Noetherian topological spaces and dimension functions (see [1] below):
1. If X is sober Noetherian, then every point specializes to a closed point.
2. If X is sober Noetherian and has a dimension function δ, then for every specialization x ⇝ y of X there is a sequence x = x_0 ⇝ x_1 ⇝ x_2 ⇝ ... ⇝ x_l = y of immediate specializations with l = δ(x) - δ(y).
3. If X is sober Noetherian with a dimension function δ and x ⇝ y is a specialization in X, then every maximal chain of specializations between x and y has the same length equal to δ(x) - δ(y) and consists of immediate specializations. (This follows immediately from the previous one.)
4. If X is sober Noetherian and has a dimension function δ such that δ(x) = 0 for all x in X closed, then dim(X) is equal to the maximal value of δ which is attained at a generic point of an irreducible component.
Let A be a finite type k-algebra where k is a field. Denote δ : Spec(A) → Z the function sending a prime ideal p to the transcendence degree of the residue field k(p) over k. We proved that for any closed point p of Spec(A) we have δ(p) = 0. Thus, if we can prove our goal, then the dimension of Spec(A) is going to equal the maximal value of the transcendence degree of a residue field (at a generic point of an irreducible component). We proved that δ(p) ≥ δ(q) if the prime p contains the prime q. We explained what we still have to prove to attain our goal. Finally, we stated Noether normalization lemma and we explained a very special case.
7. Lecture VII: We discussed and proved Noether normalization modulo two lemmas. Using Noether normalization we reformulated the goal once more. Modulo a third lemma we proved if A ⊂ B is a finite extension of domains, then any nonzero prime of B has a nonzero intersection with A. We proved that if p is a nonzero prime ideal in a UFD and there is no prime strictly between p and (0), then p is generated by a prime element. We proved that if p is a prime ideal in k[x_1, ..., x_n] which is nonzero and there is no prime strictly between p and (0), then trdeg_k kappa(p) = n - 1. To finish the proof of our goal we just need to prove going down for integral over normal. The three lemmas
1. A ring map R → S such that S is generated by a finite number of elements integral over R is finite.
2. A composition of finite ring maps is finite.
3. If R → S is a finite ring map, then every element of S is integral over R.
8. Lecture VIII We proved "going down for integral over normal". We gave an example of a finite extension of domains where going down fails. Then we started talking about tensor products.
9. Lecture IX We talked more about tensor products. Then we talked about flat modules, faithfully flat modules, flat ring maps, faithfully flat ring maps. A usful tool was the lemma: given a ring map A → B, a prime q of B lying over the prime p of A, an A-module M, and a B-module N, then we have
(M ⊗A N)q = MpAp Nq.
We used this, via Tag 00HN to show that you can check flatness of N by checking flatness of N_q over A_p for all q and p (and actually it suffices to check for q maximal). Then we finally used this to show
1. A flat ring map is faithfully flat if and only if it is surjective on spectra.
2. A flat local homomorphism of local rings is faithfully flat.
3. Going down for flat ring maps.
Next time we will deduce that flat finite type ring maps between Noetherian rings are open.
10. Lecture X We proved that flat finite type ring maps between Noetherian rings are open. We defined the completion of a module with respect to an ideal and we defined when a module is I-adically complete. We proved that the completion is complete if the ideal is finitely generated. We discussed (failure of) exactness of completion and we proved that if we have Artin-Rees, then we have exactness of completion on the category of finite modules.
11. Lecture XI We proved Artin-Rees. We proved some lemmas about complete rings. We proved the completion of a Noetherian ring is complete. We proved flatness of the completion of a Noetherian ring. We discussed the relationship between a Noetherian local ring and its completion. We stated (but did not prove) the Cohen structure theorem. We discussed its implications.
12. Lecture XII In this lecture we stated the Hilbert Syzygy Theorem. As a corollary we obtained that the Hilbert function of a finitely generated graded module over the polynomial ring is eventually a polynomial function. We introduced graded rings and graded modules. We gave an example of a module which does not have a bounded resolution by finite free module. We defined the Koszul complex in terms of a differential graded algebra. We proved a lemma stating that the Koszul complex on r elements f_1, ..., f_r is the cone of multiplication by f_r on the Koszul complex on f_1, ..., f_{r - 1}.
13. Lecture XIII We deduced from the lemma that the Koszul complex on x_1, ..., x_n is exact in positive degrees over the polynomial ring k[x_1, ..., x_n]. We proved that a finite module over a connected (A_0 = k) Noetherian graded k-algebra A has a minimal resolution by finite graded free modules. We proved that a finite module over a Noetherian local ring has a minimal resolution by finite free modules. We stated some properties of the Tor functors (existence/well definedness and symmetry) and we deduced the Hilbert Syzygy Theorem from these properties.
14. Lecture XIV We did some homological algebra. In particular we defined the left derived functors of a right exact functor F in the presence of a suitable collection P of objects such that
1. every object is a quotient of an element of P,
2. for any short exact sequence 0 → P → Q → R → 0
with Q, R in P we have P in P and 0 → F(P) → F(Q) → F(R) → 0 is exact. Then we applied this to the functor - ⊗R N for a fixed R-module N. Finally, we proved that Tor is symmetric. Thus we finished the proof of the Hilbert Syzygy Theorem.
15. Lecture XV We discussed dimension theory of Noetherian local rings and in particular the proof of 00KQ
16. Lecture XVI We defined the length of a module over a Noetherian local ring. We revisited the dimension theory of Noetherian local rings. We introduced regular local rings.
17. Lecture XVII We proved some lemmas on regular local rings. We proved regular local rings have finite global dimension. We proved that a localization of a ring of finite global dimension has finite global dimension. We proved that a minimal resolution of the residue field of a Noetherian local ring has length at least the dimension of the cotangent space.
18. Lecture XVIII We finished the proof that the localization of a regular local ring at a prime is a regular local ring. We did this by proving that if over a Noetherian local ring A we have a complex 0 → F_e → F_{e - 1} → ... → F_1 → F_0 consisting of finite free modules with maps which are zero modulo the maximal ideal which is exact in degrees e, e - 1, ..., 1, and with F_e not zero, then the dimension of A is at least e. This in turn was done by induction showing that in this case the maximal ideal of A has at least one nonzerodivisor and induction. The statement about the existence of a nonzerodivisor was done using a consideration of supports and associated primes.
19. Lecture XIX In this lecture we discussed the existence of enough injectives in the category of modules over a ring. Then we used this to define Ext functors using Hom(-, -) and injective resolutions in the second variable or projective resolutions in the first variable. Then we defined depth of a finite module over a Noetherian local ring in terms of regular sequences. At the end of the lecture we proved that depth(M) is the minimal i such that Ext^i(residue field, M) is nonzero.
20. Lecture XX In this lecture we first proved some lemmas about depth culminating in the following amazing result: the function d_M which to a prime ideal p associates d_M(p) = depth(M_p) + dim(A/p) has the following property: if p ⊃ q then d_M(q) ≥ d_M(p). We deduced that if for the maximal ideal m of A we have d_M(m) = dim(Supp(M)), then we get d_M(p) = dim(Supp(M)) for all p in the support of M. The condition d_M(m) = dim(Supp(M)) is the same as asking depth(M) = dim(Supp(M)) which is the same as asking M to be Cohen-Macaulay (CM) by definition. Thus if M is CM conclude
1. Ass(M) is the set of generic points of the closed set Supp(M)
2. dim(A/p) = dim(Supp(M)) for all generic points p of Supp(M)
3. the space Supp(M) is catenary
4. M_p is CM for all primes p
21. Lecture XXI In this lecture we discussed examples of CM and non-CM rings and modules. First we gave a precise definition of when a finite module M over a Noetherian ring R is CM and when a Noetherian ring R is CM. Next, we defined embedded associated primes of M and embedded primes of R. A CM module M does not have any embedded associated primes. This give a straightforward way to make non-CM things. A Noetherian domain of dimension 1 is CM. If k is a field then k[x, y]/(y^2) is CM. A (global) complete intersection over k is a k-algebra of the form R = k[x_1, ..., x_n]/(f_1, ..., f_c) satifying the condition that dim(R) = n - c. A complete intersection over k is CM. The quotient of a CM local ring by a regular sequence is CM. Examples k[x, y]/(xy) and k[x, y, z]/(xy) are CM. The ring k[x, y, u, v]/(xu, yu, xv, yv) is not CM. Interesting rings are often CM. At the end of the lecture I stated Serre's criterion for normality.
22. Lecture XXII Serre's criterion for normality. We first introduced the condition (S_k) for finite modules M over Noetherian rings and condition (R_k) for Noetherian rings R. Then we proved that M has (S_1) if and only if it has no embedded associated primes. Then we proved R is reduced if and only if it has (R_0) and (S_1). Next, given a Noetherian local ring (A, m , k) the following are equivalent
1. dim(A) is 1 and A is a normal domain
2. A is regular of dimension 1
3. A is a discrete valuation ring,
4. A is a valuation ring not a field
5. A is a domain and m is principal and not zero,
6. A is a PID and not a field
7. there is a t in m such that every nonzero element of A can be uniquely written as (unit) tn.
We defined a valuation ring. We proved the implication (1) implies (2) using an argument of Kollar. We stated Serre's criterion: R is normal if and only if R is (R_1) and (S_2). In the proof of Serre's criterion we observed that Kollar's argument gives that a Noetherian normal local ring of dimension ≥ 2 has depth ≥ 2. We computed some examples of normalizations to show the utility of Serre's criterion as a way to determine when you are done.
23. Lectures XXIII, XXIV, XXV We talk a little bit about sheaves on topological spaces, pushforward and pullback of sheaves, stalks of sheaves, sheafification, sheaves on bases, ringed spaces, morphisms of ringed spaces, locally ringed spaces, morphisms of ringed spaces, Spec(A) as a locally ringed space, and finally we defined schemes.
24. If you intend to continue with the next part of this course next semester, it would be great if you could read about sheaves and maybe do some exercises to familiarize yourself with that material.

Homework: (make sure you refresh the page)

1. Due Thursday, Sep 12 in lecture: All exercises from 0FJ3 and the following exercises 02CZ 02D0 02D1 02D2 from the section on the spectrum of a ring.
2. Due Thursday, Sep 19 in lecture: 02D4 02D8 02DD 02DF 0767 02DZ 0FKE
3. Due Thursday, Sep 26 in lecture: 0CVP 02DR 0E9F (affine variety means irreducible algebraic set) 0D5I
4. Due Thursday, Oct 3 in lecture: 0D5J 0D5K 0EER and carefully prove Lemma 2.7 of handout
5. Due Thursday, Oct 10 in lecture: 02CR 02CU 02CW 02DW
6. Due Thursday, Oct 17 in lecture: 02DM 078P 078R 078S
7. Due Thursday, Oct 24 in lecture: 0CYH 0CRC 0D8Y (hint: you may use that Ext functors are right derived functors of Hom; please recall that sometimes exercises are too hard and you can just say something to get partial credit)
8. Due Thursday, Oct 31 in lecture: 02LU 07DH 0D5M
9. Due Thursday, Nov 7 in lecture: 0CT2 0D1T 0D5L 0CS1
10. Due Thursday, Nov 14 in lecture: 0CR8 0CRA 02CL (the text says this should be easy, but the second part isn't).
11. Due Thursday, Nov 21 in lecture: 0CT4 0D1S 02DT
12. Due Thursday, Dec 5 in lecture: none.
Sometimes an exercise may be too hard. Then just skip it and start with the next one.

1. The Stacks project. Closely related material is listed here, but be aware that the material in the Stacks project is often more general than what we discussed in the lecture.
• Spectrum of a ring 00DY
• Localization 00CM
• Noetherian rings 00FM
• Noetherian topological spaces 0050
• Irreducible components 004U
• Chevalley's theorem 00FE
• Hilbert Nullstellensatz 00FS
• Transcendence degree of fields 030D
• Dimension of topological spaces 0054
• Dimension functions 02I8
• Finite ring maps 0562
• Integral ring maps 00GH
• Noether normalization 00OW
• Going down for integral over normal 037E
• Finite and integral ring maps handout
• Yoneda lemma 001L
• Limits and colimits 002D
• Abelian categories 00ZX
• Tensor products 00CV
• Checking zero locally 00EN
• Flat ring maps 000H9
• Mittag-Leffler and short exact sequences 03CA
• Completion 00M9
• Completion of Noetherian rings 0BNH
• Koszul complex 0621
• Regular sequences 0AUH
• Ext groups (for stuff about resolutions) 00LO
• Tor groups and flatness 00LY
• Dimension theory 00KD
• Jordan-Holder 0FCK
• Length 00IU
• Noetherian local rings 00K4
• Regular local rings 00NN
• Rings of finite global dimension 00O2
• Length of resolution of residue field 00OA
• Depth 00LE
• What makes a complex exact? 00MR
• Supports 080S
• Associated primes 00L9
• Serre's criterion 031O
• Valuation rings 00I8
• Local complete intersections 00S8
• Polynomial rings are regular 00OQ
2. The book "Algebraic Geometry" by Hartshorne (mostly for what I will say about schemes not so much about commutative algebra)
3. The book "Commutative Algebra" by Matsumura (I have the second edition published in 1980). People call this book the "old Matsumura" to distinguish from the next item
4. The book "Commutative ring theory" by Matsumura (translated by Miles Reid.
5. The book "Commutative algebra. With a view toward algebraic geometry" by David Eisenbud. I never read this myself, but I think this is a good choice to look at. It has a lot of stuff in it and it is a bit more wordy if you like that.
6. The book "Introduction to commutative algebra" by Atiyah and Macdonald. Many of those whose first introduction to commutative algebra comes from this book, swear by this book. I think this is a psychological effect (bc I swear by the old Matsumura).

[1] In the discussion in the lecture I defined a dimension function to be a function δ such that x ⇝ y ⇒ δ(x) ≥ δ(y) and if x ⇝ y is an immediate specialization, then δ(x) = δ(y) + 1. But in the Stacks project we define it to be a function such that x ⇝ y, x ≠ y ⇒ δ(x) > δ(y) and if x ⇝ y is an immediate specialization, then δ(x) = δ(y) + 1. This explains some of the difference between the discussion in the lecture and in the Stacks project. The lemmas we proved in the lecture show that the two notions are the same for sober Noetherian topological spaces. Apologies if this caused any confusion.