Math 520: Commutative and Homological Algebra (fall 2018)

Course information/syllabus

Instructor: Nick Switala (nswitala [at] uic [dot] edu).

Meeting time and place: 9:00-9:50 MWF, Taft Hall 308.

Office hours: 10:00-11:00 MWF, SEO 630, or by appointment (or drop in).

Prerequisites: Math 516-517 at UIC, or an equivalent one-year graduate sequence in abstract algebra elsewhere. (You should be familiar with the material in Chapters III and VIII of Aluffi's Algebra: chapter 0, the standard text for 516-517: rings, ideals, polynomials, modules, ring and module homomorphisms, and chain complexes.)

Text: No required text; see "References" below.

Evaluation: There will be at most ten problem sets assigned during the semester (see "Problem sets" below) to be completed and submitted for grading. Your grade will be based entirely on these problem sets: there is no attendance or participation requirement, nor are there any exams. The standard rule applies: you may collaborate on homework, but you must write the solutions up in your own words and indicate with whom you have collaborated, as well as any resources you consulted.

Accommodation: If you wish to request accommodation due to a disability, please contact the UIC Disability Resource Center at 312-413-2183.

References

Commutative algebra:

Matsumura, Commutative ring theory (unofficial text for the course; if I made you buy only one book, it would be this one);
Eisenbud, Commutative algebra with a view toward algebraic geometry;
Atiyah-Macdonald, Introduction to commutative algebra;
Altman-Kleiman, A term of commutative algebra (a companion volume to, or updated version of, Atiyah-Macdonald-- it's free);
Kaplansky, Commutative rings (very old-fashioned, but with some beautiful proofs);
Pete Clark's notes on commutative algebra;
Andreas Gathmann's notes on commutative algebra;
Akhil Mathew's notes from Jacob Lurie's Harvard course on commutative algebra;
the commutative algebra chapter of the estimable Stacks Project.

Homological algebra:

Lang, Algebra (chapter XX of the third edition is still a great reference);
Weibel, An introduction to homological algebra;
Kashiwara-Schapira, Categories and sheaves (chapter 1 is a good summary of category-theory basics).

Problem sets

Please submit hard copies of your problem sets, either in class on the due date, or in my department mailbox before the beginning of class on the due date (not by email).

PS 1 on prime ideals, and solutions

PS 2 on Noetherian and Artinian rings and modules, and solutions

PS 3 on Nakayama's lemma and localization, and solutions

PS 4 on the spectrum of a ring and dimension theory, and solutions

PS 5 on support, minimal and associated primes, and integral extensions, and solutions

PS 6 on diagram chasing and flatness, and solutions

PS 7 on colimits, limits, and completions, and solutions

Material covered

Week 1

(M 8/27) Standard "first day" stuff. Review of basic definitions (zero-divisor, unit, nilpotent element; prime ideal, maximal ideal, radical ideal; field, integral domain, reduced ring). Oka families and the Lam-Reyes prime ideal principle. Application: the radical of an ideal is the intersection of the prime ideals containing it.

(W 8/29) Nilradical and Jacobson radical of a ring. Extension and contraction of ideals; behavior of properties (maximal, prime, radical) of extended and contracted ideals. A prime ideal contains a finite product (equivalently, intersection) of ideals if and only if it contains one of the factors. A "dual" statement: prime avoidance.

(F 8/31) Products and intersections of pairwise comaximal ideals. Equivalent definitions of Noetherian and Artinian rings and modules. Examples showing that for modules, all four possibilities for (Noetherian/not Noetherian, Artinian/not Artinian) are realized. Extended discussion of an Artinian but non-Noetherian module: Macaulay's inverse polynomials.

Week 2

(M 9/3) Labor Day: no class.

(W 9/5) Many facts about Noetherian and Artinian rings and modules. Some highlights: Noetherian rings have only finitely many minimal prime ideals (proof using the prime ideal principle); Artinian rings have only finitely many maximal ideals; all prime ideals in an Artinian ring are maximal. Simple modules.

(F 9/7) Finite-length modules. Proof of Akizuki's theorem (Artinian rings are Noetherian). Key lemma: the Jacobson radical in an Artinian ring is nilpotent.

Week 3

(M 9/10) Hilbert's basis theorem. Most general version of Nakayama's lemma: if R is a ring, I an ideal, and M a finitely generated R-module, then IM=M if and only if I and the annihilator of M are comaximal. Elementary proof avoiding the "determinant trick". Several standard corollaries of Nakayama's lemma (especially useful over local rings). Vasconcelos's corollary: every surjective endomorphism of a finitely generated module is an isomorphism.

(W 9/12) Localization of a ring at a multiplicative subset: construction and universal property. Behavior of ideals under localization. Examples.

(F 9/14) More on localization. Corrected statement of the correspondence on ideals. Localization of quotient rings. Localization of modules: definition and proof that it is an exact functor. A module's being zero, or a module homomorphism's being injective, surjective, or an isomorphism, is a "local property" (true if and only if true after localizing by any prime, or even just any maximal, ideal).

Week 4

(M 9/17) Final comments on localization and exactness. Crash course in the spectrum of a ring (a topological space whose points are the ring's prime ideals). Krull dimension.

(W 9/19) Height of a prime ideal. Nagata's example of a Noetherian ring of infinite dimension (that is, in which there exist prime ideals of arbitrarily large height).

(F 9/21) Krull's principal ideal theorem: in a Noetherian ring, any prime ideal minimal over a principal ideal has height at most one. Proof due to Rees.

Week 5

(M 9/24) Proof of Krull's height theorem (Hauptidealsatz), a generalization of the principal ideal theorem (a prime ideal in a Noetherian ring minimal over an n-generated ideal has height at most n). Outline of dimension theory for polynomial rings. Highlights: a polynomial ring in one variable cannot contain a chain of three distinct prime ideals that contract to the same prime in the coefficient ring; the height of a prime ideal in a Noetherian ring remains the same after adjoining a variable and extending.

(W 9/26) Associated primes of a module. They exist for every nonzero module over a Noetherian ring; if the module is finitely generated, there are finitely many of them. Support of a module.

(F 9/28) More on associated primes, support, and localization. The associated primes of a module all belong to its support, and over a Noetherian ring, the minimal elements of the two sets coincide. Brief overview of the relationship of associated primes with the classical theory of primary decomposition of ideals.

Week 6

(M 10/1) Integral extensions and integral closures. The "determinant trick", avoided no longer.

(W 10/3) The Cohen-Seidenberg theorems for integral extensions (lying over, incomparability, and going up).

(F 10/5) Some Galois theory and the going down theorem. (A correction to the end of the lecture will be given in the next class.)

Week 7

(M 10/8) The beginning of the homological algebra component. Flat and faithfully flat modules. The Hom and tensor product functors. Finitely presented modules.

(W 10/10) Careful proof that Hom with a finitely presented source commutes with tensor product up to a flat algebra. Auxiliary lessons: diagram chasing; proving a result first for a copy of the base ring, then for finite free modules, and finally using diagrams to get the general case; doing scratch work to make sure that the isomorphisms you need actually come from the map you're using.

(F 10/12) Flatness and faithful flatness, especially in the case of a ring homomorphism. Flatness is a local property. Fibers. The going down theorem for flat ring maps.

Week 8

(M 10/15) More on flatness and faithful flatness. The ideal criterion for flatness. Projective and injective modules.

(W 10/17) More on projective, injective, and flat modules. Brief outline of the theory of injective modules over Noetherian rings. Digression on the "equational criterion for flatness" (problem 3 on PS 6).

(F 10/19) Proofs that over a local ring, finitely generated flat modules are free, and over an arbitrary ring, a module is finitely generated and projective if and only if it is finitely presented and flat. Eilenberg's swindle. Statements of things we won't prove (the Quillen-Suslin theorem; Bass's theorem that "big projective modules are free").

Week 9

(M 10/22) Proof that countably generated projective modules over a local ring are free. Well-ordered sets and transfinite induction. Statements of three results in commutative algebra where transfinite induction/recursion is used: faithfully flat descent for projectivity (Raynaud-Gruson, Perry), gonflement of a local ring (Bourbaki), and Kaplansky's 1958 theorem that any projective module over a local ring is free.

(W 10/24) Proof of Kaplansky's theorem: projective modules over local rings are free.

(F 10/26) Colimits/limits of families of modules with filtered index sets. Proof that colimit is exact. Waterhouse's example of an inverse system of sets with surjective transition maps that has empty limit.

Week 10

(M 10/29) Cofinal subsets of a directed set. Example showing that limits need not be exact; proof that they are exact in a special case. Definition of adic completion of a ring or module.

(W 10/31) Adic topologies and Cauchy sequences. If a ring is I-adically complete, I is contained in its Jacobson radical. The Artin-Rees lemma.

(F 11/2) Proof of Artin-Rees. Proof of the Krull intersection theorem in many forms. Proof that if I is contained in J, ideals of a Noetherian ring, then J-adic completeness of the ring implies I-adic completeness (a slightly tricky exercise).

Week 11

(M 11/5) Discussion of when adic completion is exact, and when the adic completion of a module coincides with the tensor product of the module with the adic completion of the base ring. Study of ideals in adic completions of rings.

(W 11/7) Proofs of properties of the completion of a Noetherian local ring with respect to its maximal ideal. Proof that any adic completion of a Noetherian ring is Noetherian (it is a quotient of a power series ring in finitely many variables over the original ring). Example (without details) of a non-complete completion in the non-Noetherian case. Chevalley's lemma.

(Remark) As of W 11/7, we have covered essentially the first three chapters (9 sections) of Matsumura, which corresponds to pretty much all of Atiyah-Macdonald except for results of a specifically algebro-geometric (Noether normalization, Nullstellensatz) or number-theoretic (Dedekind domains) flavor. Therefore we are at the end of what I consider a standard first course in commutative algebra. In particular, there will be no further problem sets. The remaining 12 lectures will focus on regular local rings.

(F 11/9) Discussion of Krull's Hauptidealsatz and its converse. Systems of parameters in Noetherian local rings; proof that they exist. Regular local rings.

Week 12

(M 11/12) Proof that regular local rings are domains, and therefore a regular system of parameters is an example of a "regular sequence". General definition of regular sequences on modules, and proof that they cannot be rearranged in general, but can be if the module is finitely generated and the ring is Noetherian and local.

(W 11/14) More on the relationship between regular sequences and systems of parameters. Depth. Projective dimension and global dimension. Minimal free resolutions of finitely generated modules over Noetherian local rings.

(F 11/16) Koszul complexes. Some calculations in the case of sequences of length two.

Week 13

(M 11/19) Koszul homology detects regularity of a sequence. End of the proof of one direction of Serre's characterization: if a Noetherian local ring is regular, then it has finite global dimension (equal to its Krull dimension).

(W 11/21) Depth-sensitivity of Ext and of Koszul homology.

(F 11/23) Thanksgiving break: no class.

Week 14

(M 11/26) More on depth-sensitivity of Ext. Ischebeck's theorem. Cohen-Macaulay modules and rings.

(W 11/28) The Auslander-Buchsbaum formula. Outline of Auslander's proof (using transfinite induction) that the global dimension of a ring can be defined using either arbitrary modules or finitely generated modules only. Digression on local cohomology.

(F 11/30) End of the proof of Serre's characterization of regular local rings. (Now) easy corollary that the property of being regular is stable under localization at a prime ideal.

Week 15

(M 12/3) Sins of omission, part 1. Overview of the hierarchy of singularities in commutative algebra: complete intersection, Gorenstein, Cohen-Macaulay, and universally catenary local rings.

(W 12/5) Sins of omission, part 2. Survey of chapter 8 ("flatness revisited") of Matsumura: generic freeness, the local criterion for flatness, miracle flatness, and Serre conditions.

(F 12/7) Valediction: the Frobenius functor in positive-characteristic commutative algebra. Frobenius modules, Cartier modules, and F-modules (a.k.a. unit Frobenius modules). Sample result: Gabber's beautiful proof that the iterated images of a Cartier-linear map on a finitely generated module over a Noetherian ring must stabilize.