Math 507: Advanced commutative algebra
Fall quarter 2022
Lectures: Thursday 9:30 - 12:20 in PDL C-401
Instructor: Jarod Alper (firstname.lastname@example.org)
Office hours: Wednesday 3-5 pm in PDL C-549
Textbooks: There is no required textbook for the class. We recommend the following texts:
- Introduction to Commutative Algebra, M.F. Atiyah and I.G. Macdonald.
- Commutative Algebra with a View Toward Algebraic Geometry, David Eisenbud
- Commutative Ring Theory, Hideyuki Matsumura
- Ideals, Varieties, and Algorithms, David A. Cox, John Little, Donal O’Shea
- Cohen-Macaulay rings, Winfried Bruns and Jürgen Herzog.
- The stacks project
Syllabus: This course is an example-based inquiry into a selection of advanced themes in commutative algebra. We will also discuss what makes an effective math lecture, and sstudents will improve their ability to communicate algebraic ideas by practicing giving short expositions. The topics covered will depend on everyone's interests.
Possible lecture topics:
- Dimension theory: Krull dimension, artinian rings, PIDs, integral extensions, Krull's Hauptidealsatz, systems of parameters, Hilbert-Samuel polynomials ...
- Flatness : Definitions, examples, Tor, going-down theorem for flatness,equational criterion for flatness, local criterion for flatness, fibral flatness, graded modules and flatness, ...
- Completions: properties of completions of noetherian local rings, Hensel's Lemma, Cohen's Structure Theorem, ...
- Artin-Rees Lemma: statement and proof, examples, applications, Krull's Intersection Theorem
- Regular sequences: definition, equivalences, examples, Koszul complex, depth, Koszul cohomology, ...
- Regular local rings: definition, examples, minimal free resolultions, projective dimension, global dimension, Auslander-Buchsbaum formula that pd(M)+depth(M)=depth(R), Auslander-Buchsbaum Theorem that regular local rings are UFDs, Serre's Theorem (or the Auslander-Buchsbaum-Serre Theorem) characterizing the regularity of a local ring by the finiteness of the global dimension, ...
- Cohen-Macaulay rings: definition, examples, non-examples, regular sequences vs systems of parameters, Miracle Flatness, ...
- Complete intersections: relationship to regularity and regular sequences, relationship to Cohen-Macaulay rings,
- Normal rings: recall definition, examples, Serre's conditions S_i, characterization of normality in terms of R_1 and S_2, Algebraic Hartog's, ...
- Gorenstein rings: injective dimension, examples, equivalences of definitions, relationship to dualizing modules, ...
- Syzygies: graded resolutions, Hilbert's Syzygy Theorem, Betti tables, examples, ...
- The geometry of rings: viewing a ring geometrically via its spectrum of prime ideals, overview of the correspondence between algebra and geometry,...
- Gröbner basis: monomial orderings, leading terms, definition and examples, Buchberger's algorithm, ideal membership problem, Effective Nullstellensatz, algorithms in commutative algebra...
- Other topics: Castelnuovo--Mumford regularity, finiteness of integral closure, elimination theory & generic freeness, valuation theory, étale and smooth ring maps, fppf descent, ...
- Week 2 (Oct 6): Henselian local rings and henselization (Ting)
- Week 3 (Oct 13): Completions (Soham and Nelson)
- Week 4 (Oct 20): Serre-Swan theorem (Haoming and Andrew)
- Week 5 (Oct 27): Ext and Tor (Jackson and Cameron)
- Week 6 (Nov 3): Dimension (Raymond and Bashir)
- Week 7 (Nov 10): Dedekind domains (Alex and Will)
- Week 8 (Nov 17): Normal rings and their characterization (Haoming), Local cohomology (Justin)
- Week 9 (Dec 1): homotopy equivalence and derived categories (Alex W), regular local rings are UFDs (Brian)
Expectations: This is not a class to sit back and nod your head. Class participation is required. You are expected to choose topics of your interest, learn these topics extremely well, and do your best to present the material effectively to your classmates. The classroom will be a welcome and informal environment where we learn from our mistakes. The expectation is that you give either a 1 hour lecture or two 30 minute lectures, but we will be flexible.
You will also be required to submit weekly "Reflections on three things." The idea is that during the previous week's lectures, you record three things (e.g definitions, theorems, themes, examples, questions) that you don't know, are confused about, or simply would like to know more about. You then look up these three things in whatever sources you find, you read about them, and then play around with them until you've improved your understanding. You should summarize what you've learned in a 1/2-1 page document. This should be submitted in class. For more background on "Three things," see Ravi's Vakil's description.
Advice on choosing topics: Anything goes!
You can choose a subtopic from the above list, e.g. Hilbert-Samuel polynomials and its relation to dimension. (You won't be able to cover everything listed in a given bullet point.) You can choose a topic that you maybe have learned in an earlier algebra course but would like to learn in more depth. You can try to summarize a research article related to commutative algebra, or you can try to give a big picture view of say applications of commutative algebra to representation theory (or algebraic topology, number theory, ...). You can also pair up with other students if you want to coordinate your lectures.
It can be challenging to develop a big picture of what commutative algebra is all about, and what type of research problems the commutative algebra community is interested in. My advice is to attend seminars and talk to others. You should also read survey articles, introductions to books, introductions to research papers, mathoverflow posts, and whatever else you find. One great resource is www.commalg.org which contains wonderful lists of survey articles and accessible papers.
Sources for improving your communication skills
For giving effective math talks, see
Improving your writing skills will also improve your speaking skills:
- How to Write Mathematics, Paul Halmos
- The Elements of Style, Strunk and White
- The Sense of Style: The Thinking Person's Guide to Writing in the 21st Century, Steven Pinker