Math 509: Advanced commutative algebra and scheme theory
Spring quarter 2025
Lectures: Mon, Wed 9:00 - 10:20 in PDL C-038
Instructor: Jarod Alper (jarod@uw.edu)
Office hours: Mon/Wed 3-4 pm in PDL C-549
Schedule:
- Weeks 1-2: All things regular: regular sequences, regular local rings, Koszul cohohomology, depth,
projective dimension,
Auslander-Buchsbaum-Serre's theorem that a noetherian local ring is regular if and only if it has finite
projective dimension.
Student presentations: schedule
Syllabus: We will cover advanced topics in commutative algebra and algebraic
geometry that are not often seen in a first-year course. The specific topics that we cover will depend on student interest. The focus of this course may deviate from
other traditional courses in the following two ways:
- examples, examples, examples! We wil explore each topic through an example-based inquiry. We may cover less material overall, but for each topic we do cover, we will get a chance to enrich our understanding by working playing around with explicit mathematical objects.
- exposition by students: each student will be responsible for choosing
a topic of their choice and present the material to their peers in a 30 minute talk. We will have discussions on what makes a good math talk versus a bad math talk. These short talks give students the opportunity their communication and speaking skills.
Possible topics:
- Flatness: Definitions, examples, Tor, going-down theorem for flatness,equational criterion for flatness, local criterion for flatness, fibral flatness, graded modules and flatness, ...
- Regular sequences: definition, equivalences, examples, Koszul complex, depth, Koszul cohomology, projective dimension, Auslander-Buchsbaum Theorem
- Algebraic groups : Group schemes, algebraic groups, affine algebraic groups, actions of algebraic groups, representations, diagonalizable groups, reductive groups.
- Geometric Invariant Theory (GIT) : Quotients of algebraic varieties by group actions, Hilbert-Mumford criterion, good vs. geometric quotients, HKKN stratification of the unstable locus.
- Toric varieties : cones and affine toric varieties, fans and toric varieties, properties of toric varieties in terms of the combinatorics of the fan.
- Syzygies: graded resolutions, Hilbert's Syzygy Theorem, Betti tables, examples, ...
- Deformation theory: embedded deformations of closed subschemes, deformations of schemes, deformations of coherent sheaves, infinitisemal deformations, obstructions, Grothendieck's Existence Theorem, Artin Algebraization, Artin's Axioms.
- Birational geometry: ampleness, bigness, nefness, base point freeness, semiampleness, ampleness criterion.
- Formalization of commutative algebra and algebraic geometry: How do you define and work with rings and varieties in Lean?
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.
Expectation 1: You present at least one 30 minute lecture on a topic of your interest.
Expectation 1: You submit weekly reflections on three things. These should be submitted on Mondays by 9:30 am on Canvas. It should be somewhere between 1/2 and 2 pages in length.
- Two research things: The idea is that from either the lectures of this course or from seminars that you attended over the last week, you record two algebro-geometric things (e.g definitions, theorems, examples, questions) that you don't know, are confused about, or simply would like to know more about. You then look up these two things in whatever sources you find, you read about them, and then play around with them until you've improved your understanding.
- One teaching thing: From the previous week's lectures (or from other courses or seminars), think of one speaking technique that you found particularly effective. Please record the speaking technique, why you find it effective, and how you plan to incorporate it into your own talks.
For more background on "Three things," see Ravi's Vakil's description.
Advice on choosing presentationtopics: Anything goes!
You can choose a subtopic from the above list, a topic you would like to learn about, ora topic that you've already been exposed to but would like to learn in more depth. You can try to summarize a research article (either a classic or a new paper), follow a textbook exposition of an article, present a big picture overview connecting disparate themes, 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 and algebraic geometry is all about, and what type of research problems the 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.
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