Seminars in Commutative Algebra

Marc Stephan: An equivariant BGG correspondence and applications to free A₄-actions

13th June, 2024

A classical question in the theory of transformation groups asks which finite groups can act freely on a product of spheres. For instance, Oliver showed that the alternating group A₄ can not act freely on any product of two equidimensional spheres.

I will report on joint projects with Henrik Rüping and Ergün Yalcin and explain that for 'most' dimensions m and n, there is no free A₄-action on S^m × Sⁿ and whenever there exists such a free action, then the corresponding cochain complex with mod 2 coefficients is rigid: its equivariant homotopy type only depends on m and n.

This involves an equivariant extension of Carlsson’s BGG correspondence in order to classify perfect complexes over 𝔽₂[A₄] with 4-dimensional total homology.

Benjamin Briggs: Koszul homomorphisms and resolutions in commutative algebra

22nd February, 2024

This is a talk about the situation in commutative algebra. A homomorphism f: S → R of commutative local rings has a derived fibre F (a differential graded algebra over the residue field k of R) and we say that f is Koszul if F is formal and its homology H(F) = Tor^S(R,k) is a Koszul algebra in the classical sense. I'll explain why this is a very good definition and how it is satisfied by many many examples.

The main application is the construction of explicit free resolutions over R in the presence of a Koszul homomorphism. These tell you about the asymptotic homological algebra of R, and so the structure of the derived category of R. This construction simultaneously generalizes the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring.

Linquan Ma: Test Ideals in Mixed Characteristic via the p-adic Riemann-Hilbert Correspondence

18th October, 2023

Multiplier ideals in characteristic zero and test ideals in positive characteristic are fundamental objects in the study of commutative algebra and birational geometry in equal characteristic. We introduced a mixed characteristic version of the multiplier/test ideal using the p-adic Riemann-Hilbert correspondence of Bhatt-Lurie. Under mild finiteness assumptions, we show that this version of test ideal commutes with localization and can be computed by a single alteration up to small perturbation.

Karthik Ganapathy: Equivariant commutative algebra in positive characteristic

25th September, 2023

In the presence of a large group action, even non-noetherian rings sometimes behave like noetherian rings. For example, Cohen proved that every symmetric ideal in the infinite variable polynomial ring is generated by finitely many polynomials (and their orbits under the infinite symmetric group). In this talk, I will give a brief introduction to equivariant commutative algebra where we systematically study such noetherian phenomena in infinite variable polynomial rings, and explain my work over fields of positive characteristic.

Amnon Yekutieli: A DG Approach to the Cotangent Complex

23rd May, 2023

Let B/A be a pair of commutative rings. We propose a DG (differential graded) approach to the cotangent complex L_B/A. Using a commutative semi-free DG ring resolution of B relative to A, we construct a complex of B-modules LCot_B/A. This construction works more generally for a pair B/A of commutative DG rings. In the talk, we will explain all these concepts. Then we will discuss the important properties of the DG B-module LCot_B/A. If time permits, we'll outline some of the proofs. It is conjectured that for a pair of rings B/A, our LCot_B/A coincides with the usual cotangent complex L_B/A, which is constructed by simplicial methods. We shall also relate LCot_B/A to modern homotopical versions of the cotangent complex.

Patrick Lank: High Frobenius pushforwards generate the bounded derived category

9th May, 2023

This talk is concerned with generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic p when the Frobenius morphism is finite. It is shown that for any compact generator G of D(X), the e-th Frobenius pushforward of G classically generates the bounded derived category whenever p^e is larger than the codepth of X, an invariant that is a measure of the singularity of X. From this, we can establish a canonical choice of strong generator when X is separated. The work is joint with Matthew R. Ballard, Srikanth B. Iyengar, Alapan Mukhopadhyay, and Josh Pollitz.

Luca Pol: Finite covers and tt-rings

8th March, 2022

Balmer initiated the study of separable commutative algebras (tt-rings in short) in tt-geometry: these are commutative algebras for which the multiplication map admits a bimodule section. Their importance has grown in recent years due to the fact that the category of modules over a tt-ring is again a tt-category, and that tt-rings allow to prove strong descent results. However, the classification of all tt-rings in a tt-category is an open problem in many cases of interest. In this talk, I will relate the notion of tt-ring to the notion of finite cover due to Mathew, and use this connection to provide classification results for tt-rings in some special cases of interest.

Tsutomu Nakamura: Flat cotorsion modules over Noether algebras and derived categories

11th November, 2021

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. We then explain several important roles of complexes of flat cotorsion modules and give some applications.

Aron Simis: Some conjectures in commutative algebra

28th October, 2021

There are "big" conjectures and not-so-big ones in the field. Some of the first have either been solved (often by unexpected tools) or are still pending like a fruit on the top of a tree with delicate branches, making it often hard for a layperson like some of us. This talk is about more modest conjectures, at anyone's reach and pending from trees with more stable branches. Some of these may have some interest in algebraic geometry.

Yves André: On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)

29th April, 2021

Up to a finite covering, a sequence of nested subvarieties of an affine algebraic variety just looks like a flag of vector spaces (Noether); understanding this "up to" is a primary motivation for a fine study of finite coverings.

The aim of this talk is to give a bird's-eye view of some fundamental questions about them, which took root in Algebraic Geometry (descent problems etc.), then motivated major trends in Commutative Algebra (F-singularities etc.), and recently found complete solutions using p-adic methods (perfectoids). Rather than going into detail of the latter, the emphasis will be on synthesizing, from the geometric viewpoint, a rather scattered theme.

This is based on joint work with Luisa Fiorot.