Commutative Algebra Seminar

This is a report on joint work with Mark Walker. Classical algebraic Chern-Weil theory provides a formula for the Chern character of a projective module P over a commutative ring in terms of a connection on P. In this talk, I will discuss an analogous formula for the Chern character of a matrix factorization. Along the way, I will provide background on matrix factorizations, and also on classical Chern-Weil theory.

L-S category is a numerical invariant originating in differential geometry, which has since found an important place in rational homotopy theory. In the early 80s commutative algebraists and rational homotopy theorists began to uncover deep parallels between their disciplines, greatly enriching both sides. Constructions and theorems on one side were found to have "looking-glass" analogues on the other. For example, L-S category can be defined for local rings or homomorphisms, and this invariant has been used to prove important results in local commutative algebra. Nonetheless, L-S category has not been well studied in commutative algebra in the years since. I will talk about an attempt to develop some of the properties of this invariant. In particular, I'll present a looking glass analogue of the mapping theorem, which is invaluable for computing L-S category in rational homotopy theory. I will also be very careful to explain any topology I might want to use.

We explore the behavior of prime characteristic invariants, such as F-signature and Hilbert-Kunz multiplicity, under birational morphisms. We will discuss both positive and negative behavior.

Let R be a standard graded polynomial ring that is finitely generated over a field, and let I be a homogenous prime ideal. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/I^t, as t goes to infinity, which led to the development of an asymptotic invariant by Dao and Montanõ. I will discuss their results, and give concrete examples of the calculation of this new invariant in the case of determinantal rings.

A fundamental theorem of Rickard states for a pair of coherent rings that the categories of perfect complexes are triangle equivalent if and only if their bounded derived categories of finitely presented modules are triangle equivalent. The proofs of both directions are somewhat delicate and not treated properly in any text book. The aim of this talk is to propose new proofs, which should be of independent interest; they involve the notion of Cauchy completion for one direction, and the notion of a distinguished t-structure for the other direction.

We define the test ideal of a general closure operation cl, and give some of its properties.  We highlightconnections to the trace ideal and interior operations, and the applications of these viewpoints to the study of singularities of commutative rings. In all characteristics, test ideals coming from big Cohen-Macaulay modules or algebras can take on the role of the tight closure test ideal used in characteristic $p>0$ to study singularities.

This talk concerns the finite generation conjecture for finite tensor categories. One is free to think only of representation categories of finite-dimensional Hopf algebras here. The conjecture proposes that for any finite tensor category C, the self-extension algebra of the unit is a finitely generated algebra, and for any object V in C the extensions from the unit to V provide a finitely generated module over this algebra. This conjecture was proved for finite groups (in finite characteristic) by Golod, and Evans and Venkov, in which case C is rep(G), and for finite group schemes by Friedlander and Suslin. I will discuss how this finite generation property for cohomology behaves under certain ``fundamental operations" for tensor categories. I will spend much of the time discussing what these fundamental operations actually are, their general significance, and the specific examples of quantum groups and algebras of functions on group schemes. (For functions on a group scheme, we essentially want to understand how cohomology behaves under Hopf deformation.) This is joint work with Eric Friedlander and Julia Plavnik.

In the 1970s Kunz proved that a Noetherian ring of positive characteristic is regular if and only if the Frobenius is flat. I will discuss various extensions of this theorem. Then I will give a version of Kunz's theorem in mixed characteristic (joint with Bhatt and Iyengar), and a version in characteristic zero (joint with Schwede).

Two algebraic varieties are said to be linked if their union is a particularly nice kind of variety, a complete intersection. Thus, linked varieties are in some sense complementary: one carries an imprint of the other. Linkage has an algebraic incarnation that is used to study ideals. In the talk I will explain how the idea that linked ideals are ``complementary'' can be used to unwind the detailed structure of a classification of local rings of codimension 3. What that structure is only became clear to us---Oana Veliche, Jerzy Weyman, and myself---after extensive experimentation, and justifying it rigorously felt, at times, like solving a solitaire.

This is joint work with Irena Swanson found on arXiv:1806.03545. Given a polynomial ring C over a field and proper ideals I and J whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of I+J into a collection of primes described in terms of the associated primes of select powers of I and of J. We discuss applications to constructing primary decompositions for powers of I+J, and to attacking the persistence problem for associated primes of powers of an ideal.