Commutative Algebra Seminar

A finitely generated graded module over a polynomial ring can have the property that it is generated in a single degree. Further, it may happen that all of its relations have the same degree, and so on with all subsequent relations (syzygies). Such modules are called pure, and remarkably, the numeric data of the minimal free resolution of a pure module is known precisely, up to a scalar multiple. Motivated by some classical questions concerning syzygies, I will discuss how these lead naturally to systems of diophantine equations and will share some interesting observations. This is joint work with two undergraduate students, Noah Huang and Harrison Wolf.

The well-known trace map on matrices can be generalized to a map on any module over a commutative ring. The image of this map is a trace ideal. The modern theory of trace ideals has found far-reaching applications, for instance, in calculating centers of endomorphism rings, the classification of rings, and in approaching conjectures concerning Ext-rigidity. This talk will serve as a gentle introduction to trace ideals with some applications.

Let X be a projective variety, and let L be an ample line bundle on X. For a point x in X, the Seshadri constant of L at x is defined as the infimum, taken over all curves C passing through x, of the ratios \frac{L.C}{m}, where L.C denotes the intersection product of L and C, and m is the multiplicity of C at x. This concept was introduced by J.-P. Demailly in 1990, inspired by Seshadri’s ampleness criterion. Seshadri constants provide insights into both the local behavior of L at x and certain global properties of X.

The notion of Seshadri constants has been extended in various directions. Two important generalizations include replacing L with a vector bundle of arbitrary rank and replacing the point x with a finite collection of points. We will give an overview of the current research in this area and discuss some recent results.

This is joint work with Srikanth Iyengar. The derived category of a commutative local noetherian ring and the module category of a modular group algebra are tensor triangulated categories. A dualizable object in such a category is one that has a dual that is compatible with the tensor structure. The question that we address in this paper is whether the subcategory of dualizable objects in certain colocal subcategories is the idempotent closure of image of the compact objects under the local cohomology functor associated to the subcategory. In this lecture, I will try to explain what all of these words mean, why one might care about such a question and how we get a negative answer is certain cases.

Two years ago I spoke on work with Hailong Dao on the summands in syzygies of modules over artinian Burch rings. I'll review that story, and explain what we now know and conjecture about the direct sum decompositions of syzygies more generally. This is work -- very much "in progress" -- with Dao, Polini and Ulrich; and, for part of it, Cuong and Kobayashi.

There is a longstanding question about the perfectoidness of Shimura varieties at infinite level, which would lead to a profound connection between étale cohomology and coherent cohomology. In this talk, we will develop a p-adic Hodge theory for general valuation rings, generalizing Tate's work on discrete valuation rings. This enables us to establish “stalkwise perfectoidness” for Shimura varieties at infinite level, which is sufficient for such a connection. As an application, we prove that the integral completed cohomology groups vanish in higher degrees, verifying a conjecture of Calegari-Emerton for arbitrary Shimura varieties. This is my recent work arXiv.2407.14488.