Commutative Algebra Seminar

If R is a local ring of dimension d and x = x_1, ..., x_d is a maximal regular sequence, then R/(x) is an R-module of finite length and finite projective dimension. There exist at least three open questions which stipulate that such quotients of regular sequences are the "simplest" or "smallest" modules amongst all modules of finite length and finite projective dimension. I will talk about some evidence supporting this philosophy. I will also sketch a proof from a joint work with Josh Pollitz where we solve the Loewy length version of this problem over strict Cohen-Macaulay rings.

I discuss joint work with Briggs, Grifo, and Pollitz, in which we investigate a question due to Avramov: Does every central element of degree two in the homotopy Lie algebra of a local ring come from an embedded deformation? I'll define the terms appearing in this question and answer it. (Spoiler alert: No)

We will survey some recent results about a class of rings called *perfectoid Tate algebras*. These rings do not satisfy Hilbert's nullstellensatz, as shown by Gleason. We suggest an alternative, based on the theory of Berkovich spaces. We conclude by showing our statement in the case of two variables.

The arithmetic rank of a variety is the minimal number of equations needed to define it set-theoretically, i.e., the smallest number of polynomials generating the defining ideal upto radical. Computing this invariant is notoriously difficult: the minimal generators up to radical often bear little relation to the given ideal generators and can vary unpredictably across characteristics.

Residual intersections provide a natural extension of the classical notion of algebraic links. We establish a general upper bound for the arithmetic rank of any residual intersection of a complete intersection ideal in an arbitrary Noetherian ring, and we show that this bound is sharp under specific characteristic assumptions. This work is joint with Kesavan Mohana Sundaram, Taylor Murray, and Vaibhav Pandey.