Commutative Algebra Seminar

Title

Bernstein's inequality and sandwich Bernstein-Sato polynomials

Abstract

Rings of differential operators and their module theory have a number of important applications in Commutative Algebra. Much of the power of D-module theory stems from the fact that many “large” modules over polynomial rings exhibit striking finiteness properties as D-modules. The fountain of finiteness is a fundamental result called Bernstein’s inequality; this Bernstein inequality also readily implies the existence of the well-studied Bernstein-Sato polynomial.

Motivated by the study of Bernstein’s inequality on singular rings, we introduce a two-sided or “sandwich” analogue of the Bernstein-Sato polynomial, which has a closer connection to Bernstein’s inequality. We will discuss this new notion and its connections to simplicity of rings of differential operators and Bernstein’s inequality. If time permits, we will encounter some odd numerical F-invariants along the way.

This is based on joint work with David Lieberman (University of Nebraska).