Generalized Lyubeznik numbers

TL;DR

This paper introduces generalized Lyubeznik numbers, a new family of invariants for local rings that extend classical Lyubeznik numbers by capturing finer algebraic information through D-module theory.

Contribution

The paper defines generalized Lyubeznik numbers using D-modules, develops the necessary functorial framework, and explores their properties and computations for specific classes of ideals.

Findings

Generalized Lyubeznik numbers include classical invariants and provide finer algebraic information.

They serve as indicators of F-regularity and F-rationality in positive characteristic.

Explicit computations are provided for monomial and determinantal ideals.

Abstract

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers, but that captures finer information. These "generalized Lyubeznik numbers" are defined as lengths of certain iterated local cohomology modules in a category of D-modules, and in order to define them, we develop the theory of a functor Lyubeznik utilized in proving that his original invariants are well defined. In particular, this functor gives an equivalence of categories with a category of D-modules. These new invariants are indicators of F-regularity and F-rationality in characteristic p>0, and have close connections with characteristic cycle multiplicities in characteristic zero. We compute the generalized Lyubeznik numbers associated to monomial ideals using interpretations as lengths in a category of straight modules, as well as provide examples of these invariants associated to certain determinantal ideals.