Generalized Lyubeznik numbers

Abstract

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers but captures finer information. These generalized Lyubeznik numbers are defined in terms of $D$-modules and are proved well defined using a generalization of the classical version of Kashiwara’s equivalence for smooth varieties; we also give a definition for finitely generated $K$-algebras. These new invariants are indicators of $F$-singularities in characteristic $p>0$ and have close connections with characteristic cycle multiplicities in characteristic zero. We characterize the generalized Lyubeznik numbers associated to monomial ideals and compute examples of those associated to determinantal ideals.