Lyubeznik numbers, $F$-modules and modules of generalized fractions

TL;DR

This paper introduces an algorithm to compute Lyubeznik numbers of local rings in prime characteristic using $F$-modules and modules of generalized fractions, with implementation in Macaulay2.

Contribution

It develops a novel algorithm leveraging $F$-module structures and modules of generalized fractions to calculate Lyubeznik numbers, implemented in Macaulay2.

Findings

Algorithm successfully computes Lyubeznik numbers for certain local rings.

Modules of generalized fractions can be endowed with $F$-module structures.

The method connects local cohomology, $F$-modules, and generalized fractions for computational purposes.

Abstract

This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring $R$ of prime characteristic. The methods used employ Lyubeznik's $F$-modules over $R$, particularly his $F$-finite $F$-modules, and also the modules of generalized fractions of Sharp and Zakeri. It is shown that many modules of generalized fractions over $R$ have natural structures as $F$-modules; these lead to $F$-module structures on certain local cohomology modules over $R$, which are exploited, in conjunction with $F$-module structures on injective $R$-modules that result from work of Huneke and Sharp, to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2.