Localization, local cohomology, and the b-function of a D-module with respect to a polynomial

TL;DR

This paper explores the relationship between the b-function, localization, and local cohomology of D-modules generated by a single element, providing algorithms for their computation without additional assumptions.

Contribution

It introduces general algorithms to compute the b-function, localization, and local cohomology modules of D-modules, reformulating existing localization algorithms.

Findings

The b-function, when it exists, governs the structure of related modules.

Algorithms are provided for computing these modules and the b-function.

The approach does not require extra assumptions beyond the existence of the b-function.

Abstract

Given a $D$-module $M$ generated by a single element, and a polynomial $f$, one can construct several $D$-modules attached to $M$ and $f$ and can define the notion of the (generalized) $b$-function following M. Kashiwara. These modules are closely related to the localization and the local cohomology of $M$. We show that the $b$-function, if it exists, controls these modules and present general algorithms for computing these modules and the $b$-function (if it exists) without any further assumptions. For these algorithms, we reformulate the localization algorithm for $D$-modules.