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VOL. 77 | 2018 Localization, local cohomology, and the $b$-function of a $D$-module with respect to a polynomial

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. We also give some examples of multiplicity computation of such $D$-modules including a possibly well-known explicit formula for the localization of the polynomial ring by a hyperplane arrangement.

Information

Published: 1 January 2018
First available in Project Euclid: 21 September 2018

zbMATH: 07034257
MathSciNet: MR3839714

Digital Object Identifier: 10.2969/aspm/07710353

Subjects:
Primary: 13D45, 13P10, 14B15, 14F10, 52C35

Rights: Copyright © 2018 Mathematical Society of Japan

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