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

摘要

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.