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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 definethe notion of the (generalized) $b$-function following M. Kashiwara. Thesemodules are closely related to the localization and the local cohomology of$M$. We show that the $b$-function, if it exists, controls these modules andpresent general algorithms for computing these modules and the $b$-function (ifit exists) without any further assumptions. For these algorithms, wereformulate the localization algorithm for $D$-modules.

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