$D$-modules, Bernstein-Sato polynomials and $F$-invariants of direct summands

TL;DR

This paper investigates the structure of D-modules over rings that are direct summands of polynomial or power series rings, establishing finiteness, existence of Bernstein-Sato polynomials, and relations between F-invariants and D-module properties.

Contribution

It demonstrates the finiteness of D-module length for localizations and local cohomology, proves the existence of Bernstein-Sato polynomials in this setting, and links F-jumping numbers with D-module invariants.

Findings

Localization and local cohomology have finite D-module length.

Bernstein-Sato polynomials exist for elements in R.

F-jumping numbers are rational and discrete, related to Bernstein-Sato polynomials.

Abstract

We study the structure of $D$-modules over a ring $R$ which is a direct summand of a polynomial or a power series ring $S$ with coefficients over a field. We relate properties of $D$-modules over $R$ to $D$-modules over $S$. We show that the localization $R_f$ and the local cohomology module $H^i_I(R)$ have finite length as $D$-modules over $R$. Furthermore, we show the existence of the Bernstein-Sato polynomial for elements in $R$. In positive characteristic, we use this relation between $D$-modules over $R$ and $S$ to show that the set of $F$-jumping numbers of an ideal $I\subseteq R$ is contained in the set of $F$-jumping numbers of its extension in $S$. As a consequence, the $F$-jumping numbers of $I$ in $R$ form a discrete set of rational numbers. We also relate the Bernstein-Sato polynomial in $R$ with the $F$-thresholds and the $F$-jumping numbers in $R$.