Bernstein's inequality and holonomicity for certain singular rings

TL;DR

This paper extends Bernstein's inequality and holonomic D-module theory to invariant rings and strongly F-regular graded algebras, showing these modules have finite length and are holonomic in these contexts.

Contribution

It introduces a generalized Bernstein filtration approach to establish holonomicity and finite length properties for D-modules over new classes of rings.

Findings

Holonomic D-modules have finite length in these settings.

Rings of invariants and strongly F-regular algebras are holonomic.

Bernstein inequalities hold for these rings.

Abstract

In this manuscript we prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic D-modules, in this context, have finite length. We obtain these results using a more general version of Bernstein filtrations.