On the injective dimension of F-finite modules and holonomic D-modules

TL;DR

This paper investigates the injective dimension of F-finite modules over regular local rings in characteristic p and holonomic D-modules in characteristic zero, establishing bounds relating dimension and injective dimension.

Contribution

It proves a lower bound for the injective dimension of F-finite modules and extends similar results to certain holonomic D-modules in characteristic zero.

Findings

Injective dimension of F-finite modules is at least dimension minus one.

Analogous bounds are established for holonomic D-modules in characteristic zero.

Results unify understanding of module dimensions across different characteristics.

Abstract

Let $R$ be a regular local ring containing a field $k$ of characteristic $p$ and $M$ be an $\mathscr{F}$-finite module. In this paper, we study the injective dimension of $M$. We prove that $\operatorname{dim}_R(M) -1 \leq\operatorname{inj.dim}_R(M)$. If $R = k[[x_1,\ldots,x_n]]$ where $k$ is a field of characteristic $0$ we prove the analogous result for a class of holonomic $\mathscr{D}$-modules which contains local cohomology modules.