On the injective dimension of $\mathscr{F}$-finite modules and holonomic $\mathscr{D}$-modules

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.