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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 theinjective 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 afield of characteristic $0$ we prove the analogous result for a class ofholonomic $\mathscr{D}$-modules which contains local cohomology modules.

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