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.
Categories: math.AC