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Let $M$ be either an $F$-finite $F$-module over a noetherian regular ring ofcharacteristic $p > 0$ or a holonomic $D$-module over a formal power seriesring over a field of characteristic zero. We prove that $\injdim_R M$ enjoys adichotomy property: it has only two possible values, $\dim \Supp_R M - 1$ or$\dim \Supp_R M$.

A dichotomy for the injective dimension of $F$-finite $F$-modules and holonomic $D$-modules