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

TL;DR

This paper establishes a dichotomy in the injective dimension of certain algebraic modules, showing it can only take two specific values related to the support dimension, for both $F$-finite $F$-modules and holonomic $D$-modules.

Contribution

It proves a new dichotomy result for the injective dimension of $F$-finite $F$-modules and holonomic $D$-modules, linking it directly to the support dimension.

Findings

Injective dimension is either support dimension minus one or support dimension.

The result applies to $F$-finite $F$-modules over regular rings of characteristic $p > 0$.

The result applies to holonomic $D$-modules over formal power series rings in characteristic zero.

Abstract

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