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

TL;DR

This paper establishes that the injective dimension of certain algebraic modules in positive and zero characteristic equals the dimension of their support, linking module theory with geometric properties.

Contribution

It proves that for F-finite F-modules and holonomic D-modules, the injective dimension matches the support dimension, extending known results across characteristics.

Findings

Injective dimension equals support dimension for F-finite F-modules in characteristic p.

Injective dimension equals support dimension for holonomic D-modules in characteristic 0.

Results unify understanding of module injective dimensions across different algebraic contexts.

Abstract

We investigate injective dimension of $F$-finite $F$-modules in characteristic $p$ and holonomic $D$-modules in characteristic 0. One of our main results is the following. If, either $R$ is a regular ring of finite type over an infinite field of characteristic $p>0$ and $M$ is an $F_R$-finite $F_R$-module, or $R=k[x_1,\dots,x_n]$ where $k$ is a field of characteristic 0 and $M$ is a holonomic $D(R,k)$-module, then the injective dimension of $M$ is the same as the dimension of its support.