The category of F-modules has finite global dimension

TL;DR

This paper proves that under certain conditions, the category of F-modules over a regular ring of characteristic p has finite global dimension, providing new insights into their homological properties.

Contribution

It establishes finite global dimension for the category of F-modules over regular rings under mild conditions, extending Hochster's results.

Findings

The category of F-modules has finite global dimension d+1.

Examples show Ext^1 groups can be infinite for F-finite F-modules.

The results apply when R is essentially of finite type over an F-finite regular local ring.

Abstract

Let R be a regular ring of characteristic p. Hochster showed that the category of Lyubeznik's F-modules has enough injectives, so that every F-module has an injective resolution in this category. We show that under mild conditions on R, for example when R is essentially of finite type over an F-finite regular local ring, the category of F-modules has finite global dimension d+1 where d=dimR. We also give examples to show that for F-finite F-modules, $\Ext_{F_R}^1(M,N)$ need not be finite.