Cartier Crystals have finite global dimension

TL;DR

This paper proves that the category of quasi-coherent Cartier crystals on an F-finite noetherian ring has finite global dimension, extending known results for regular rings to a broader class.

Contribution

It establishes the finite global dimension of the category of quasi-coherent Cartier crystals, generalizing previous results for regular rings to F-finite noetherian rings.

Findings

Category of Cartier crystals is equivalent to unit Cartier modules.

Finite global dimension is achieved with a uniform bound on injective resolutions.

Generalizes Ma's result for regular rings to F-finite noetherian rings.

Abstract

We show that the category of quasi-coherent Cartier crystals is equivalent to the category of unit Cartier modules on an F-finite noetherian ring R, and that these equivalent categories have finite global dimension, by showing that every quasi-coherent Cartier crystal has a finite injective resolution. The length of the resolution is uniformly bounded by a bound only depending on R. Our result should be viewed as a generalization of a result of Ma showing that the category of unit R[F]-modules over a F-finite regular ring R has finite global dimension dim R + 1.