Cartier Crystals Have Finite Global Dimension

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 the main result of Ma [Ma14] showing that the category of unit $\mathit{R}[\mathit{F}]$-modules over an F-finite regular ring R has finite global dimension $dim\mathit{R}+1$.