On the injective dimension of unit Cartier and Frobenius modules

TL;DR

This paper establishes a uniform upper bound of the injective dimension for unit Cartier and Frobenius modules over noetherian F-finite rings of prime characteristic, linking it to the support dimension.

Contribution

It proves that the injective dimension of unit Frobenius and Cartier modules is at most the support dimension plus one, providing a new bound in prime characteristic algebra.

Findings

Injective dimension of unit Frobenius modules is at most support dimension plus one.

The same bound applies to unit Cartier modules over noetherian F-finite rings.

Provides a uniform bound of dim A + 1 for injective dimension over such rings.

Abstract

Let $R$ be a regular $F$-finite ring of prime characteristic $p$. We prove that the injective dimension of every unit Frobenius module $M$ in the category of unit Frobenius modules is at most $\operatorname{dim}(\operatorname{Supp}_R(M))+1$. We further show that for unit Cartier modules the same bound holds over any noetherian $F$-finite ring $A$ of prime characteristic $p$. This shows that $\dim A+1$ is a uniform upper bound for the injective dimension of any unit Cartier module over a noetherian $F$-finite ring $A$.