Minimal $\gamma$-sheaves

Abstract

In a seminal work Lyubeznik [1997] introduces a category $F$-finite modules in order to show various finiteness results of local cohomology modules of a regular ring $R$ in positive characteristic. The key notion on which most of his arguments rely is that of a generator of an $F$-finite module. This may be viewed as an $R$ finitely generated representative for the generally nonfinitely generated local cohomology modules. In this paper we show that there is a functorial way to choose such an $R$-finitely generated representative, called the minimal root, thereby answering a question that was left open in Lyubeznik’s work. Indeed, we give an equivalence of categories between $F$-finite modules and a category of certain $R$-finitely generated modules with a certain Frobenius operation which we call minimal $\gamma$-sheaves.

As immediate applications we obtain a globalization result for the parameter test module of tight closure theory and a new interpretation of the generalized test ideals of Hara and Takagi [2004] which allows us to easily recover the rationality and discreteness results for $F$-thresholds of Blickle et al. [2008].