Minimal γ sheaves

In the seminal work [Lyu97] Lyubeznik 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 non-finitely 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 [Lyu97]. 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 γ-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 [HT04] which allows us to easily recover the rationality and discreteness results for F-thresholds of [BMS07].