In a seminal work Lyubeznik [1997] introduces a category
-finite modules
in order to show various finiteness results of local cohomology modules of a regular ring
in positive
characteristic. The key notion on which most of his arguments rely is that of a
generator of an
-finite module. This
may be viewed as an
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
-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
-finite modules and a
category of certain
-finitely
generated modules with a certain Frobenius operation which we call minimal
-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
-thresholds
of Blickle et al. [2008].
