Minimal \(\gamma\)--sheaves

In this note we show that finitely generated unit \(O_X[\sigma]\)--modules for \(X\) regular and \(F\)--finite have a minimal root (in the sense of [Lyubeznik, F-modules] Definition~3.6). This problem was posed by Lyubeznik and answered by himself in the case that \(X\)=Spec \(R\) is a complete local ring.
One immediate consequence of this result is that the parameter test module of tight closure theory commutes with localization. As a further application of the methods in this paper we give new proofs of the results on discreteness and rationality of \(F\)--thresholds [arXiv:0705.1210] and on \(D\)-module generation [arXiv:math/0502405v1]. The new proofs are valid in a slightly more general setting such that they also party cover the generalizations recently obtained in [arXiv:0706.3028].