Minimal $\gamma$--sheaves

TL;DR

This paper proves the existence of minimal roots for finitely generated unit $O_X[\sigma]$-modules on regular, $F$-finite schemes, with implications for tight closure theory and $F$-thresholds.

Contribution

It establishes the existence of minimal roots for these modules, extending Lyubeznik's results and providing new proofs for key properties in tight closure and $F$-module theory.

Findings

Parameter test modules commute with localization

Discreteness and rationality of $F$-thresholds proven in a new way

$D$-module generation results extended

Abstract

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].