  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].
