Finiteness Properties of Local Cohomology for F-Pure Local Rings

In this paper, we show that for an F-pure local ring
$(R, \mathfrak m)$
, all local cohomology modules
$H^i_{\mathfrak m}(R)$
have finitely many Frobenius compatible submodules. This answers positively the open question raised by Enescu and Hochster [3] (see also [2], where it was stated as a conjecture when R is Cohen–Macaulay). We also prove that when
$(R, \mathfrak m)$
is excellent and F-pure on the punctured spectrum, all local cohomology modules
$H^i_{\mathfrak m}(R)$
have finite length in the category of R-modules with Frobenius action. Finally, we show that the property that all
$H^i_{\mathfrak m}(R)$
have finitely many Frobenius compatible submodules passes to localizations.