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In this paper, we show that for an  F -pure local ring, all local cohomology moduleshave 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 whenis excellent and  F -pure on the punctured spectrum, all local cohomology moduleshave finite length in the category of  R -modules with Frobenius action. Finally, we show that the property that allhave finitely many Frobenius compatible submodules passes to localizations.

Finiteness Properties of Local Cohomology for F -Pure Local Rings