Given a local ring of positive prime characteristic there is a natural Frobenius action
on its local cohomology modules with support at its maximal ideal. In this paper we
study the local rings for which the local cohomology modules have only finitely many
submodules invariant under the Frobenius action. In particular we prove that F-pure
Gorenstein local rings as well as the face ring of a finite simplicial complex localized
or completed at its homogeneous maximal ideal have this property. We also introduce
the notion of an antinilpotent Frobenius action on an Artinian module over a local
ring and use it to study those rings for which the lattice of submodules of the local
cohomology that are invariant under Frobenius satisfies the ascending chain
condition.
Keywords
local cohomology, Frobenius action, Frobenius functor,
F-pure ring, Gorenstein ring, antinilpotent module, tight
closure, face ring, FH-finite ring, finite FH-length