A finiteness condition on local cohomology in positive characteristic (Q659901)

Let \((R, m, k)\) be a local ring of positive characteristic \(p\) and dimension \(d\). For an \(R\)-module \(M\) we say that \(F: M \to M\) is a Frobenius action on \(M\) if \(F\) is additive and \(F(rm)=r^pF(m)\) for all \(r \in R\) and \(m \in M\). An \(R\)-submodule \(N\) of \(M\) is called \(F\)-compatible if \(F(N) \subseteq N\). The Frobenius homomorphism \(F: R\to R\) defined by \(F(r)=r^p\) induces an action on the local cohomology modules \(H_m^{i}(R)\) and the ring \(R\) is said to be \(FH\)-finite if all the modules \(H_m^{i}(R)\) (\(1 \leq i \leq d\)) contain finitely many \(F\)-compatible submodules. In this paper the author proves that if \(R\) is a local \(F\)-injective Cohen-Macaulay ring of prime characteristic \(p > 2\) with canonical ideal \(I\) such that \(R/I\) is \(F\)-injective, then \(R\) is \(FH\)-finite. The author also conjectures that the same conclusion holds under the weaker assumption that \(R\) is a Cohen-Macaulay \(F\)-pure local ring of positive characteristic.