Finiteness properties of local cohomology for \(F\)-pure local rings
Let \((R, \mathfrak{m}, k)\) be a local ring of positive characteristic \(p>0\). Frobenius acts naturally on the ring and its local cohomology modules \(H^i_{\mathfrak{m}} (R)\), \(0 \leq i \leq d=\dim(R)\). The ring \(R\) is called \(FH\)-finite if its local cohomology modules \(H^i_{\mathfrak{m}} (R)\) have only finitely many Frobenius compatible submodules, \(0 \leq i \leq \dim(R)\). The main result of the paper shows that an \(F\)-pure local ring is \(FH\)-finite. This answers positively an open question of \textit{M. Hochster} and the reviewer [Algebra Number Theory 2, No. 7, 721--754 (2008; Zbl 1190.13003)] and a conjecture of the reviewer in the Cohen-Macaulay case [J. Pure Appl. Algebra 216, No. 1, 115--118 (2012; Zbl 1238.13013)]. Another interesting result shows that an excellent local ring that is \(F\)-pure on the punctured spectrum has \(FH\)-finite length, which means that its local cohomology modules \(H^i_{\mathfrak{m}} (R)\) have finite length in the category of modules with a Frobenius action, for all \(0 \leq i \leq \dim(R)\). The author also shows that the property that \(R\) is \(FH\)-finite, respectively that \(R\) has \(FH\)-finite length, localizes.
scientific article; zbMATH DE number 6369439