Local cohomology properties of direct summands

In this article, we prove that if R→S is a homomorphism of Noetherian rings that splits, then for every i≥0 and ideal I⊂R, is finite when is finite. In addition, if S is a Cohen–Macaulay ring that is finitely generated as an R-module, such that all the Bass numbers of , as an S-module, are finite, then all the Bass numbers of , as an R-module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik [5]. As a consequence, we exhibit a Gorenstein F-regular UFD of positive characteristic that is not a direct summand, not even a pure subring, of any regular ring.