Local cohomology properties of direct summands

TL;DR

This paper investigates the properties of local cohomology modules under ring homomorphisms, establishing finiteness of associated primes and Bass numbers for direct summands, with applications to Gorenstein $F$-regular UFDs.

Contribution

It proves new finiteness results for local cohomology modules of direct summands and extends these to a broader class of functors, with implications for ring theory.

Findings

Finiteness of associated primes for direct summands.

Finiteness of Bass numbers under certain conditions.

Existence of a Gorenstein $F$-regular UFD not a direct summand of a regular ring.

Abstract

In this article, we prove that if $R\to S$ is a homomorphism of Noetherian rings that splits, then for every $i\geq 0$ and ideal $I\subset R$, $\Ass_R H^i_I(R)$ is finite when $\Ass_S H^i_{IS}(S)$ 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 $H^i_{IS}(S)$, as an $S$-module, are finite, then all the Bass numbers of $H^i_{I}(R)$, as an $R$-module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik. 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.