Finiteness of the set of associated primes for local cohomology modules of ideals via properties of almost factorial rings

TL;DR

This paper explores conditions under which the set of associated primes of local cohomology modules is finite, using properties of almost factorial rings to establish key equivalences and results.

Contribution

It introduces new criteria linking the finiteness of associated primes of local cohomology modules to properties of almost factorial rings, advancing understanding in this area.

Findings

Finiteness of Ass H_I^{d+1}(J) is equivalent to Ass H_I^d(R/J) under certain conditions.

Properties of almost factorial rings enable comparison of local cohomology modules.

Conditions are provided for the finiteness of Ass H_I^i(J) for all i.

Abstract

We investigate the finiteness of the set of associated primes for local cohomology modules $H_I^{i}(J)$ of an ideal $J$ generated by an $R$-sequence, through the comparison of $H_I^{d+1}(J)$ and $H_I^d(R/J)$, where $d = \mathrm{depth}_I(R)$. The properties of almost factorial rings play a key role in enabling this comparison. Under suitable conditions, we prove that the finiteness of $\mathrm{Ass} H_I^{d+1}(J)$ is equivalent to that of $\mathrm{Ass} H_I^d(R/J)$. Moreover, we give a few conditions under which the finiteness of $\mathrm{Ass} H_I^i(J)$ holds for all $i$.