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

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 $i$.