Associate primes of local cohomology modules over certain quotients of regular rings

Abstract

Let $R$ be a regular ring containing a field $k$. Let $\mathbf{x} = x_1, \ldots, x_r$ be a regular sequence in $R$ such that $R/(\mathbf{x})$ is a regular ring. Fix $m \geq 1$. Set $A_m = R/(\mathbf{x})^m$. We show that for any ideal $Q$ of $A_m$ the set $\text{Ass} \ H^i_Q(A_m)$ is a finite set for $i \geq 0$, in the following cases: 1. $\text{char}\ k = p > 0$. 2. $\text{char} \ k = 0$, $R$ is local or a smooth affine algebra over $k$.