Associate primes of local cohomology modules over certain quotients of regular rings
Tony J. Puthenpurakal
TL;DR
The paper addresses the finiteness of associated primes of local cohomology modules for quotients $A_m=R/(oldsymbol{x})^m$ of a regular ring $R$ by a regular sequence $oldsymbol{x}$ with $R/(oldsymbol{x})$ regular. It reduces finiteness from $A_m$ to $A_1$ via an extended Rees algebra and analyzes two regimes: $ ext{char}\,k=p>0$ using $F$-finite module techniques and $ ext{char}\,k=0$ with $R$ local or smooth over $k$ using $D$-module (Weyl algebra) methods. The core contribution is establishing an equivalence: $ ext{Ass}_R H^i_I(A_1)$ finite iff $ ext{Ass}_R H^i_I(A_m)$ finite for all $m\, ext{and}\,i$, thereby proving finiteness for $A_m$ from the known $A_1$ case. This yields new large examples where local cohomology has finite associated primes, leveraging graded $F$-modules, holonomic $D$-modules, and the extended Rees construction.
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 $k$.