Associated primes of local cohomology of flat extensions with regular fibers and $\Sigma$-finite $D$-modules

TL;DR

This paper investigates the finiteness of associated primes of local cohomology modules in flat extensions with regular fibers, using $ ext{ extSigma}$-finite $D$-modules, and provides positive results for polynomial and power series extensions.

Contribution

It establishes finiteness results for associated primes in certain flat extensions and introduces $ ext{ extSigma}$-finite $D$-modules as a key tool, extending previous understanding.

Findings

Finiteness of associated primes holds for polynomial and power series extensions when $ ext{dim}(R/Iigcap R) extless= 1.

Reduction of the problem to power series extensions is analyzed.

Examples of $ ext{ extSigma}$-finite $D$-modules are provided with applications to local cohomology.

Abstract

In this article, we study the following question raised by Mel Hochster: let $(R,m,K)$ be a local ring and $S$ be a flat extension with regular closed fiber. Is $\cV(mS)\cap\Ass_S H^i_I(S)$ finite for every ideal $I\subset S$ and $i\in \NN?$ We prove that the answer is positive when $S$ is either a polynomial or a power series ring over $R$ and $\dim(R/I\cap R)\leq 1.$ In addition, we analyze when this question can be reduced to the case where $S$ is a power series ring over $R$. An important tool for our proof is the use of $\Sigma$-finite $D$-modules, which are not necessarily finitely generated as $D$-modules, but whose associated primes are finite. We give examples of this class of $D$-modules and applications to local cohomology.