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

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.