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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 closedfiber. 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 apolynomial or a power series ring over $R$ and $\dim(R/I\cap R)\leq 1.$ Inaddition, we analyze when this question can be reduced to the case where $S$ isa 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 thisclass of $D$-modules and applications to local cohomology.

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