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

In this manuscript, 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 V(mS)∩AssSHIi(S) finite for every ideal I⊂S and i∈N? We prove that the answer is positive when S is either a polynomial or a power series ring over R and dim(R/I∩R)⩽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 Σ-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.