Associated primes of local cohomology and S2-ification

AbstractLet R be commutative Noetherian, I⊂R an ideal, and M a finitely generated R-module. We prove that if R/P has an S2-ification for all P∈Spec(R) then the set of primes associated to the second local cohomology module HI2(M) is finite when ht(IR/P)≥2 for all P∈AssRM and AssRM⊆AssRR. We use that to show that if dim(R)=3 and the ideal transform of R with respect to any height 2 ideal generated by non-zerodivisors is a finitely generated module, then AssRHIi(M) is finite for any I with ht (IR/P)≥2. We also reduce the problem of showing AssRHIi(M) is finite for local four dimensional rings to an extremely concrete case