The local cohomology of a parameter ideal with respect to an arbitrary ideal

Let R be a regular ring, let J an ideal generated by sequence of codimension at least 2, and I containing J. We give example module HI3(J) with infinitely many associated primes, answering question Hochster Núñez-Betancourt in the negative. In fact, for i≤4, we show that under suitable hypotheses on R/J, AssHIi(J) is finite if only AssHIi−1(R/J) finite. Our proof this statement involves novel generalization isomorphism Hellus, which may some independent interest. The finiteness comparison between tends to improve as our R/J become more restrictive. To illustrate extreme end phenomenon, prime characteristic p>0 setting, regular, then all i≥0.