Let R be a regular ring of dimension d containing a field K of characteristic zero. If E is an R-module let Assi E = {Q ∈ AssE | heightQ = i}. Let P be a prime ideal in R of height g. We show that if R/P satisfies Serre’s condition Ri then Assg+i+1 Hg+1P(R) is a finite set. As an application of our techniques we prove that if P is a prime ideal in R such that (R/P)q is regular for any non-maximal prime ideal q then HiP(R) has finitely many associate primes for all i.

Mathematics Subject Classification. Primary 13D45; Secondary 13C13 .