ASSOCIATED PRIMES OF LOCAL COHOMOLOGY MODULES OVER REGULAR RINGS

Let R be an excellent regular ring of dimension d containing a field K of characteristic zero. Let I be an ideal in R. We show that Ass H-I(d-1)(R)is a finite set. As an application, we show that if I is an ideal of height g with height Q = g for all minimal primes of I then for all but finitely many primes P superset of I with height P >= g + 2, the topological space Spec degrees (R-P / IR (P)) is connected. We also show that to prove a conjecture of Lyubeznik (regarding finiteness of associate primes for local cohomology modules) for all excellent regular rings of dimension <= d containing a field of characteristic zero, it suffices to prove AssS H-J(g+1) (S) is finite for all ideals J in S of height g ( here 0 <= g <= d), where S is an excellent regular domain of dimension <= d containing an uncountable field of characteristic zero.