Let
be an excellent
regular ring of dimension
containing a field
of
characteristic zero. Let
be an ideal in
.
We show that
is a finite set. As an application, we show that if
is an ideal of
height
with
for all minimal primes
of
then for all but
finitely many primes
with
, the
topological space
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
containing a field of characteristic zero, it suffices to prove
is finite for
all ideals
in
of height
(here
), where
is an excellent regular
domain of dimension
containing an uncountable field of characteristic zero.