Let R0 be any domain, let R = R0[U1,...,Us]/I, where U1,..., Us are indeter-minates of positive degrees d1,...,ds, and I C R0[U1,...,Us] is a homogeneous ideal. The main theorem in this paper is Theorem 2.6, a generalization of The-orem 1.5 in [KS], which slates that all the associated primes of H:= HSR+ (R) contain a certain non-zero ideal c(I) of Ro called the "content" of I (see Definition 2.4.) It follows that the support of H is simply V(c(I)R + R +) (Corollary 1.8) and. in particular. H vanishes if and only if c(I) is the unit ideal. These results raise the question of whether local cohomology modules have finitely many minimal associated primes - this paper provides further evidence in favor of such a result (Theorem 2.10 and Remark 2.12.) Finally, we give a very short proof of a weak version of the monomial conjecture based on Theorem 2.6.
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