ABSTRACT
Let R0 be any domain, let R R0U1 UsI, where U1 Us are indeterminates of positive degrees d1 ds, and I R0U1 Us is a homogeneous ideal. The main theorem in this paper is Theorem 2.6, a generalization of Theorem 1.5 in [KS], which states that all the associated primes of H : HsR
R contain a certain non-zero ideal cI of R0 called the “content” of I (see Definition 2.4.) It follows that the support of H is simply VcIRR
(Corollary 1.8) and, in particular, H vanishes if and only if cI 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.
