The support of top graded local cohomology modules

Original · EN

Let R₀ be any domain, let R=R₀[U₁,..., Uₛ]/I, where U₁,..., Uₛ are indeterminates of some positive degrees, and I⊂ R₀[U₁,..., Uₛ] is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of H:=Hˢᵣ₊(R) contain a certain non-zero ideal c(I) of R₀ called the ``content'' of I. It follows that the support of H is simply V((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 favour of such a result. Finally, we give a very short proof of a weak version of the monomial conjecture based on these results.