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## The support of top graded local cohomology modules

### Abstract

Let \$R_0\$ be any domain, let \$R=R_0[U_1, ..., U_s]/I\$, where \$U_1, ..., U_s\$ are indeterminates of some positive degrees, and \$I\subset R_0[U_1, ..., U_s]\$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of \$H:=H^s_{R_+}(R)\$ contain a certain non-zero ideal \$c(I)\$ of \$R_0\$ called the\ud ``content'' of \$I\$. It follows that the support of \$H\$ is simply \$V(\content(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.\u

Year: 2003
OAI identifier: oai:eprints.whiterose.ac.uk:10161

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