The support of top graded local cohomology modules

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