  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
``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.
