Associated primes of graded components of local cohomology modules

The $i$-th local cohomology module of a finitely generated graded module $M$ over a standard positively graded commutative Noetherian ring $R$, with respect to the irrelevant ideal $R_+$, is itself graded; all its graded components are finitely generated modules over $R_0$, the component of $R$ of degree 0. This paper is concerned with the asymptotic behaviour of $\Ass_{R_0}(H^i_{R_+}(M)_n)$ as $n \to -\infty$. The smallest $i$ for which such study is interesting is the finiteness dimension $f$ of $M$ relative to $R_+$, defined as the least integer $j$ for which $H^j_{R_+}(M)$ is not finitely generated. Brodmann and Hellus have shown that $\Ass_{R_0}(H^f_{R_+}(M)_n)$ is constant for all $n < < 0$ (that is, in their terminology, $\Ass_{R_0}(H^f_{R_+}(M)_n)$ is asymptotically stable for $n \to -\infty$). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that $R$ is a homomorphic image of a regular ring): our answer is precisely the set of contractions to $R_0$ of certain relevant primes of $R$ whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when $i > f$. They noted that Singh's study of a particular example (in which $f = 2$) shows that $\Ass_{R_0}(H^3_{R_+}(R)_n)$ need not be asymptotically stable for $n \to -\infty$. The second main aim of this paper is to determine, for Singh's example, $\Ass_{R_0}(H^3_{R_+}(R)_n)$ quite precisely for every integer $n$, and, thereby, answer one of the questions raised by Brodmann and Hellus.