Associated primes of graded components of local cohomology modules

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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 R₀, the component of R of degree 0. This paper is concerned with the asymptotic behaviour of ᵣ₀(Hⁱᵣ₊(M)ₙ) as n → -∞. 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 Hʲᵣ₊(M) is not finitely generated. Brodmann and Hellus have shown that ᵣ₀(Hᶠᵣ₊(M)ₙ) is constant for all n < < 0 (that is, in their terminology, ᵣ₀(Hᶠᵣ₊(M)ₙ) is asymptotically stable for n → -∞). 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 R₀ 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 ᵣ₀(H³ᵣ₊(R)ₙ) need not be asymptotically stable for n → -∞. The second main aim of this paper is to determine, for Singh's example, ᵣ₀(H³ᵣ₊(R)ₙ) quite precisely for every integer n, and, thereby, answer one of the questions raised by Brodmann and Hellus.