Primes associated to multigraded modules

Let R = ⊕n∈Nt Rn be a Noetherian Nt -graded ring, and let M = ⊕n∈Nt Mn be a finitely generated Nt -graded R-module. We investigate the behavior of the set of associated primes AssR0Mn as n varies, under various hypotheses on R.;If R is 'standard', i.e., generated in total degree one, we show that AssR0Mn is independent of n for all sufficiently large n, just as in the N -graded case. We later give a construction for finding a specific k(M) ∈ Nt such that { AssR0Mn } is stable for n ≥ k(M). We then show, given any homogeneous ideals I1, ..., Is of R, that k( M/Im11&cdots;Ims SM ) grows linearly with (m1,..., ms).;We prove a number of applications of the stability seen in the standard case. Among these, we give a new, unified, proof of the asymptotic stability of AssA( N/In11&cdots;Int tN ) for a finitely generated A-module N and ideals I1,..., It of an arbitrary Noetherian ring A.;We consider several nonstandard cases as well. First, we show that for any Noetherian N -graded ring, the sequence { AssR0Mn } must be eventually periodic, but need not be eventually stable. This type of behavior also holds for a class of Nt -graded rings, t > 1, which we call 'quasi-standard'.;Our most interesting new result arises from the following situation. If S is a standard graded ring, and I a homogeneous ideal of S, then the associated graded ring R := GrI(S) has a natural bigraded structure, which is typically not standard. For modules over these rings, we show that there is a 'positive cone' of indices in which AssR0Mn is stable.