Asymptotic stability of primes associated to homogeneous components of multigraded modules

Let A B be a homogeneous extension of Noetherian standard -graded rings with A0=B0=R. Let M be a finitely generated -graded B-module and N M a finitely generated graded A-submodule of M. In this paper, we investigate the asymptotic behavior of the set of primes associated to the module Mn/Nn and prove that for all sufficiently large , the set AssR(Mn/Nn) is stable. We also give a certain inequality for the spread of standard multigraded rings, which is a natural generalization of Burchʹs inequality for the analytic spread of an ideal.