Primes associated to multigraded modules

Abstract

Let R=⊕ n∈ N t R n be a Noetherian multigraded ring, and let M be a finitely generated multigraded R-module. We investigate the asymptotic behavior of Ass R 0 (M n ) . In case R is generated in total degree one, we show that the expected stability occurs. We also consider several non-standard cases. For general N -graded R, we show that {Ass R 0 ( M n )} is eventually periodic, but need not be stable. For rings graded by N t , with t⩾2, we show that in some cases a form of periodicity holds, while in others there is a “cone” of stability.