Asymptotic stability of 𝐴𝑠𝑠(𝑀/𝐼ⁿ𝑀)

Abstract

Let R be a noetherian ring, I an ideal of R and M a finitely generated R-module. Let A be the map from N into the subsets of Spec ( R ) {\text {Spec}}(R) defined by A ( n ) = Ass R ( M / I n M ) A(n) = {\text {Ass}_R}(M/{I^n}M) . We shall prove: (1) For n sufficiently large, A ( n ) A(n) is independent of n. We moreover give two examples concerning the behavior of A ( n ) A(n) .