ON THE ASYMPTOTIC STABILITY OF CERTAIN SETS OF PRIME IDEALS

Abstract

Let (R, m) be a Noetherian local ring, I,J two ideals of R, and M a

finitely generated R−module. It is shown that the set Ass

R

(HI d−1(N

n

))∪

{m} is stable for large n, where we use N

n

to denote R−module

J nM/J n+1M or R−module M/J nM and d is the eventual value of

dim N

n

. We also give a counter-example for stability of certain sets of

associated prime ideals of Ext −modules and of attached prime ideals of

local cohomology modules w.r.t. m