Attached primes of local cohomology modules under localization and completion

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. Following I.G. Macdonald [8], the set of all attached primes of the Artinian local cohomology module Hmi(M) is denoted by AttR(Hmi(M)). In [13, Theorem 3.7], R.Y. Sharp proved that if R is a quotient of a Gorenstein local ring then the shifted localization principle holds true for any local cohomology modules Hmi(M), i.e.(1)AttRp(HpRpi−dim⁡(R/p)(Mp))={qRp|q∈AttRHmi(M),q⊆p} for any p∈Spec(R). In this paper, we improve Sharp's result as follows: the shifted localization principle holds true if and only if R is universally catenary and all its formal fibers are Cohen-Macaulay, if and only if the shifted completion principle(2)AttRˆ(Hmi(M))=⋃p∈AttR(Hmi(M))AssRˆ(Rˆ/pRˆ) holds true for any local cohomology module Hmi(M). This also improves the main result of the paper by T.D.M. Chau and the first author [2].