2014/04/01 by Nhan, Le Thanh, Quy, Pham Hung · 2 citations
#13D45 #13E05 #13H10 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1404.0111
Let (R,\m) be a Noetherian local ring and M a finitely generated R-module. Following I. G. Macdonald \citeMac, the set of all attached primes of the Artinian local cohomology module Hi\m(M) is denoted by \AttR(Hi\m(M)). In \cite[Theorem 3.7]Sh, R. Y. Sharp proved that if R is a quotient of a Gorenstein local ring then the shifted localization principle always holds true, i.e. \Att_R\p(Hi-dim (R/\p)_\p R\p(M\p))=\\q R\p| \q∈\AttRHi\m(M), \q⊆ \p\ (1) for any local cohomology modules Hi\m(M) and any \p∈\Spec (R). In this paper, we improve Sharp's result as follows: the shifted localization principle always holds true if and only if R is universally catenary and all its formal fibers are Cohen-Macaulay, if and only if \Att\R(Hi\m(M))=\bigcup_\p∈\AttR(Hi\m(M))\Ass\R(\R/\p\R) (2) holds true for any finitely generated R-module M and any integer i≥ 0. This also improves the main result of the paper \citeCN.