On the annihilators of formal local cohomology modules

Abstract

Let a denote an ideal in a commutative Noetherian local ring (R,m) and M a non-zero finitely generated R-module of dimension d. Let d := dim(M/aM). In this paper we calculate the annihilator of the top formal local cohomology module Fda(M). In fact, we prove that AnnR(F d a(M)) = AnnR(M/UR(a,M)), where UR(a,M) := ∪{N : N ⩽ M and dim(N/aN) < dim(M/aM)}. We give a description of UR(a,M) and we will show that AnnR(F d a(M)) = AnnR(M/ ∩pj∈AsshRM∩V(a) Nj), where 0 = ∩n j=1 Nj denotes a reduced primary decomposition of the zero submodule 0 in M and Nj is a pj-primary submodule of M , for all j = 1, . . . , n. Also, we determine the radical of the annihilator of Fda(M). We will prove that √ AnnR(Fa(M)) = AnnR(M/GR(a,M)), where GR(a,M) denotes the largest submodule of M such that AsshR(M) ∩ V(a) ⊆ AssR(M/GR(a,M)) and AsshR(M) denotes the set {p ∈ AssM : dimR/p = dimM}.