On the annihilators and attached primes of top local cohomology modules

Abstract

Let \frak a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that {\rm Ann}_R(H_{\frak a}^{{\dim M}({\frak a}, M)}(M))= {\rm Ann}_R(M/T_R({\frak a}, M)), where T_R({\frak a}, M) is the largest submodule of M such that {\rm cd}({\frak a}, T_R({\frak a}, M))< {\rm cd}({\frak a}, M). Several applications of this result are given. Among other things, it is shown that there exists an ideal \frak b of R such that {\rm Ann}_R(H_{\frak a}^{\dim M}(M))={\rm Ann}_R(M/H_{\frak b}^{0}(M)). Using this, we show that if H_{\frak a}^{\dim R}(R)=0, then {\rm Att}_RH^{{\dim R}-1}_{\frak a}(R)=\{{\frak p}\in {\rm Spec}\,R|\,{\rm cd}({\frak a}, R/{\frak p})={\dim R}-1\}. These generalize the main results of \cite[Theorem 2.6]{BAG}, \cite[Theorem 2.3]{He} and \cite[Theorem 2.4]{Lyn}.