On the annihilators and attached primes of top local cohomology modules

Abstract

Let \({\mathfrak{a}}\) be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that \({{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}\) , where \({T_R(\mathfrak{a}, M)}\) is the largest submodule of M such that \({{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}\) . Several applications of this result are given. Among other things, it is shown that there exists an ideal \({\mathfrak{b}}\) of R such that \({{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}\) . Using this, we show that if \({ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}\) , then \({{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}\) These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]).