Czechoslovak Mathematical Journal, first online, pp. 1-7

Annihilators and attached primes of top local cohomology modules

Shahram Rezaei

Received November 5, 2019.   Published online September 30, 2020.

Abstract:  Let $\frak{a}$ be an ideal of Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper we determine ${\rm Ann}_R({\rm H}_{\frak{a}}^{{\rm cd} (\frak{a},M)}(M))$ and ${\rm Att}_R({\rm H}_{\frak{a}}^{{\rm cd}(\frak{a},M)}(M))$, which are two important problems concerning the last nonzero local cohomology module ${\rm H}_{\frak{a}}^{{\rm cd}(\frak{a},M)}(M)$. We show that ${\rm Ann}_R({\rm H}_{\frak{a}}^{{\rm cd} (\frak{a},M)}(M))= {\rm Ann}_R(M/T_R(\frak{a},M))$, where $T_R(\frak{a},M)$ is the largest submodule of $M$ such that ${\rm cd}(\frak{a},T_R(\frak{a},M))<{\rm cd}(\frak{a},M)$. Using the above result we determine the attached primes of the top local cohomology module ${\rm Att}_R({\rm H}_{\frak{a}}^{{\rm cd} (\frak{a},M)}(M))$. In fact, we show that ${\rm Att}_R({\rm H}_{\frak{a}}^{{\rm cd}(\frak{a},M)}(M))=\{ \frak{p} \in{\rm Supp}_R M \colon{\rm cd} (\frak{a},R/\frak{p})= {\rm cd} (\frak{a},M) \}$. Then by using these, we obtain some main results of A. Atazadeh, M. Sedghi, R. Naghipour (2014), K. Bahmanpour, J. A'zami, G. Ghasemi (2012) and K. Divaani-Aazar (2009).
Keywords:  annihilator; attached prime; local cohomology
Classification MSC:  13D45, 14B15, 13E05
DOI:  10.21136/CMJ.2020.0479-19

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Affiliations:   Shahram Rezaei, Department of Mathematics, Faculty of Science, Payame Noor University, Nakhl St, Lashkarak Highway, Tehran 19569, Iran, e-mail:

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