Attached primes and annihilators of top local cohomology modules defined by a pair of ideals

S. Karimi ; Sh. Payrovi

Algebra and discrete mathematics, Tome 29 (2020) no. 2, pp. 211-220.

Résumé

Assume that $R$ is a complete Noetherian local ring and $M$ is a non-zero finitely generated $R$-module of dimension $n=\dim(M)\geq 1$. It is shown that any non-empty subset $T$ of $\mathrm{Assh}(M)$ can be expressed as the set of attached primes of the top local cohomology modules $H_{I,J}^n(M)$ for some proper ideals $I,J$ of $R$. Moreover, for ideals $I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p$ and $J'$ of $R$ it is proved that $T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))$ if and only if $J'\subseteq J$. Let $H_{I,J}^n(M)\neq 0$. It is shown that there exists $Q\in \mathrm{Supp}(M)$ such that $\dim(R/Q)=1$ and $H_Q^n(R/{\mathfrak p})\neq 0$, for each $\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))$. In addition, we prove that if $I$ and $J$ are two proper ideals of a Noetherian local ring $R$, then $\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})$, where $T_R(I,J,M)$ is the largest submodule of $M$ with $\mathrm{cd}(I,J,T_R(I,J,M))\mathrm{cd}(I,J,M)$, here $\mathrm{cd}(I,J,M)$ is the cohomological dimension of $M$ with respect to $I$ and $J$. This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6].

Détail

Keywords: associated prime ideals, attached prime ideals, top local cohomology modules.

Bibliographie

[1] A. Atazadeh, M. Sedghi and R. Naghipour, “On the annihilators and attached primes of top local cohomology modules”, Arch. Math., 102 (2014), 225–236

[2] K. Bahmanpour, J. A'zami and Gh. Ghasemi, “On the annihilators of local cohomology modules”, J. Algebra, 363 (2012), 8–13

[3] M. H. Bijan-Zadeh, “Torsion theories and local cohomology over commutative Noetherian rings”, J. London Math. Soc., 19 (1979), 402–410

[4] N. Bourbaki, Commutative algebra, Herman, Paris, 1972

[5] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge studies in advanced mathematics, 60, Cambridge University Press, Cambridge, 1998

[6] L. Chu, “Top local cohomology modules with respect to a pair of ideals”, Proc. Amer. Math. Soc., 139 (2011), 777–782

[7] L. Chu and Q. Wang, “Some results on local cohomology modules defined by a pair of ideals”, J. Math. Kyoto Univ., 49 (2009), 193–200

[8] M. T. Dibaei and R. Jafari, “Top local cohomology modules with specified attached primes”, Algebra Colloq., 15 (2008), 341–344

[9] M. T. Dibaei and S. Yassemi, “Attached primes of the top local cohomology modules with respect to an ideal”, Arch. Math., 84:4 (2005), 292–297

[10] M. T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal (II), arXiv: 0501063v1 [math.AC]

[11] H. Matsumura, Commutative ring theory, Cambridge University Press, 1986

[12] I. G. MacDonald, “Secondary representations of modules over a commutative rings”, Symp. Math., XI (1973), 23–43

[13] R. Takahashi, Y. Yoshino and T. Yoshizawa, “Local cohomology based on a nonclosed support defined by a pair of ideals”, J. Pure Appl. Algebra, 213 (2009), 582–600