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 Mat. Zametki, 2011, Volume 89, Issue 1, Pages 109–119 (Mi mz8919)

Cofiniteness with Respect to a Serre Subcategory

A. Hajikarimi

Abstract: Let $\Phi$ be a system of ideals in a commutative Noetherian ring $R$, and let $\mathscr S$ be a Serre subcategory of $R$-modules. We set
$$H^i_\Phi( \cdot , \cdot )=\varinjlim_{\mathfrak b\in\Phi}\operatorname{Ext}^i_R(R/\mathfrak b\otimes_R \cdot , \cdot ).$$
Suppose that $\mathfrak a$ is an ideal of $R$, and $M$ and $N$ are two $R$-modules such that $M$ is finitely generated and $N \in \mathscr S$. It is shown that if the functor $D_\Phi( \cdot )=\varinjlim_{\mathfrak b\in\Phi}\operatorname{Hom}_R(\mathfrak b, \cdot )$ is exact, then, for any $\mathfrak b\in\Phi$, $\operatorname{Ext}^j_R(R/\mathfrak b,H^i_\Phi(M,N))\in\mathscr S$ for all $i,j\ge 0$. It is also proved that if there is a non-negative integer $t$ such that $H^i_{\mathfrak a}(M,N)\in\mathscr S$ for all $i<t$, then $\operatorname{Hom}_R(R/\mathfrak a,H^t_{\mathfrak a}(M,N))\in\mathscr S$, provided that $\mathscr S$ is contained in the class of weakly Laskerian $R$-modules. Finally, it is shown that if $L$ is an $R$-module and $t$ is the infimum of the integers $i$ such that $H^i_{\mathfrak a}(L)\notin\mathscr S$, then $\operatorname{Ext}^j_R(R/\mathfrak a,H^t_{\mathfrak a}(M,L))\in\mathscr S$ if and only if $\operatorname{Ext}^j_R(R/\mathfrak a,\operatorname{Hom}_R(M,H^t_{\mathfrak a}(L)))\in\mathscr S$ for all $j\ge 0$.

Keywords: cofinite modules, generalized local cohomology modules, Serre subcategory.

DOI: https://doi.org/10.4213/mzm8919

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English version:
Mathematical Notes, 2011, 89:1, 121–130

Bibliographic databases:

UDC: 517

Citation: A. Hajikarimi, “Cofiniteness with Respect to a Serre Subcategory”, Mat. Zametki, 89:1 (2011), 109–119; Math. Notes, 89:1 (2011), 121–130

Citation in format AMSBIB
\Bibitem{Haj11} \by A.~Hajikarimi \paper Cofiniteness with Respect to a Serre Subcategory \jour Mat. Zametki \yr 2011 \vol 89 \issue 1 \pages 109--119 \mathnet{http://mi.mathnet.ru/mz8919} \crossref{https://doi.org/10.4213/mzm8919} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2841498} \transl \jour Math. Notes \yr 2011 \vol 89 \issue 1 \pages 121--130 \crossref{https://doi.org/10.1134/S0001434611010135} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000288653100013} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79952429298} 

• http://mi.mathnet.ru/eng/mz8919
• https://doi.org/10.4213/mzm8919
• http://mi.mathnet.ru/eng/mz/v89/i1/p109

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This publication is cited in the following articles:
1. Tran Tuan Nam, Nguyen Minh Tri, “Serre subcategories and the cofiniteness of generalized local cohomology modules with respect to a pair of ideals”, Int. J. Algebr. Comput., 26:6 (2016), 1267–1282
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