Geodesic
Cofiniteness with Respect to a Serre Subcategory
Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 109-119 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, 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 $j\ge 0$.
Mots-clés : cofinite modules
Keywords: generalized local cohomology modules, Serre subcategory. 
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     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a10/}
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A. Hajikarimi. Cofiniteness with Respect to a Serre Subcategory. Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 109-119. http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a10/

[1] M. H. Bijan-Zadeh, “A common generalization of local cohomology theories”, Glasgow Math. J., 21:2 (1980), 173–181 | DOI | MR | Zbl

[2] J. Asadollahi, P. Schenzel, Cofiniteness of Generalized Local Cohomology Modules for Principal Ideals, Preprint

[3] J. Asadollahi, K. Khashyarmanesh, Sh. Salarian, “On the finiteness properties of the generalized local cohomology modules”, Comm. Algebra, 30:2 (2002), 859–867 | DOI | MR | Zbl

[4] K. Khashyarmanesh, M. Yassi, “On the finiteness property of generalized local cohomology modules”, Algebra Colloq., 12:2 (2005), 293–300 | MR | Zbl

[5] L. Chu, “The Artinianness of local cohomology modules”, Northeast. Math. J., 23:1 (2007), 87–94 | MR | Zbl

[6] M. P. Brodmann, R. Y. Sharp, Local Cohomology. An Algebric Introduction with Geometric Applications, Cambridge Stud. Adv. Math., 60, Cambridge Univ. Press, 1998 | MR | Zbl

[7] K. Divaani-Aazar, A. Mafi, “Associated primes of local cohomology modules of weakly Laskerian modules”, Comm. Algebra, 34:2 (2006), 681–690 | DOI | MR | Zbl

[8] H. Zöschinger, “Minimax-moduln”, J. Algebra, 102:1 (1986), 1–32 | DOI | MR | Zbl

[9] H. Saremi, Matlis Duality and Finiteness Properties of Generalized Local Cohomology Modules, Preprint