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Cofiniteness with Respect to a Serre Subcategory
A. Hajikarimi Islamic Azad University
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.
UDC:
517 Received: 27.10.2009
DOI:
10.4213/mzm8919
Fulltext:
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