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Local homology, finiteness of Tor modules and cofiniteness

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 نشر من قبل Kamran Divaani-Aazar
 تاريخ النشر 2017
  مجال البحث
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Let $frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $V(fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{fa}_i(M) eq 0$. We show that $M$ is $fa$-cofinite if and only if the $R$-module $Tor^R_i(R/fa,M)$ is finitely generated for every $0leq ileq n$. This provides a hands-on and computable finitely-many-steps criterion to examine $mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.


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