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arXiv · 1701.07721

Local homology, finiteness of Tor modules and cofiniteness

Kamran Divaani-Aazar·Hossein Faridian·Massoud Tousi
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Abstract

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 $H^{\fa}_i(M)\neq 0$. 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 $0\leq i\leq n$. 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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Kamran Divaani-Aazar, Hossein Faridian, Massoud Tousi. 2017-01-26. Local homology, finiteness of Tor modules and cofiniteness. https://arxiv.org/abs/1701.07721

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