Acyclicity test of complexes modulo Serre subcategories using the residue fields

Abstract

Let $R$ be a commutative noetherian ring, and let $\mathcal{S}$(resp. $\mathcal{L}$) be a Serre(resp. localizing) subcategory of the category of $R$-modules. If $\mathbb{F}$ is an unbounded complex of $R$-modules Tor-perpendicular to $\mathcal{S}$ and $d$ is an integer, then $\HH{i\geqslant d}{S\otimes_R \Bbb F}$ is in $\mathcal{L}$ for each $R$-module $S$ in $\mathcal{S}$ if and only if $\HH{i\geqslant d}{k(\fp)\otimes_R \Bbb F}$ is in $\mathcal{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathcal{S}$, where $k(\fp)$ is the residue field at $\fp$. As an application, we show that for any $R$-module $M$, $\Tor_{i\geqslant 0}^R(k(\fp),M)$ is in $\mathcal{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathcal{S}$ if and only if $\Ext^{i \geqslant 0}_R(S,M)$ is in $\mathcal{L}$ for each cyclic $R$-module $S$ in $\mathcal{S}$. We also obtain some new characterizations of regular and Gorenstein rings in the case of $\mathcal{S}$ consists of finite modules with supports in a specialization-closed subset $V(I)$ of $\Spec R$.