Acyclicity test of complexes modulo Serre subcategories using the residue fields

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 $\text{HH}i⩾dS{\otimes }_R\mathbb{F}$ is in $\mathcal{L}$ for each $R$-module $S$ in $\mathcal{S}$ if and only if $\text{HH}i⩾dk(\text{fp}){\otimes }_R\mathbb{F}$ is in $\mathcal{L}$ for each prime ideal $\text{fp}$ such that $R\mathrm{/}\text{fp}$ is in $\mathcal{S}$, where $k(\text{fp})$ is the residue field at $\text{fp}$. As an application, we show that for any $R$-module $M$, ${\text{Tor}}_{i⩾0}^R(k(\text{fp}),M)$ is in $\mathcal{L}$ for each prime ideal $\text{fp}$ such that $R\mathrm{/}\text{fp}$ is in $\mathcal{S}$ if and only if ${\text{Ext}}_R^{i⩾0}(S,M)$ 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 $\text{Spec}R$.