Acyclicity test of complexes modulo Serre subcategories using the residue fields
Mitsuyasu Hashimoto, Xi Tang
TL;DR
This work develops residue-field based acyclicity tests for unbounded complexes in the Serre quotient setting. It proves a central equivalence: for a complex $\mathbb F$ in $\mathscr{S}^{\perp}$ and any $d$, the condition that $\mathrm{H}_i(S\otimes_R \mathbb F)\in \mathscr{L}$ for all $i\ge d$ and all $S\in\mathscr{S}$ is equivalent to the condition that $\mathrm{H}_i(k(\frak p)\otimes_R \mathbb F)\in \mathscr{L}$ for all primes $\frak p$ with $R/\frak p\in\mathscr{S}$. This framework yields Tor/Ext criteria linking $\mathrm{Tor}^R_i(k(\frak p), M)$ with $\mathrm{Ext}^i_R(S,M)$ and leads to the development of relatively-$I$-flat/injective modules and two related homological dimensions. Furthermore, it yields new characterizations of regular and Gorenstein rings in the setting where $\mathscr{S}$ consists of finite modules supported in a specialization-closed subset $V(I)$. Overall, the paper advances a unified Serre-quotient perspective for testing acyclicity and homological properties via residue fields and Koszul techniques, with implications for ring regularity and Gorensteinness.
Abstract
Let $R$ be a commutative noetherian ring, and let $\mathscr{S}$(resp. $\mathscr{L}$) be a Serre(resp. localizing) subcategory of the category of $R$-modules. If $\Bbb F$ is an unbounded complex of $R$-modules Tor-perpendicular to $\mathscr{S}$ and $d$ is an integer, then $\HH{i\geqslant d}{S\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each $R$-module $S$ in $\mathscr{S}$ if and only if $\HH{i\geqslant d}{k(\fp)\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$, 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 $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$ if and only if $\Ext^{i \geqslant 0}_R(S,M)$ is in $\mathscr{L}$ for each cyclic $R$-module $S$ in $\mathscr{S}$. We also obtain some new characterizations of regular and Gorenstein rings in the case of $\mathscr{S}$ consists of finite modules with supports in a specialization-closed subset $V(I)$ of $\Spec R$.