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This is a Quantitative Study study.

September 28, 2025Open Access

Acyclicity test of complexes modulo Serre subcategories using the residue fields

Key Points

Acyclicity of complexes modulates the interplay between Serre and localizing subcategories.
Residue fields play a crucial role in establishing equivalence conditions for modules across subcategories: $ ext{Ext}^{i eq 0}$ and $ ext{Tor}^{i eq 0}$ validate local embeddings.
Characterizations of regular and gorenstein rings emerge from specific conditions on Serre subcategories of finite modules.
The framework built here extends to complexes of $ ext{R}$-modules, demonstrating powerful results in homological algebra.

Abstract

Let R be a commutative noetherian ring, and let S (resp. L) be a Serre (resp. localizing) subcategory of the category of R-modules. If F is an unbounded complex of R-modules Tor-perpendicular to S and d is an integer, then i dSR F is in L for each R-module S in S if and only if i dk () R F is in L for each prime ideal such that R/ is in S, where k () is the residue field at. As an application, we show that for any R-module M, ₈ ₀R (k (), M) is in L for each prime ideal such that R/ is in S if and only if ^i 0R (S, M) is in L for each cyclic R-module S in S. We also obtain some new characterizations of regular and Gorenstein rings in the case of S consists of finite modules with supports in a specialization-closed subset V (I) of R.

Acyclicity test of complexes modulo Serre subcategories using the residue fields

Key Points

Abstract

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