Acyclicity test of complexes modulo Serre subcategories using the residue fields

2025/03/08 by Xi Tang, Hashimoto, Mitsuyasu, Tang, Xi
Computer Science · Chemistry · Mathematics · #Computational Drug Discovery Methods #Molecular spectroscopy and chirality #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2503.06354

Abstract

Let R be a commutative noetherian ring, and let \mathscrS(resp. \mathscrL) 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 \mathscrS and d is an integer, then \HHi\geqslant dS⊗_R \Bbb F is in \mathscrL for each R-module S in \mathscrS if and only if \HHi\geqslant dk(\fp)⊗_R \Bbb F is in \mathscrL for each prime ideal \fp such that R/\fp is in \mathscrS, 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 \mathscrL for each prime ideal \fp such that R/\fp is in \mathscrS if and only if \Ext^{i \geqslant 0}_R(S,M) is in \mathscrL for each cyclic R-module S in \mathscrS. We also obtain some new characterizations of regular and Gorenstein rings in the case of \mathscrS consists of finite modules with supports in a specialization-closed subset V(I) of \Spec R.

Related

• Cofiniteness with respect to extension of Serre subcategories
• Melkersson condition on Serre subcategories
• Cofiniteness with respect to extension of Serre subcategories at small dimensions
• Local cohomology of ideals and the R_n condition of Serre
• Cofiniteness with respect to extension of Serre subcategories