Exactness of cochain complexes via additive functors

TL;DR

This work analyzes $e$-exactness of cochain complexes over rings and its relationship to the spectral-category functor $P$ and singular-torsion localization. It establishes precise correspondences between $e$-exactness in $ ext{Mod-}R$ and exactness in the spectral category and extends homological lemmas to noncommutative settings using a Goodearl-type localization, while generalizing the framework via Gabriel topologies to define $rak{F}$-exactness. A key negative result shows that no additive functor to an abelian category can capture $e$-exactness exactly in general, highlighting intrinsic limitations of functorial detection. The paper clarifies the hierarchy between $e$-exactness and $rak{G}$-exactness, providing noncommutative localization tools and a unified approach to exactness under localization and Gabriel topologies.

Abstract

We investigate the relation between the notion of $e$-exactness, recently introduced by Akray and Zebary, and some functors naturally related to it, such as the functor $P\colon\operatorname{Mod} R\to \operatorname{Spec}(\operatorname{Mod} R)$, where $\operatorname{Spec}(\operatorname{Mod} R)$ denotes the spectral category of $\operatorname{Mod} R$, and the localization functor with respect to the singular torsion theory.

Theorems & Definitions (24)

• Definition 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• Corollary 2.3
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Lemma 3.4
• proof