Cotorsion pairs and a $K$-theory Localization Theorem

TL;DR

This paper develops a new approach to $K$-theory localization in exact categories using cotorsion pairs to construct Waldhausen categories, enabling a generalized Quillen Localization Theorem without requiring Serre subcategories.

Contribution

It introduces a novel method of constructing Waldhausen categories via cotorsion pairs to prove a generalized $K$-theory localization theorem in exact categories.

Findings

Established a Waldhausen structure from cotorsion pairs in exact categories.

Proved a new version of Quillen's Localization Theorem relating $K$-theory of categories and cofibers.

Extended Quillen's Resolution Theorem to a homotopical setting with weak equivalences.

Abstract

We show that a complete hereditary cotorsion pair $(\C,\C^\bot)$ in an exact category $\E$, together with a subcategory $\Z\subseteq\E$ containing $\C^\bot$, determines a Waldhausen category structure on the exact category $\C$, in which $\Z$ is the class of acyclic objects. This allows us to prove a new version of Quillen's Localization Theorem, relating the $K$-theory of exact categories $\A\subseteq\B$ to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require $\A$ to be a Serre subcategory, which produces new examples. Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.