Localisation theorems for the connective K-theory of exact categories
Christoph Winges
TL;DR
The paper develops a unified localisation theorem for the connective algebraic K-theory $K$ of exact $\infty$-categories, showing that strong left filtering subcategories induce cofibre sequences $K(\mathcal{U}) \to K(\mathcal{C}) \to K((\mathcal{C}/\mathcal{U})_W)$. The authors implement Waldhausen's generic fibration theorem together with two models for the relative cofibre, relate quotients via the Gabriel–Quillen embedding, and derive a non-idempotent-complete generalisation using $\mathcal{U}^{\natural{K_0(\mathcal{C})}}$. This framework recovers classical localisation theorems (Quillen, Schlichting, Barwick) as corollaries and provides a single structural reason behind their validity, with applications to cotorsion theories and t-structures. The results offer a robust, universal approach to localisation in K-theory across stable, exact, and abelian settings, clarifying when Waldhausen and exact quotients agree and enabling broad generalisations.
Abstract
We prove a localisation theorem for the K-theory of filtering subcategories of exact $\infty$-categories which subsumes the localisation theorem for stable $\infty$-categories, Quillen's localisation theorem for abelian categories, and Schlichting's localisation theorem for s-filtering subcategories.