Devissage and Localization for the Grothendieck Spectrum of Varieties

TL;DR

This paper develops a new categorical framework called CGW-categories to study the K-theory of varieties, extending classical theorems like dévissage and localization to this setting, and proves their applicability to schemes of finite type.

Contribution

It introduces CGW- and ACGW-categories as generalizations of exact and abelian categories, enabling new K-theoretic computations for varieties and schemes.

Findings

Defined CGW- and ACGW-categories with Qullen Q-construction.

Proved dévissage and localization theorems for ACGW-categories.

Established the equivalence of different definitions of the Grothendieck spectrum of varieties.

Abstract

We introduce a new perspective on the $K$-theory of exact categories via the notion of a CGW-category. CGW-categories are a generalization of exact categories that admit a Qullen $Q$-construction, but which also include examples such as finite sets and varieties. By analyzing Quillen's proofs of d\'evissage and localization we define ACGW-categories, an analogous generalization of abelian categories for which we prove theorems analogous to d\'evissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying d\'evissage and localization allows us to calculate a filtration on the $K$-theory of schemes of finite type. As an application of this theory we construct a comparison map showing that the two authors' definitions of the Grothendieck spectrum of varieties are equivalent.