Devissage and Localization for 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.