Dévissage for Waldhausen K-theory

Abstract

A devissage--type theorem in algebraic $K$-theory is a statement that identifies the $K$-theory of a Waldhausen category $\mathscr{C}$ in terms of the $K$-theories of a collection of Waldhausen subcategories of $\mathscr{C}$ when a devissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between devissage theorems of \emph{single type} and of \emph{multiple type} depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original devissage theorem for abelian categories (single type) and Waldhausen's theorem on spherical objects for more general Waldhausen categories (multiple type). In this paper, we study some general aspects of devissage--type theorems and prove a general devissage theorem of single type and a general devissage theorem of multiple type.