D\'{e}vissage for Waldhausen K-theory

Abstract

A d\'evissage-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 d\'evissage condition about the existence of appropriate finite filtrations is satisfied. We distinguish between d\'evissage theorems of single type and of multiple type, depending on the number of Waldhausen subcategories and their properties. The main representative examples of such theorems are Quillen's original d\'evissage 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 d\'evissage-type theorems and prove a general d\'evissage theorem of single type and a general d\'evissage theorem of multiple type.