A d\'evissage theorem of non-connective $K$-theory

Abstract

The purpose of this article is to show a version of d\'evissage theorem of non-connective $K$-theory. Our theorem contains Quillen's d\'evissage theorem, Waldhausen's cell filtration theorem and theorem of heart as special cases. In this sense, we give an affirmative answer to Thomason's problem in Thomason-Trobaugh's paper. We introduce the notions of cell structures and d\'evissage spaces and our main theorem states a structure of non-connective $K$-theory of d\'evissage spaces in terms of non-connective $K$-theory of heart of cell structures. A specific feature in our proof is 'motivic' in the sense that properties of $K$-theory which we will utilze to prove the theorem are only categorical homotopy invariance, localization and co-continuity. On the other hands, it is well-known that the analogue of the d\'evissage theorem for $K$-theory does not hold for Hochschild homology theory. In this point of view, we could say that d\'evissage theorem is not 'motivic' over dg-categories. To overcome this dilemma, the notion of d\'evissage spaces should not be expressed by the language of dg-categories. First three sections are devoted to the foundation of our model of stable $(\infty,1)$-categories which we will play on to give a description of d\'evissage spaces.