A dévissage theorem of nonconnective $K$theory
Abstract
The purpose of this article is to show a version of dévissage theorem of nonconnective $K$theory. Our theorem contains Quillen's dévissage 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 ThomasonTrobaugh's paper. We introduce the notions of cell structures and dévissage spaces and our main theorem states a structure of nonconnective $K$theory of dévissage spaces in terms of nonconnective $K$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 cocontinuity. On the other hands, it is wellknown that the analogue of the dévissage theorem for $K$theory does not hold for Hochschild homology theory. In this point of view, we could say that dévissage theorem is not 'motivic' over dgcategories. To overcome this dilemma, the notion of dévissage spaces should not be expressed by the language of dgcategories. 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 dévissage spaces.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 DOI:
 10.48550/arXiv.1906.01589
 arXiv:
 arXiv:1906.01589
 Bibcode:
 2019arXiv190601589M
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Category Theory