2019/06/04 by Satoshi Mochizuki, Mochizuki, Satoshi
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.1906.01589
openalex publication_date 2019/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this article is to show a version of dévissage theorem of non-connective 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 Thomason-Trobaugh's paper. We introduce the notions of cell structures and dévissage spaces and our main theorem states a structure of non-connective K-theory of dévissage 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 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 dg-categories. To overcome this dilemma, the notion of dévissage spaces should not be expressed by the language of dg-categories. First three sections are devoted to the foundation of our model of stable (∞,1)-categories which we will play on to give a description of dévissage spaces.