The fundamental theorem via derived Morita invariance, localization, and \(\mathbb A^{1}\)-homotopy invariance
type: Item
instance of: scholarly article
zbMATH Open document ID: 1257.18007
author: Gonçalo Tabuada
publication date: 2012-08-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.5936
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 6072566
zbMATH Keywords: fundamental theorem
zbMATH Keywords: dg categories
zbMATH Keywords: homotopy algebraic \(K\)-theory
zbMATH Keywords: periodic cyclic homology
MaRDI profile type: Publication
OpenAlex ID: W2051296535
arXiv ID: 1103.5936
cites work: Higher \(K\)-theory via universal invariants
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cites work: Uniqueness of enhancement for triangulated categories
cites work: Invariance and localization for cyclic homology of DG algebras
cites work: Cyclic homology, comodules, and mixed complexes
cites work: Q3378982
DOI: 10.1017/IS011004009JKT155
published in: Journal of \(K\)-Theory
title: The fundamental theorem via derived Morita invariance, localization, and \(\mathbb A^{1}\)-homotopy invariance
review text: Let \(E\) be a functor from the category of dg categories to a triangulated category. The author says that \(E\) ``satisfies the fundamental theorem'' if, for any dg category \(A\), NEWLINE\[NEWLINEE(A[t,t^{-1}]) \cong E(A) \oplus \Sigma (E(A)).NEWLINE\]NEWLINE Here \(A[t,t^{-1}]\) is the tensor product of \(A\) with the algebra of Laurent polynomials.NEWLINENEWLINEThe author shows that if \(E\) is derived Morita invariant, localizing, and \(\mathbb A ^{1}\)-homotopy invariant, \(E\) satisfies the fundamental theorem. This result unifies and simplifies the proof of the fundamental theorem for homotopy algebraic \(K\)-theory in \textit{C. A. Weibel} [Contemp. Math. 83, 461--488 (1989; Zbl 0669.18007)] and the proof of the fundamental theorem for periodic cyclic homology in \textit{C. Kassel} [J. Algebra 107, 195--216 (1987; Zbl 0617.16015)]. The author notes applications to quasi-compact, quasi-separated schemes. The author's proof of the fundamental theorem is based on ideas introduced by him in [Int. Math. Res. Not. 2005, No. 53, 3309--3339 (2005; Zbl 1094.18006)].
reviewed by: Peter Haskell
description: scientific article; zbMATH DE number 6072566