Hermitian \(K\)-theory on a theorem of Giffen
type: Item
instance of: scholarly article
title: Hermitian \(K\)-theory on a theorem of Giffen
zbMATH Open document ID: 1070.19006
author: Marco Schlichting
published in: \(K\)-Theory
publication date: 2005-04-07T00:00:00Z
review text: Let \(R\) be a ring with involution \(a \mapsto \overline{a}\), and let \(\varepsilon\) be a central element in \(R\), so that \(\varepsilon\overline{\varepsilon} = 1\). Let \(_{\varepsilon}K_0^h(R)\) denote the Grothendieck group of isometry classes of finitely generated projective right \(R\)-modules equipped with a non-degenerate \(\varepsilon\)-hermitian form. In analogy to algebraic \(K\)-theory \textit{M. Karoubi} defined in ``Périodicité de la \(K\)-théorie hermitienne'' [Lect. Notes Math. 343, 301--411 (1973; Zbl 0274.18016)] the higher hermitian \(K\)-theory groups of \(R\) as the homotopy groups of the space \( _{\varepsilon}K^h(R) \cong\, _{\varepsilon}K_0^h(R) \times B_{\varepsilon}O(R).\) \smallskip Let \(h:K(R) \rightarrow \,_{\varepsilon}K^h(R)\) denote the hyperbolic functor from algebraic \(K\)-theory to hermitian \(K\)-theory. Giffen constructed a category \(_\varepsilon \mathcal W(R)\), which should yield a homotopy fibration \[ K(R) \rightarrow \,_{\varepsilon}K^h(R) \rightarrow \,_\varepsilon \mathcal W(R). \] The author points out that the proof of this result given by \textit{R. Charney} and \textit{R. Lee} [``On a theorem of Giffen'', Mich. Math. J. 33, 169--186 (1986; Zbl 0609.55004)] contains an error, and he provides a different proof, which applies more generally to split exact categories with duality, provided that \(2\) is invertible in the category. The author remarks that in the meantime he has been able to remove the assumption on the invertibility of \(2\).
reviewed by: Manfred Kolster
Mathematics Subject Classification ID: 19G38
Mathematics Subject Classification ID: 19D06
Mathematics Subject Classification ID: 11E70
zbMATH DE Number: 2153593
zbMATH Keywords: Hermitian \(K\)-theory
zbMATH Keywords: exact categories with duality
zbMATH Keywords: \(Q\)-construction
MaRDI profile type: Publication
full work available at URL: s10977-004-0929-5
OpenAlex ID: W2004118324
cites work: On a theorem of Giffen
cites work: LOCALIZATION IN HERMITIAN $K$-THEORY OF RINGS
cites work: Le théorème fondamental de la K-théorie hermitienne
cites work: Chain complexes and stable categories
cites work: Classifying spaces and fibrations
cites work: Higher algebraic K-theory: I
cites work: Hermitian \(\mathcal{U}\)-theory of exact categories with duality functors
cites work: First quadrant spectral sequences in algebraic k-theory via homotopy colimits
DOI: 10.1007/S10977-004-0929-5
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description: scientific article; zbMATH DE number 2153593