On the relative Wall-Witt groups
type: Item
instance of: scholarly article
title: On the relative Wall-Witt groups
zbMATH Open document ID: 0828.19002
DOI: 10.1007/BF02783212
author: Yongjin Song
published in: Israel Journal of Mathematics
publication date: 1995-12-12T00:00:00Z
review text: Let \(R_*\) be a simplicial ring with involution and \(\varepsilon = \pm 1\). In ``Hermitian algebraic \(K\)-theory of simplicial rings and topological spaces'' [J. Math. Pures Appl., IX. Sér. 64, 175-235 (1985; Zbl 0597.18006)], \textit{D. Burghelea} and \textit{Z. Fiedorowicz} introduced hermitian \(K\)-theory \(_\varepsilon L (R_*)\), which splits \({1 \over 2}\)-locally into the product of the symmetric part \(_\varepsilon L^s (R_*)\) and the antisymmetric part \(_\varepsilon L^a (R_*)\), and they also proved that the antisymmetric part \(_\varepsilon L^a_n (R_*)\) is naturally isomorphic to the Wall-Witt group \(_\varepsilon W_n (\pi_0 R_*)\) of the discrete ring \(\pi_0 R_*\) for \(n \geq 1\). Using results of \textit{T. G. Goodwillie} from ``Relative algebraic \(K\)-theory and cyclic homology'' [Ann. Math., II. Ser. 124, 347-402 (1986; Zbl 0627.18004)] the author proves the following vanishing result for relative Wall-Witt groups: Suppose \(f : R \to S\) is a map of simplicial rings with involution such that the induced map \(\pi_0 R \to \pi_0 S\) is surjective with nilpotent kernel. Then \(_\varepsilon W_n (f) \otimes \mathbb{Q} = 0\) for \(n \geq 1\).
Mathematics Subject Classification ID: 19G38
zbMATH DE Number: 784917
zbMATH Keywords: hermitian \(K\)-theory
zbMATH Keywords: cyclic homology
zbMATH Keywords: Wall-Witt groups
reviewed by: Manfred Kolster
MaRDI profile type: Publication
cites work: K-Theory of Forms. (AM-98)
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full work available at URL: https://doi.org/10.1007/bf02783212
OpenAlex ID: W2075159685
description: scientific article; zbMATH DE number 784917