On the relative Wall-Witt groups
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 \(n \geq 1\).
scientific article; zbMATH DE number 784917