Hermitian \(K\)-theory on a theorem of Giffen
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 \(2\).
scientific article; zbMATH DE number 2153593