The algebraic \(K\)-theory of the classical groups and some twisted forms. (English) Zbl 0798.11009

This paper appeared in preprint form in the MSRI preprint series in 1990.
From the author’s introduction: “Section 1 gives some spectral sequence machinery for computing the homotopy groups on an \(n\)-cube of spectra and applies this to a spectral sequence computing the \(G\)-theory of the complement of closed subsets \(Z_ 1, \dots, Z_ n\) of a scheme \(Z\) in terms of the \(G\)-theory of the \(Z_ i\). This is a generalization of Quillen’s \(G\)-theory localization sequence. We use this in Section 2 to give a spectral sequence for the \(G\)-theory of principal \((n\)- dimensional) torus bundle \(X\) over a scheme \(Y\) in terms of the \(G\)- theory of \(Y\) as a module over the representation ring \(R(T)\) of the torus \(T:=Gm^ n \dots \). Using the topological results of Atiyah- Hirzebruch-Hodgkin [M. Atiyah and F. Hirzebruch, Proc. Sympos. Pure Math. 3, 7-38 (1961; Zbl 0108.177); L. Hodgkin, Topology 6, 1-36 (1967; Zbl 0186.571)], the \(G\)-theory of \(G/T\) is generated over \(G_ * (S)\) by the classes of the homogeneous line bundles, which enables us to compute the spectral sequence, and thus the \(G\)-theory of \(G \dots\). In Section 3, we set up the machinery of Brauer- Severi descent which we then use in Section 4 to extend our computation of the \(K\)-theory of split forms of \(GL_ n\), \(SL_ n\), and \(Sp_ n\) to the nonsplit case (for inner forms) and to some of the principal homogeneous spaces”.


11E70 \(K\)-theory of quadratic and Hermitian forms
19G99 \(K\)-theory of forms
Full Text: DOI


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