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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”.

MSC:

11E70 \(K\)-theory of quadratic and Hermitian forms
19G99 \(K\)-theory of forms
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[1] J. F. Adams, Lectures on Lie groups , W. A. Benjamin, Inc., New York-Amsterdam, 1969. · Zbl 0206.31604
[2] S. Araki, Hopf structures attached to \(K\)-theory; Hodgkin’s theorem , Ann. of Math. (2) 85 (1967), 508-525. JSTOR: · Zbl 0171.43903
[3] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces , Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 7-38. · Zbl 0108.17705
[4] H. Gillet, Homological descent for the \(K\)-theory of coherent sheaves , Algebraic \(K\)-theory, number theory, geometry and analysis (Bielefeld, 1982), Lecture Notes in Math., vol. 1046, Springer, Berlin, 1984, pp. 80-103. · Zbl 0557.14009
[5] L. Hodgkin, On the \(K\)-theory of Lie groups , Topology 6 (1967), 1-36. · Zbl 0186.57103
[6] M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces , Invent. Math. 92 (1988), no. 3, 479-508. · Zbl 0651.18008
[7] M. Levine, The algebraic \(K\)-theory of the classical groups and some twisted forms , Math. Sci. Res. Inst. preprint 06125-90, 1990. · Zbl 0798.11009
[8] M. Levine, V. Srinivas, and J. Weyman, \(K\)-theory of twisted Grassmannians , \(K\)-Theory 3 (1989), no. 2, 99-121. · Zbl 0705.14007
[9] A. Merkurjev and A. A. Suslin, \(K\)-cohomology of Severi-Brauer varieties and norm residue homomorphism , Math. USSR-Izv. 21 (1983), 307-340. · Zbl 0525.18008
[10] A. Merkurjev and A. A. Suslin, On the norm residue symbol of degree three , LOMI preprint, 1986. · Zbl 1200.14041
[11] I. Panin, \(K\)-theory of Grassmann fibre bundles and its twisted forms , LOMI preprint E-5-89, 1989.
[12] I. A. Panin, On algebraic \(K\)-theory of generalized flag fiber bundles and some of their twisted forms , Algebraic \(K\)-theory, Adv. Soviet Math., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 21-46. · Zbl 0744.18007
[13] I. A. Panin, A splitting principle and the algebraic \(K\)-theory of some homogeneous varieties , preprint, 1991.
[14] D. Quillen, Higher algebraic \(K\)-theory. I , Algebraic \(K\)-theory, I: Higher \(K\)-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, 85-147. Lecture Notes in Math., Vol. 341. · Zbl 0292.18004
[15] M. Rost, Hilbert theorem 90 for \(K_3\) for degree-two extensions , preprint, 1986.
[16] A. A. Suslin, \(K\)-theory and \(K\)-cohomology of certain group varieties , Algebraic \(K\)-theory, Adv. Soviet Math., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 53-74. · Zbl 0746.19004
[17] R. G. Swan, \(K\)-theory of quadric hypersurfaces , Ann. of Math. (2) 122 (1985), no. 1, 113-153. JSTOR: · Zbl 0601.14009
[18] R. W. Thomason and T. Trobaugh, Higher algebraic \(K\)-theory of schemes and of derived categories , The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435. · Zbl 0731.14001
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