##
**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”.

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”.

Reviewer: K.Szymiczek (Tychy)

### Keywords:

\(K\)-theory of nonsplit forms; \(G\)-theory; spectral sequence; homogeneous line bundles; Brauer-Severi descent
Full Text:
DOI

### References:

[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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.