Skip to main content Accessibility help

1-Homotopy of Chevalley Groups

  • Matthias Wendt (a1)


In this paper, we describe the sheaves of 1-homotopy groups of a simply-connected Chevalley group G; these sheaves can be identified with the sheafification of certain unstable Karoubi-Villamayor K-groups.



Hide All
Abe83.Abe, E.. Whitehead groups of Chevalley groups over polynomial rings. Comm. Alg. 11 (1983), 12711307.
BCW76.Bass, H., Connell, E.H. and Wright, D.L.. Locally polynomial algebras are symmetric algebras. Invent. Math. 38 (1976), 279299.
BG73.Brown, K.S. and Gersten, S.M.. Algebraic K-theory as generalized sheaf cohomology. In: Algebraic K-theory I: Higher K-theories, Lecture Notes in Math. 341, pp. 266292. Springer (1973).
Bor91.Borel, A.. Linear algebraic groups. Graduate texts in mathematics 126. Springer (1991).
Che55.Chevalley, C.. Sur certains groupes simples. Tohoku Math. J.(2) 7(1955), 1466.
CHSW08.Cortiñas, G., Haesemeyer, C., Schlichting, M. and Weibel, C.A.. Cyclic homology, cdh-cohomology and negative K-theory. Ann. Math. 167 (2008), 125.
Con02.Conrad, B.D.. A modern proof of Chevalley's theorem on algebraic groups. J. Ramanujan Math. Soc. 17 (2002), 118.
CTS87.Colliot-Thélène, J.-L. and Sansuc, J.-J.. Principal homogeneous spaces under flasque tori: Applications. J. Algebra 106 (1987), 148205.
Dut99.Dutta, S.P.. A theorem on smoothness – Bass-Quillen, Chow groups and intersection multiplicities of Serre. Trans. Amer. Math. Soc. 352 (1999), 16351645.
Ger73.Gersten, S.M.. Higher K-theory of rings. In: Algebraic K-theory I: Higher Ktheories, Lecture Notes in Math. 341, 342. Springer (1973).
Gil08.Gille, S.. The first Suslin homology group of a split simply-connected semisimple algebraic group. Preprint (2008).
GJ99.Goerss, P.G. and Jardine, J.F.. Simplicial Homotopy Theory. Progress in Mathematics 174, Birkhäuser (1999).
GMV91.Grunewald, F.J., Mennicke, J. and Vaserstein, L.. On symplectic groups over polynomial rings. Math. Z. 206 (1991), 3556.
Hir03.Hirschhorn, P.S.. Model categories and their localizations. Mathematical Surveys and Monographs 99. American Mathematical Society (2003).
HT08.Hutchinson, K. and Tao, L.. A note on the Milnor-Witt K-theory and a theorem of Suslin. Comm. Alg. 36 (2008), 27102718.
Jar81.Jardine, J.F.. Algebraic homotopy theory, groups and K-theory. PhD thesis, University of British Columbia, 1981. Available as
Jar83.Jardine, J.F.. On the homotopy groups of algebraic groups. J. Algebra 81 (1983), 180201.
Jar00.Jardine, J.F.. Motivic symmetric spectra. Doc. Math. 5 (2000), 445552.
Jou73.Jouanolou, J.-P.. Une suite exacte de Mayer-Vietoris en K-théorie algébrique. In: Algebraic K-theory I: Higher K-theories, Lecture Notes in Math. 341, 293316. Springer (1973).
Kar73.Karoubi, M.. Périodicité de la K-théorie hermitienne. In: Algebraic K-theory III: Hermitian K-theory and Geometric Applications, Lecture Notes in Math. 343, 301411. Springer (1973).
KS82.Kopeiko, V.I. and Suslin, A.A.. Quadratic modules and the orthogonal group over polynomial rings. J. Math. Sci. 20 (1982), 26652691.
KV69.Karoubi, M. and Villamayor, O.. Foncteurs Kn en algèbre et en topologie. C.R. Acad. Sci. Paris Sér. A-B 269 (1969), A416A419.
Lam06.Lam, T.-Y.. Serre's problem on projective modules. Monographs in Mathematics. Springer (2006).
Lin81.Lindel, H.. On the Bass-Quillen conjecture concerning projective modules over polynomial rings. Invent. Math. 65 (1981), 319323.
Mat69.Matsumoto, H.. Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. École Norm. Sup. (4) 2 (1969), 162.
Mor06a.Morel, F.. 1-algebraic topology. In: International Congress of Mathematicians II, pp. 10351059. European Mathematical Society (2006).
Mor06b.Morel, F.. 1-algebraic topology over a field. Preprint (2006).
Mor07.Morel, F.. 1-classification of vector bundles over smooth affine schemes. Preprint (2007).
MV99.Morel, F. and Voevodsky, V.. 1-homotopy theory of schemes. Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45143.
Plo93.Plotkin, E.. Surjective stabilization of the K 1-functor for some exceptional Chevalley groups. J. Soviet. Math. 64 (1993), 751766.
Pop89.Popescu, D.. Polynomial rings and their projective modules. Nagoya Math. J. 113 (1989), 121128.
PV96.Plotkin, E. and Vavilov, N.. Chevalley groups over commutative rings I: Elementary calculations. Acta Appl. Math. 45 (1996), 73113.
Qui76.Quillen, D.G.. Projective modules over polynomial rings. Invent. Math. 36 (1976), 167171.
Ste79.Stein, M.R.. Stability theorems for K 1, K 2 and related functors modeled on Chevalley groups. Japan. J. Math.(N.S.) 4 (1978), 77108.
Sus77.Suslin, A.A.. The structure of the special linear group over rings of polynomials. Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 235252.
Sus87.Suslin, A.A.. Torsion in K 2 of fields. K-Theory 1 (1987), 529.
Vor81.Vorst, A.C.F.. The general linear group of polynomial rings over regular rings. Comm. Alg. 9 (1981), 499509.
Wei89.Weibel, C.A.. Homotopy algebraic K-theory. In: Algebraic K-theory and Algebraic Number Theory, Contemporary Mathematics 83, 461488. American Mathematical Society (1989).
Wen07.Wendt, M.. On fibre sequences in motivic homotopy theory. PhD thesis, Universität Leipzig (2007).



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed