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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2018, Issue 739


Homology of SL2 over function fields I: Parabolic subcomplexes

Matthias Wendt
Published Online: 2015-09-23 | DOI: https://doi.org/10.1515/crelle-2015-0047


The present paper studies the group homology of the groups SL2(k[C]) and PGL2(k[C]), where C=C¯{P1,,Ps} is a smooth affine curve over an algebraically closed field k. It is well known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of rank two vector bundles on the curve C¯. There is a natural subcomplex consisting of cells with suitably non-trivial isotropy group. The paper provides explicit formulas for the equivariant homology of this “parabolic subcomplex”. These formulas also describe group homology of SL2(k[C]) above degree s, generalizing a result of Suslin in the case s=1.


  • [1]

    P. Abramenko and K. S. Brown, Buildings, Grad. Texts in Math. 248, Springer, New York 2008. Google Scholar

  • [2]

    M. F. Atiyah, On the Krull–Schmidt theorem with applications to sheaves, Bull. Soc. Math. France 84 (1956), 307–317. Google Scholar

  • [3]

    M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), no. 1, 181–207. CrossrefGoogle Scholar

  • [4]

    K. S. Brown, Cohomology of groups. Corrected reprint of the 1982 original, Grad. Texts in Math. 87, Springer, New York 1994. Google Scholar

  • [5]

    J. L. Dupont, Scissors congruences, group homology and characteristic classes, Nankai Tracts Math. 1, World Scientific, Hackensack 2001. Google Scholar

  • [6]

    E. M. Friedlander and G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 59 (1984), 347–361. CrossrefGoogle Scholar

  • [7]

    R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. Google Scholar

  • [8]

    K. Hutchinson, A Bloch–Wigner complex for SL2, J. K-theory 12 (2013), no. 1, 15–68. Google Scholar

  • [9]

    K. Hutchinson, A refined Bloch group and the third homology of SL2 of a field, J. Pure Appl. Algebra 217 (2013), 2003–2035. Google Scholar

  • [10]

    K. Hutchinson, On the low-dimensional homology of SL2(k[T,T-1]), J. Algebra 425 (2015), 324–366. Google Scholar

  • [11]

    K. P. Knudson, Homology of linear groups, Progr. Math. 193, Birkhäuser, Basel 2001. Google Scholar

  • [12]

    G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl. 56 (1994), no. 3, 293–300. CrossrefGoogle Scholar

  • [13]

    A. D. Rahm, The homological torsion of PSL2 of the imaginary quadratic integers, Trans. Amer. Math. Soc. 365 (2013), no. 3, 1603–1635. Google Scholar

  • [14]

    A. D. Rahm, Accessing the cohomology of discrete groups above their virtual cohomological dimension, J. Algebra 404 (2014), no. C, 152–175. CrossrefWeb of ScienceGoogle Scholar

  • [15]

    A. D. Rahm and M. Wendt, On Farrell–Tate cohomology of SL2 over S-integers, preprint (2014), http://arxiv.org/abs/1411.3542.

  • [16]

    M. Rosen, S-units and S-class group in algebraic function fields, J. Algebra 26 (1973), 98–108. CrossrefGoogle Scholar

  • [17]

    J.-P. Serre, Trees, Springer, Berlin 1980. Google Scholar

  • [18]

    U. Stuhler, Zur Frage der endlichen Präsentierbarkeit gewisser arithmetischer Gruppen im Funktionenkörperfall, Math. Ann. 224 (1976), no. 3, 217–232. CrossrefGoogle Scholar

  • [19]

    U. Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980), no. 3, 263–281. CrossrefGoogle Scholar

  • [20]

    A. A. Suslin, K3 of a field and the Bloch group (Russian), Galois theory, rings, algebraic groups and their applications. Collected papers, Tr. Mat. Inst. Steklova 183, Nauka, Leningrad (1990), 180–199, 229; translation in Proc. Steklov Inst. Math. 183 (1991), 217–239. Google Scholar

About the article

Received: 2014-04-29

Published Online: 2015-09-23

Published in Print: 2018-06-01

This work has partially been supported by the Alexander-von-Humboldt-Stiftung.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 739, Pages 159–205, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0047.

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