Homology of SL2 over function fields I: parabolic subcomplexes

2014/04/23 by Matthias Wendt, Wendt, Matthias
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #math.KT #msc:20E42 #msc:20G10

paper · pdf · doi:10.48550/arxiv.1404.5825

38 pages

arxiv created 2014/04/23 · arxiv updated 2014/04/24

Abstract

The present paper studies the homology of the groups SL₂(k[C]) and GL₂(k[C]) where C=C∖\P₁,…,P_s\ 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 vector bundles on the complete curve. There is a natural subcomplex of cells with non-unipotent isotropy group. The paper provides explicit formulas for the equivariant homology of this "parabolic subcomplex". These formulas also describe the homology of SL₂(k[C]) above degree s, with finite coefficients away from the characteristic of k, generalizing a result of Suslin for the case s=1.

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