Homology of SL2 over function fields I: parabolic subcomplexes

Original · EN

The present paper studies the homology of the groups SL₂(k[C]) and GL₂(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 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.