# Homology of SL2 over function fields I: Parabolic subcomplexes

Homology of SL2 over function fields I: Parabolic subcomplexes AbstractThe present paper studies the group homology of the groups SL2⁡(k⁢[C]){\operatorname{SL}_{2}(k[C])}and PGL2⁡(k⁢[C]){\operatorname{PGL}_{2}(k[C])}, where C=C¯∖{P1,…,Ps}{C=\overline{C}\setminus\{P_{1},\dots,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 rank two vector bundles on the curve C¯{\overline{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]){\operatorname{SL}_{2}(k[C])}above degree s, generalizing a result of Suslin in the case s=1{s=1}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik de Gruyter

# Homology of SL2 over function fields I: Parabolic subcomplexes

, Volume 2018 (739): 47 – Jun 1, 2018
47 pages

/lp/degruyter/homology-of-sl2-over-function-fields-i-parabolic-subcomplexes-d33vkx2WIo
Publisher
De Gruyter
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5345
eISSN
1435-5345
D.O.I.
10.1515/crelle-2015-0047
Publisher site
See Article on Publisher Site

### Abstract

AbstractThe present paper studies the group homology of the groups SL2⁡(k⁢[C]){\operatorname{SL}_{2}(k[C])}and PGL2⁡(k⁢[C]){\operatorname{PGL}_{2}(k[C])}, where C=C¯∖{P1,…,Ps}{C=\overline{C}\setminus\{P_{1},\dots,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 rank two vector bundles on the curve C¯{\overline{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]){\operatorname{SL}_{2}(k[C])}above degree s, generalizing a result of Suslin in the case s=1{s=1}.

### Journal

Journal für die reine und angewandte Mathematikde Gruyter

Published: Jun 1, 2018

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