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Euler class groups and the homology of elementary and special linear groups
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Schlichting, Marco (2017) Euler class groups and the homology of elementary and special linear groups. Advances in Mathematics, 320 . pp. 1-81. doi:10.1016/j.aim.2017.08.034 ISSN 0001-8708.
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WRAP-Euler-class-groups-homology-elementary-special-linear-groups-Schlichting-2018.pdf - Accepted Version - Requires a PDF viewer. Available under License Creative Commons Attribution Non-commercial No Derivatives 4.0. Download (921Kb) | Preview |
Official URL: https://doi.org/10.1016/j.aim.2017.08.034
Abstract
We improve homology stability ranges for elementary and special linear groups over rings with many units. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor-Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor-Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the Euler class is necessary and sufficient for an oriented projective module P of rank d to split off a rank 1 free direct summand. Along the way we obtain a new presentation of Milnor-Witt K-theory and of symplectic K2 simplifying the classical Matsumoto-Moore presentation.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Homology theory, Linear algebraic groups, Rings (Algebra) | ||||||||
| Journal or Publication Title: | Advances in Mathematics | ||||||||
| Publisher: | Academic Press | ||||||||
| ISSN: | 0001-8708 | ||||||||
| Official Date: | 7 November 2017 | ||||||||
| Dates: |
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| Volume: | 320 | ||||||||
| Page Range: | pp. 1-81 | ||||||||
| DOI: | 10.1016/j.aim.2017.08.034 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 14 August 2017 | ||||||||
| Date of first compliant Open Access: | 4 September 2018 |
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