On the cohomology of linear groups over imaginary quadratic fields

Dutour Sikirić, Mathieu; Gangl, Herbert; Gunnells, Paul E.; Hanke, Jonathan; Schürmann, Achill; Yasaki, Dan

Abstract

Let Γ be the group GLN(OD), where OD is the ring of integers in the imaginary quadratic field with discriminant D<0. In this paper we investigate the cohomology of Γ for N=3,4 and for a selection of discriminants: D≥−24 when N=3, and D=−3,−4 when N=4. In particular we compute the integral cohomology of Γ up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for Γ developed by Ash [4, Ch. II] and Koecher [24]. Our results extend work of Staffeldt [40], who treated the case N=3, D=−4. In a sequel [15] to this paper, we will apply some of these results to computations with the K -groups K4(OD), when D=−3,−4.

Citation

Dutour Sikirić, M., Gangl, H., Gunnells, P. E., Hanke, J., Schürmann, A., & Yasaki, D. (2016). On the cohomology of linear groups over imaginary quadratic fields. Journal of Pure and Applied Algebra, 220(7), 2564-2589. https://doi.org/10.1016/j.jpaa.2015.12.002

• Journal Article Type: Article
• Acceptance Date: Dec 4, 2015
• Online Publication Date: Jan 19, 2016
• Publication Date: Jul 1, 2016
• Deposit Date: Dec 14, 2015
• Publicly Available Date: Jan 19, 2017
• Journal: Journal of Pure and Applied Algebra
• Print ISSN: 0022-4049
• Electronic ISSN: 1873-1376
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 220
• Issue: 7
• Pages: 2564-2589
• DOI: https://doi.org/10.1016/j.jpaa.2015.12.002
• Public URL: https://durham-repository.worktribe.com/output/1396231