On the cohomology of linear groups over imaginary quadratic fields

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• 作者：Mathieu Dutour Sikirić^a ; ^{mathieu.dutour@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ; Herbert Gangl^b ; ^{herbert.gangl@durham.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; ; Paul E. Gunnells^c ; ^{gunnells@math.umass.edu" class="auth_mail" title="E-mail the corresponding author} ; ; Jonathan Hanke^d ; ^{jonhanke@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ; Achill Schü ; rmann^e ; ^{achill.schuermann@uni-rostock.de" class="auth_mail" title="E-mail the corresponding author} ; ; Dan Yasaki^f ; ^{d_yasaki@uncg.edu" class="auth_mail" title="E-mail the corresponding author} ;
• 关键词：Primary ; 11F75 ; secondary ; 11F67 ; 20J06
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：220
• 期：7
• 页码：2564-2589
• 全文大小：735 K

文摘

Let Γ be the group GL_N(O_D)${\mathrm{GL}}_N({\mathcal{O}}_D)$, where O_D${\mathcal{O}}_D$ is the ring of integers in the imaginary quadratic field with discriminant D<0$D<0$. In this paper we investigate the cohomology of Γ for N=3,4$N=3,4$ and for a selection of discriminants: D≥−24$D\ge -24$ when 207a0652db0ec9b156d96dda83dbb" title="Click to view the MathML source">N=3$N=3$, and D=−3,−4$D=-3,-4$ when N=4$N=4$. 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 207a0652db0ec9b156d96dda83dbb" title="Click to view the MathML source">N=3$N=3$, D=−4$D=-4$. In a sequel [15] to this paper, we will apply some of these results to computations with the K  -groups 20583f83e8d6e593fa" title="Click to view the MathML source">K₄(O_D)$K_4({\mathcal{O}}_D)$, when D=−3,−4$D=-3,-4$.