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The p-periodicity of the groups GL (n, Os(K)) and SL(n, Os(K))

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  26 February 2010

B. Bürgisser  and
B. Eckmann
B. Bürgisser
Affiliation:
Eidgenössische Technische Hochschule, Mathematik, ETH-Zentrum, CH-8092 Zürich, Switzerland.
B. Eckmann
Affiliation:
Eidgenössische Technische Hochschule, Mathematik, ETH-Zentrum, CH-8092 Zürich, Switzerland.
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Extract

In this paper we investigate the p-periodicity of the S-arithmetic groups G = GL(n, Os(K)) and G1 = SL(n, Os(K)) where Os(K) is the ring of S-integers of a number field K (cf. [12, 13]; S is a finite set of places in K including the infinite places). These groups are known to be virtually of finite (cohomological) dimension, and thus the concept of p-periodicity is defined; it refers to a rational prime p and to the p-primary component Ĥi(G, A, p) of the Farrell-Tate cohomology Ĥi(G, A) with respect to an arbitrary G-module A. We recall that Ĥi coincides with the usual cohomology Hi for all i above the virtual dimension of G, and that in the case of a finite group (i.e., a group of virtual dimension zero) the Ĥi, i ∈ℤ, are the usual Tate cohomology groups. The group G is called p-periodic if Ĥi(G, A, p) is periodic in i, for all A, and the smallest corresponding period is then simply called the p-period of G. If G has no p-torsion, the p-primary component of all its Ĥi is 0, and thus G is trivially p-periodic.

MSC classification

Secondary: 20G10: Cohomology theory
Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 89 - 97
Copyright
Copyright © University College London 1984

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