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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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References

1.Bass, H.. Algebraic K-theory (Benjamin, New York-Amsterdam, 1968).Google Scholar
2.Bürgisser, B.. Gruppen virtuell endlicher Dimension und Periodizitüt der Cohomologic. Diss. ETH, Nr. 6425 (Zürich, 1979).Google Scholar
3.Bürgisser, B.. On the p-periodicity of arithmetic subgroups of general linear groups. Comment. Math. Helv., 55 (1980), 499–509.CrossRefGoogle Scholar
4.Bürgisser, B.. Finite p-periodic quotients of general linear groups. Math. Ann., 256 (1981), 121–132.CrossRefGoogle Scholar
5.Bürgisser, B.. Yoneda product in Farrell Tate cohomology and periodicity. Preprint FIM, ETH Zürich (1981).Google Scholar
6.Bürgisser, B.. The p-torsion of the Farrell Tate cohomology of GL(Φk (p), O (K)), and SL(Φk(p), O(K)). Preprint F1M, ETH Zürich (1981).Google Scholar
7.Bürgisser, B.. On the projectivc class group of arithmetic groups. Math. Zeilschrift, 184 (1983), 339–357.CrossRefGoogle Scholar
8.Cartan, H. and Eilenberg, S.. Homological algebra (Princeton University Press, 1956).Google Scholar
9.Farrell, F. T.. An extension of Tate cohomology to a class of infinite groups. J. Pure Appl. Algebra, 10 (1977), 153–161.CrossRefGoogle Scholar
10.Huppert, B.. Endliche Gruppen I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellung, 134 (Springer, Berlin-Heidelberg -New York, 1967).Google Scholar
11.Janusz, G. J.. Algebraic number fields. Pure and Appl. Math., 55 (Academic Press, New York, 1973).Google Scholar
12.Serre, J.-P.. Cohomologie des groupes discrets. Prospects in Math., Ann. Math. Study, 70 (1971), 77–169.Google Scholar
13.Serre, J.-P.. Arithmetic groups. Proc. of the Sept. 1977 Durham conference on homological and combinatorial techniques in group theory. Edited by Wall, C. T. C.. (Cambridge University Press, 1979).Google Scholar
14.Swan, R. G.. The p-period of a finite group. Illinois J. Math., 4 (1960), 341–346.CrossRefGoogle Scholar
15.Swan, R. G. and Evans, E. G.. K-theory of finite groups and orders, Lecture Notes in Math., 149 (Springer, Berlin Heidelberg New York, 1970).CrossRefGoogle Scholar