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Homology stability for unitary groups over S-arithmetic rings

Published online by Cambridge University Press:  05 November 2010

G. Collinet
G. Collinet
Affiliation:
Institut de Recherche Mathématiques Avancées, UMR 7501 de l'Université de Strasbourg et du CNRS. gael.collinet@math.unistra.fr
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Abstract

We prove that the homology of unitary groups over rings of S-integers in number fields stabilizes. Results of this kind are well known to follow from the high acyclicity of ad-hoc polyhedra. Given this, we exhibit two simple conditions on the arithmetic of hermitian forms over a ring A relatively to an anti-automorphism which, if they are satisfied, imply the stabilization of the homology of the corresponding unitary groups. When R is a ring of S-integers in a number field K, and A is a maximal R-order in an associative composition algebra F over K, we use the strong approximation theorem to show that both of these properties are satisfied. Finally we take a closer look at the case of On(ℤ[½]).

Keywords

Homology stabilityunitary groupsS-arithmetic groupsadjacency relationsstrong approximation
Type
Research Article
Information
Journal of K-Theory , Volume 8 , Issue 2 , October 2011 , pp. 293 - 322
Copyright
Copyright © ISOPP 2010

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