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Acyclic coefficient systems on buildings

Published online by Cambridge University Press:  21 April 2005

Elmar Grosse-Klönne
Affiliation:
Mathematisches Institut der Universität Münster, Einsteinstrasse 62, D-48149 Münster, [email protected]
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Abstract

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For cohomological (respectively homological) coefficient systems ${\mathcal F}$ (respectively ${\mathcal V}$) on affine buildings X with Coxeter data of type $\widetilde{A}_d$, we give for any $k\ge1$ a sufficient local criterion which implies $H^k(X,{\mathcal F})=0$ (respectively $H_k(X,{\mathcal V})=0$). Using this criterion we prove a conjecture of de Shalit on the acyclicity of coefficient systems attached to hyperplane arrangements on the Bruhat–Tits building of the general linear group over a local field. We also generalize an acyclicity theorem of Schneider and Stuhler on coefficient systems attached to representations.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2005