Acyclic coefficient systems on buildings

[摘要] 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 $kge1$ 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.

[学科分类] 数学（综合）