Acyclic coefficient systems on buildings

TL;DR

This paper establishes local criteria for the vanishing of cohomology and homology groups of coefficient systems on affine buildings, proving a conjecture related to hyperplane arrangements and generalizing existing acyclicity theorems.

Contribution

It introduces new local criteria for acyclicity of coefficient systems on affine buildings and proves a conjecture of de Shalit, extending previous results by Schneider and Stuhler.

Findings

Proved a conjecture of de Shalit on acyclicity of hyperplane arrangement coefficient systems.

Established criteria ensuring vanishing of cohomology and homology groups for affine buildings.

Generalized existing acyclicity theorems to broader classes of coefficient systems.

Abstract

For cohomological (resp. homological) coefficient systems ${\mathcal F}$ (resp. ${\mathcal V}$) on affine buildings $X$ with Coxeter data of type $\widetilde{A}_d$ we give for any $k\ge1$ a sufficient local criterion which implies $H^k(X,{\mathcal F})=0$ (resp. $H_k(X,{\mathcal V})=0)$. 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.