Generalized blueprints and \(p\)-adic buildings. I
type: Item
instance of: scholarly article
title: Generalized blueprints and \(p\)-adic buildings. I
zbMATH Open document ID: 0935.20019
DOI: 10.1023/A:1004968205498
author: Michael A. Abramson
published in: Geometriae Dedicata
publication date: 2000-05-04T00:00:00Z
review text: \textit{M. A. Ronan} and \textit{J. Tits} [Math. Ann. 278, 291-306 (1987; Zbl 0628.51001)] constructed buildings using the notion of a blueprint, which is a kind of labelled fundamental domain (in the geometric sense) of the structure of all rank 2 residues of the building (it is a labelling of a foundation). Not every building can be constructed using a blueprint, and not every blueprint gives rise to a building (it always gives rise to a certain chamber system, though). In the affine case (of rank \(\geq 4\)), all examples of buildings constructed with blueprints are of Laurent series type (they are Moufang and the characteristic of the residue field is equal to the characteristic of the field). No blueprint construction is known for the \(p\)-adic affine buildings. In the paper under review, the author defines a generalized blueprint (a clever and simple generalization of a blueprint) and shows that explicit examples give rise to \(p\)-adic affine buildings (over simply laced diagrams). The idea is to start with an ordinary blueprint and get certain maps into play in order to ``twist'' this ordinary blueprint. When all maps are trivial, the generalized blueprint is equivalent to the original ordinary one. For generalized blueprints, one can then define a chamber system as in the ordinary blueprint case. The criterion to have a building is the same as in the ``non-twisted'' case. Then the author constructs a generalized blueprint for any simply-laced affine diagram, any field \(K\) with discrete valuation and any uniformizer in that field, and he proves that this particular generalized blueprint defines an affine building (which will be proved to be the building over \(K\) with corresponding diagram only in Part II). The construction uses heavily some specific properties of affine Coxeter reflection groups.
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 1335047
zbMATH Keywords: \(p\)-adic affine buildings
zbMATH Keywords: simply laced diagrams
zbMATH Keywords: generalized blueprints
zbMATH Keywords: chamber systems
zbMATH Keywords: affine Coxeter reflection groups
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
full work available at URL: a:1004968205498
OpenAlex ID: W90820618
description: scientific article; zbMATH DE number 1335047