Generalized blueprints and \(p\)-adic buildings. I (Q1302101)

\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.