Editing Building (mathematics) (section)

===Affine building===
Let {{mvar|K}} be a field lying between {{math|'''Q'''}} and its [[p-adic number|{{mvar|p}}-adic completion]] {{math|'''Q'''<sub>''p''</sub>}} with respect to the usual [[Archimedean property|non-Archimedean]] [[p-adic norm|{{mvar|p}}-adic norm]] {{math|{{norm|''x''}}<sub>''p''</sub>}} on {{math|'''Q'''}} for some prime {{mvar|p}}. Let {{mvar|R}} be the [[subring]] of {{mvar|K}} defined by

:{{math|''R'' {{=}} { ''x'' : {{norm|''x''}}<sub>''p''</sub> ≤ 1 } }}

When {{math|''K'' {{=}} '''Q'''}}, {{mvar|R}} is the [[Localization of a ring|localization]] of {{math|'''Z'''}} at {{mvar|p}} and, when {{math|''K'' {{=}} '''Q'''<sub>''p''</sub>}}, {{math|''R'' {{=}} '''Z'''<sub>''p''</sub>}}, the [[p-adic integer|{{mvar|p}}-adic integers]], i.e. the closure of {{math|'''Z'''}} in {{math|'''Q'''<sub>''p''</sub>}}.

The vertices of the building {{mvar|X}} are the {{mvar|R}}-lattices in {{math|''V'' {{=}} ''K''<sup>''n''</sup>}}, i.e. {{mvar|R}}-[[submodules]] of the form

:{{math|''L'' {{=}} ''R''·''v''<sub>1</sub> ⊕ ··· ⊕  ''R''·''v''<sub>''n''</sub>}}

where {{math|(''v''<sub>''i''</sub>)}} is a basis of {{mvar|V}} over {{mvar|K}}. Two lattices are said to be ''equivalent'' if one is a scalar multiple of the other by an element of the multiplicative group {{math|''K''*}} of {{mvar|K}} (in fact only integer powers of {{mvar|p}} need be used). Two lattices {{math|''L''<sub>1</sub>}} and {{math|''L''<sub>2</sub>}} are said to be ''adjacent'' if some lattice equivalent to {{math|''L''<sub>2</sub>}} lies between {{math|''L''<sub>1</sub>}} and its sublattice {{math|''p''·''L''<sub>1</sub>}}: this relation is symmetric. The {{mvar|k}}-simplices of {{mvar|X}} are equivalence classes of {{math|''k'' + 1}} mutually adjacent lattices, The {{math|(''n'' − 1)}}-simplices correspond, after relabelling, to chains

:{{math|''p''·''L''<sub>''n''</sub> ⊂ ''L''<sub>1</sub> ⊂ ''L''<sub>2</sub> ⊂ ··· ⊂ ''L''<sub>''n'' – 1 </sub> ⊂ ''L''<sub>''n''</sub>}}

where each successive quotient has order {{mvar|p}}. Apartments are defined by fixing a basis {{math|(''v''<sub>''i''</sub>)}} of {{mvar|V}} and taking all lattices with basis {{math|(''p''<sup>''a''<sub>''i''</sub></sup> ''v''<sub>''i''</sub>)}} where {{math|(''a''<sub>''i''</sub>)}} lies in {{math|'''Z'''<sup>''n''</sup>}} and is uniquely determined up to addition of the same integer to each entry.

By definition each apartment has the required form and their union is the whole of {{mvar|X}}. The second axiom follows by a variant of the Schreier refinement argument. The last axiom follows by a simple counting argument based on the orders of finite Abelian groups of the form

:{{math|''L'' + ''p''<sup>''k''</sup> ·''L''<sub>''i''</sub> / ''p''<sup>''k''</sup> ·''L''<sub>''i''</sub>}}

A standard compactness argument shows that {{mvar|X}} is in fact independent of the choice of {{mvar|K}}. In particular taking {{math|''K'' {{=}} '''Q'''}}, it follows that {{mvar|X}} is countable. On the other hand, taking {{math|''K'' {{=}} '''Q'''<sub>''p''</sub>}}, the definition shows that {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} admits a natural simplicial action on the building.

The building comes equipped with a ''labelling'' of its vertices with values in {{math|'''Z''' / ''n'''''Z'''}}. Indeed, fixing a reference lattice {{mvar|L}}, the label of {{mvar|M}} is given by

:{{math|label(''M'') {{=}} log<sub>''p''</sub> {{abs|''M'' / ''p''<sup>''k''</sup> ''L''}} modulo ''n''}}

for {{mvar|k}} sufficiently large. The vertices of any {{math|(''n'' – 1)}}-simplex in {{mvar|X}} has distinct labels, running through the whole of {{math|'''Z''' / ''n'''''Z'''}}. Any simplicial automorphism {{mvar|φ}} of {{mvar|X}} defines a permutation {{mvar|π}} of {{math|'''Z''' / ''n'''''Z'''}} such that {{math|label(''φ''(''M'')) {{=}} ''π''(label(''M''))}}. In particular for {{mvar|g}} in {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}},

:{{math|label(''g''·''M'') {{=}} label(''M'') + log<sub>''p''</sub> {{norm|det ''g''}}<sub>''p''</sub> modulo ''n''}}.

Thus {{mvar|g}} preserves labels if {{mvar|g}} lies in {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}.