Editing Building (mathematics) (section)

===Spherical building===
Let {{mvar|F}} be a [[field (mathematics)|field]] and let {{mvar|X}} be the simplicial complex with vertices the non-trivial vector subspaces of {{math|''V'' {{=}} ''F''<sup>''n''</sup>}}. Two subspaces {{math|''U''<sub>1</sub>}} and {{math|''U''<sub>2</sub>}} are connected if one of them is a subset of the other. The {{mvar|k}}-simplices of {{mvar|X}} are formed by sets of {{math|''k'' + 1}} mutually connected subspaces. Maximal connectivity is obtained by taking {{math|''n'' − 1}} proper non-trivial subspaces and the corresponding {{math|(''n'' − 1)}}-simplex corresponds to a ''[[Flag (linear algebra)|complete flag]]''

: {{math|(0) ⊂ ''U''<sub>1</sub> ⊂ ··· ⊂ ''U''<sub>''n'' – 1 </sub> ⊂ ''V''}}

Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces {{math|''U''<sub>''i''</sub>}}.

To define the apartments in {{mvar|X}}, it is convenient to define a ''frame'' in {{mvar|V}} as a basis ({{math|''v''<sub>''i''</sub>}}) determined up to scalar multiplication of each of its vectors {{math|''v''<sub>''i''</sub>}}; in other words a frame is a set of one-dimensional subspaces {{math|''L''<sub>''i''</sub> {{=}} ''F''·''v''<sub>''i''</sub>}} such that any {{mvar|k}} of them generate a {{mvar|k}}-dimensional subspace. Now an ordered frame {{math|''L''<sub>1</sub>, ..., ''L''<sub>''n''</sub>}} defines a complete flag via

: {{math|''U''<sub>''i''</sub> {{=}} ''L''<sub>1</sub> ⊕ ··· ⊕ ''L''<sub>''i''</sub>}}

Since reorderings of the various {{math|''L''<sub>''i''</sub>}} also give a frame, it is straightforward to see that the subspaces, obtained as sums of the {{math|''L''<sub>''i''</sub>}}, form a simplicial complex of the type required for an apartment of a spherical building. The axioms for a building can easily be verified using the classical [[Schreier refinement theorem|Schreier refinement argument]] used to prove the uniqueness of the [[Jordan–Hölder decomposition]].