Suppose \(M\) is a free module of finite rank over a Dedekind domain \(R\). Let \({\mathcal S}(M)\) be the partially ordered set whose elements are \(\{(M_ 1, M_ 2)\mid M_ 1\), \(M_ 2\) are non-zero submodules of \(M\), \(M=M_ 1 \oplus M_ 2\}\) with inclusion relation given by \((M_ 1,M_ 2)\leq (M_ 1', M_ 2')\) if \(M_ 1\subseteq M_ 1'\) and \(M_ 2\supseteq M_ 2'\). R. M. Charney [Invent Math. 56, 1-17 (1980; Zbl 0427.18013)] showed that the simplicial object corresponding to \({\mathcal S}(M)\) has the homotopy type of a bouquet of spheres and therefore is spherical – i.e., has reduced homology which vanishes except in the top dimension. Our starting point is the observation that when \(R\) is a field, this result may by interpreted as a statement about a certain fibre space over the Tits building of \(\text{GL}(n,R)\).
In this work we define the notion of split building \({\mathcal S}(V)\) for any finite dimensional vector space \(V\) over a field, which has a sesquilinear (or bilinear) form. In case the form is zero \({\mathcal S}(V)\) coincides with the notion above. If \(G\) is any connected reductive linear algebraic group defined over a field \(k\), we also define the split building \({\mathcal S}(G,k)\) of \(G\) with respect to \(k\). Both \({\mathcal S}(V)\) and \({\mathcal S}(G,k)\) are locally finite posets.