On the representation of subspaces of chain geometries (Q1194374)

Let us consider a unitary commutative and associative algebra \(\mathbb{A}\) over a field \(\mathbb{F}\) with \(\text{char}(\mathbb{F})\neq 2\) and the chain space \(\Sigma(\mathbb{F},\mathbb{A})\) [i.e. the projective line \(\mathbb{P}(\mathbb{A})\)] defined over \(\mathbb{A}\). The author characterizes in geometrical terms subspaces of \(\Sigma(\mathbb{F},\mathbb{A})\) determined by subalgebras of \(\mathbb{A}\). A subspace of \(\Sigma(\mathbb{F},\mathbb{A})\) is any set \(S\) of points such that for every chain \(C\) either \(C\subseteq S\) or \(| C\cap S|<3\). \(S\) is connected if any two of its points can be joined by a finite sequence of intersecting chains contained in \(S\). Assume \(\mathbb{A}\) to contain for every \(a\) at least two \(\lambda\) such that \(a+\lambda\) is invertible. Then every nontrivial connected subspace of \(\Sigma(\mathbb{F},\mathbb{A})\) is the connected component of \(\gamma(\infty)\) in \(\gamma(S)\), where \(S=\mathbb{P}(U)\) is determined by a subalgebra \(U\) of \(\mathbb{A}\) and \(\gamma\) is in \(\mathbf{PGL}(2,\mathbb{A})\).