Localizations of quasi-algebraic extensions

The article is part of a project aimed at describing the subgroup lattice of a Chevalley group over a ring. Namely, we consider the subgroups of a Chevalley group \(G(\Phi ,A)\) $G(\mathrm{\Phi },A)$ , containing the elementary subgroup \(E(\Phi ,R)\) $E(\mathrm{\Phi },R)$ over a subring R of A. Conjecturally, if \(\Phi \) $\mathrm{\Phi }$ is a simply laced root system, then the lattice is standard if and only if the ring extension \(R\subseteq A\) $R\subseteq A$ is quasi-algebraic. Roughly speaking, a ring extension \(R\subseteq A\) $R\subseteq A$ is quasi-algebraic if it cannot be reduced to the extension \(F\subseteq F[\hspace{0.55542pt}t]\) $F\subseteq F[t]$ , where F is a field and t is an independent variable, by taking subrings of A and quotients. To prove the conjecture we plan to use a localization method. The first step is to consider the case when R is semilocal. We show that in this case all finitely generated R-subalgebras of A are semilocal as well. In fact, the latter condition is even equivalent, thus, we obtain a new characterization of quasi-algebraic extensions.