ABSTRACT
One summarizes the essential definitions and results of reductive S-groups schemes which are useful for our purpose of defining Schubert Schemes and their associated canonical and functorial Smooth Resolutions and for extending to this setting the buildings constructions. In fact, it is remarked, that both A. Grothendieck reductive group schemes and Tits Buildings are both inspired by the fundamental Cl. Chevalley's Tohoku paper. The principal objects associated with a building become in the schematic context twisted locally constant finite S-schemes. The representability of the (R)-subgroups functor implies the representability of functors of more restricted classes of subgroups allowing to introduce the analogues of Convex Hull subcomplexes of a Building in the relative frame. The Parabolic Subgroups Functor is obtained naturally from the definition and is representable by a smooth and projective S-scheme with integral geometric fibers.
