Extension theorems for reductive group schemes

Abstract

We prove several basic extension theorems for reductive group schemes via extending Lie algebras and via taking schematic closures. We also prove that, for each scheme $Y$, the category in groupoids of adjoint group schemes over $Y$ whose Lie algebra ${\mathcal{O}}_Y$-modules have perfect Killing forms is isomorphic, via the differential functor, to the category in groupoids of Lie algebra ${\mathcal{O}}_Y$-modules which have perfect Killing forms and which, as ${\mathcal{O}}_Y$-modules, are coherent and locally free.