$B$-structures on $G$-bundles and local triviality

Abstract

Let G be a split reductive group scheme over Z (recall that for any algebraically closed field k there is a bijection G G k between isomorphism classes of such group schemes and isomorphism classes of connected reductive algebraic groups over k). Let B be a Borel subgroup of G. Let S be a scheme and X a smooth proper scheme over S with connected geometric fibers of pure dimension 1. Our goal is to prove the following theorems.