$B$-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.Theorem 1.Any G-bundle on X admits a B-structure after a suitable surjective etale base change S → S.Theorem 2. Any G-bundle on X becomes Zariski-locally trivial after a suitable etale base change S → S.Then for any G-bundle F on X its restriction to U becomes trivial after a suitable faithfully flat base change S → S with S being locally of finite presentation over S. If S is a scheme over Z[n -1 ] where n is the order of π 1 (G(C)) then S can be chosen to be etale over S.2. Remarks.a) Theorem 2 follows from Theorem 1 because a Bbundle on any scheme is Zariski-locally trivial.b) If S is the spectrum of an algebraically closed field Theorems 1-3 are well known (of course in this case base change is not necessary).In this situation Theorem 3 was proved in [6], while Theorems 1 and 2 follow from the triviality of G-bundles over the generic point of X.The triviality of the Galois cohomology H 1 (k(X), G) was conjectured by J. P. Serre and proved by R. Steinberg [9] and A. Borel and T. A. Springer [3].Note that Steinberg's result is for a perfect field of dimension 1-inconvenient here since k(X) is not perfect in characteristic p (whereas of course it is of dimension 1).The restriction to perfect fields was due to the need to have G

Locations

• Mathematical Research Letters - View - PDF