Affine Grassmannians of group schemes and exotic principal bundles over A^1

Abstract

Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.

Introduction re-written. Other minor improvements. Final version

References in corpus (1)

• A proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields

Cited by in corpus (4)

• On the Grothendieck-Serre conjecture on principal bundles in mixed characteristic
• On the Grothendieck-Serre Conjecture about principal bundles and its generalizations
• Counting Parabolic Principal G-bundles with Nilpotent Sections over ${\mathbb{P}}^1$
• R-equivalence on group schemes