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

TL;DR

This paper investigates the triviality of principal G-bundles over the affine line, introducing affine Grassmannians for group schemes and providing criteria for triviality, with implications for exotic bundles over A^1.

Contribution

It defines affine Grassmannians for group schemes, analyzes their Bruhat decompositions, and establishes new criteria for the triviality of principal bundles over A^1_U.

Findings

Principal G-bundles over A^1_U are not necessarily trivial.

Affine Grassmannians can be used to study bundle triviality.

Examples of non-trivial bundles that are not pull-backs from U.

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.