## $$B$$-structures on $$G$$-bundles and local triviality.(English)Zbl 0874.14043

Let $$G$$ be a split reductive group scheme over $$\mathbb{Z}$$ (recall that for any algebraically closed field $$k$$ there is a bijection $$G\mapsto G\otimes k$$ 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 étale base change $$S'\to S$$.
Theorem 2. Any $$G$$-bundle on $$X$$ becomes Zariski-locally trivial after a suitable étale base change $$S'\to S$$.
Theorem 3. Suppose that $$G$$ is semisimple. Let $$D$$ be a subscheme of $$X$$ such that the projection $$D\to S$$ is an isomorphism. Set $$U:=X\setminus D$$. Then for any $$G$$-bundle $$F$$ on $$X$$ its restriction to $$U$$ becomes trivial after a suitable faithfully flat base change $$S'\to S$$ with $$S'$$ being locally of finite presentation over $$S$$. If $$S$$ is a scheme over $$\mathbb{Z}[n^{-1}]$$ where $$n$$ is the order of $$\pi_1(G(\mathbb{C}))$$ then $$S'$$ can be chosen to be étale over $$S$$.

### MSC:

 14L30 Group actions on varieties or schemes (quotients) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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