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\(B\)-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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