1 results
On Grothendieck–Serre’s conjecture concerning principal
$G$-bundles over reductive group schemes: I
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 3 / March 2015
- Published online by Cambridge University Press:
- 28 October 2014, pp. 535-567
- Print publication:
- March 2015
-
- Article
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Let
$k$ be an infinite field. Let 
$R$ be the semi-local ring of a finite family of closed points on a 
$k$-smooth affine irreducible variety, let 
$K$ be the fraction field of 
$R$, and let 
$G$ be a reductive simple simply connected 
$R$-group scheme isotropic over 
$R$. Our Theorem 1.1 states that for any Noetherian 
$k$-algebra 
$A$ the kernel of the map induced by the inclusion of
$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$
$R$ into 
$K$ is trivial. Theorem 1.2 for 
$A=k$ and some other results of the present paper are used significantly in Fedorov and Panin [A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field, Preprint (2013), arXiv:1211.2678v2] to prove the Grothendieck–Serre’s conjecture for regular semi-local rings 
$R$ containing an infinite field.
