Journal Article

On Grothendieck–Serre’s conjecture concerning principal $G$-bundles over reductive group schemes: I

Vavilov, N., Stavrova, A., Panin, I.

2015

Overview

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 $$\\begin{eqnarray}H_{\\acute{\\text{e}}\\text{t}}^{1}(R\\otimes _{k}A,G)\\rightarrow H_{\\acute{\\text{e}}\\text{t}}^{1}(K\\otimes _{k}A,G)\\end{eqnarray}$$ induced by the inclusion of $R$ 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.

• Publisher: London Mathematical Society,Cambridge University Press
• Subject: Algebraic group theory