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

Abstract

AbstractLet$k$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.