On the Grothendieck-Serre conjecture about principal bundles and its generalizations

Abstract

Let $U$ be a regular connected affine semilocal scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W{\times }_{k}U$ is trivial if it is trivial over the generic fiber of the projection $W{\times }_{k}U\to U$.

We also simplify the proof of the Grothendieck–Serre conjecture: let $U$ be a regular connected affine semilocal scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$.

We generalize some other related results from the simple simply connected case to the case of arbitrary reductive group schemes.