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.

让$ü$是一个域上的规则连接仿射半局部方案$ķ$. 让$G$是一个约简群方案$ü$. 假如说$G$有一个适当的抛物子群方案，我们证明以下陈述。给定一个仿射$ķ$-方案$W$， 一个校长$G$- 捆绑$W{\times }_{ķ}ü$如果它对投影的通用纤维微不足道，则微不足道$W{\times }_{ķ}ü\to ü$.

我们还简化了格洛腾迪克-塞尔猜想的证明：让$ü$是一个域上的规则连接仿射半局部方案$ķ$. 让$G$是一个约简群方案$ü$. 一个校长$G$- 捆绑 $ü$如果它在通用点上是微不足道的，那么它就是微不足道的$ü$.

我们将一些其他相关结果从简单的简单连接情况推广到任意约简群方案的情况。