The constant case of the Grothendieck-Serre conjecture is proved for reductive group schemes over any geometrically regular local algebra over a mixed-characteristic DVR.
On the Gille theorem for the relative projective line
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abstract
Let $X$ be a Noetherian separated scheme. Let $G$ be a reductive $X$-group scheme, and let $E$ be a principal $G$-bundle over $\mathbb{P}^1_X$. We prove that if the restriction of $E$ to $\infty\times X$ is Zariski locally trivial, then $E$ is itself Zariski locally trivial.
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Constant case of the Grothendieck-Serre conjecture in mixed characteristic
The constant case of the Grothendieck-Serre conjecture is proved for reductive group schemes over any geometrically regular local algebra over a mixed-characteristic DVR.