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The Grothendieck--Serre conjecture predicts that every generically trivialtorsor under a reductive group scheme $G$ over a regular local ring $R$ istrivial. The mixed characteristic case of the conjecture is widely open. Weconsider the following setup. Let $A$ be a mixed characteristic DVR, $G$ areductive group scheme over $A$, $X$ an irreducible smooth projective$A$-scheme, $\mathcal G$ a principal $G$-bundle over $X$. Suppose $\mathcal G$is generically trivial. We prove that in this case $\mathcal G$ is Zariskilocally trivial. This result confirms the conjecture.

On the Grothendieck--Serre conjecture for projective smooth schemes over a DVR