Geodesic
Proof of~the Grothendieck--Serre conjecture on principal bundles over regular local rings containing a~field
Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 780-795

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Let $R$ be a regular local ring containing a field. Let $\mathbf{G}$ be a reductive group scheme over $R$. We prove that a principal $\mathbf{G}$-bundle over $R$ is trivial if it is trivial over the field of fractions of $R$. In other words, if $K$ is the field of fractions of $R$, then the map $$ H^1_{\mathrm{et}}(R,\mathbf{G})\to H^1_{\mathrm{et}}(K,\mathbf{G}) $$ of the non-Abelian cohomology pointed sets induced by the inclusion of $R$ in $K$ has trivial kernel. This result was proved in [1] for regular local rings $R$ containing an infinite field.
Keywords: reductive group schemes, principal bundles, Grothendieck–Serre conjecture. 
@article{IM2_2020_84_4_a6,
     author = {I. A. Panin},
     title = {Proof of~the {Grothendieck--Serre} conjecture on principal bundles over regular local rings containing a~field},
     journal = {Izvestiya. Mathematics },
     pages = {780--795},
     publisher = {mathdoc},
     volume = {84},
     number = {4},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a6/}
}
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I. A. Panin. Proof of~the Grothendieck--Serre conjecture on principal bundles over regular local rings containing a~field. Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 780-795. http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a6/