Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field
- Authors: Panin I.A.1
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Affiliations:
- St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 84, No 4 (2020)
- Pages: 169-186
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/142298
- DOI: https://doi.org/10.4213/im8982
- ID: 142298
Cite item
Abstract
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 setsinduced by the inclusion of $R$ in $K$ has trivial kernel. This result was proved in [1] for regularlocal rings $R$ containing an infinite field.
About the authors
Ivan Alexandrovich Panin
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Email: paniniv@gmail.com
Doctor of physico-mathematical sciences
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