A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields

Fedorov, Roman; Panin, Ivan

doi:10.1007/s10240-015-0075-z

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A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields

Fedorov, Roman ¹ ; Panin, Ivan ²

¹ Mathematics Department, Kansas State University 138 Cardwell Hall 66506 Manhattan KS USA
² Steklov Institute of Mathematics at St.-Petersburg Fontanka 27 191023 St.-Petersburg Russia

Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 169-193

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Résumé

Let $R$ be a regular local ring containing an infinite field. Let $G$ be a reductive group scheme over $R$ . We prove that a principal $G$ -bundle over $R$ is trivial if it is trivial over the fraction field of $R$ . In other words, if $K$ is the fraction field of $R$ , then the map of non-abelian cohomology pointed sets

H_{\overset{´}{e} t}^{1} (R, G) \to H_{\overset{´}{e} t}^{1} (K, G)

induced by the inclusion of

R

into

K

has a trivial kernel.

DOI : 10.1007/s10240-015-0075-z

Keywords: Algebraic Group, Group Scheme, Principal Bundle, Monic Polynomial, Closed Subscheme

Affiliations des auteurs :

Fedorov, Roman ¹ ; Panin, Ivan ²

¹ Mathematics Department, Kansas State University 138 Cardwell Hall 66506 Manhattan KS USA
² Steklov Institute of Mathematics at St.-Petersburg Fontanka 27 191023 St.-Petersburg Russia

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     title = {A proof of the {Grothendieck{\textendash}Serre} conjecture on principal bundles over regular local rings containing infinite fields},
     journal = {Publications Math\'ematiques de l'IH\'ES},
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Fedorov, Roman; Panin, Ivan. A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields. Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 169-193. doi: 10.1007/s10240-015-0075-z

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