Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field,Izvestiya: Mathematics

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Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field
Izvestiya: Mathematics ( IF 0.8 ) Pub Date : 2020-09-01 , DOI: 10.1070/im8982
I. A. Panin ₁

Affiliation

Let ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$R$} be a regular local ring containing a field. Let ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn2.gif] {$\mathbf{G}$} be a reductive group scheme over ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$R$} . We prove that a principal ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn2.gif] {$\mathbf{G}$} -bundle over ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$R$} is trivial if it is trivial over the field of fractions of ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$R$} . In other words, if ##IMG## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn3.gif] {$K$} is the field of fractions of ##IMG##

中文翻译：

关于包含场的规则局部环上主束的格洛腾迪克-塞勒猜想的证明

令## IMG ## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$ R $}为包含字段的常规本地环。让## IMG ## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn2.gif] {$ \ mathbf {G} $}是##上的归约组方案。 IMG ## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$ R $}。我们证明了委托人## IMG ## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn2.gif] {$ \ mathbf {G} $}-捆绑在＃＃IMG ## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif]如果在$＃的分数字段上微不足道，{$ R $}是微不足道的。 IMG ## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn1.gif] {$ R $}。换句话说，如果## IMG ## [http://ej.iop.org/images/1064-5632/84/4/780/IZV_84_4_780ieqn3.gif] {$ K $}是##的分数字段IMG ##

更新日期：2020-09-02

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