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Zap. Nauchn. Sem. POMI, 2019, Volume 485, Pages 176–186 (Mi znsl6875)  

A short exact sequence

I. Panin

St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Russia

Abstract: Let $R$ be a semi-local integral Dedekind domain and $K$ be its fraction field. Let $\mu: \mathbf{G} \to \mathbf{T}$ be an $R$-group schemes morphism between reductive $R$-group schemes, which is smooth as a scheme morphism. Suppose that $T$ is an $R$-torus. Then the map $\mathbf{T}(R)/\mu(\mathbf{G}(R)) \to \mathbf{T}(K)/\mu(\mathbf{G}(K))$ is injective and certain purity theorem is true. These and other results are derived from an extended form of Grothendieck–Serre conjecture proven in the present paper for rings $R$ as above.

Key words and phrases: semi-simple algebraic group, principal bundle, Grothendieck–Serre conjecture, purity theorem.

Funding Agency Grant Number
Russian Foundation for Basic Research 19-01-00513
The author acknowledges support of the RFBR grant No. 19-01-00513.


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Received: 23.10.2019
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Citation: I. Panin, “A short exact sequence”, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Zap. Nauchn. Sem. POMI, 485, POMI, St. Petersburg, 2019, 176–186

Citation in format AMSBIB
\Bibitem{Pan19}
\by I.~Panin
\paper A short exact sequence
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXXI
\serial Zap. Nauchn. Sem. POMI
\yr 2019
\vol 485
\pages 176--186
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6875}


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