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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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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}