An extended form of the Grothendieck–Serre conjecture

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Abstract

Let $R$ be a regular semi-local integral domain containing a field,$K$ the fraction field of $R$, and $\mu\colon \mathbf{G} \to \mathbf{T}$ an$R$-group scheme morphism between reductive $R$-group schemes which issmooth as a scheme morphism. Suppose that $\mathbf{T}$ 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 a purity theorem holds. These andother results can be derived from an extended form of the Grothendieck–Serre conjectureproven in the present paper for any such ring $R$.

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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