A short exact sequence

TL;DR

This paper proves an extended form of the Grothendieck--Serre conjecture for regular semi-local integral domains containing a field, establishing injectivity of certain group scheme maps and a purity theorem.

Contribution

It introduces a new proof of the Grothendieck--Serre conjecture for a broad class of rings, extending previous results in algebraic geometry and group schemes.

Findings

Injectivity of T(R)/mu(G(R)) into T(K)/mu(G(K))

Establishment of a purity theorem for the given class of rings

Extension of Grothendieck--Serre conjecture to semi-local integral domains containing a field

Abstract

Let R be a regular semi-local integral domain containing a field and K be its fraction field. Let mu: G --> 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 T(R)/mu(G(R)) --> T(K)/mu(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.