Rost injectivity for classical groups over function fields of curves over local fields

TL;DR

This paper proves the injectivity of the Rost invariant for certain classical groups over function fields of curves over local fields, extending understanding of Galois cohomology and algebraic group invariants.

Contribution

It establishes the injectivity of the Rost invariant for classical groups over specific function fields, under conditions on the residue field and algebraic structures, advancing cohomological classification.

Findings

Rost invariant is injective for groups of outer type A_n under specified conditions.

Injectivity holds for classical groups over function fields of curves over local fields.

Results connect algebraic group properties with Galois cohomology invariants.

Abstract

Let F be a complete discretely valued field with residue field a global field or a local field with no real orderings. Let G be an absolutely simple simply connected group of outer type A_n. If 2 and the index of the underlying algebra of G are coprime to the characteristic of the residue field of F, then we prove that the Rost invariant map from the first Galois cohomology set of G to the degree three Galois cohomology group is injective. Let L be the function field of a curve over a local field K and G an absolutely simple simply connected linear algebraic group over L of classical type. Suppose that the characteristic of the residue field of K is a good prime for G. As a consequence of our result and some known results we conclude that the Rost invariant of G is injective.