Rost invariant


In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraic group G over a field k, which associates an element of the Galois cohomology group H3 to a principal homogeneous space for G. Here the coefficient group Q/Z is the tensor product of the group of roots of unity of an algebraic closure of k with itself. first introduced the invariant for groups of type F4 and later extended it to more general groups in unpublished work that was summarized by.
The Rost invariant is a generalization of the Arason invariant.

Definition

Suppose that G is an absolutely almost simple simply connected algebraic group over a field k. The Rost invariant associates an element a of the Galois cohomology group H3 to a G-torsor P.
The element a is constructed as follows. For any extension K of k there is an exact sequence
where the middle group is the étale cohomology group and Q/Z is the geometric part of the cohomology.
Choose a finite extension K of k such that G splits over K and P has a rational point over K. Then the exact sequence splits canonically as a direct sum so the étale cohomology group contains Q/Z canonically. The invariant a is the image of the element 1/ of Q/Z under the trace map from H to H, which lies in the subgroup H3.
These invariants a are functorial in field extensions K of k; in other words the fit together to form an element of the cyclic group Inv3 of cohomological invariants of the group G, which consists of morphisms of the functor K→H1 to the functor K→H3. This element of Inv3 is a generator of the group and is called the Rost invariant of G.