Rost invariant
# 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 $H^3(k,$ Q/Z(2)) to a principal homogeneous space for G. Here the coefficient group Q/Z(2) is the tensor product of the group of roots of unity of an algebraic closure of k with itself. Markus Rost (1991) first introduced the invariant for groups of type $F_4$ and later extended it to more general groups in unpublished work that was summarized by Serre (1995).

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(P) of the Galois cohomology group $H^3(k,Q/Z(2))$ to a G-torsor P.

The element a(P) is constructed as follows. For any extension K of k there is an exact sequence

    $0\rightarrow H^{3}(K,\mathbf {Q} /\mathbf {Z} (2))\rightarrow H_{et}^{3}(P_{K},\mathbf {Q} /\mathbf {Z} (2))\rightarrow \mathbf {Q} /\mathbf {Z}$

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(P) is the image of the element 1/[K:k] of Q/Z under the trace map from $H^3$ $_{et}(P_K,Q/Z(2))$ to $H^3$ $_{et}(P,Q/Z(2)),$ which lies in the subgroup $H^3(k,Q/Z(2)).$

These invariants a(P) are functorial in field extensions K of k; in other words the fit together to form an element of the cyclic group $Inv^3(G,Q/Z(2))$ of cohomological invariants of the group G, which consists of morphisms of the functor $K→H^1(K,G)$ to the functor $K→H^3(K,Q/Z(2)).$ This element of $Inv^3(G,Q/Z(2))$ is a generator of the group and is called the Rost invariant of G.

