Rost invariant

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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(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Markus Rost (1991) first introduced the invariant for groups of type F4 and later extended it to more general groups in unpublished work that was summarized by Serre (1995).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[edit]

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 H3(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

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 H3
et
(PK,Q/Z(2)) to H3
et
(P,Q/Z(2)), which lies in the subgroup H3(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 Inv3(G,Q/Z(2)) of cohomological invariants of the group G, which consists of morphisms of the functor K→H1(K,G) to the functor K→H3(K,Q/Z(2)). This element of Inv3(G,Q/Z(2)) is a generator of the group and is called the Rost invariant of G.

References[edit]