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Rost (1991) first introduced the invariant for groups of type F₄ and later extended it to more general groups in unpublished work that was summarized by Serre (1995).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

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

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 H3
et(P_K,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

Garibaldi, Ryan Skip (2001), "The Rost invariant has trivial kernel for quasi-split groups of low rank", Comment. Math. Helv., 76 (4): 684–711, doi:10.1007/s00014-001-8325-8, MR 1881703
Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), "Rost invariants of simply connected algebraic groups", Cohomological invariants in Galois cohomology, University Lecture Series, 28, Providence, RI: American Mathematical Society, ISBN 0-8218-3287-5, MR 1999383, Zbl 1159.12311
Rost, Markus (1991), "A (mod 3) invariant for exceptional Jordan algebras", C. R. Acad. Sci. Paris Sér. I Math., 313 (12): 823–827, MR 1138557
Serre, Jean-Pierre (1995), "Cohomologie galoisienne: progrès et problèmes", Astérisque, Séminaire Bourbaki Exp. No. 783, 227 (4): 229–257, MR 1321649

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