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. {{harvs|txt|first=Markus |last=Rost|authorlink=Markus Rost|year=1991}} first introduced the invariant for groups of type F₄ and later extended it to more general groups in unpublished work that was summarized by {{harvs|txt|last=Serre|year=1995}}.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

: $0\rightarrow H^3(K,\mathbf{Q}/\mathbf{Z}(2)) \rightarrow H^3_{et}(P_K, \mathbf{Q}/\mathbf{Z}(2)) \rightarrow \mathbf{Q}/\mathbf{Z}$

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{{su|b=et|p=3}}(P_K,Q/Z(2)) to H{{su|b=et|p=3}}(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

{{citation|mr=1881703|last=Garibaldi|first= Ryan Skip|title=The Rost invariant has trivial kernel for quasi-split groups of low rank|journal=Comment. Math. Helv. |volume=76 |year=2001|issue=4|pages= 684–711|doi=10.1007/s00014-001-8325-8|arxiv=math/0205305}}
{{citation | chapter=Rost invariants of simply connected algebraic groups| last1=Garibaldi | first1=Skip | author1-link=Skip Garibaldi | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Serre | first3=Jean-Pierre | author3-link=Jean-Pierre Serre | title=Cohomological invariants in Galois cohomology | series=University Lecture Series | volume=28 | location=Providence, RI | publisher=American Mathematical Society | year=2003 | isbn=0-8218-3287-5 | zbl=1159.12311 | mr=1999383 }}
{{citation|mr=1138557|last=Rost|first= Markus|title=A (mod 3) invariant for exceptional Jordan algebras|journal=Comptes Rendus de l'Académie des Sciences, Série I |volume=313 |year=1991|issue= 12|pages= 823–827}}
{{citation|mr=1321649 |last=Serre|first= Jean-Pierre|title=Cohomologie galoisienne: progrès et problèmes|series=Séminaire Bourbaki Exp. No. 783|journal=Astérisque |volume= 227 |year=1995|issue= 4|pages= 229–257|url= http://www.numdam.org/item?id=SB_1993-1994__36__229_0}}

Category:Algebraic groups