In mathematics, the Rost invariant is a
cohomological invariant In mathematics, a cohomological invariant of an algebraic group ''G'' over a field (mathematics), field is an invariant of forms of ''G'' taking values in a Galois cohomology group.
Definition
Suppose that ''G'' is an algebraic group defined over ...
of an
absolutely simple simply connected
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
''G'' over a field ''k'', which associates an element of the
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...
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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of the group of
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
of an algebraic closure of ''k'' with itself. 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4 and later extended it to more general groups in unpublished work that was summarized by .
The Rost invariant is a generalization of the
Arason invariant In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field ''k'' of characteristic not 2, taking values in H3(''k'',Z/2Z). It was intr ...
.
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
:
where the middle group is the
étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjecture ...
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(''P''
''K'',Q/Z(2)) to H(''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''.
References
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*{{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
Algebraic groups