In mathematics, a cohomological invariant of an

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. ...

''G'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

is an invariant of forms of ''G'' taking values in a

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.

Definition

Suppose that ''G'' is an

defined over a

''K'', and choose a separably closed field containing ''K''. For a finite extension ''L'' of ''K'' in let Γ_''L'' be the

absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...

of ''L''. The first cohomology H¹(''L'', ''G'') = H¹(Γ_''L'', ''G'') is a set classifying the ``G''-torsors over ''L'', and is a functor of ''L''. A cohomological invariant of ''G'' of dimension ''d'' taking values in a Γ_''K''-module ''M'' is a natural transformation of functors (of ''L'') from H¹(L, ''G'') to H^''d''(L, ''M''). In other words a cohomological invariant associates an element of an abelian cohomology group to elements of a non-abelian cohomology set. More generally, if ''A'' is any functor from finitely generated extensions of a field to sets, then a cohomological invariant of ''A'' of dimension ''d'' taking values in a Γ-module ''M'' is a natural transformation of functors (of ''L'') from ''A'' to H^''d''(L, ''M''). The cohomological invariants of a fixed group ''G'' or functor ''A'', dimension ''d'' and

Galois module In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ...

''M'' form an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

denoted by Inv^''d''(''G'',''M'') or Inv^''d''(''A'',''M'').

Examples

*Suppose ''A'' is the functor taking a field to the isomorphism classes of dimension ''n'' etale algebras over it. The cohomological invariants with coefficients in Z/2Z is a free module over the cohomology of ''k'' with a basis of elements of degrees 0, 1, 2, ..., ''m'' where ''m'' is the integer part of ''n''/2. *The

Hasse−Witt invariant In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form ''Q'' over a field ''K'' takes values in the Brauer group Br(''K''). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt. The quadratic form ''Q'' ma ...

of a quadratic form is essentially a dimension 2 cohomological invariant of the corresponding spin group taking values in a group of order 2. *If ''G'' is a quotient of a group by a smooth finite central subgroup ''C'', then the boundary map of the corresponding exact sequence gives a dimension 2 cohomological invariant with values in ''C''. If ''G'' is a special orthogonal group and the cover is the spin group then the corresponding invariant is essentially the

. *If ''G'' is the orthogonal group of a quadratic form in characteristic not 2, then there are Stiefel–Whitney classes for each positive dimension which are cohomological invariants with values in Z/2Z. (These are not the topological

Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets ...

es of a real vector bundle, but are the analogues of them for vector bundles over a scheme.) For dimension 1 this is essentially the discriminant, and for dimension 2 it is essentially the

. *The Arason invariant ''e''₃ is a dimension 3 invariant of some even dimensional quadratic forms ''q'' with trivial discriminant and trivial Hasse−Witt invariant. It takes values in Z/2Z. It can be used to construct a dimension 3 cohomological invariant of the corresponding spin group as follows. If ''u'' is in H¹(''K'', Spin(''q'')) and ''p'' is the quadratic form corresponding to the image of ''u'' in H¹(''K'', O(''q'')), then ''e''₃(''p''−''q'') is the value of the dimension 3 cohomological invariant on ''u''. *The Merkurjev−Suslin invariant is a dimension 3 invariant of a special linear group of a central simple algebra of rank ''n'' taking values in the tensor square of the group of ''n''th roots of unity. When ''n''=2 this is essentially the Arason invariant. *For absolutely simple simply connected groups ''G'', the

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 homogene ...

is a dimension 3 invariant taking values in Q/Z(2) that in some sense generalizes the Arason invariant and the Merkurjev−Suslin invariant to more general groups.

References

* * *{{citation, mr=1321649 , last=Serre, first= Jean-Pierre , title=Cohomologie galoisienne: progrès et problèmes , series=Séminaire Bourbaki, Vol. 1993/94. Exp. No. 783 , journal=Astérisque , volume= 227 , year=1995, pages= 229–257, url=http://www.numdam.org/item?id=SB_1993-1994__36__229_0 Algebraic groups