Suppose that G is an algebraic group defined over a field K, and choose a separably closed field containing K. For a finite extensionL of K in let ΓL be the absolute Galois group of L. The first cohomology H1 = H1 is a set classifying the forms of G over L, and is a functor of L. A cohomological invariant of G of dimension d taking values in a ΓK-module M is a natural transformation of functors from H1 to Hd. 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 from A to Hd. The cohomological invariants of a fixed group G or functor A, dimension d and Galois moduleM form an abelian group denoted by Invd or Invd.
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 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 subgroupC, 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 Hasse−Witt invariant.
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. For dimension 1 this is essentially the discriminant, and for dimension 2 it is essentially the Hasse−Witt invariant.
The Arason invariante3 is a dimension 3 invariant of some even dimensional quadratic formsq 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 and p is the quadratic form corresponding to the image ofu in H1, then e3 is the value of the dimension 3 cohomological invariant on u.
For absolutely simple simply connected groups G, the Rost invariant is a dimension 3 invariant taking values in Q/Z that in some sense generalizes the Arason invariant and the Merkurjev−Suslin invariant to more general groups.
