Cohomological invariant

In mathematics, a cohomological invariant of an algebraic group G over a field is an invariant of forms of G taking values in a Galois cohomology group.

Suppose that G is an algebraic group defined over a field K, and choose a separably closed field K containing K. For a finite extension L of K in K let Γ_L be the absolute Galois group of L. The first cohomology H¹(L, G) = H¹(Γ_L, G) 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 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 M form an abelian group denoted by Inv^d(G,M) or Inv^d(A,M).

...
Wikipedia