Nonabelian H1 and Galois 1-cocycles

Let a group $\Gamma$ act on a group $A$ . A nonabelian $1$ -cocycle is a map $z:\Gamma\to A$ satisfying

z_{\sigma\tau} = z_\sigma\,{}^\sigma z_\tau.

Two cocycles are cohomologous when they differ by twisted conjugacy:

z'_\sigma = a^{-1}z_\sigma\,{}^\sigma a

for some $a\in A$ . Their classes form the pointed set

H^1(\Gamma,A),

whose basepoint is the trivial cocycle. Unless $A$ is abelian, this is not naturally a group.

Galois form

For an algebraic $k$ -group $G$ , one writes

H^1(k,G) = H^1(\Gamma_k,G(k_s)),

using continuous cocycles for the absolute Galois group. It classifies $G$ -torsors over $k$ up to isomorphism. If $K/k$ is finite Galois, then $H^1(\operatorname{Gal}(K/k),G(K))$ records torsors split by $K$ .

Forms require a different coefficient group

Forms of $G$ are classified by $H^1(k,\operatorname{Aut}(G_{k_s}))$ . Inner forms come from $H^1(k,G_{\mathrm{ad}})$ , while a pure inner twist uses a lift in $Z^1(k,G)$ . Suppressing the coefficient group loses the mathematical content.

Stable conjugacy

For a strongly regular semisimple $\gamma$ with torus centralizer $G_\gamma$ , rational conjugacy classes inside its stable class are parametrized by

\ker\!\left[ H^1(k,G_\gamma)\to H^1(k,G) \right].

This is the cohomological obstruction whose Fourier analysis leads to endoscopy and $\kappa$ -orbital integrals.

Relation to the letter

The letter's inner twisting is a class in the appropriate inner automorphism group. “Locally trivial at almost all places” means that its restriction to the corresponding local cohomology sets is the basepoint.

References

Jean-Pierre Serre, Galois Cohomology, Springer, 1997.
Robert E. Kottwitz, “Stable trace formula: cuspidal tempered terms,” Duke Mathematical Journal 51 (1984), 611–650.