Appendix on Galois cohomology

Abstract

For a connected reductive group G over F, let Ĝ denote the algebraic group over C, which is the connected component of the Langlands L-group of G. Consider the category of reductive groups over F, whose morphisms are the group homomorphism G → H, which are defined over F. The Borovoi fundamental group π(G) is a functor from the category of reductive groups over F to the category of finitely generated Abelian groups with Γ_F -action. The Borovoi fundamental group is defined as follows. For tori T over F we have a canonical isomorphism $\pi (T) = Hom_{\bar{F}-alg}(G_m, T )$, and $\hat{T}=\pi(T)\otimes_Z C^\ast$ with induced Γ_F -action. For semisimple G over F we have a canonical isomorphism π(G) = π₁(G) (algebraic fundamental group). In general, let G_sc be the simply connected covering of the derived group G_der of G. Choose maximal F-tori T_sc → T_der → T. Then π(G) = π(T)/image(π(T_sc)).