Remarks on non-abelian cohomology of proalgebraic groups

In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709–769 to study the unipotent section conjecture. The non-abelian cohomology $H^1_{\mathrm {nab}}(\mathcal {G},\mathcal {P})$ is a scheme. The argument $\mathcal {G}$ is a proalgebraic group; the coefficient group $\mathcal {P}$ is prounipotent with trivial center and endowed with an outer action of $\mathcal {G}$. This outer action uniquely determines an extension $\widehat {\mathcal {G}}$ of $\mathcal {G}$ by $\mathcal {P}$. With suitable hypotheses, the scheme $H^1_{\mathrm {nab}} (\mathcal {G},\mathcal {P})$ parametrizes the $\mathcal {P}$ conjugacy classes of sections of $\widehat {\mathcal {G}} \to \mathcal {G}$.