Extensions of étale by connected group spaces

Abstract:

The main theorem, in rough terms, asserts the following. Let $K$ and $D$ be group spaces over a scheme $S$. Assume that $K$ has connected fibers and that $D$ is finite and étale over $S$ . Assume that there exists a single finite, surjective, étale, Galois morphism $\overline S \to S$ which decomposes (scheme-theoretically) every extension of $D$ by $K$. Let $\pi = \operatorname {Aut}(\overline S /S)$. Then group space extensions of $D$ with kernel $K$ are in bijective correspondence with pairs $(\xi ,\chi )$ consisting of a $\pi$-group extension \[ \xi :1 \to K(\overline S) \to X \to D(\overline S ) \to 1\] and a $\pi$-group homomorphism $\chi :X \to \operatorname {Aut}(\overline K )$ which lifts the conjugation map $X \to \operatorname {Aut}(K(\overline S ))$ and which agrees with the conjugation map $K(\bar S) \to \operatorname {Aut}(\overline K )$. In this way, the calculation of group space extensions is reduced to a purely group-theoretic calculation.