Affine smooth and connected properties in exact sequences of algebraic groups

The morphism $(g,n)\mapsto (g,gn)$ is an isomorphism $G\times N\cong G{\times }_{Q}G$; its inverse sends $(g,h)$ to $(g,g^{-1}h)$, which factors through the kernel by the group law. Thus base change of $q$ by itself is projection from $G\times N$. If $N$ is affine this projection is affine, and [F1] gives that $q$ is affine. If $Q$ also is affine, its inverse image $G$ is affine. Conversely if $G$ is affine, [F1] supplies affine $Q$; its exact quotient agrees with the represented normal quotient by the fppf lifting and kernel-pair identity just proved.

For each $z\in Q$, choose an algebraic closure $\mathrm{\Omega }$ of $\kappa (z)$ by [F4]. The nonempty finite-type fibre $G_z{\times }_{\kappa (z)}\mathrm{\Omega }$ has an $\mathrm{\Omega }$-point by [F4], and translation by it identifies that fibre with $N_{\mathrm{\Omega }}$ using step 1.1. If $N$ is smooth, $N_{\mathrm{\Omega }}$ is smooth by [F3]; hence its affine chart rings are geometrically regular. Field descent in [F3] makes $G_z$ geometrically regular over $\kappa (z)$. The given flatness and finite presentation of $q$ now make $q$ smooth by [F3]. If $Q$ is smooth too, composition in [F3] makes $G$ smooth. Conversely if $G$ is smooth, it is geometrically reduced. Faithful flat pullback injects the local coordinate sections of $Q$ into those of $G$, so every nilpotent section of the geometric $Q$ is zero. Thus $Q$ is geometrically reduced, and [F2] makes it smooth.

Surjectivity makes a quotient of connected $G$ connected. If both $N$ and $Q$ are connected, [F2] and the fibre identification in step 2.1 make every fibre of $q$ connected. For a decomposition of $G$ into two disjoint open-and-closed subsets, each connected fibre lies entirely in one; their images are disjoint opens in $Q$ by [F3] and cover $Q$. Connectedness of $Q$ forces one image empty and hence one original subset empty. Thus $G$ is connected. AC is inherited from [F1]–[F3].