Rosenlicht almost-complements to abelian subvarieties

By [F1], $A$ is central and $q:G\to Q=G\mathrm{/}A$ is a smooth torsor over a smooth geometrically integral group. Its generic fibre is a smooth $A_{k(Q)}$-torsor. Apply [F2] there to obtain $\psi$ with $\psi (ag)=na+\psi (g)$ for a positive integer $n$. Its finitely many defining coefficients spread to a nonempty open of $Q$, giving a rational map $G⇢A$. The extension theorem in [F2] gives a morphism $\varphi :G\to A$. The covariance extends over $A\times G$: this product is geometrically integral and the two morphisms agree on a dense open; the target is separated. Translate $\varphi$ by $-\varphi (e)$, retaining covariance and obtaining a pointed morphism. It is a homomorphism by [F2], and $\varphi {\mathrm{\mid }}_A=[n]$.

Let $N=\ker \varphi$, a closed normal subgroup. Multiplication $m:A\times N\to G$ is the pullback of $[n]:A\to A$ along $\varphi$: the isomorphism with $G{\times }_{\varphi ,A,[n]}A$ sends $(a,h)$ to $(ah,a)$, with inverse $(g,a)\mapsto (a,a^{-1}g)$. It is a homomorphism by centrality, finite faithfully flat by [F3], and its kernel is $A[n]$ diagonally embedded as stated. Put $H=N^0$, the open and closed identity component. It is normal: after algebraic closure conjugation by $G$ preserves the component containing the identity, and the factorization descends; equivalently, the conjugation map from the geometrically connected $G\times N^0$ lands in that open and closed component. The restriction $m_H:A\times H\to G$ is finite and flat, being the restriction of $m$ to an open and closed subscheme. Its image is closed by finiteness and open by flat finite presentation. It contains the identity, so connectedness of $G$ makes it surjective. Therefore it is finite faithfully flat. Its kernel is exactly $A\cap H$, a closed subgroup of $A[n]$, so finite.

If $k$ is perfect, replace $H$ with $R=H_{\mathrm{r}\mathrm{e}\mathrm{d}}$, which is smooth connected and normal by [F3]. Over an algebraic closure $R$ has the same points as $H$, so $A\times R\to G$ is surjective on closed points. Its image is closed, since it is a closed restriction of the finite $m_H$, hence the map is surjective. Let $J=A\cap R$, finite as a subgroup of $A[n]$. The normal quotient $(A\times R)\mathrm{/}J$ exists by [F1]. The induced homomorphism to $G$ has trivial scheme kernel, so is a monomorphism on all test schemes. By [F3] it is a closed immersion. This closed immersion is surjective and has reduced target $G$; its ideal is nilpotent and thus zero. It is an isomorphism. Hence $m_R$ is the quotient torsor and is faithfully flat as well as finite, with the claimed kernel. This proves the perfect-field clause without asserting that an arbitrary surjective morphism is flat. AC and DC are inherited from [F1]–[F3].