Pseudo-abelian varieties over perfect fields are complete

An algebraic closure of a perfect field is separable algebraic over it. By [F1] we may extend to that closure, retain pseudo-abelianness, prove properness there, and descend properness afterward. Work henceforth over algebraically closed $k$. Let $Z=Z(G)$ and $A=(Z_{\mathrm{r}\mathrm{e}\mathrm{d}}{)}^0$. By [F2], $A$ is a smooth connected central subgroup. If $A$ were nonproper, the dichotomy in [F2] would produce a smooth connected affine subgroup $U\subset A$ of positive dimension. Since $A$ is central, $U$ is normal in $G$, contradicting pseudo-abelianness. Thus $A$ is proper, and is an abelian variety, including when it is the trivial group.

Choose a sufficiently large finite identity jet so that its conjugation representation $\rho :G\to \mathrm{GL}(V)$ has scheme kernel $Z$, by [F3]. The quotient $G\mathrm{/}Z$ exists and $\rho$ factors through it. The induced map has trivial scheme kernel: fppf locally any quotient point lifts to $G$, and a lift with trivial representation lies in $Z$, hence represents the identity quotient point. Therefore [F3] embeds $G\mathrm{/}Z$ as a closed subgroup of the affine $\mathrm{GL}(V)$, so $G\mathrm{/}Z$ is affine. The quotient $Z\mathrm{/}A$ is finite. Indeed every connected component of $Z$ has reduction a translate of $A$: reduction is a smooth group and its components are translates of its identity component. On geometric points, dividing each component by $A$ leaves one point. Thus $Z\mathrm{/}A$, a separated finite-type scheme, has finitely many geometric points and dimension zero. A finite-type zero-dimensional scheme over a field is finite: on finitely many affine charts its Noetherian rings have dimension zero and are Artinian; its finitely many points are open and closed, their local Artinian affine neighbourhoods form a disjoint finite cover, and the resulting finite-dimensional rings give finiteness. The quotient maps now form the exact sequence $1⟶Z\mathrm{/}A⟶G\mathrm{/}A⟶G\mathrm{/}Z⟶1.$ Its kernel and fppf surjectivity follow directly by lifting quotient representatives, and [F4] makes $G\mathrm{/}A$ affine, since $Z\mathrm{/}A$ is finite and hence affine. This includes zero-dimensional centres; no global-function assertion about $G$ is needed.

By [F4] choose a smooth connected normal almost-complement $H$ to $A$. The map $H\to G\mathrm{/}A$ has finite kernel $J=A\cap H$, and induces an isomorphism $H\mathrm{/}J\cong G\mathrm{/}A$: every local representative in $G$ is fppf locally $ah$ by the almost-complement, and two $H$ representatives give the same coset precisely when their ratio belongs to $J$. The represented quotient in [F3] therefore gives an exact sequence $1\to J\to H\to G\mathrm{/}A\to 1$. Both outer terms are affine, so [F4] makes $H$ affine. It is smooth, connected, and normal in $G$, hence pseudo-abelianness makes it trivial. The finite faithfully flat multiplication $A\times H\to G$ becomes the closed immersion $A\hookrightarrow G$; a faithfully flat closed immersion has zero defining ideal, by faithful flatness of its quotient ring, so it is an isomorphism. Thus $G=A$ is proper over the algebraic closure. Descend properness by [F1]. Smoothness and geometric connectedness then make $G$ an abelian variety under the stated definition. AC and DC are inherited from [F1]–[F4].