Barsotti-Chevalley existence over an arbitrary field, allowing nonsmooth affine kernel

First suppose $G$ smooth. In characteristic zero the field is perfect and [F1] already proves the assertion. In characteristic $p>0$, choose an algebraic closure by [F4] and let $k_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}$ be the union of its finite purely inseparable extensions of $k$. This is a perfect field. By [F1], $G_{k_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}}$ has a smooth connected affine closed normal subgroup $N_{\mathrm{\infty }}$ with abelian quotient. Descend this subgroup to a finite purely inseparable $K\mathrm{/}k$ inside $k_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}}$, as follows. On a finite affine cover of $G$, its ideal is finitely generated. Include in $K$ the finitely many coefficients of its generators, generators of the relations identifying the ideals on a finite affine cover of each overlap, and the finite equations making multiplication, inverse, identity and conjugation factor through it. Equality of ideals and vanishing of these equations hold after faithful scalar extension, hence already at the finite stage after enlarging $K$. Also include an affine finite-presentation model of $N_{\mathrm{\infty }}$ and the finitely many chart maps and inverse equations identifying it with this descended closed subscheme. Thus the subgroup $N^{\mathrm{\prime }}\subset G_K$ is affine and normal. Smoothness descends by [F2]; connectedness descends since a disconnection remains one after scalar extension. The quotient $G_K\mathrm{/}{N}^{\mathrm{\prime }}$ becomes the given abelian quotient after scalar extension, because both represent the same fppf coset sheaf, by [F3]. Its properness therefore descends by [F2]. This supplies a finite purely inseparable stage with all the required properties.

Apply the power-ideal descent in [F2] to $N^{\mathrm{\prime }}$ to obtain a connected affine closed normal $N\subset G$ for which $N_K$ contains $N^{\mathrm{\prime }}$ as a nilpotent closed subscheme. Let $Q=G\mathrm{/}N$, represented by [F3]. The map $G_K\to Q_K$ factors through $P=G_K\mathrm{/}{N}^{\mathrm{\prime }}$; the induced map $P\to Q_K$ is surjective because the projection from $G_K$ is surjective. This map has proper source and separated finite-type target and is proper: its graph is closed in $P{\times }_K{Q}_K$, whose projection to $Q_K$ is proper, by the definition of properness and base change. Hence $Q_K$ is proper over $K$. Explicitly, for every $K$-scheme $T$ and closed $Z\subset Q_K{\times }_{K}T$, its preimage in $P{\times }_{K}T$ is closed and its image in $T$ is closed by properness of $P$; surjectivity, retained by base change, makes that image exactly the image of $Z$. Finite type and separatedness of $Q_K$ come from [F3], giving properness by [F4]. Descend properness to $Q$ by [F2]. Since $G$ is smooth connected, [F3] makes $Q$ smooth connected and geometrically connected. It is therefore an abelian variety by [F5]. This proves the smooth-source case without descending a smooth subgroup to $k$.

For arbitrary $G$ in characteristic zero, [F1] reduces to the smooth-source case. In positive characteristic let $f:G\to S$ be a sufficiently high relative Frobenius with smooth image, as in [F4]. It is finite and a universal homeomorphism onto $S$, which is connected. Its scheme kernel $F$ is finite, hence affine, and connected: a universal homeomorphism has a single geometric point in the fibre over the identity. By [F3], $f$ is the represented exact quotient $G\mathrm{/}F$ and is faithfully flat of finite presentation. Apply the smooth-source case to $S$ to obtain a connected affine normal $M\subset S$ with abelian quotient $A=S\mathrm{/}M$. Define $N=G{\times }_{S}M$, a closed normal subgroup. The restricted projection is an exact sequence $1\to F\to N\to M\to 1$, so [F3] makes $N$ affine. The projection $N\to M$ is a base change of the finite universal homeomorphism $f$ and is onto; hence $N$ is connected.

The composite $G\to S\to A$ has scheme kernel $N$ and is fppf surjective: both factors are faithfully flat of finite presentation. It therefore identifies its coset sheaf with $G\mathrm{/}N$. Indeed every point of $A$ lifts fppf locally first to $S$ and then to $G$, and two lifts differ precisely by a point of $N$; these assertions after arbitrary test-scheme base change identify the sheaves. The represented quotient in [F3] is thus $A$, an abelian variety. Its projectivity and commutativity follow from [F5]. The Frobenius step used the smooth image $S$, not the whole twist of $G$, which can remain nonsmooth for every exponent; and the inseparable descent in step 2.1 retained nilpotent subgroup structure. Thus no smoothness or uniqueness of the arbitrary-field kernel was introduced. AC and DC are inherited from [F1]–[F5].