Let $K$ be a perfect field. In what follows, an algebraic group $G/K$ is by definition a group scheme of finite type over $K$.

The following seems to be well-known:

**Theorem:** Let $G/K$ be a connected smooth algebraic group. Then there is a connected smooth *affine* normal closed subgroup $N$ of $G$, an abelian variety $A/K$ and a homomorphism $G\to A$ with kernel $N$ such that the sequence
$$0\to N\to G\to A\to 0$$
is exact for the fppf-topology (say).

Can someone give me a proper reference or a hint why this is true? (Checking the literature I find on the one hand plenty of references treating affine algebraic groups, and on the other hand references containing the theory of abelian varieties, but I was surprised not to find a reference containing a proof of this Theorem about the "mixed case".)

notwell-known but deserves to be! First, Chevalley's thm is false over every imperfect field. Second, over fields of char. $> 0$ there is abetterresult (even for finite fields): there's an exact sequence $1 \rightarrow Z \rightarrow G \rightarrow H \rightarrow 1$ with smooth affine $H$ and central $Z$ that is semi-abelian (i.e., extension of abelian variety by a torus). Its buried in Demazure-Gabriel & godsend for cohomological purposes, since commutative term is on left rather than right. See Example A.3.8 and Thm. A.3.9 in "Pseudo-reductive groups" for more. – BCnrd Oct 25 '10 at 16:06