TITLE: Is there a group-scheme equivalent of the theorem that any Lie group is diff. to a compact one cross R^n?
QUESTION [5 upvotes]: I'm rather ignorant in both fields, but I would still like to endeavor asking this question. I've just learned that any Lie group is diffeomorphic to a compact Lie group cross $\mathbb{R}^n$. While there is no (to my knowledge) equivalent to diffeomorphic in the group-schemes language, this does have obvious implications about the cohomology of Lie groups (which has an analog in the group-scheme language.)
So: Is there an analog of this theorem for group-schemes? What is it? What can we say?

REPLY [3 votes]: For algebraic groups over a perfect field $k$, one has Chevalley's Theorem. It says that every algebraic group $G$ over $k$ contains a unique closed normal linear subgroup $H$ such that $G/H$ is an abelian variety. The abelian variety is the analogue of the compact Lie group, and the linear group $H$ is the analogue of affine space.