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?