Is there a group-scheme equivalent of the theorem that any Lie group is diff. to a compact one cross R^n? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:30:43Z http://mathoverflow.net/feeds/question/40834 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40834/is-there-a-group-scheme-equivalent-of-the-theorem-that-any-lie-group-is-diff-to Is there a group-scheme equivalent of the theorem that any Lie group is diff. to a compact one cross R^n? James D. Taylor 2010-10-02T13:47:10Z 2010-10-02T16:11:54Z <p>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.)</p> <p>So: Is there an analog of this theorem for group-schemes? What is it? What can we say?</p> http://mathoverflow.net/questions/40834/is-there-a-group-scheme-equivalent-of-the-theorem-that-any-lie-group-is-diff-to/40844#40844 Answer by cfranc for Is there a group-scheme equivalent of the theorem that any Lie group is diff. to a compact one cross R^n? cfranc 2010-10-02T16:11:54Z 2010-10-02T16:11:54Z <p>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.</p>