Brian Conrad posted:
Every group scheme over a field is separated: rational points are closed immersions, and the diagonal is the base change of the identity section. Also, a connected group locally of finite type over field k is of finite type (use geometric connectedness and pass to the algebraic closure of k), whence smoothness follows for characteristic 0 in the locally of finite type case. (The proof of Cartier's theorem works in the locally of finite type case over a field of characteristic 0, so this reasoning is silly.) Any noetherian group scheme over field of characteristic 0 is formally smooth: the completion at 1 is a formal group of finite dimension, and Cartier's proof works in formal case (use formal Lie theory without a smoothness hypothesis!), or use Theorem 3.3ff Exp. VII of SGA3. Then translate and extend base field. QED