A Theorem of Cartier (e.g.  [Mumford](http://books.google.com/books?id=-5weWX_YD6sC&printsec=frontcover&dq=curves+on+an+algebraic+surface&source=bl&ots=1r4O86OtJy&sig=PIH5b5EipbZYJWXrWy77NvltSP4&hl=en&ei=4PjUS9uMKMOC8gbbi_mCDA&sa=X&oi=book_result&ct=result&resnum=3&ved=0CBIQ6AEwAg#v=onepage&q&f=false), Lecture 25) states that every separated, finite type group scheme $G/k$ over a field $k$ of characteristic $0$ is reduced. Does this result remain valid if we drop the assumption that $G/k$ is separated and of finite type?


Frans Oort (MR0206005) observed that one can use limit formalism to argue that every affine group scheme over $k$ is reduced.


**Edit:** BCnrd pointed out that group schemes over a field are automatically separated. Furthermore, the proof of Cartier's Theorem in Mumford's book remains valid for a locally Noetherian $k$-group scheme.



Are group schemes in Char 0 reduced? (YES)
The answer is yes - *every* group scheme over a field of characterstic zero is reduced: see [Schémas en groupes quasi-compacts sur un corps et groupes henséliens](http://www.math.u-psud.fr/~biblio/numerisation/docs/P_PERRIN-109/pdf/P_PERRIN-109.pdf) (especially Thm. 2.4 in part II and Thm. 1.1 and Cor. 3.9 in part V of the 1st part), and for a summary of the relevant results see 4.2 (in particular 4.2.8) of [Approximation des schémas en groupes, quasi compacts sur un corps](http://archive.numdam.org/article/BSMF_1976__104__323_0.djvu).


