Are group schemes in Char 0 reduced? (YES)

A Theorem of Cartier (e.g. Mumford, 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.

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Every gp scheme over field is sep'td: ratl pts are closed immersions, and diag is base change of id. section. Also, conn'd gp lft over field k is f.t. (use geometric conn'dness and pass to kbar), whence smooth for char 0 in lft case. (Pf of Cartier's thm works in lft case over field of char. 0, so this reasoning is silly.) Any noetherian gp scheme over fld of char. 0 is formally smooth: completion at 1 is formal gp of finite dim, and Cartier's pf works in formal case (use formal Lie theory w/o smoothness hypothesis!) or use Thm 3.3ff Exp. VII SGA3. Then translate and extend base field. QED – BCnrd Apr 26 '10 at 2:51
BCnrd, Im srry t sy tht Im strting to hv pbs rding u! :/ – Mariano Suárez-Alvarez Apr 26 '10 at 3:03
I posted the comment community-wiki style with vowels added back in! – Harry Gindi Apr 26 '10 at 3:05
By the way, BCnrd, would you prefer that I quote you as BCnrd or Brian Conrad? – Harry Gindi Apr 26 '10 at 3:09
For the record, I have so far very much enjoyed your ramblings! :) – Mariano Suárez-Alvarez Apr 26 '10 at 4:19