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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
up vote 12 down vote accepted

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 (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.

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Thanks! The paper looks interesting. – jlk Apr 28 '10 at 22:25

BCnrd 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

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Thanks Harry! Much more pleasant for the eyes! – François G. Dorais Apr 29 '10 at 22:09

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