A Theorem of Cartier (e.g. <a href="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"> Mumford</a>, 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.

I do not even see how to construct (non-trivial) examples of non-affine group schemes over $k$ that fail to be locally of finite type, and answers describing such constructions are also welcome.

**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.

Examples of non-Noetherian $k$-group schemes are  the ``universal covering spaces" of abelian varieties found in a <a href="http://arxiv.org/abs/0902.3464"> paper</a>  by Vakil and Wickelgren.

@BCnrd, Txk 4 rspns.  Let me know if I misunderstood anything.