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.

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 paper by Vakil and Wickelgren.

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

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.

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.

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 paper by Vakil and Wickelgren.

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

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.

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 paper by Vakil and Wickelgren.

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

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.

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.

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.

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 paper by Vakil and Wickelgren.

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