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I agree with Johnathan Jonathan Wise's answer, but what about possibly non-quasi-separated algebraic spaces, i.e. according to Raynaud-Gruson or To\"en-Vaqui\'e?

It seems that the answer is amazingly... Yes, that quotient is an algebraic space! Actually, you've already given the proof: R is a scheme and is an etale equivalence relation. It's a rather crazy algebraic space in that it doesn't have an open subscheme. The theorem in Knutson that says open subschemes always exist in Knutson uses quasi-separatedness, as it apparently must. So it seems you've discovered a group object in the category of algebraic spaces which is not quasi-separated or a scheme. Well done! I never would have guessed such a thing would exist, much less be so simple. (An even slightly simpler example would be the additive group modulo Z, at least in characteristic 0.)

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I agree with Johnathan Wise's answer, but what about not possibly non-quasi-separated algebraic spaces, i.e. according to Raynaud-Gruson or To\"en-Vaqui\'e?

It seems that the answer is amazingly... Yes, that quotient is an algebraic space! Actually, you've already given the proof: R is a scheme and is an etale equivalence relation. It's a rather crazy algebraic space in that it doesn't have an open subscheme. The theorem that says open subschemes always exist in Knutson uses quasi-separatedness, as it apparently must. So it seems you've discovered a group object in the category of algebraic spaces which is not quasi-separated or a scheme. Well done! I never would have guessed such a thing would exist, much less be so simple. (An even slightly simpler example would be the additive group modulo Z, at least in characteristic 0.)

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I agree with Johnathan Wise's answer, but what about not possibly non-quasi-separated algebraic spaces, i.e. according to Raynaud-Gruson or To\"en-Vaqui\'e?

It seems that the answer is amazingly... Yes, that quotient is an algebraic space! Actually, you've already given the proof: R is a scheme and is an etale equivalence relation. It's a rather crazy algebraic space in that it doesn't have an open subscheme. The theorem that says open subschemes always exist in Knutson uses quasi-separatedness, as it apparently must. So it seems you've discovered a group object in the category of algebraic spaces which is not quasi-separated or a scheme. Well done! I never would have guessed such a thing would exist, much less be so simple. (An even slightly simpler example would be the additive group modulo Z, at least in characteristic 0.)