Let $X$ be a variety (i.e. a reduced scheme of finite type over a field) and let $G$ be an abstract group, finitely generated, acting of $X$ algebraically freely. The example I have in mind is $\mathbb Z$ acting by shifts on $\mathbb A^1$. The quotient in this example clearly does not exist as a noetherian scheme since the fibres are discrete infinite. Is it possible to still make sense of the quotient algebraically? I guess the best way to put it formally is:


Is it possible to put the category of varieties over the field k into a bigger "algebraic" category such that the functor of points of the quotient (i.e. orbits of the values of the usual functior of points) is always representable? What I mean by "algebraic" is that I am definitely not interested in a quotient in the sense of complex geometry.


Is it possible to achieve this goal by considering a suitable category of non-Noetherian schemes?



quotients of varieties as non-noetherian schemes?
The quotient exists as an algebraic space (if we ignore the literature that defines algebraic spaces to be separated). The equivalence relation is given by the standard action groupoid. This is étale, since both the projection and the action map from $X \times G$ to $X$ are locally finitely presented and formally étale (indeed, they are locally isomorphisms). I'm not sure what you are looking for by passing to non-Notherian schemes, beyond allowing the "morphism space" $X \times G$ to be used.


