Question

Let G be an algebraic group acting on a scheme X. Then f: X --> Y is a categorical quotient if it is constant on G-orbits and any other G-invariant morphism factors through it in a unique fashion. We say f is a 'good' categorical quotient if:

1) f is a surjective open submersion (i.e. the topology on Y is induced from X).

2) for any open U ⊂ Y, the induced map from functions on U to G-invariant functions on f^-1(U) is an isomorphism.

Does anyone know an example of a 'bad' categorical quotient (by which I mean...well...a not good one).

Answer 1

Note that if f: X→Y is a categorical quotient in the category of schemes which is stable under base change by open immersions, then the second condition (ie. O_Y→(f_* O_X)^G is an isomorphism) is automatically satisfied.

In the paper "Examples and counterexamples for existence of categorical quotients" by A'Campo-Neuen and Hausen, there is an example of a categorical quotient f: X→A¹ such that f^-1(A¹ - 0)→A¹ - 0 is not a categorical quotient. I haven't checked but I believe this should also give an example where condition (2) fails.

I don't know of example of a categorical quotient where condition (1) fails.