Let $G$ be an algebraic group acting on a scheme $X$. Then $f: X \to Y$ is called a categorical quotient if it is constant on $G$-orbits and every $X \to Z$ constant on $G$-orbits factors through it in a unique fashion. We call $f$ a 'good' categorical quotient if:
1) $f$ is a surjective open submersion (i.e. $Y$ has the quotient topology).
2) for any open $U \subset Y$, the induced map $\mathcal O_U \to (\mathcal O_{f^{-1}(U)})^G$ is an isomorphism.
Does anyone know an example of a 'bad' categorical quotient (by which I mean…well…a not good one).
Best Answer
Note that if $f: X \to Y$ is a categorical quotient in the category of schemes which is stable under base change by open immersions, then the second condition (ie. $\mathcal O_Y \to (f_* \mathcal O_X)^G$ 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 4.6 of a categorical quotient $f: X \to \mathbb A^1$ such that $f^{-1}(\mathbb A^1-0) \to \mathbb A^1 - 0$ 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.