TITLE: Are GIT's good categorical quotients just locally ringed space coequalizers?
QUESTION [12 upvotes]: Introduction: The definition of "good categorical quotient" in geometric invariant theory (given below) seems fairly ad hoc to me, except that it looks very similar to the coequalizer of the action in the category of locally ringed spaces. I'm trying to understand this similarity.
Some background: Suppose we have a group action $G\times X \to X$ in the category $\bf Sch$ of schemes.  The category $\bf LRS$ of locally ringed spaces is cocomplete (see propositiion 1.12 in A. Vezzani's thesis), so in particular, $G\times X \rightrightarrows X$ has a $\bf LRS$-coequalizer; call it $Q$.  One can construct it as the coequalizer in the category of topological spaces equipped with the sheaf of $G$-invariant functions to make it a locally ringed space, as Vezzani's proposition explains more carefully.
If $G\times X \rightrightarrows X$ has a $\bf Sch$-coequalizer $Y$, then GIT calls $Y$ a "categorical quotient"; see, for example, R. Birkner's Introduction to GIT.  Note that here we get a universal map $u: Q\to Y$ in $\bf LRS$.  
$Y$ is called a "good categorical quotient" if it satisfies 3 properties, also found in Birkner's notes, which I prefer to re-express here in terms of the universal map $u$:
i) $u^\sharp : {\cal O}_Y \to u _* {\cal O}_Q$ is an isomorphism.
ii) $u: Q\to Y$ is a closed map (at the level of topological spaces).
iii) $u$ takes disjoint closed sets to disjoint closed sets.
(These together imply that $X\to Y$ is a categorical quotient, i.e. $\bf Sch$-coequalizer; see Mumford's GIT I.2 remark 6.)
All of these properties are clearly satisfied if $u$ is an isomorphism.  That is, if the $\bf LRS$-coequalizer is a scheme, that scheme is a good categorical quotient.
Towards a converse, (i) implies that $u$ must be dominant, and hence by (ii) it is surjective.  And (iii) implies it is injective on closed points, so it's looking a lot like an isomorphism.  Can anything more be said?

1)  When is a good categorical quotient an $\bf LRS$-coequalizer of the group action?  I.e., are there nice and general conditions where $u$ must be an isomorphism?
2) If the answer is not always, what's a counterexample? (Answered by Anton)

Thanks!

REPLY [4 votes]: The answer to the title question is NO. If $X\rightrightarrows Y\to Z$ is a coequalizer in $\bf LRS$, then the underlying topological space of $Z$ is the coequalizer of the underlying topological spaces (and the structure sheaf is the equalizer of pushforwards of the structure sheaves). On the other hand, good categorical quotients identify any two points whose orbit closures intersect. So one of my favorite good quotients, $\mathbb A^1/\mathbb G_m$, provides a counterexample. The good categorical quotient in schemes has a single point as its underlying topological space, but the quotient in $\bf LRS$ has two points.
I think the bad behavior on underlying topological spaces is probably the only problem. That is, a good categorical quotient is a coequalizer in $\bf LRS$ precisely when it is a geometric quotient (i.e. when we impose the additional condition that each fiber is a single orbit).