coarse moduli space and $\pi_0$

I've been reading this really nice paper by Alper http://math.columbia.edu/~jarod/good_moduli_spaces.pdf, and there's a question that doesn't seem to be answered (perhaps it's not relevant).

Any stack F has a corresponding sheaf of connected components' (or sheaf of isomorphism classes), by taking $\pi_0^{pr}(F)(S) = \pi_0 (F(S))$ and then sheafifying. (where $\pi_0$ of a groupoid, or more generally a category, is the set of isomorphism classes)

If $X$ is an Artin stack (although I'm currently more interested in DM stacks) and $X$ admits a good moduli space, then is $X \to \pi_0(X)$ a good moduli space?

Also, when is the good moduli space a scheme (and not just an algebraic space)?

And finally, if $X$ = $Spec R$ is affine and $G$ acts on it (I'm mainly interested in the $G$ finite case), is $\pi_0([X/G]) = Spec R^G$?

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Why do you expect there to be any kind of relation between moduli spaces and connected components ? If $X$ is a connected scheme over a field $k$, then I guess that $X$ is a good moduli space of itself while $\pi_0(X)$ is something like $Spec(k)$`. –  Matthieu Romagny Mar 14 '12 at 20:18
apologies for the confusing notation: if X is a sheaf on th, say, etale site of affine schemes. Then X(R) is a set for all rings R. Understand this set as a discrete topological space, therefore it makes sense to take the connected components of it. This gives me back the same set!(as you wrote: a scheme is a good moduli space of itself). If X is a stack of groupoids, then X(R) is a groupoid which (by taking nerves) can be interpreted as a (1-connected) topological space: it makes sense to consider $\pi_1$ (automorphisms), $\pi_0$ (connected components), so $\pi_0(X)$ is the shf closest to X. –  Yosemite Sam Mar 14 '12 at 20:42
I'm guessing the OP means pi_0 in the groupoid sense, so that pi_0 of a scheme is itself. –  David Roberts Mar 14 '12 at 20:43
Beat me to it... –  David Roberts Mar 14 '12 at 20:44
I think the question is yes, but I'll check properly before I answer. –  David Roberts Mar 16 '12 at 0:06

A gerbe with structure morphism $X \to Y$ is fppf locally on $Y$ of the form $B_YG := [Y/G]$, where $G$ is a group-algebraic space which is fppf over $Y$, and the action on $Y$ is trivial. (In fact, it is even étale locally on this form, since the structure morphism $X \to Y$ of a gerbe is smooth.)
Although gerbes certainly have at lot of good properties, the structure morphisms of a gerbe need not be a good moduli space in the sense of Alper. It is, exactly when $G$ above is linearly reductive.
The property of being a gerbe as a very strong property. Thus good (or a coarse) moduli spaces are seldom coarse sheaves. In particular $Spec\ R^G$ will usually not be the coarse sheaf of $[Spec\ R / G]$. Taking the stack quotient and then taking the associated sheaf, is the same as taking the sheaf quotient directly. The result is seldom (never?) representable unless the action of $G/N$ is free, where $N$ is the kernel of the action.