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 $\pi_0([X/G]) = Spec R^G$?

`$X$`

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