# If X is the coarse moduli space of the algebraic stack M, is there a nice description of Hom(_,X)?

Let $\mathcal{M}$ be an algebraic stack, and let $X$ be its coarse moduli space (assume it exists as a scheme).

We know that $h_X(Spec(k))=\mathcal{M}(Spec(k))$ if $k$ is algebraically closed. Is there anything intelligent we can say about $h_X(U)$ for a general scheme $U$?

For example, can you come up with an algorithm for knowing what $h_X(U)$ is that would be considerably easier than constructing $X$?

• You didn't need to make that question Community Wiki. It's a perfectly valid question. – André Henriques Jul 16 '11 at 21:54
• I'm no expert, but maybe is it something like $h_X(U)=\pi_0(\mathcal{M}(U))$ i.e. the set of isomorphism classes of objects of the groupoid $\mathcal{M}(U)$? Or is my guess totaly mistaken? – Qfwfq Jul 16 '11 at 22:08
• @Andre: I think I can't change it now. The reason I made it community wiki is because the question is pretty vague. There is no precise definition of an algorithm that is "considerably easier" than some other algorithm. – James D. Taylor Jul 16 '11 at 22:17
• @unknowngoogle: that sounds interesting! Can anyone confirm? – James D. Taylor Jul 16 '11 at 22:18

• If you take the plane instead of the line, with the involution $(x,y) \mapsto (-x,-y)$, the quotient map is not flat anymore. – Angelo Jul 17 '11 at 21:03
What André says is absolutely correct. The functor represented by the moduli space does not have any reasonable description, except in very particular cases. Given a map $U \to X$, it can be hard work to decide whether it comes from an object of $\mathcal M(U)$.