# Finiteness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid.

Are there known results that tell us anything about the finitness of $\pi_0(X(Y))$?

To explain what I have in mind, if $X$ is a zero dimensional scheme of finite type over a field $k$, then $\pi_0(X(k))$ is $X(k)$ itself and it is finite.

• isn't that just the coarse moduli scheme/space? – Will Chen Apr 8 '15 at 18:16
• mathoverflow.net/questions/90975/coarse-moduli-space-and-pi-0 – bananastack Apr 8 '15 at 20:31
• @oxeimon: Typically not, since it is typically not a sheaf. To get a coarse moduli space in this way you (at the very least) have to sheafify it. bananastack's link says more in this direction. – tracing Apr 9 '15 at 12:09
• oxeimon was referring to the previous form of my question, were I mention the sheafification. However after bananastack's comment, I decided to focus the question only on the finitness part as the answer in the link already dealt with the other part. – Bear Apr 9 '15 at 15:00