# Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.)

Under which conditions is $\# X(k) = \# X_c(k)$?

Here we take the groupoid cardinality/stacky sum/mass formula to define $\#X(k)$: $$\# X(k) := \sum_{x\in X(k)} \frac{1}{\# Aut(x)}.$$

I think this equality holds if $X$ is a separated finite type DM stack whose coarse moduli space is a scheme (and not just an algebraic space).

Does it hold more generally, ie, only assuming the existence of $X_c$ and not any other properties?

• How do you prove it in the case you mention? – Mattia Talpo Feb 5 '15 at 7:41