# Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?

By a "nice" stack I mean a smooth finite type separated DM-stack over $\mathbf C$. If this isn't "nice" enough, also assume the stack to be the quotient stack of a smooth projective variety by the action of a smooth finite type separated group scheme. If this still isn't "nice" enough, also assume the coarse moduli spaces to be schemes (and not just algebraic spaces). But let's not assume the stacks to be representable. That would be too "nice" for me.

Note that the notion of a quasi-finite morphism makes sense for "nice" stacks, because quasi-finiteness is local for the etale topology.

Also, the extra conditions of "niceness" mentioned above might actually be automatic under the initial hypotheses of "niceness".

Perhaps one can make easy counterexamples by considering stacks $X$ with small coarse moduli spaces, e.g., $X^c = \{pt\}$. Then any map from $X_c$ to a scheme is quasi-finite (clearly), but if $X$ itself is a positive-dimensional stack, this will not be the case. If correct, the question remains then how to make a nice stack with this property.

• For any two finite type separated DM stacks $X$ and $Y$ over a field $k$, a $k$-morphism $f:X \rightarrow Y$ is quasi-finite if and only if the induced maps between sets of (isom. classes of) $\overline{k}$-points has finite fibers. That is a basic exercise with the very definition of "finite type" for DM-stacks and ways of checking quasi-finiteness for finite-type schemes over a field. So that answers your question affirmatively since $\overline{k}$-points of the coarse space are the same as (isom. classes of) such points of the given separated DM stack. – user76758 Apr 15 '14 at 20:11