For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified covers of $X^\text{an}$.

If $X$ is instead a Deligne Mumford stack over Spec $\mathbb{C}$ corresponding to a complex orbifold $X^\text{an}$, is there an analogous theorem providing an equivalence between finite etale morphisms from DM stacks to $X$ and finite orbifold coverings of $X^\text{an}$? (hence etale fundamental group of such a DM stack is just the profinite completion of the orbifold fundamental group?)

In particular, I'm thinking of the situation where the stack is the moduli stack of elliptic curves over $\mathbb{C}$ and the orbifold is just $\mathcal{H}/SL_2(\mathbb{Z})$.

Can anyone provide a reference to a precise statement?

  • $\begingroup$ A finite etale cover of a DM stack is "the same" as descent datum on a finite etale cover of a scheme that is etale over the DM stack, so what is the difficulty in pulling down the result from the scheme case via descent theory? $\endgroup$
    – user74230
    Dec 20, 2014 at 6:01

1 Answer 1


The answer is yes. See Theorem 20.4 on p. 80 in Behrang Noohi's



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