Let $\mathcal{X}$ be a *canonical* stack (**edit:** I forgot to say I also want $\mathcal{X}$ smooth), and $\pi : \mathcal{X}\to X$ its coarse moduli space morphism. Let also $D$ be a prime divisor (i.e. just one reduced component) on $X$, and $\mathcal{D}=\pi^{-1}(D)$ the pulled back divisor on $\mathcal{X}$, which will also be prime.



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> Is the isomorphism of $\mathcal{O}\_X$-modules
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> $$\pi\_\* \mathcal{O}(\mathcal{D})\cong \mathcal{O}(D)$$
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> always true?
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> 



Question about canonical DM stacks
I'm writing an answer that expands on my last comment:


I assume that your $D$ on $X$ is an effective Cartier divisor, that I see as an invertible sheaf $\mathcal{O}(D)$ on $X$ with a global section $1\_D$.


Then its pullback $\mathcal{D}$ as a divisor will be given by $(\pi^\*\mathcal{O}(D),\pi^\*1\_D)$. If the stack $\mathcal{X}$ is tame, for example by Lemma 2.8 of <http://arxiv.org/pdf/0811.1949v2.pdf> there is an isomorphism $\pi\_\*\mathcal{O}(\mathcal{D})=\pi\_\*\pi^\*\mathcal{O}(D)\cong \mathcal{O}(D)$.


