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.

Is the isomorphism of $\mathcal{O}_X$-modules

 

$$\pi_* \mathcal{O}(\mathcal{D})\cong \mathcal{O}(D)$$

 

always true?

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.

Is the isomorphism of $\mathcal{O}_X$-modules

$$\pi_* \mathcal{O}(\mathcal{D})\cong \mathcal{O}(D)$$

always true?

Let $\mathcal{X}$ be a canonical stack, 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.

Is the isomorphism of $\mathcal{O}_X$-modules

$$\pi_* \mathcal{O}(\mathcal{D})\cong \mathcal{O}(D)$$

always true?

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.

Is the isomorphism of $\mathcal{O}_X$-modules

$$\pi_* \mathcal{O}(\mathcal{D})\cong \mathcal{O}(D)$$

always true?