Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In particular I would like two know if there is an isomorphism
$$\pi_{*}\mathcal{E}xt^{1}_{\mathcal{X}}(\mathcal{F},\pi^{*}\mathcal{G})\cong \mathcal{E}xt^{1}_{X}(\pi_{*}\mathcal{F},\mathcal{G})$$
where $\mathcal{F}$ is a coherent sheaf on $\mathcal{X}$ and $\mathcal{G}$ is a coherent sheaf on $X$.
By Corollary $2.10$ in http://arxiv.org/pdf/0811.1955.pdf
Let $f:\mathcal{X}\rightarrow \mathcal{Y}$ be a proper morphism of Deligne-Mumford stacks and $\mathcal{F}\in D^+_c(\mathcal{X})$, $\mathcal{G}\in D^+(\mathcal{Y})$. The the morphsim
$$Rf_{*}R\mathcal{H}om_{\mathcal{X}}(\mathcal{F},f^{!}\mathcal{G})\rightarrow R\mathcal{H}om_{\mathcal{X}}(Rf_{*}\mathcal{F},Rf_{*}f^{!}\mathcal{G})\rightarrow R\mathcal{H}om_{\mathcal{Y}}(Rf_{*}\mathcal{F},\mathcal{G}).$$
is an isomorphism.
If $f:\mathcal{X}\rightarrow \mathcal{Y}$ is a representable finite étale morphism of noetherian algebraic stacks, then the functor $f^{!}$ is the same as $f^{*}$. Therefore we have an isomorphism
$$Rf_{*}R\mathcal{H}om_{\mathcal{X}}(\mathcal{F},f^{*}\mathcal{G})\rightarrow R\mathcal{H}om_{\mathcal{X}}(Rf_{*}\mathcal{F},Rf_{*}f^{*}\mathcal{G})\rightarrow R\mathcal{H}om_{\mathcal{Y}}(Rf_{*}\mathcal{F},\mathcal{G}).$$
Finally, $f$ finite implies $R^if_*\mathcal{F} = 0$ for $i\geq 1$. Therefore, if $\mathcal{H}om_{\mathcal{X}}(\mathcal{F},f^{*}\mathcal{G}) = 0$ we get
$$f_{*}\mathcal{E}xt^{1}_{\mathcal{X}}(\mathcal{F},f^{*}\mathcal{G})\cong \mathcal{E}xt^{1}_{\mathcal{Y}}(f_{*}\mathcal{F},\mathcal{G}).$$
