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 $X$.



Grothendieck duality for stacks
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


$$f\_{\*}\mathcal{E}xt^{1}\_{\mathcal{X}}(\mathcal{F},f^{\*}\mathcal{G})\cong \mathcal{E}xt^{1}\_{\mathcal{Y}}(f\_{\*}\mathcal{F},\mathcal{G}).$$


