Let $\cal X$ be a DM stack of finite type over a field (if necessary, I will assume that $k=\mathbb{C}$ and $\cal X$ is a scheme, or even a variety) and $G$ be a finite group. Then we have a quotient stack $[\mathcal{X}/G]$ and in fact we have a natural morphism $\phi:\mathcal{X}\to [\mathcal{X}/G]$.


Is $\phi$ always flat? If not, under what conditions is $\phi$ flat?



Are quotients of stacks flat?
The map $\phi$ is in fact étale. By definition a map $B \to [\mathcal X/G]$ (say $B$ is a scheme) is a $G$-torsor $E \to B$ and a $G$-equivariant map $E \to \mathcal X$. The base change of $\phi$ is exactly $E \to B$. Since the property of being étale is stable under base change and local, it suffices to show that $E\to B$ is étale. But part of the definition of being a torsor is being locally trivial in whatever topology you are considering, so now it suffices to check that the map from $G$ to a point is étale, which is clear.


The general principle here is that properties of a group scheme $G$ should carry over to properties of the universal map $t \colon \mathrm{pt} \to BG$, e.g. if $G$ is smooth then $t$ is smooth. 


