I am trying to understand the statement that a Deligne-Mumford stack is locally a quotient $[U/G]$, where $G$ is a finite group. I don't understand why you can make $G$ a finite group, instead of a general group scheme. For example, if $G$ is a nonconstant étale group scheme over $S$, and $\mathcal{X}=[S/G]$ a quotient stack over $S$, how to make $\mathcal{X}$ a group quotient?


Intuitively, I feel the difference between a finite group and a finite étale group scheme is like the difference between a constant sheaf and a local system. But I am not sure whether it is true. Hope someone can help me to understand these group scheme things. It would be helpful if you can provide some reference.



The difference between an étale finite group scheme and a finite group
The way I think about finite etale group schemes over a connected scheme S, is that it's a group object in the category of schemes finite etale over S. This latter category is well-known to be equivalent to the category of finite sets X together with a action of $\pi\_1(S,s)$ (pick some geometric point $s\in S$). This is described in many sources, though the canonical reference would probably be SGA 1, Expose V. Thus, a finite etale group scheme is just a finite (abstract) group $G$, together with an action of $\pi\_1(S,s)$ acting on $G$ via group automorphisms. Expose XI of SGA 1 also talks a little about group schemes.


A finite group is just the group $G$ without the action of $\pi\_1(S,s)$. Of course, you can always have $\pi\_1(S,s)$ act on any group trivially, giving $G$ the structure of a constant group scheme over $S$ (ie, just $|G|$ disjoint copies of $S$, corresponding to a trivial degree $|G|$ cover of $S$), so in that sense any abstract group can be thought of as a constant group scheme.


As an example, you can consider the difference between $\mathbb{Z}/N\mathbb{Z}$ (integers mod $N$) and $\mu\_N$ ($N$th roots of unity). They can both be thought of as group schemes over $\mathbb{Q}$, but whereas the former doesn't have a natural action by $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, the latter does. Indeed, usually people think of $\mu\_N$ over $\mathbb{Q}$ as $\text{Spec }\mathbb{Q}[x]/(x^N-1)$, and $x^N-1$ will only factor into degree 1 polynomials over $\mathbb{Q}$ if $N \le 2$. When $N\ge 3$, the irreducible factors of degree $> 1$ correspond to connected components of the cover $\mu\_N\rightarrow \text{Spec }\mathbb{Q}$ which themselves are nontrivial covers of $\mathbb{Q}$, also showing that $\mu\_N$ is nonconstant. The action of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ can be seen as acting on the geometric fiber of $\mu\_N$ over $\overline{\mathbb{Q}}$, where you can see explicitly that the action is just the typical action sending primitive $N$th roots of unity to other primitive $N$th roots of unity. Contrast this with the situation over $\mathbb{C}$, where for any $N$, $x^N-1$ will factor into linear polynomials (corresponding to the fact that $\pi\_1(\text{Spec }\mathbb{C})$ is trivial), making $\mu\_N/\mathbb{C}$ the trivial degree $N$ cover, hence constant.


A DM stack can be thought of as locally being $[U/G]$ ($G$ a finite group) because etale locally, any finite etale group scheme is constant (and quotienting out by a constant group scheme is the same as quotienting by a group - think about what the quotient means on functors of points). To see that every finite etale group scheme is locally constant, note that an action of $\pi\_1(S)$ on $G$ corresponding to some group scheme $\mathcal{G}$ is the same as a homomorphism $\pi\_1(S)\rightarrow S\_{|G|}$. The kernel of this homomorphism is a finite index subgroup of $\pi\_1(S)$, corresponding to a scheme $T$ finite etale over $S$. The pullback of this homomorphism gives an exact sequence
$$\pi\_1(T)\rightarrow\pi\_1(S)\rightarrow S\_{|G|}$$
 since $T$ was defined by the property that $\pi\_1(T)$ is the kernel, so the composition is 0. Thus the homomorphism $\pi\_1(T)\rightarrow S\_{|G|}$ thought of as the pullback of the homomorphism $\pi\_1(S)\rightarrow S\_{|G|}$ to $T$ corresponds to the pullback of $\mathcal{G}$ to $T$, which is constant because the associated $\pi\_1(T)$-action is trivial.


