In the stacks project a Deligne-Mumford stack is an algebraic stack X such that there exists a scheme U and a surjective etale morphism U —> X. An algebraic stack X is said to be DM if the diagonal Δ : X —> X x X is unramified. In fact Theorem Tag 06N3 says:
X is DM if and only if X is Deligne-Mumford.
An algebraic stack X is said to be quasi-DM if the diagonal Δ : X —> X x X is locally quasi-finite. The analogue of the theorem above is Theorem Tag 06MF which says:
X is quasi-DM if and only if there exists a scheme U and a surjective, flat, locally finitely presented, and locally quasi-finite morphism U —> X.
The proofs of these theorems are completely parallel. Assume X is DM (resp. quasi-DM). We try to construct etale (resp. loc fp + flat + loc quasi-finite) maps from schemes toward X. In both cases the strategy is the following:
- Pick a smooth morphism U —> X,
- choose a suitable point x of X,
- let F be the fibre of U over x, and
- “slice” U, i.e., find a complete intersection V(f_1, …, f_d) ⊂ U such that f_1, …, f_d form a regular system of parameters (resp. regular sequence) at some point of F
In both cases the proof shows that, after possibly shrinking U, the morphism V(f_1, …, f_d) —> X is flat, locally finitely presented, and unramified (resp. locally quasi-finite). A bit of care is needed in choosing the point x on X. I decided to use “finite type points”; in both cases one then has to do a bit of work to show that the “fibre F” has desirable properties: in the DM case one need to produce x such that F —> U is unramified and in the quasi-DM case such that F —> U is locally quasi-finite.
The reasoning above is completely standard. However, there is a way to deduce the first theorem from the second. I decided against arguing like this in the stacks project as it is perhaps a little nonstandard. Here is the argument. Let X be DM. By the second theorem we can find U —> X which is surjective, flat, locally of finite presentation, and locally quasi-finite. Let H_{d, lci}(U/X) be the LCI locus in the relative degree d Hilbert stack of U over X (see Section Tag 06CJ). Then H_{d, lci}(U/X) —> X is smooth (this is explained in the proof of Theorem Tag 06DC). But of course it is clear that H_{d, lci}(U/X) —> X has relative dimension 0, hence it is etale. This doesn’t quite finish the proof because H_{d, lci}(U/X) is (as defined in the stacks project) an algebraic stack and not an algebraic space; but a straightforward argument shows (because X is DM) that the disjoint union for varying d of the open substacks of H_{d, lci}(U/X) having trivial inertia surjects onto X.