### 111.5.6 Existence of finite covers by schemes

The existence of finite covers of Deligne-Mumford stacks by schemes is an important result. In intersection theory on Deligne-Mumford stacks, it is an essential ingredient in defining proper push-forward for non-representable morphisms. There are several results about $\overline{\mathcal{M}}_ g$ relying on the existence of a finite cover by a *smooth* scheme which was proven by Looijenga. Perhaps the first result in this direction is [Theorem 6.1, seshadri_quotients] which treats the equivariant setting.

Vistoli:

*Intersection theory on algebraic stacks and on their moduli spaces*[vistoli_intersection]If $\mathcal{X}$ is a Deligne-Mumford stack with a moduli space (ie. a proper morphism which is bijective on geometric points), then there exists a finite morphism $X \to \mathcal{X}$ from a scheme $X$.

Laumon, Moret-Bailly: [Chapter 16, LM-B]

As an application of Zariski's main theorem, Theorem 16.6 establishes: if $\mathcal{X}$ is a Deligne-Mumford stack finite type over a Noetherian scheme, then there exists a finite, surjective, generically étale morphism $Z \to \mathcal{X}$ with $Z$ a scheme. It is also shown in Corollary 16.6.2 that any Noetherian normal algebraic space is isomorphic to the algebraic space quotient $X'/G$ for a finite group $G$ acting a normal scheme $X$.

Edidin, Hassett, Kresch, Vistoli:

*Brauer Groups and Quotient stacks*[ehkv]Theorem 2.7 states: if $\mathcal{X}$ is an algebraic stack of finite type over a Noetherian ground scheme $S$, then the diagonal $\mathcal{X} \to \mathcal{X} \times _ S \mathcal{X}$ is quasi-finite if and only if there exists a finite surjective morphism $X \to F$ from a scheme $X$.

Kresch, Vistoli:

*On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map*[kresch-vistoli]It is proved here that any smooth, separated Deligne-Mumford stack finite type over a field with quasi-projective coarse moduli space admits a finite, flat cover by a smooth quasi-projective scheme.

Olsson:

*On proper coverings of Artin stacks*[olsson_proper]Proves that if $\mathcal{X}$ is an Artin stack separated and finite type over $S$, then there exists a proper surjective morphism $X \to \mathcal{X}$ from a scheme $X$ quasi-projective over $S$. As an application, Olsson proves coherence and constructibility of direct image sheaves under proper morphisms. As an application, he proves Grothendieck's existence theorem for proper Artin stacks.

Rydh:

*Noetherian approximation of algebraic spaces and stacks*[rydh_approx]Theorem B of this paper is as follows. Let $X$ be a quasi-compact algebraic stack with quasi-finite and separated diagonal (resp. a quasi-compact Deligne-Mumford stack with quasi-compact and separated diagonal). Then there exists a scheme $Z$ and a finite, finitely presented and surjective morphism $Z \to X$ that is flat (resp. étale) over a dense quasi-compact open substack $U \subset X$.

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