The Stacks project

112.5.6 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 $U \subset X$.

Comments (2)

Comment #4317 by AAK on

See also Theorem B in David Rydh's paper [rydh_approx].

Comment #4475 by on

It is kinda impossible to keep this chapter up to date. I've added your suggestion. Thanks and see here.

There are also:

  • 4 comment(s) on Section 112.5: Papers in the literature

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04V1. Beware of the difference between the letter 'O' and the digit '0'.