# Is a stack that is finite etale over an algebraic stack also algebraic?

Here by "algebraic stack" I mean a stack in groupoids over $(\textbf{Sch}/S)_\text{etale}$ (for some scheme $S$) whose diagonal morphism is representable (by schemes), and which is covered by a surjective etale morphism from a scheme.

The question is, if $\mathcal{X}\rightarrow\mathcal{Y}$ is a morphism of stacks in groupoids which is representable finite etale, and and $\mathcal{Y}$ is an algebraic stack, then is $\mathcal{X}$ also an algebraic stack?

You can certainly pull back the scheme cover of $\mathcal{Y}$ to get a surjective etale scheme cover of $\mathcal{X}$, so really I suppose I'm asking if the diagonal morphism of $\mathcal{X}$ must be representable.

The stacks project has a similar result (Lemmas 67.15.3, 67.15.4), but their definition of an algebraic stack only requires the diagonal morphism to be representable by algebraic spaces, but their result seems to rely on the fact that a stack covered by an algebraic space is an algebraic space, which of course isn't true if you replace algebraic spaces by schemes.

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The answer is yes. I think that you haven't looked at the right place: see Tag 05XY (currently Lemma 71.8.2) in the Stacks Project. –  Matthieu Romagny Jun 22 at 9:07
If you're talking about stacks.math.columbia.edu/tag/05XY , then that lemma still only refers to "algebraic stacks" as defined by the stacks project, which is not the same as my definition (I require the diagonal to be representable by schemes. The stacks project only requires it be representable by algebraic spaces). –  oxeimon Jun 22 at 10:50
@WillChen: You should just write out the diagram and check this for yourself. Factor the diagonal morphism of $\mathcal{X}$ as the composition of the relative diagonal morphism from $\mathcal{X}$ to $\mathcal{X}\times_{\mathcal{Y}}\mathcal{X}$ followed by the natural morphism from $\mathcal{X}\times_{\mathcal{Y}}\mathcal{X}$ to $\mathcal{X}\times \mathcal{X}$. What (2-)Cartesian diagrams contain each of these factor morphisms? –  Jason Starr Jun 23 at 14:15
Also, what you are calling an "algebraic stack" is called a "Deligne-Mumford stack" by most authors. –  Jason Starr Jun 23 at 14:16