I'm wondering what the precise relationship is between an algebraic stack being locally of finite presentation and being limit preserving. Under some mild hypotheses on the diagonal (in force throughout the book), Prop. 4.18 of the book by Laumon--Moret-Bailly shows that a stack locally of finite presentation over some base scheme is limit preserving; I don't think the hypotheses on the diagonal are used here, and indeed I think I can piece together a proof pretty easily using the assumptions of the stacks project.

Now, I'm wondering if the converse is true. Some reasons to think it might be: in Artin's "Versal Deformations and Algebraic Stacks", he defines limit preserving, and says that it is what he had previously referred to as locally of finite presentation. Also, for algebraic spaces we have http://stacks.math.columbia.edu/tag/05N0 which shows that the two conditions are equivalent (and equivalent to being limit preserving on objects, unsurprisingly).

A reason to think they might not be equivalent: I haven't seen this statement anywhere in the literature.


1 Answer 1


Suppose that $\mathcal{X}$ is an algebraic stack which is limit preserving on objects over $S$. Suppose that $U \to \mathcal{X}$ is a smooth surjective map from a scheme. Then $U \to S$ is limit preserving by the results of Section Tag 06CT. Thus $U$ is locally of finite presentation over S (for example by the reference you gave). This exactly means that $\mathcal{X} \to S$ is locally of finite presentation.

I didn't check the direction you said was OK, but if it is, then this also means there is no difference between "limit preserving on objects" and "limit preserving" for algebraic stacks over schemes. There is a difference in general (for stacks in groupoids over schemes for example). Anyway, this is one of the many things missing from the Stacks project (thanks for pointing it out). The Stacks project takes contributions (by email or via git pull requests) if you are so inclined.

  • $\begingroup$ Thank you very much - this is great. I will try to find the time soon to check the other direction, and maybe even write it up for the Stacks project. $\endgroup$
    – beginner
    Jan 19, 2014 at 16:49

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