Pointwise surjective presentations of stacks

Abstract

We show that any stack $\mathfrak{X}$ of finite type over a Noetherian scheme has a presentation $X\to \mathfrak{X}$ by a scheme of finite type such that $X(F)\to \mathfrak{X}(F)$ is onto, for every finite or real closed field F. Under some additional conditions on $\mathfrak{X},$ we show the same for all perfect fields. We prove similar results for (some) Henselian rings. We give two applications of the main result. One is to counting isomorphism classes of stacks over the rings $\mathbb{Z}/p^n;$ the other is about the relation between real algebraic and Nash stacks.

Keywords:

• Nash stacks
• stacks
• zeta functions

2020 Mathematics Subject Classifications:

• 14A20
• 14G05
• 11G25

Acknowledgments

We thank Shahar Carmeli, Raf Cluckers, and Ofer Gabber for fruitful discussions. We thank Angelo Vistoli for answering a question of ours on MathOverFlow, proving Lemma 3.2.

Notes

1 The definition in [11, §2.3] is slightly more restrictive, though we believe the result is true without the restriction.

2 The proof there uses implicitly [12, 60.13.2]

Additional information

Funding

A.A. was partially supported by ISF grants 687/13, and grant 249/17. N.A. was partially supported by NSF grant DMS-1902041. Both authors were partially supported by BSF grants 2012247 and 2018201.