Abstract
We show that any stack of finite type over a Noetherian scheme has a presentation
by a scheme of finite type such that
is onto, for every finite or real closed field F. Under some additional conditions on
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
the other is about the relation between real algebraic and Nash stacks.
Keywords:
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]