Lemma 101.45.7 (0DUD)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.45: Stabilizer preserving morphisms
Lemma 101.45.7 (cite)

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Lemma 101.45.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Assume $f$ is étale, $f$ induces an isomorphism between automorphism groups at points (Remark 101.19.5), and for every algebraically closed field $k$ the functor

$f : \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{X}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{Spec}}(k), \mathcal{Y})$

is an equivalence. Then $f$ is an isomorphism.

Proof. By Lemma 101.14.5 we see that $f$ is universally injective. Combining Lemmas 101.45.1 and 101.45.3 we see that $f$ is representable by algebraic spaces. Hence $f$ is an open immersion by Morphisms of Spaces, Lemma 67.51.2. To finish we remark that the condition in the lemma also guarantees that $f$ is surjective. $\square$

Comments (2)

Comment #7856 by Rachel Webb on October 26, 2022 at 17:41

How is it used that $f$ is separated?

Comment #8075 by Stacks Project on January 21, 2023 at 15:23

THanks and fixed here.

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