Lemma 66.4.12 (03E1)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.4: Points of algebraic spaces
Lemma 66.4.12 (cite)

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Lemma 66.4.12. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Consider the map

\[ \{ \mathop{\mathrm{Spec}}(k) \to X \text{ monomorphism where }k\text{ is a field}\} \longrightarrow |X| \]

This map is injective.

Proof. This follows from Lemma 66.4.11. $\square$

Comments (2)

Comment #7758 by Laurent Moret-Bailly on September 13, 2022 at 08:51

In connection with Comment #7747, what is really proved here is this: if $\varphi:\mathrm{Spec}(k)\to X$ is a monomorphism, then every $\mathrm{Spec}(K)\to X$ defining the same point factors (uniquely) through $\varphi$ . The stated result follows immediately.

Comment #8004 by Stacks Project on January 16, 2023 at 12:48

Very good and fixed here.

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