The Stacks project

Lemma 66.4.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $p : \mathop{\mathrm{Spec}}(K) \to X$ and $q : \mathop{\mathrm{Spec}}(L) \to X$ be morphisms where $K$ and $L$ are fields. Assume $p$ and $q$ determine the same point of $|X|$ and $p$ is a monomorphism. Then $q$ factors uniquely through $p$.

Proof. Since $p$ and $q$ define the same point of $|X|$, we see that the scheme

\[ Y = \mathop{\mathrm{Spec}}(K) \times _{p, X, q} \mathop{\mathrm{Spec}}(L) \]

is nonempty. Since the base change of a monomorphism is a monomorphism this means that the projection morphism $Y \to \mathop{\mathrm{Spec}}(L)$ is a monomorphism. Hence $Y = \mathop{\mathrm{Spec}}(L)$, see Schemes, Lemma 26.23.11. We conclude that $q$ factors through $p$. Uniqueness comes from the fact that $p$ is a monomorphism. $\square$

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