The Stacks project

Lemma 61.10.8. Let $S$ be a scheme. Let $k$ be a field and let $Z \to \mathop{\mathrm{Spec}}(k)$ be a monomorphism of algebraic spaces over $S$. Then either $Z = \emptyset $ or $Z = \mathop{\mathrm{Spec}}(k)$.

Proof. By Lemmas 61.10.3 and 61.4.9 we see that $Z$ is a separated algebraic space. Hence there exists an open dense subspace $Z' \subset Z$ which is a scheme, see Properties of Spaces, Proposition 60.13.3. By Schemes, Lemma 25.23.11 we see that either $Z' = \emptyset $ or $Z' \cong \mathop{\mathrm{Spec}}(k)$. In the first case we conclude that $Z = \emptyset $ and in the second case we conclude that $Z' = Z = \mathop{\mathrm{Spec}}(k)$ as $Z \to \mathop{\mathrm{Spec}}(k)$ is a monomorphism which is an isomorphism over $Z'$. $\square$


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