The Stacks project

Lemma 69.21.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-separated and locally of finite type and $Y$ is locally Noetherian. The following are equivalent:

  1. The morphism $f$ is separated.

  2. For any diagram ( there is at most one dotted arrow.

  3. For all diagrams ( with $A$ a discrete valuation ring there is at most one dotted arrow.

  4. For all diagrams ( where $A$ is a discrete valuation ring and where the image of $\mathop{\mathrm{Spec}}(K) \to X$ is a point of codimension $0$ on $X$ there is at most one dotted arrow.

Proof. We have (1) $\Rightarrow $ (2) by Morphisms of Spaces, Lemma 66.43.1. The implications (2) $\Rightarrow $ (3) and (3) $\Rightarrow $ (4) are immediate. It remains to show (4) implies (1).

Assume (4). We have to show that the diagonal $\Delta : X \to X \times _ Y X$ is a closed immersion. We already know $\Delta $ is representable, separated, a monomorphism, and locally of finite type, see Morphisms of Spaces, Lemma 66.4.1. Choose an affine scheme $U$ and an étale morphism $U \to X \times _ Y X$. Set $V = X \times _{\Delta , X \times _ Y X} U$. It suffices to show that $V \to U$ is a closed immersion (Morphisms of Spaces, Lemma 66.12.1). Since $X \times _ Y X$ is locally of finite type over $Y$ we see that $U$ is Noetherian (use Morphisms of Spaces, Lemmas 66.23.2, 66.23.3, and 66.23.5). Note that $V$ is a scheme as $\Delta $ is representable. Also, $V$ is quasi-compact because $f$ is quasi-separated. Hence $V \to U$ is separated and of finite type. Consider a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & V \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & U } \]

of morphisms of schemes where $A$ is a discrete valuation ring with fraction field $K$ and where $K$ is the residue field of a generic point of the Noetherian scheme $V$. Since $V \to X$ is étale (as a base change of the étale morphism $U \to X \times _ Y X$) we see that the image of $\mathop{\mathrm{Spec}}(K) \to V \to X$ is a point of codimension $0$, see Properties of Spaces, Section 65.10. We can interpret the composition $\mathop{\mathrm{Spec}}(A) \to U \to X \times _ Y X$ as a pair of morphisms $a, b : \mathop{\mathrm{Spec}}(A) \to X$ agreeing as morphisms into $Y$ and equal when restricted to $\mathop{\mathrm{Spec}}(K)$ and that this restriction maps to a point of codimension $0$. Hence our assumption (4) guarantees $a = b$ and we find the dotted arrow in the diagram. By Limits, Lemma 32.15.3 we conclude that $V \to U$ is proper. In other words, $\Delta $ is proper. Since $\Delta $ is a monomorphism, we find that $\Delta $ is a closed immersion (Étale Morphisms, Lemma 41.7.2) as desired. $\square$

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