The Stacks project

Proof. We have (1) $\Rightarrow $ (2) by Morphisms of Spaces, Lemma 67.43.1. 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 67.4.1. 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 67.12.1). Since $X \times _ Y X$ is locally of finite type over $Y$ we see that $U$ is Noetherian (use Morphisms of Spaces, Lemmas 67.23.2, 67.23.3, and 67.23.5). 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

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 66.10. 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. $\square$