The Stacks project

Proposition 67.16.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact, and $X$ is decent. Then $f$ is universally closed if and only if the existence part of the valuative criterion holds.

Proof. In Morphisms of Spaces, Lemma 66.42.1 we have seen one of the implications. To prove the other, assume that $f$ is universally closed. Let

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Y } \]

be a diagram as in Morphisms of Spaces, Definition 66.41.1. Let $X_ A = \mathop{\mathrm{Spec}}(A) \times _ Y X$, so that we have

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[rd] & X_ A \ar[d] \\ & \mathop{\mathrm{Spec}}(A) } \]

By Morphisms of Spaces, Lemma 66.8.4 we see that $X_ A \to \mathop{\mathrm{Spec}}(A)$ is quasi-compact. Since $X_ A \to X$ is representable, we see that $X_ A$ is decent also, see Lemma 67.5.3. Moreover, as $f$ is universally closed, we see that $X_ A \to \mathop{\mathrm{Spec}}(A)$ is universally closed. Hence we may and do replace $X$ by $X_ A$ and $Y$ by $\mathop{\mathrm{Spec}}(A)$.

Let $x' \in |X|$ be the equivalence class of $\mathop{\mathrm{Spec}}(K) \to X$. Let $y \in |Y| = |\mathop{\mathrm{Spec}}(A)|$ be the closed point. Set $y' = f(x')$; it is the generic point of $\mathop{\mathrm{Spec}}(A)$. Since $f$ is universally closed we see that $f(\overline{\{ x'\} })$ contains $\overline{\{ y'\} }$, and hence contains $y$. Let $x \in \overline{\{ x'\} }$ be a point such that $f(x) = y$. Let $U$ be a scheme, and $\varphi : U \to X$ an ├ętale morphism such that there exists a $u \in U$ with $\varphi (u) = x$. By Lemma 67.7.3 and our assumption that $X$ is decent there exists a specialization $u' \leadsto u$ on $U$ with $\varphi (u') = x'$. This means that there exists a common field extension $K \subset K' \supset \kappa (u')$ such that

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d] & U \ar[d] \\ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[rd] & X \ar[d] \\ & \mathop{\mathrm{Spec}}(A) } \]

is commutative. This gives the following commutative diagram of rings

\[ \xymatrix{ K' & \mathcal{O}_{U, u} \ar[l] \\ K \ar[u] & \\ & A \ar[lu] \ar[uu] } \]

By Algebra, Lemma 10.50.2 we can find a valuation ring $A' \subset K'$ dominating the image of $\mathcal{O}_{U, u}$ in $K'$. Since by construction $\mathcal{O}_{U, u}$ dominates $A$ we see that $A'$ dominates $A$ also. Hence we obtain a diagram resembling the second diagram of Morphisms of Spaces, Definition 66.41.1 and the proposition is proved. $\square$

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