The Stacks project

Lemma 66.42.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and separated. Then the following are equivalent

  1. $f$ is universally closed,

  2. the existence part of the valuative criterion holds as in Definition 66.41.1, and

  3. given any commutative solid diagram

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

    where $A$ is a valuation ring with field of fractions $K$, there exists a dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition 26.20.3.

Proof. Since $f$ is separated parts (2) and (3) are equivalent by Lemma 66.41.5. The equivalence of (3) and (1) follows from Lemma 66.42.2. $\square$


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