The Stacks project

Lemma 67.43.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and quasi-separated. Then the following are equivalent

  1. $f$ is separated and universally closed,

  2. the valuative criterion holds as in Definition 67.41.1,

  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 unique dotted arrow, i.e., $f$ satisfies the valuative criterion as in Schemes, Definition 26.20.3.

Proof. Since $f$ is quasi-separated, the uniqueness part of the valutative criterion implies $f$ is separated (Lemma 67.43.2). Conversely, if $f$ is separated, then it satisfies the uniqueness part of the valuative criterion (Lemma 67.43.1). Having said this, we see that in each of the three cases the morphism $f$ is separated and satisfies the uniqueness part of the valuative criterion. In this case the lemma is a formal consequence of Lemma 67.42.3. $\square$

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