The Stacks project

Lemma 66.4.10. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces over $S$.

  1. If $g \circ f$ is separated then so is $f$.

  2. If $g \circ f$ is locally separated then so is $f$.

  3. If $g \circ f$ is quasi-separated then so is $f$.

Proof. Consider the factorization

\[ X \to X \times _ Y X \to X \times _ Z X \]

of the diagonal morphism of $g \circ f$. In any case the last morphism is a monomorphism. Hence for any scheme $T$ and morphism $T \to X \times _ Y X$ we have the equality

\[ X \times _{(X \times _ Y X)} T = X \times _{(X \times _ Z X)} T. \]

Hence the result is clear. $\square$

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