The Stacks project

Lemma 68.17.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. We have the following implications among the conditions on $f$:

\[ \xymatrix{ \text{representable} \ar@{=>}[rd] & & & & \\ & \text{very reasonable} \ar@{=>}[r] & \text{reasonable} \ar@{=>}[r] & \text{decent} \ar@{=>}[r] & (\beta ) \\ \text{quasi-separated} \ar@{=>}[ru] & & & & } \]

Proof. This is clear from the definitions, Lemma 68.5.1 and Morphisms of Spaces, Lemma 67.4.12. $\square$

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