The Stacks project

Lemma 76.10.1. Let $S$ be a scheme. Let $(f, f') : (X \subset X') \to (Y \subset Y')$ be a morphism of thickenings of algebraic spaces over $S$. Then

  1. $f$ is an affine morphism if and only if $f'$ is an affine morphism,

  2. $f$ is a surjective morphism if and only if $f'$ is a surjective morphism,

  3. $f$ is quasi-compact if and only if $f'$ quasi-compact,

  4. $f$ is universally closed if and only if $f'$ is universally closed,

  5. $f$ is integral if and only if $f'$ is integral,

  6. $f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,

  7. $f$ is universally injective if and only if $f'$ is universally injective,

  8. $f$ is universally open if and only if $f'$ is universally open,

  9. $f$ is representable if and only if $f'$ is representable, and

  10. add more here.

Proof. Observe that $Y \to Y'$ and $X \to X'$ are integral and universal homeomorphisms. This immediately implies parts (2), (3), (4), (7), and (8). Part (1) follows from Limits of Spaces, Proposition 70.15.2 which tells us that there is a 1-to-1 correspondence between affine schemes étale over $X$ and $X'$ and between affine schemes étale over $Y$ and $Y'$. Part (5) follows from (1) and (4) by Morphisms of Spaces, Lemma 67.45.7. Finally, note that

\[ X \times _ Y X = X \times _{Y'} X \to X \times _{Y'} X' \to X' \times _{Y'} X' \]

is a thickening (the two arrows are thickenings by Lemma 76.9.8). Hence applying (3) and (4) to the morphism $(X \subset X') \to (X \times _ Y X \to X' \times _{Y'} X')$ we obtain (6). Finally, part (9) follows from the fact that an algebraic space thickening of a scheme is again a scheme, see Lemma 76.9.5. $\square$

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