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 ├ętale over $X$ and $X'$ and between affine schemes ├ętale 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09ZY. Beware of the difference between the letter 'O' and the digit '0'.