The Stacks project

Lemma 67.4.8. All of the separation axioms listed in Definition 67.4.2 are stable under composition of morphisms.

Proof. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of algebraic spaces to which the axiom in question applies. The diagonal $\Delta _{X/Z}$ is the composition

\[ X \longrightarrow X \times _ Y X \longrightarrow X \times _ Z X. \]

Our separation axiom is defined by requiring the diagonal to have some property $\mathcal{P}$. By Lemma 67.4.5 above we see that the second arrow also has this property. Hence the lemma follows since the composition of (representable) morphisms with property $\mathcal{P}$ also is a morphism with property $\mathcal{P}$, see Section 67.3. $\square$

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Your email address will not be published.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 03KQ. Beware of the difference between the letter 'O' and the digit '0'.