The Stacks project

Lemma 29.14.5. Let $P$ be a property of ring maps. Assume $P$ is local and stable under composition. The composition of morphisms locally of type $P$ is locally of type $P$.

Proof. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms locally of type $P$. Let $x \in X$. Choose an affine open neighbourhood $W \subset Z$ of $g(f(x))$. Choose an affine open neighbourhood $V \subset g^{-1}(W)$ of $f(x)$. Choose an affine open neighbourhood $U \subset f^{-1}(V)$ of $x$. By Lemma 29.14.4 the ring maps $\mathcal{O}_ Z(W) \to \mathcal{O}_ Y(V)$ and $\mathcal{O}_ Y(V) \to \mathcal{O}_ X(U)$ satisfy $P$. Hence $\mathcal{O}_ Z(W) \to \mathcal{O}_ X(U)$ satisfies $P$ as $P$ is assumed stable under composition. $\square$


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Your email address will not be published.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 01SV. Beware of the difference between the letter 'O' and the digit '0'.