Lemma 29.14.3. Let $f : X \to S$ be a morphism of schemes. Let $P$ be a property of ring maps. Let $U$ be an affine open of $X$, and $V$ an affine open of $S$ such that $f(U) \subset V$. If $f$ is locally of type $P$ and $P$ is local, then $P(\mathcal{O}_ S(V) \to \mathcal{O}_ X(U))$ holds.

**Proof.**
As $f$ is locally of type $P$ for every $u \in U$ there exists an affine open $U_ u \subset X$ mapping into an affine open $V_ u \subset S$ such that $P(\mathcal{O}_ S(V_ u) \to \mathcal{O}_ X(U_ u))$ holds. Choose an open neighbourhood $U'_ u \subset U \cap U_ u$ of $u$ which is standard affine open in both $U$ and $U_ u$, see Schemes, Lemma 26.11.5. By Definition 29.14.1 (1)(b) we see that $P(\mathcal{O}_ S(V_ u) \to \mathcal{O}_ X(U'_ u))$ holds. Hence we may assume that $U_ u \subset U$ is a standard affine open. Choose an open neighbourhood $V'_ u \subset V \cap V_ u$ of $f(u)$ which is standard affine open in both $V$ and $V_ u$, see Schemes, Lemma 26.11.5. Then $U'_ u = f^{-1}(V'_ u) \cap U_ u$ is a standard affine open of $U_ u$ (hence of $U$) and we have $P(\mathcal{O}_ S(V'_ u) \to \mathcal{O}_ X(U'_ u))$ by Definition 29.14.1 (1)(a). Hence we may assume both $U_ u \subset U$ and $V_ u \subset V$ are standard affine open. Applying Definition 29.14.1 (1)(b) one more time we conclude that $P(\mathcal{O}_ S(V) \to \mathcal{O}_ X(U_ u))$ holds. Because $U$ is quasi-compact we may choose a finite number of points $u_1, \ldots , u_ n \in U$ such that

By Definition 29.14.1 (1)(c) we conclude that $P(\mathcal{O}_ S(V) \to \mathcal{O}_ X(U))$ holds. $\square$

## 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Your email address will not be published.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.

## Comments (0)