The Stacks project

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

\[ U = U_{u_1} \cup \ldots \cup U_{u_ n}. \]

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

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