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. $\square$

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