The Stacks project

Lemma 29.14.6. Let $P$ be a property of ring maps. Assume $P$ is local and stable under base change. The base change of a morphism locally of type $P$ is locally of type $P$.

Proof. Let $f : X \to S$ be a morphism locally of type $P$. Let $S' \to S$ be any morphism. Denote $f' : X_{S'} = S' \times _ S X \to S'$ the base change of $f$. For every $s' \in S'$ there exists an open affine neighbourhood $s' \in V' \subset S'$ which maps into some open affine $V \subset S$. By Lemma 29.14.4 the open $f^{-1}(V)$ is a union of affines $U_ i$ such that the ring maps $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U_ i)$ all satisfy $P$. By the material in Schemes, Section 26.17 we see that $f^{-1}(U)_{V'} = V' \times _ V f^{-1}(V)$ is the union of the affine opens $V' \times _ V U_ i$. Since $\mathcal{O}_{X_{S'}}(V' \times _ V U_ i) = \mathcal{O}_{S'}(V') \otimes _{\mathcal{O}_ S(V)} \mathcal{O}_ X(U_ i)$ we see that the ring maps $\mathcal{O}_{S'}(V') \to \mathcal{O}_{X_{S'}}(V' \times _ V U_ i)$ satisfy $P$ as $P$ is assumed stable under base change. $\square$

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