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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