Lemma 35.27.2. Then property $\mathcal{P}(f : X \to Y)=$“for every $x \in X$ the map of local rings $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is injective” is fpqc local on the source.

Proof. Omitted. This is just a (probably misguided) attempt to be playful. $\square$

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