Lemma 67.17.1. Let S be a scheme. Let W \subset S be a scheme theoretically dense open subscheme (Morphisms, Definition 29.7.1). Let f : X \to S be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Then f^{-1}(W) is scheme theoretically dense in X.
Proof. We will use the characterization of Morphisms, Lemma 29.7.5. Assume V \subset X is an open and g \in \Gamma (V, \mathcal{O}_ V) is a function which restricts to zero on f^{-1}(W) \cap V. We have to show that g = 0. Assume g \not= 0 to get a contradiction. By More on Morphisms, Lemma 37.45.6 we may shrink V, find an open U \subset S fitting into a commutative diagram
\xymatrix{ V \ar[r] \ar[d]_\pi & X \ar[d]^ f \\ U \ar[r] & S, }
a quasi-coherent subsheaf \mathcal{F} \subset \mathcal{O}_ U, an integer r > 0, and an injective \mathcal{O}_ U-module map \mathcal{F}^{\oplus r} \to \pi _*\mathcal{O}_ V whose image contains g|_ V. Say (g_1, \ldots , g_ r) \in \Gamma (U, \mathcal{F}^{\oplus r}) maps to g. Then we see that g_ i|_{W \cap U} = 0 because g|_{f^{-1}W \cap V} = 0. Hence g_ i = 0 because \mathcal{F} \subset \mathcal{O}_ U and W is scheme theoretically dense in S. This implies g = 0 which is the desired contradiction. \square
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