Lemma 37.47.4. Let \pi : X \to Y be a finite morphism. Let x \in X with y = \pi (x) such that \pi ^{-1}(\{ y\} ) = \{ x\} . Then
For every neighbourhood U \subset X of x in X, there exists a neighbourhood V \subset Y of y such that \pi ^{-1}(V) \subset U.
The ring map \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x} is finite.
If \pi is of finite presentation, then \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x} is of finite presentation.
For any quasi-coherent \mathcal{O}_ X-module \mathcal{F} we have \mathcal{F}_ x = \pi _*\mathcal{F}_ y as \mathcal{O}_{Y, y}-modules.
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