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.

## Comments (0)