The Stacks project

Remark 29.51.10. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type. There are (at least) two properties that we could use to define generically finite morphisms. These correspond to whether you want the property to be local on the source or local on the target:

  1. (Local on the target; suggested by Ravi Vakil.) Assume every quasi-compact open of $Y$ has finitely many irreducible components (for example if $Y$ is locally Noetherian). The requirement is that the inverse image of each generic point is finite, see Lemma 29.51.1.

  2. (Local on the source.) The requirement is that there exists a dense open $U \subset X$ such that $U \to Y$ is locally quasi-finite.

In case (1) the requirement can be formulated without the auxiliary condition on $Y$, but probably doesn't give the right notion for general schemes. Property (2) as formulated doesn't imply that the fibres over generic points are finite; however, if $f$ is quasi-compact and $Y$ is as in (1) then it does.

Comments (2)

Comment #5550 by Laurent Moret-Bailly on

So, SP does not define generically finite morphisms?

Comment #5735 by on

Well, I would like to see more votes for either definition or people clamoring for having at least some definition.

There are also:

  • 2 comment(s) on Section 29.51: Generically finite morphisms

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