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:

(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.

(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

Comment #5735 by Johan on

There are also: