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[II Definition 6.2.3, EGA]

Definition 29.20.1. Let $f : X \to S$ be a morphism of schemes.

  1. We say that $f$ is quasi-finite at a point $x \in X$ if there exist an affine neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and an affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ such that $f(U) \subset V$, the ring map $R \to A$ is of finite type, and $R \to A$ is quasi-finite at the prime of $A$ corresponding to $x$ (see above).

  2. We say $f$ is locally quasi-finite if $f$ is quasi-finite at every point $x$ of $X$.

  3. We say that $f$ is quasi-finite if $f$ is of finite type and every point $x$ is an isolated point of its fibre.

Comments (4)

Comment #2714 by Ariyan Javanpeykar on

A reference: This definition is made in EGA II, Definition 6.2.3

Comment #10278 by Jaehyeon Lee on

I think the definition of 'quasi-finite at x' does not come from EGA II (6.2.3).

Corollarie ([EGA II, (6.2.2)]) Soit un morphisme de type fini. Les conditions suivantes sont équivalentes : 1. Tout point est isolé dans sa fibre (autrement dit le sous-espace est discret). 2. Pour tout , le préschéma est un -préschéma fini. 3. Pour tout , l’anneau est un -module quasi‑fini.

Definition ([EGA II, (6.2.3)]) Lorsque est un morphisme de type fini vérifiant les conditions équivalentes de (6.2.2), on dit que est quasi-fini, ou que est quasi-fini sur .

I believe the correct reference is [EGA I, 2nd edition, Definition 6.11.3] ((6.11.2) of [EGA I, 2nd edition] is the same as (6.2.2) of [EGA II])

Definition ([EGA I 2nd edition, (6.11.3)]) Lorsque est un morphisme de type fini vérifiant les conditions équivalentes de (6.11.2), on dit que est quasi-fini, ou que est quasi-fini sur . On dit qu'un morphisme est quasi-fini en un point s'il existe un voisinage ouvert affine de et un voisinage ouvert affine de tels que , et que le morphisme , restriction de , soit quasi-fini. On dit qu'un morphisme est localement quasi-fini (ou que est localement quasi-fini sur ) s'il est quasi-fini en tout point de .

Comment #10664 by on

Yes,OK, but the refence does work for the notion of a quasi-finite morphism. So I am going to leave as is.

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View Definition 29.20.1 as pdf