nLab
morphism of finite presentation

Contents

Definition

A homomorphism of schemes f:YXf \colon Y\to X is

  1. finitely presented at xXx\in X if there is an affine open neighborhood UxU\ni x and an affine open set VYV\subset Y, f(V)Uf(V)\subset U such that 𝒪 Y(V)\mathcal{O}_Y(V) is finitely presented as an 𝒪 X(U)\mathcal{O}_X(U)-algebra.

  2. locally finitely presented if it is finitely presented at each xXx\in X.

  3. finitely presented if it is locally finitely presented, quasicompact and quasiseparated.

  4. essentially finitely presented if it is a localization of a finitely presented morphism.

Example

A standard open Spec(R[{s} 1])Spec(R)Spec(R[\{s\}^{-1}]) \longrightarrow Spec(R) (Zariski topology) is of finite presentation. More generallly, an étale morphism of schemes is of finite presentation (though essentially by definition so).

References

Revised on November 25, 2013 01:27:26 by Urs Schreiber (89.204.137.196)