nLab morphism of finite presentation




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

  1. finitely presented at xXx\in X if there is an affine open neighborhood UU containing xx and an affine open set VYV\subseteq Y with f(U)Vf(U)\subseteq V such that 𝒪 X(U)\mathcal{O}_X(U) is finitely presented as an 𝒪 Y(V)\mathcal{O}_Y(V)-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.


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


Last revised on December 12, 2020 at 02:49:11. See the history of this page for a list of all contributions to it.