morphism of finite presentation



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

  1. finitely presented at xXx\in X if there is an affine

open neighborhoodUxU\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.

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

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

  3. 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 July 3, 2016 at 14:32:03. See the history of this page for a list of all contributions to it.