nLab morphism of finite presentation

Definition

A homomorphism of schemes $f \colon X\to Y$ is

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

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

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

Related concepts

• morphism of finite type

References

• The Stacks Project, Section 01TO: Morphisms of finite presentation

• wikipedia: Finite, quasi-finite, finite type, and finite presentation morphisms

• David Rydh, Why are unramified maps not required to be locally of finite presentation?, MO/206333/2503.