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Ingredients

Concepts

Constructions

Examples

Theorems

Contents

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).

References

Last revised on December 11, 2020 at 21:49:11. See the history of this page for a list of all contributions to it.