# Contents

## Definition

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

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

open neighborhood$U\ni x$ and an affine open set $V\subset Y$, $f(V)\subset U$ such that $\mathcal{O}_Y(V)$ is finitely presented as an $\mathcal{O}_X(U)$-algebra.

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

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.

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