# nLab affinoid domain

Contents

### Context

#### Analytic geometry

analytic geometry (complex, rigid, global)

## Basic concepts

analytic function

analytification

GAGA

# Contents

## Idea

In the context of non-archimedean analytic geometry, affinoid domains are basic model spaces: a Berkovich analytic space is, in particular, a topological space equipped with an atlas by (analytic spectra underlying) affinoid domains.

## Definition

###### Definition

An affinoid domain in an affinoid space $X = Spec_{an} A$ is a closed subset $V \subset X$ such that there is a homomorphism of $k$-affinoid spaces

$\phi : Spec_{an} A_V \to X$

for some $A_V$, whose image is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.

###### Definition

A morphism $f\colon X\to Y$ of affinoid spaces is an affionoid domain embedding if it induces an isomorphism of $X$ with an affinoid domain in $Y$

These are the “admissible morphisms” in the site of affinoid domains. (…)

## References

• Vladimir Berkovich, section 2.2 of Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

• Doosung Park, Affinoid domains, lecture notes (pdf)

Last revised on July 17, 2014 at 23:09:15. See the history of this page for a list of all contributions to it.