nLab affinoid domain

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.

(Berkovich 09, def. 2.2.1)

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 $Y$

(Berkovich 09, def. 2.2.7)

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

Properties

• Tate's acyclicity theorem

Related concepts

• closed cover

• Berkovich space

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)