nLab
affinoid domain

Contents

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 anAX = Spec_{an} A is a closed subset VXV \subset X such that there is a homomorphism of kk-affinoid spaces

ϕ:Spec anA VX \phi : Spec_{an} A_V \to X

for some A VA_V, whose image is VV, and such that every other morphism of kk-affinoid spaces into XX whose image is contained in VV uniquely factors through this morphism.

(Berkovich 09, def. 2.2.1)

Definition

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

(Berkovich 09, def. 2.2.7)

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

Properties

References

  • Vladimir Berkovich, section 2.2 of Non-archimedean analytic spaces, lectures at the Advanced School on pp-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.