---
tag: F6VBCVP13K
kind: concept
name: Adic space
msc: [14G22, 13A18, 32P05]
sources:
  - url: https://en.wikipedia.org/wiki/Rigid_analytic_space
---

## Summary

An adic space is a geometric space, introduced by Roland Huber, whose points are continuous valuations on topological rings, see the [[definition|5AXZG4PXTE]].
Adic spaces give one framework for non-Archimedean geometry that contains the [[rigid analytic spaces|YAH6MZVG7K]] over a complete non-Archimedean field and also locally Noetherian formal schemes.
Their points include the classical points and extra points of higher rank, which make the topology work well.
[[Perfectoid spaces|Q9Y4CJE1X5]] are adic spaces of a special kind.

## Explanations

### L1 human

In ordinary geometry, a point of a space is a way to evaluate functions.
In an adic space, a point is a way to measure the size of functions, given by a valuation.
The model case is the $p$-adic closed unit disc, built from power series with $p$-adic coefficients tending to $0$, which converge on the disc.
Besides the usual points, it has points that measure functions by their largest size on a smaller disc, and some infinitesimal refinements of these.
With these extra points, coverings behave as they should, so functions defined piece by piece can be glued.