---
tag: 5AXZG4PXTE
kind: definition
name: Adic space
msc: [14G22, 13A18, 32P05]
defines: F6VBCVP13K
uses: [TSF8RZDD5B, 2VF2TSJ3PY, B85T5VKR3M, G0V86SJBQW, TB6TM6R4MD]
sources:
  - url: https://en.wikipedia.org/wiki/Rigid_analytic_space
---

## Definition

A *Huber ring* is a [[topological ring|TSF8RZDD5B]] $A$ with an open subring $A_0$ and a finitely generated ideal $I$ of $A_0$.
The topology of $A_0$ must be the [[$I$-adic topology|2VF2TSJ3PY]].
An element $a$ of $A$ is *power-bounded* if the set $\set{a^n : n \ge 1}$ is bounded, and $A^{\circ}$ denotes the subring of power-bounded elements.
A *Huber pair* $(A, A^+)$ is a Huber ring $A$ with an open subring $A^+ \subseteq A^{\circ}$ that is [[integrally closed|G0V86SJBQW]] in $A$.

A *valuation* on $A$ is a map $v \colon A \to \Gamma \cup \set{0}$, where $\Gamma$ is a totally ordered abelian group written multiplicatively, with $v(0) = 0$, $v(1) = 1$, $v(ab) = v(a) v(b)$ and $v(a + b) \le \max(v(a), v(b))$.
It is *continuous* if $\set{a \in A : v(a) < \gamma}$ is open for every $\gamma$ in the value group of $v$.
The *adic spectrum* $\operatorname{Spa}(A, A^+)$ is the set of equivalence classes of continuous [[valuations|B85T5VKR3M]] $v$ on $A$ with $v(f) \le 1$ for all $f \in A^+$.
Its topology is generated by the rational subsets
$$U\!\left(\tfrac{f_1, \dots, f_n}{g}\right) = \set{v : v(f_i) \le v(g) \ne 0 \text{ for } i = 1, \dots, n},$$
where $f_1, \dots, f_n$ generate an open ideal of $A$.
It carries a structure presheaf $\mathcal{O}$ of complete topological rings, built from completions of the localizations $A[1/g]$.

An *adic space* is a triple $(X, \mathcal{O}_X, (v_x)_{x \in X})$ such that:

1. $X$ is a topological space and $\mathcal{O}_X$ is a [[sheaf|TB6TM6R4MD]] of complete topological rings on $X$;
2. for each $x \in X$, $v_x$ is an equivalence class of valuations on the stalk $\mathcal{O}_{X,x}$;
3. every point has an open neighbourhood isomorphic, with these structures, to $\operatorname{Spa}(A, A^+)$ for a Huber pair whose structure presheaf is a sheaf.

## Remarks

Example: $\operatorname{Spa}(\Q_p, \Z_p)$ is a single point.
The $p$-adic valuation is continuous and bounded by $1$ on $\Z_p$.
The trivial valuation, with $v(a) = 1$ for $a \ne 0$, is not continuous, since $\set{a : v(a) < 1} = \set{0}$ is not open in $\Q_p$.
Other valuations of $\Q_p$ bounded on $\Z_p$ are equivalent to one of these two.

Non-example of a Huber ring: $\R$ with its usual topology.
An open subring contains an interval $(-\varepsilon, \varepsilon)$, which generates $\R$ as an additive group, so the only open subring is $\R$.
Its topology is neither discrete nor indiscrete, so it is not the $I$-adic topology for any ideal $I$ of $\R$.

The structure presheaf of $\operatorname{Spa}(A, A^+)$ need not be a sheaf in general; this is why clause 3 asks for it.
Huber proved that it is a sheaf when, for example, $A$ is a strongly Noetherian Tate ring, which covers the affinoid algebras of rigid geometry.