I.2: Huber rings

In this post we define affinoid adic spaces.

Let ${A}$ be a topological ring. We say that ${A}$ is non-archimedean if it admits a base of neighbourhoods of ${0}$ consisting of subgroups of the additive group ${(A,+)}$ underlying ${A}$. Typical examples include non-archimedean fields ${k}$, ${k}$-affinoid algebras and Witt vectors ${W(\mathcal{O}_F)}$ of the valuation ring ${\mathcal{O}_F}$ of a non-archimedean field of characteristic ${p>0}$.

Let ${v: A \rightarrow \Gamma \cup \{0\}}$ be a valuation on a non-archimedean topological ring ${A}$. We say ${v}$ is continuous if for all elements ${a \in A}$ and ${\gamma \in \Gamma}$ the subsets of ${A}$ of the form ${\{x \in A | v(x-a) < \gamma \}}$ are open (enough to consider ${a=0}$).

We can define ${A _{\leq \gamma} = \{x \in A| v(x) \leq \gamma\}}$ for ${\gamma \in \Gamma}$. If ${\Gamma \not = 1 }$ then ${v}$ is continuous if and only if ${A_{\leq \gamma}}$ are open for all ${\gamma \in \Gamma}$. Similarly we can define ${A_{< \gamma}}$. When ${\Gamma = 1}$ and ${v}$ is trivial then ${\mathrm{supp}(v) = A_{<1} = \{ v <1 \}}$ and hence ${v}$ is continuous is and only if ${\mathrm{supp}(v)}$ is an open prime ideal.

We define the continuous spectrum of a non-archimedean topological ring ${A}$ by

$\displaystyle \mathrm{Cont}(A):= \{ v\in \mathrm{Spv}(A) | \ \ v \textrm{ continuous}\} \subset \mathrm{Spv}(A)$

If ${f: A' \rightarrow A}$ is a continuous map between non-archimedean topological rings and ${v\in \mathrm{Cont}(A)}$, then ${v' = v \circ f}$ is in ${\mathrm{Cont}(A')}$. This follows by remarking that for any ${\gamma ' \in \Gamma '}$ the subset ${A' _{< \gamma'} = f^{-1}(A_{< \gamma '})}$ is open.

Huber rings

Let ${I}$ be an ideal in a topological ring ${A}$. We say that ${I}$ defines the topology of ${A}$ if ${\{I^n\} _{n \in \mathbb{N}}}$ is a basis of neighbourhoods of ${0}$. We call such ${A}$ an ${I}$-adic ring.

Definition 5

1. ${A}$ is a Huber ring if there exists an open subring ${A_0 \subset A}$ with the topology defined by a finitely generated ideal ${I}$ of ${A_0}$. We call any such ${A_0}$ a ring of definition of ${A}$.

2. ${A}$ is a Tate ring if it is a Huber ring and if there exists a topologically nilpotent unit in ${A}$.

Out of Huber rings we will construct adic spaces. We remark here that Tate rings are basically non-archimedean rings which behave like Banach algebras. When ${A}$ is a topological ${k}$-algebra, for a non-archimedean field ${k}$, then the following conditions are equivalent

1. ${A}$ is a Huber ring.
2. ${A}$ is a Tate ring.
3. There exists a subring ${A_0 \subset A}$ such that ${a A_0}$ (${a \in k^{\times}}$) forms a basis of open neighbourhoods of ${0}$.

In fact a normed ${k}$-algebra is automatically a Tate ${k}$-algebra.

Definition 6

1. A subset ${M \subset A}$ is called bounded if for all neighbourhoods ${U}$ of ${0}$, there exists a neighbourhood ${V}$ of ${0}$ such that ${M \cdot V \subset U}$.

2. An element ${a \in A}$ is power-bounded if ${\{a^n | n \in \mathbb{N}\}}$ is bounded. We denote by ${A^{\circ}}$ the set of power-bounded elements in ${A}$.

Definition 7 An affinoid ring is a pair ${(A,A^+)}$ where ${A}$ is a Huber ring and ${A^+ \subset A^{\circ}}$ is an open and integrally closed subring of ${A}$. A morphism of affinoid rings ${f: (A,A^+) \rightarrow (B,B^+)}$ is a continuous morphism ${f: A \rightarrow B}$ such that ${f(A^+) \subset B^+}$.

Having this, we can finally define an affinoid adic space out of an affinoid ring ${(A,A^+)}$ by

$\displaystyle \mathrm{Spa}(A,A^+) = \{v \in \mathrm{Cont}(A) | \ \ v(a) \leq 1 \textrm{ for all } a\in A^+ \} / \simeq$

where ${\simeq}$ denotes the equivalence relation on valuations as defined before.

We can equip ${\mathrm{Spa}(A,A^+)}$ with the topology generated by open subsets ${\{v \in \mathrm{Spa}(A,A^+) | \ \ v(a)\leq v(b) \not = 0\}}$ where ${a,b \in A}$. In fact, we define a rational subset

$\displaystyle U(\frac{f_1,...,f_n}{g}) = \{ v \in \mathrm{Spa}(A,A^+) | \ \ v(f_i) \leq v(g) \not = 0, \ \ i=1,...,n\}$

for ${f_i}$‘s which define an open ideal of ${A}$. Then such subsets form a basis of neighbourhood of ${\mathrm{Spa}(A,A^+)}$ stable under finite intersections.

We will return to topological properties of ${\mathrm{Spa}(A,A^+)}$ in the next post.