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.

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