In this post we define affinoid adic spaces.

Let be a topological ring. We say that is **non-archimedean** if it admits a base of neighbourhoods of consisting of subgroups of the additive group underlying . Typical examples include non-archimedean fields , -affinoid algebras and Witt vectors of the valuation ring of a non-archimedean field of characteristic .

Let be a valuation on a non-archimedean topological ring . We say is **continuous** if for all elements and the subsets of of the form are open (enough to consider ).

We can define for . If then is continuous if and only if are open for all . Similarly we can define . When and is trivial then and hence is continuous is and only if is an open prime ideal.

We define the **continuous spectrum** of a non-archimedean topological ring by

If is a continuous map between non-archimedean topological rings and , then is in . This follows by remarking that for any the subset is open.

** Huber rings **

Let be an ideal in a topological ring . We say that defines the topology of if is a basis of neighbourhoods of . We call such an -adic ring.

Definition 5

- is a
Huber ringif there exists an open subring with the topology defined by a finitely generated ideal of . We call any such a ring of definition of .- is a
Tate ringif it is a Huber ring and if there exists a topologically nilpotent unit in .

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 is a topological -algebra, for a non-archimedean field , then the following conditions are equivalent

- is a Huber ring.
- is a Tate ring.
- There exists a subring such that () forms a basis of open neighbourhoods of .

In fact a normed -algebra is automatically a Tate -algebra.

Definition 6

- A subset is called
boundedif for all neighbourhoods of , there exists a neighbourhood of such that .- An element is
power-boundedif is bounded. We denote by the set of power-bounded elements in .

Definition 7Anaffinoid ringis a pair where is a Huber ring and is an open and integrally closed subring of . A morphism of affinoid rings is a continuous morphism such that .

Having this, we can finally define an **affinoid adic space** out of an affinoid ring by

where denotes the equivalence relation on valuations as defined before.

We can equip with the topology generated by open subsets where . In fact, we define a rational subset

for ‘s which define an open ideal of . Then such subsets form a basis of neighbourhood of stable under finite intersections.

We will return to topological properties of in the next post.