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.
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.
- is a Huber ring if 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 ring if 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.
- A subset is called bounded if for all neighbourhoods of , there exists a neighbourhood of such that .
- An element is power-bounded if is bounded. We denote by the set of power-bounded elements in .
Definition 7 An affinoid ring is 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.