Let be an affinoid ring. Let us define a rational subset

where the ‘s define an open ideal in .

Recall that denotes a ring of formal power series

such that for .

We define

This is the completion of with respect to the ideal of definition of .

Let be the integral closure of in . Denote by its completon. Then one can check that is an affinoid ring and also is an affinoid ring.

We can endow the topological space with presheaves , by putting for any rational subset of :

Hence depends only on and if is a rational subset of with then we get a natural continuous ring homomorphism . Thus for any open subset of we put

where the limit is taken over the rational subsets and similarly for . Those are presheaves of complete topological rings.

Let be the standard map. We have the following lemma

Lemma 14The map factors through . Moreover for any continuous homomorphism from to a complete affinoid ring such that factors through there exists a unique continuous homomorphism such that .

*Proof:* See Lemma 8.1 in [Wed].

Let . If then can be extended to , thus also to the stalk . Observe that is a local ring with the maximal ideal . Write for the residue field at . It is equipped with a valuation . We write for its valuation ring. Huber proves that

It turns out that are not in general sheaves. We say that is **sheafy**, when they are. When is **strongly noetherian**, i.e. is noetherian for all , then is sheafy. Also is sheafy when is a perfectoid algebra. We shall return to this later on.

** 4.1. General adic spaces **

Let us define the category of tuples where

- is a topological space
- is a sheaf of complete topological rings on such that the stalk of is a local ring.
- is an equivalence class of valuations on the stalk .

The morphisms are pairs such is a continuous map of topological spaces and is a morphism of pre-sheaves of topological rings (i.e. for all open, is a continuous ring homomorphism) such that for all the induced ring homomorphism is compatible with the valuation and , that is (this implies that is local and ).

Definition 15Anaffinoid adic spaceis an object of which is isomorphic to for some affinoid pair .

Observe that a difference with what we had before is that we *demand* that be a sheaf.

Definition 16Anadic spaceis an object of which is locally an affinoid adic space.

Morphisms between adic spaces are the morphisms in . For example the morphisms of adic spaces correspond bijectively to the continuous ring homomorphisms .

Reference:

[Wed] T. Wedhorn, “`Adic spaces”‘, unpublished notes