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 .
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 14 The 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 15 An affinoid adic space is 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 16 An adic space is 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 .
[Wed] T. Wedhorn, “`Adic spaces”‘, unpublished notes