An important subclass of adic spaces are perfectoid spaces which form a basis for the pro-etale topology of locally noetherian adic spaces (Proposition 4.8 in Scholze’s “p-adic Hodge theory for rigid-analytic varieties”). Perfectoid spaces are glued from perfectoid algebras, hence I will start by explaining a notion of perfectoid rings. The word “`perfectoid”‘ refers to their “`almost perfect structure”‘ – the Frobenius map is surjective and there is a “`tilt construction”‘ which allows for a passage between algebras in characteristic and characteristic .
We recall that a ring is uniform if its subring of power-bounded elements is bounded.
Definition 27 A complete Tate ring is perfectoid if is uniform and there exists a pseudo-uniformizer such that in and such that the -th power Frobenius map
is an isomorphism.
In fact, for any complete Tate ring and as above is necessarily injective. Indeed, if satisfies for some then the element lies in since its -th power does. Hence, we really demand surjectivity in the definition of a perfectoid ring. In fact, the surjectivity is equivalent to the surjectivity of the Frobenius map
for any . The statement follows by induction from .
In what follows we denote by any map induced by where is a ring and are two ideals containing such that .
Remark 28 Discretely valued non-archimedean fields of residue characteristic are not perfectoid (hence, for example ). Indeed, and are Artin local rings of different lengths and hence cannot be isomorphic.
Standard examples of perfectoid rings include:
- – the completion of .
- The -adic completion of , which we denote by .
- , where the completion is -adic.
A natural question is whether one can give a more general definition of perfectoid Huber rings which do not have to be Tate. So that for example , the -adic completion of , would be perfectoid.
Let be a topological ring with . The following are equivalent:
- is perfectoid.
- is a perfect uniform complete Tate ring.
Here, perfect means that the Frobenius is an isomorphism.
Proof: If is perfectoid then is an isomorphism and hence also by induction. We conclude by taking inverse limit, using completeness and inverting .
If is perfect, then is satisfied automatically. If then so that is an isomorphism. Thus is surjective.
Definition 30 A perfectoid field is a perfectoid ring which is a non-archimedean field.
Let us note the following criterion
Proposition 31 Let be a non-archimedean field. Then is perfectoid if and only if the following conditions hold:
- is not discretely valued.
- is surjective, where is the ring of integers of .
Let us finish with another characterisation of perfectoid rings
Proposition 32 Let be a complete uniform Tate ring.
- If there exists a pseudo-uniformiser such that and the Frobenius map is surjective, then is a perfectoid ring.
- Conversely, if is a perfectoid ring then is surjective under the additional assumption that the ideal is closed.
Proof: (1) is easy and (2) comes down to showing an isomorphism