I.7: Perfectoid rings

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 {0} and characteristic {p}.

 

We recall that a ring {R} is uniform if its subring of power-bounded elements {R ^{\circ} \subset R} is bounded.

Definition 27 A complete Tate ring {R} is perfectoid if {R} is uniform and there exists a pseudo-uniformizer {\varpi \in R} such that {\varpi ^p | p} in {R^{\circ}} and such that the {p}-th power Frobenius map

\displaystyle \Phi: R ^{\circ} / \varpi \rightarrow R^{\circ} / \varpi ^{p}

is an isomorphism.

In fact, for any complete Tate ring {R} and {\varpi} as above {\Phi: R ^{\circ} / \varpi \rightarrow R^{\circ} / \varpi ^{p}} is necessarily injective. Indeed, if {x \in R^{\circ}} satisfies {x^p = \varpi ^p y} for some {y \in R^{\circ}} then the element {x/ \varpi \in R} lies in {R^{\circ}} since its {p}-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

\displaystyle R^{\circ} / (p, \varpi ^n) \rightarrow R^{\circ} / (p, \varpi ^{np})

for any {n \geq 1}. The statement follows by induction from {n=1}.

 

In what follows we denote by {\Phi} any map {R /I \rightarrow R / J } induced by {x \mapsto x^{p}} where {R} is a ring and {I,J \subset R} are two ideals containing {p} such that {I^p \subset J}.

Remark 28 Discretely valued non-archimedean fields {K} of residue characteristic {p} are not perfectoid (hence, for example {\mathbb{Q}_p}). Indeed, {K^{\circ}/\varpi} and {K^{\circ} / \varpi ^{p}} are Artin local rings of different lengths and hence cannot be isomorphic.

Standard examples of perfectoid rings include:

  1. {\mathbb{Q}_p ^{cyc}} – the completion of {\mathbb{Q}_p(\mu _{p^{\infty}})}.
  2. The {t}-adic completion of {\mathbb{F}_p((t))(t^{1/p^{\infty}})}, which we denote by {\mathbb{F}_p((t^{1/p^{\infty}}))}.
  3. {\mathbb{Q}_p ^{cyc}\langle T^{1/p^{\infty}} \rangle = (\mathbb{Z}_p ^{cyc}[ T^{1/p^{\infty}}])^{\wedge}[1/p]}, where the completion is {p}-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 {\mathbb{Z} _p ^{cyc}[[T^{1/p^{\infty}}]]}, the {(p,T)}-adic completion of {\mathbb{Z}_p ^{cyc}[T^{1/p^{\infty}}]}, would be perfectoid.

    Proposition 29

    Let {R} be a topological ring with {pR =0}. The following are equivalent:

    1. {R} is perfectoid.
    2. {R} is a perfect uniform complete Tate ring.

     

    Here, perfect means that the Frobenius {\Phi: R \rightarrow R} is an isomorphism.

     

    Proof: If {R} is perfectoid then {\Phi : R ^{\circ} / \varpi \rightarrow R^{\circ} / \varpi ^{p}} is an isomorphism and hence also {R ^{\circ} / \varpi ^n \rightarrow R^{\circ} / \varpi ^{np}} by induction. We conclude by taking inverse limit, using completeness and inverting {\varpi}.

     

    If {R} is perfect, then {\varpi ^p |p=0} is satisfied automatically. If {x \in R^{\circ}} then {x ^p \in R^{\circ}} so that {\Phi : R^{\circ} \rightarrow R^{\circ}} is an isomorphism. Thus {\Phi : R ^{\circ} / \varpi \rightarrow R^{\circ} / \varpi ^{p}} is surjective. \Box

    Definition 30 A perfectoid field is a perfectoid ring which is a non-archimedean field.

    Let us note the following criterion

    Proposition 31 Let {K} be a non-archimedean field. Then {K} is perfectoid if and only if the following conditions hold:

    1. {K} is not discretely valued.
    2. {|p| <1}
    3. {\Phi: \mathcal{O} _K / p \rightarrow \mathcal{O} _K /p} is surjective, where {\mathcal{O} _K} is the ring of integers of {K}.

     

    Let us finish with another characterisation of perfectoid rings

    Proposition 32 Let {R} be a complete uniform Tate ring.

    1. If there exists a pseudo-uniformiser {\varpi \in R} such that {\varpi ^p |p} and the Frobenius map {\Phi: R ^{\circ} / p \rightarrow R^{\circ} /p}is surjective, then {R} is a perfectoid ring.
    2. Conversely, if {R} is a perfectoid ring then {\Phi: R^{\circ} /p \rightarrow R^{\circ} /p} is surjective under the additional assumption that the ideal {pR^{\circ} \subset R^{\circ}} is closed.

     

    Proof: (1) is easy and (2) comes down to showing an isomorphism

    \displaystyle R^{\circ}/p \rightarrow \varprojlim _n R^{\circ} / (p, \varpi ^n)

    \Box

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