Introductionpeople.math.harvard.edu/~zyao/paper/Perfd_AV.pdfBhargav Bhatt for proposing the project, for his guidance and for his constant encouragement, and we would like to thank

PERFECTOID COVERS OF ABELIAN VARIETIES

CLIFFORD BLAKESTAD, DAMIAN GVIRTZ, BEN HEUER, DARIA SHCHEDRINA, KOJI SHIMIZU,

PETER WEAR, AND ZIJIAN YAO

Abstract. For an abelian variety A over an algebraically closed non-archimedean field of residuecharacteristic p, we show that there exists a perfectoid space which is the tilde-limit of lim←−[p]

A.

1. Introduction

Let p be a prime and let K be a non-archimedean field which is either p-adic or of characteristicp. Let us moreover assume that K is algebraically closed. For an abelian variety A over K weconsider the inverse system of A under the multiplication by p morphism:

. . .[p]−→ A

[p]−→ A[p]−→ A

Via the rigid analytification functor, we may see this as an inverse system of analytic adic spacesover Spa(K,OK). The primary goal of this article is to show that the “inverse limit” of this towerexists in some way and is a perfectoid space: Since inverse limits rarely exist in the category of adicspaces (even for affinoid spaces), in [14] the authors introduce the weaker notion of tilde-limits toremedy this problem. This is the notion of ”limits” we are going to use. More precisely, we prove:

Theorem 1. Let A be an abeloid variety over K, for instance an abelian variety seen as a rigidspace. Then there is a unique perfectoid space A∞ over K such that A∞ ∼ lim←−[p]

A is a tilde-limit.

The possibility of results in this direction is mentioned in §7 and §13 of [12], and in the case ofabelian varieties A with good reduction, this theorem was proven already in [7] (Lemme A.16). Inorder to motivate our strategy for the general case, let us sketch the proof in the good reductioncase (we follow Exercise 4 – 6 in [1], the proof is spelt out in detail in §2.3, Corollary 2.17 below):

Let A be an abelian variety of good reduction over K. Let OK be the ring of integers of K and letπ ∈ OK be a pseudo-uniformiser such that p ∈ πOK . Then we can consider A as an abelian schemeover OK . Let A be its π-completion, a formal scheme over OK . Its adic generic fibre is the rigidanalytification of A that we denote by the same letter. The mod π special fibre As = A×OK/π isa group scheme over OK/π, so the map [p] : As → As factors through the relative Frobenius map.The inverse limit lim←−[p]

As in the category of schemes is thus relatively perfect over OK/π. We can

similarly form the inverse limit A∞ = lim←−[p]A in the category of formal schemes. Its adic generic

fibre A∞ is a tilde-limit of lim←−[p]A, and it is perfectoid since lim←−[p]

As is relatively perfect.

This gives the construction of A∞ in the case of good reduction. The overall strategy of our proofof Theorem 1 is similar: We construct tilde-limits of various spaces via formal models and showthat these tilde-limits are perfectoid via a Frobenius factorisation property on the special fibre.

Next, we consider the case that A is an abelian variety with bad reduction. The assumption thatK is algebraically closed assures that A is then semi-stable. In this case, the theory of Raynaud

1

PERFECTOID COVERS OF ABELIAN VARIETIES 2

extensions provides us with a short exact sequence

0→ T → E → B → 0

of rigid groups, where T = (Ganm )d is a split rigid torus and B is the analytification of an abelian

variety with good reduction, such that A = E/M for a discrete lattice M ⊂ E. This short exactsequence is split locally on B, allowing us to locally write E as a product of T and an open subspaceof B. Our strategy, which more generally applies to any abeloid variety over K, is now the following:

(1) Use formal models to show that there exists a perfectoid tilde-limit T∞ ∼ lim←−[p]T .

(2) Use the formal models from (1) to construct a perfectoid tilde-limit E∞ ∼ lim←−[p]E.

(3) Study the quotient map E → A in the limit over [p] to construct the desired space A∞.

More precisely, this article is organised as follows: In §2 we develop a notion of a [p]-F -formaltower for a commutative rigid group G over K, which is roughly an inverse system of formal modelsfor [p] : G→ G that factor through the relative Frobenius mod π. This is an axiomatisation of thedata that one uses to constructA∞ in the case of good reduction. In particular, the same proof showsthat if G admits [p]-F -formal tower, then there is a unique perfectoid tilde-limit G∞ ∼ lim←−[p]

G.

In chapter §3 we give a [p]-F -formal tower for a split rigid torus T in terms of a family of formalmodels Tq1/n , thus showing that a perfectoid tilde-limit T∞ of the inverse system of [p] on T exists.

Crucially, this tower of formal models has the extra structure of an action by the formal torus T .We can therefore in §4 use the language of formal fibre bundles to construct a [p]-F -formal tower

for E: The Raynaud extension of A arises from a short exact sequence of formal group schemes

0→ T → E → B → 0

by taking generic fibres and forming the pushout with respect to the open immersion T η → T .

Equivalently, since the sequence is locally split, we can see E → B as a principal T -bundle andformation of E amounts to a change of fibre from T η to T . Since the formal model tower of §3receives a T -action, we now obtain a [p]-F -formal tower for E from the formal models of §4 by thechange of fibre from T to the formal models Tq1/n . This gives the desired perfectoid tilde-limit E∞.

In chapter §5 we finish the proof of the main theorem by constructing A∞ from E∞ as fol-lows: After choosing lattices M ⊆ M1/pn ⊆ E that are sent to M under [pn] : E → E, the[p]-multiplication tower of A = E/M naturally factors into two separate towers: One is the tower

of maps E/M1/pn+1 → E/M1/pn induced from [p]-multiplication of E, the other is induced fromthe projection maps E/M → E/M1/pn . By choosing certain charts of E that give local splittingsof the projection E → E/M = A, we show that the first of these towers has a perfectoid tilde-limitE/M1/p∞ that is locally isomorphic to open subsets of E∞. Since the second tower is easy todescribe in terms of the chosen charts, it is then straightforward to find the desired space A∞.

In chapter §6 we finally use our explicit construction to say something about the local geometryof A∞. In particular, we describe the perfectoid group homomorphism E∞ → A∞ in terms ofperfectoid principal bundles. The goal of this last section is to give an alternative description ofA∞ in terms of a quotient of perfectoid groups: There are a lattice M∞ ⊆ E∞ and a pro-finiteperfectoid group D∞ over K for which there is a short exact sequence of perfectoid groups

0→M∞ → D∞ × E∞ → A∞ → 0.

This may be seen as an analogue in the limit of the sequence 0→M → E → A→ 0.At the end there is an appendix on fibre bundles and associated fibre bundle constructions in

the context of formal, rigid and perfectoid spaces.


Acknowledgements

This work started as a group project at the 2017 Arizona Winter School. We would like to thankBhargav Bhatt for proposing the project, for his guidance and for his constant encouragement,and we would like to thank Matthew Morrow for his help during the Arizona Winter School. Inaddition we would like to thank the organizers of the Arizona Winter School for setting up a greatenvironment for us to participate in this project.

During this work Damian Gvirtz and Ben Heuer were supported by the Engineering and PhysicalSciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometryand Number Theory (The London School of Geometry and Number Theory), University CollegeLondon. During the Arizona Winter School, Daria Shchedrina was supported by Prof. Dr. PeterScholze and the DFG. Peter Wear was supported by NSF grant DMS-1502651 and UCSD and wouldlike to thank Kiran Kedlaya for helpful discussions.

Notation

We fix a prime p. Let K be a perfectoid field that is either p-adic or of characteristic p, withring of integers OK and a fixed pseudo-uniformiser π ∈ OK such that p ∈ πOK . From §4 on we aregoing to assume that K is algebraically closed, but we don’t need to make this assumption yet.

By an adic space over Spa(K,OK), we mean an adic spaces in the sense of [14], and we adopt thenotion of perfectoid spaces defined in §2 ibid. In their language, adic spaces in the sense of Huberare referred to as honest adic spaces. Throughout the article, we make no distinction between rigidspaces and their corresponding honest adic spaces. In particular, by a cover of a rigid space weshall always mean a cover of the associated adic space, and therefore the cover of the rigid spacewill automatically be an admissible cover in the sense of rigid analytic geometry. If a rigid space isobtained from a K-scheme X via rigid-analytification X 7→ Xan, we will often denote both by thesame symbol X in order to simplify the notation.

In a similar spirit, we are often going to see formal schemes as adic spaces via the fully faithfuladification functor, and denote both by the same letter. For a formal scheme X over Spf(OK)with the π-adic topology, we denote by Xη := X×Spa(OK ,OK) Spa(K,OK) its adic generic fibre. We

denote by X = X×Spf OK Spf OK/π its mod π special fibre, considered as a scheme over SpecOK/π.Finally, let us recall the following standard terminology:

(1) Let X be a rigid space over K. A formal model of X is an admissible topologically finite

type formal scheme X over OK together with an isomorphism of its generic fibre Xη∼−→ X

(which is often suppressed from notation).(2) Let φ : X → Y be a morphism of rigid spaces over K, with formal models X and Y

respectively. A morphism of formal schemes Φ : X→ Y is a formal model of φ if it agreeswith φ on the adic generic fiber via the identifications Xη

∼−→ X and Yη∼−→ Y .

2. Tilde-limits of rigid groups

2.1. Tilde-limits and formal models. Inverse limits often do not exist in the category of adicspaces, and neither do they in rigid spaces. Instead we use the notion of tilde-limits from [14]:

Definition 2.1. Let (Xi)i∈I be a filtered inverse system of adic spaces with quasi-compact andquasi-separated transition maps, let X be an adic space with a compatible system of morphismsfi : X → Xi. We write X ∼ lim←−Xi and say that X is a tilde-limit of the inverse system (Xi)i∈I if


the map of underlying topological spaces |X| → lim←−|Xi| is a homeomorphism, and there exists an

open cover of X by affinoids Spa(A,A+) ⊂ X such that the map

lim−→Spa(Ai,A

+i )⊂Xi

Ai → A

has dense image, where the direct limit runs over all i ∈ I and all open affinoid subspacesSpa(Ai, A

+i ) ⊂ Xi through which the morphism Spa(A,A+) ⊆ X → Xi factors.

Remark 2.2. As pointed out after Proposition 2.4.4 of [14], tilde-limits (if they exist) are in generalnot unique. For example, consider the inverse system consisting of a single affinoid pre-perfectoidspace X = Spa(A,A+) in the sense of Definition 2.3.4 in [14]. Then both the space X as well as its

strong completion X = Spa(A, A+) are tilde-limits of X. However, we’ll see in Corollary 2.6 thatthe perfectoid tilde-limits we construct are unique.

Let us recall a few results from [14], §2.4 about the construction of tilde-limits that we will usefrequently throughout:

Proposition 2.3 ([14], Proposition 2.4.2). Let (Ai, A+i ) be a direct system of affinoids over (OK ,OK)

with compatible rings of definition Ai,0 carrying the π-adic topology. Let (A,A+) = (lim−→Ai, lim−→A+i )

be the affinoid algebra equipped with the topology making lim−→Ai,0 with the π-adic topology a ring ofdefinition. Then

Spa(A,A+) ∼ lim←− Spa(Ai, A+i ).

Proposition 2.4 ([14], Proposition 2.4.3). Let X ∼ lim←−i∈I Xi be a tilde-limit and let Ui ↪→ Xi be

an open immersion for some i ∈ I. Let Uj := Ui ×Xi Xj for j ≥ i and U := Ui ×Xi X, then

U ∼ lim←−j≥i

Uj .

Proposition 2.5 ([14], Proposition 2.4.5). Let (Xi)i∈I be an inverse system of adic spaces over(K,OK) and assume that there is a perfectoid space X such that X ∼ lim←−i∈I Xi. Then for any

perfectoid (K,OK)-algebra (B,B+),

X(B,B+) = lim←−i∈I

Xi(B,B+)

Corollary 2.6. If a perfectoid tilde-limit X ∼ lim←−Xi exists, it is unique up to unique isomorphism.

In the situation of the Corollary, we will also refer to X as the perfectoid tilde-limit of lim←−Xi. Ofcourse perfectoid tilde-limit don’t always exists. One case in which this they exist is the following:

Proposition 2.7. Let (Spa(Ai, A+i ))i∈I be an inverse system of affinoid perfectoid spaces over

(K,OK). Let (A,A+) = (lim−→Ai, lim−→A+i ) be the affinoid algebra endowed with the topology making

A0 = lim−→A◦i with the π-adic topology an open subring of definition, and denote by (A, A+) the

completion. Then Spa(A,A+) ∼ lim←− Spa(Ai, A+i ) is a perfectoid tilde-limit.

Proof. The space Spa(A,A+) is a tilde-limit of (A,A+) by Proposition 2.3. It is then clear from the

definition that this remains true for (A, A+). It remains to see that this is a perfectoid (K,OK)-

algebra: But A0 is a flat OK-algebra and in the direct limit the Frobenius isomorphisms A◦i /π1/p ∼=

A◦i /π induce an isomorphism

A0/π1/p ∼= A0/π.

Thus by Lemma 5.6 in [8], (A, A+) is a perfectoid (K,OK)-algebra. �


In the situation that the Xi are rigid space, one way to construct tilde-limits is by constructingwell-behaved formal models. The reason for this is the combination of the following two Lemmas:

Lemma 2.8. Let (Xi, φij)i∈I be an inverse system of formal schemes Xi over OK with affinetransition maps φij : Xj → Xi. Then the inverse limit X = lim←−Xi exists in the category of formalschemes over OK . If all the Xi are flat formal schemes, so is X.

Proof. In the affine case, if the inverse system is Spf Ai, take A to be the π-adic completion oflim−→Ai, then Spf A is the inverse limit of the Spf Ai. If the Ai are flat, then so is A because it istorsion-free over the valuation ring OK .

In general, ones uses that the transition maps are affine to reduce to the affine case. �

The proof also shows that in the situation of the Lemma, X is in particular a tilde-limit X ∼ lim←−Xiwhen considered as adic spaces. This remains true after passing to adic generic fibres:

Lemma 2.9. Let (Xi, φij)i∈I be an inverse system of formal schemes Xi over OK with affinetransition maps φij and let X = lim←−φj Xi be the limit. Let Xi = (Xi)η and X = (X)η be the adic

generic fibres. Then X ∼ lim←−Xi.

Proof. This is a consequence of Proposition 2.3: The transition maps in the system are affine,hence quasi-compact quasi-separated. In order to prove the Lemma, we can restrict to an affineopen subset Spf(A) of X that arises as the inverse limit of affine open subsets Spf(Ai) ⊆ Xi. Here allformal schemes are considered with the π-adic topology and A is the π-adic completion of lim−→Ai. On

the generic fibre, Ai with the π-adic topology is an open subring of definition of Ai[1/π]. Therefore,Proposition 2.3 applies and shows that Spf(A)η ∼ lim←− Spf(Ai)η as desired. �

Remark 2.10. This lemma essentially says that one can always construct a tilde-limit of aninverse system of rigid spaces Xi if it arises from an inverse system of formal schemes Xi with affinetransition maps. This is precisely what Scholze uses in [10] to construct the space XΓ0(p∞)(ε)a (seeCorollary III.2.19 in [10] and its proof).

Remark 2.11. Let us recall Raynaud’s theory of formal models: Under mild assumptions, one canalways find formal models of rigid spaces, and (possibly after admissible blow ups) of morphismsbetween them. More precisely, Raynaud’s theorem [2, section 8.4] states that

(1) Let X be a quasi-separated quasi-paracompact rigid space over K. Then there exist anadmissible quasi-paracompact formal model X for X.

(2) If X′ → X is an admissible blow-up of admissible formal schemes, then it induces an

isomorphism on the generic fibre X′η∼−→ Xη.

(3) Let X and Y be admissible quasi-paracompact formal schemes over OK and let f : Xη → Yη

be a morphism of their associated rigid spaces. Then there exist an admissible blow-upπ : X′ → X and a map f : X′ → Y such that fη = f ◦ πη.

In particular, given an inverse system (Xi, φij) of rigid spaces, one can always choose formal modelsXi for the Xi, and by successive admissible blow-ups one can also find models for the transitionmaps φij . This theorem doesn’t immediately allow us to construct a tilde-limit using Lemma 2.9as the morphisms might not be affine.

We will need the following basic lemma later on.

Lemma 2.12. Let (Ai, A+i ) and (Bi, B

+i ) be direct systems of affinoids over (K,OK) with compati-

ble rings of definition Ai,0 and Bi,0 carrying the π-adic topology. Let Spa(A,A+) ∼ lim−→ Spa(Ai, A+i )


and Spa(B,B+) ∼ lim−→ Spa(Bi, B+i ) (note that we are not necessarily using the tilde-limits con-

structed in Proposition 2.3). Then

Spa(A,A+)× Spa(B,B+) ∼ lim←−(Spa(Ai, A+i )× Spa(Bi, B

+i ))

Proof. The map of underlying topological spaces is a homeomorphism as fibre products of topolog-ical spaces commute with inverse limits. We are therefore reduced to checking that if lim−→Ai → A

has dense image and lim−→Bi → B has dense image, then lim−→(Ai ⊗ Bi) → A ⊗ B has dense image.

As direct limits commute with tensor products, we have lim−→(Ai ⊗ Bi) = (lim−→Ai) ⊗ (lim−→Bi). Nowdensity can be checked directly on elements. �

As local tilde-limits glue, the above lemma instantly extends to the case where our tilde-limitscome from the setup of Lemma 2.9.

2.2. Tilde-limits for rigid groups. Let G be a rigid group over K, that is, a group object in thecategory of rigid spaces over K. Throughout we will always consider commutative rigid groups.The main topic of study of this work is the [p]-multplication tower

. . .[p]−→ G

[p]−→ G.

Question 2. When is there a perfectoid space G∞ such that G∞ ∼ lim←−[p]G is a tilde-limit?

We are primarily interested in the following examples:

(1) Analytifications of finite type group schemes over K. Examples include the analytificationof an abelian variety A over K and of tori T over K.

(2) Generic fibres of topologically finite type formal group schemes over OK .(3) Raynaud’s covering space E of an abelian variety with semi-stable reduction.

Lemma 2.8 and 2.9 motivate the following definition.

Definition 2.13. Let G be a rigid group. A [p]-formal tower for G is a family of formal models{Gn}n∈N of G, together with affine transition maps [p]n+1 : Gn+1 → Gn which are formal modelsof [p] : G→ G. Sometimes we suppress notation and write [p] for [p]n on Gn.

The following Corollary is an immediate consequence of our discussion in the previous subsection:

Corollary 2.14. Let G be a rigid group. If G admits a [p]-formal tower, then there exists an adicspace G∞ such that G∞ ∼ lim←−[p]

G.

For example, for any formal group scheme G for which [p] is affine, the [p]-multiplication towerof G gives rise to a [p]-formal tower for its generic fibre Gη.

2.3. Perfectoidness. We now introduce the formalism of [p]-F -formal towers to axiomatise theapproach mentioned in the introduction which shows that A∞ exists and is perfectoid in the caseof good reduction.

Definition 2.15. A [p]-F -formal tower for a rigid analytic group G is a [p]-formal tower

({Gn}n∈N, [p]n+1 : Gn+1 → Gn)

such that on the mod π special fibre Gn, each [p]n+1 factors through the relative Frobenius map:


G(p)n+1

Gn+1 Gn[p]n+1

Frel

Proposition 2.16. Let G be a rigid group over a perfectoid field K. If G admits a [p]-F -formaltower, then there exists a perfectoid space G∞ such that G∞ ∼ lim←−[p]

G.

Proof. Let ({Gn}n∈N, [p]n+1 : Gn+1 → Gn) be a [p]-F -formal tower for G. By Lemma 2.9 we have

G∞ := (lim←−[p]

G)η ∼ lim←−[p]

G.

To see that G∞ is perfectoid, we proceed as the proof of [10], Corollary III.2.19. It suffices toprove that G∞ = lim←−[p]

G can be covered by formal schemes of the form Spf(S) where S is a flat

OK-algebra such that the Frobenius map

S/π1/p → S/π

is an isomorphism. Lemma 5.6 of [8] then shows that S[1/π] is perfectoid.

By assumption, on the mod π special fibre Gn, [p]n+1 factors through the relative Frobenius. Nowtake any affine open subspace Spf(S0) ⊆ G0. Let [pi] : Gi → G0 be the composition [p]i ◦ · · · ◦ [p]1,and let Spf Si ⊂ Gi be the pullback of Spf S0 via [pi]. Then we have a commutative diagram:

S(p)i S

(p)i+1

. . . Si−1 Si Si+1 . . .

Frel FrelV V

where the horizontal maps are induced from [p] mod π.

From this we can check on elements that relative Frobenius is an isomorphism on S∞ := lim−→iSi.

Since K is perfectoid, we moreover have an isomorphism OK/π1/p → OK/π from the absoluteFrobenius on OK/π. Therefore absolute Frobenius on S∞/π induces an isomorphism

S∞/π1/p ∼−→ S∞/π.

Since each Gi is flat, so are the Si and thus so is S∞. Thus S∞[1/π] is a perfectoid K-algebra.Since G∞ is covered by affinoids of the form Spf(S∞)η, this shows that G∞ is perfectoid. �

In particular this gives in more detail the proof of the result that we sketched in the introduction:

Corollary 2.17. Let A be an abelian variety of good reduction over a perfectoid field K. Then A∞exists and is perfectoid.

More generally, if G is a flat commutative formal group scheme such that [p]-multiplication isaffine, then the maps [p] : G→ G define a [p]-F -formal tower for the rigid analytic group G = Gη,and G∞ := (lim←−[p]

G)η is the perfectoid tilde-limit of lim←−[p]G.


2.4. Examples.

Example. Let G be the p-adic completion of the affine group scheme Gm over OK . The underlyingformal scheme of G is Spf OK〈X±1〉. Multiplication by p on Gm gives a [p]-F -formal tower for G,so for the generic fibre G = Gη we obtain the perfectoid tilde-limit G∞ := (G∞)η ∼ lim←−[p]

G. More

precisely, multiplication by p corresponds to the homomorphism

[p] : OK〈X±1〉 → OK〈X±1〉, X → Xp.

In the direct limit, we obtain (lim−→[p]OK〈X±1〉)∧ = OK〈X±1/p∞〉. Therefore, taking the generic

fiber we get

G∞ = Spa(K〈X±1/p∞〉,OK〈X±1/p∞〉)and one can verify directly that we indeed have G∞ ∼ lim←−[p]

G.

Example. An example of a very different flavour is the p-adic completion G of the affine groupscheme Ga over OK . Note that G = Gη is not equal to Gana , but is the closed unit disc in the latter.

The underlying formal scheme of G is Spf S with S = OK〈X〉, and the [p]-multiplication is nowgiven by

[p] : OK〈X〉 → OK〈X〉, X → pX.

In the direct limit, we first obtain the algebra S′∞ = OK〈 1p∞X〉, consisting of power series f =∑∞

n=0 anXn ∈ OK [[X]] for which there exists an m ∈ Z≥0 so that |pnman| → 0. Next we need to

take the p-adic completion to form S∞. But we have

pnOK〈1

p∞X〉 = pnOK +XOK〈

1

p∞X〉

and therefore S′∞/πn = OK/πnOK . Consequently, the completion is just S∞ = OK and thus its

adic generic fiber is G∞ = Spa(K,OK). This is still perfectoid but is just one point!One can explain this observation geometrically: On the level of K-points, the formal scheme G

is the closed unit disc and [p] is scaling by p. A K-point in lim←−[p]G(K) therefore corresponds to a

sequence of K-points of the closed unit disc of the form

(x,1

px,

1

p2x, . . . ).

But for this to be contained in the closed unit disc, we must have x = 0.

2.5. group structure of the limit. One reason why perfectoid limits along group morphisms areparticularly interesting is that the perfectoidness ensures that the limit has again a group structure:

Definition 2.18. A perfectoid group is a group object in the category of perfectoid spaces.

Note that the category of perfectoid spaces over K has finite products, so the notion of a groupobject makes sense.

Proposition 2.19. Let G be a rigid group with a perfectoid tilde-limit G∞ ∼ lim←−[p]G. Then

(1) there is a unique way to endow G∞ with the structure of a perfectoid group in such a waythat all projections G∞ → G are group homomorphisms

(2) given another rigid group H with perfectoid tilde-limit H∞ ∼ lim←−[p]H and a group homo-

morphism H → G, there is a unique group homomorphism H∞ → G∞ commuting with allprojection maps. In particular, formation of the perfectoid lim←−[p]

is functorial.


Proof. These are all consequences of the universal property of the perfectoid tilde-limit, Proposi-tion 2.5, which shows that one can argue like in the case of usual limits. �

3. Formal models for tori

In this section we want to show that for a split rigid torus T over K, a tilde-limit T∞ exists andis perfectoid. We do this by exhibiting a [p]-F -model of T .

As a preparation, we consider the torus Ganm over K. Recall that it arises from rigid analytification

of the affine torus Gm over K. Note however that Ganm is not affinoid (and not even quasi-compact).

It contains the generic fibre of the p-adic completion of Gm as an open subspace. If we see Ganm

as the rigid affine line with the origin removed, this subspace Gm can be identified with the unitcircle. In other words, on the level of points it corresponds to O×K ⊆ K×.

Finally, recall that for every x ∈ K× we have a translation map

Ganm

x·−→ Ganm

that is an isomorphism of rigid spaces sending the point 1 to x.

3.1. A family of explicit covers. We briefly recall how Ganm is constructed: The following is taken

from [2], §9.2 with slightly different notation. Throughout we use the following standard shorthandnotation: For any 0 6= a, b ∈ K with |a| ≤ |b| consider the topologically finite type K-algebra

K〈X/b〉 :=

{ ∞∑n=0

cn(X/b)n ∈ K[[X]] where cn ∈ K such that |cn| → 0

}which is isomorphic to K〈X〉 via X 7→ X/b, as well as

K〈a/X,X/b〉 := K〈X/b〉〈Z〉/(ZX − a).

The associated affinoid space B(a, b) is the annulus of radii |a| and |b| inside AanK :

B(a, b) = Sp(La,b), where La,b := K〈a/X,X/b〉.When |a| < |b|, then by the boundary of B(a, b) we mean the disjoint union of the affinoid subdo-mains B(a, a) and B(b, b), which we refer to as the inner and outer boundary, respectively. Explicitly,

(1)

B(a, b)←↩B(a, a)

La,b = K〈a/X,X/b〉 →K〈a/X,X/a〉 = La,a

a/X,X/b 7→a/X,a

b(X/a)

and similarly

(2)

B(a, b)←↩B(b, b)

La,b = K〈a/X,X/b〉 →K〈b/X,X/b〉 = Lb,b

a/X,X/b 7→ a

b(b/X), X/b.

Since we are mainly interested in the p-multiplication map, we will construct a family of coversof Gan

m on which [p] can be seen directly. Fix q ∈ K× with |q| < 1. Then the annuli {B(qi, qi−1)}i∈Zform an admissible affinoid covering of Gan

m and are glued together by identifying the inner boundaryof B(qi, qi−1) with the outer boundary of B(qi+1, qi). We denote this cover by Uq.

Assume now that q has a p-th root q1/p in K. The above then gives a finer cover Uq1/p of Ganm .

Using both covers Uq and Uq1/p , we can easily see the [p]-multiplication [p] : Ganm → Gan

m as follows:


Consider the affinoid open subsets B(qi/p, q(i−1/p)) of the source and B(qi, qi−1) of the target. Then[p] restricts to

(3)

B(qi, qi−1)[p]←−B(qi/p, q(i−1)/p)

K〈qi/X,X/qi−1〉 → K〈qi/p/X,X/q(i−1)/p〉

qi/X,X/qi−1 7→ (qi/p/X)p, (X/q(i−1)/p)p.

These further restrict to morphisms B(qi/p, q(i−1)/p) → B(qi, qi) on the inner boundary, and sim-ilarly for the inner boundary. These maps are by construction compatible with the glue maps (1)and (2). For the map (2), this is reflected by the fact that the following diagram commutes:

(4)

B(qi, qi−1) B(qi, qi) X/qi−1 q · (X/qi)

B(qi/p, q(i−1)/p) B(qi/p, qi/p) (X/qi−1)p q · (X/qi)p,

[p] [p]

and similarly with qi/X 7→ (qi/X)p. The case of the inner boundary (1) is essentially identical.

3.2. A family of formal models. At this point we have constructed a cover Uq of Ganm depending

on a choice of q ∈ K× with |q| < 1. The affinoid subspaces B(qi, qi−1) that we have used for thisadmit natural formal models: Namely, consider the topologically finite type OK-algebras

OK〈X/a〉 =

{ ∞∑n=0

cn(X/b)n ∈ K[[X]] where cn ∈ OK such that |cn| → 0

},

L◦qi,qi−1 := OK〈qi/X,X/qi−1〉 = OK〈X/qi−1〉〈Z〉/(Y Z − qi).It is shown in [2] §9.2 that L◦qi,qi−1 is flat over OK and thus gives rise to a flat formal OK-scheme

B(qi, qi−1) = Spf(L◦qi,qi−1) which is a formal OK-model of B(qi, qi−1). The open formal subschemes

B(qi−1, qi−1) = B(qi, qi−1)((X/qi−1)−1),

B(qi, qi) = B(qi, qi−1)((qi/X)−1)

are formal models for B(qi−1, qi−1),B(qi, qi) respectively. We conclude:

Lemma 3.1. The affine formal schemes {B(qi, qi−1)}i∈Z glue together to an OK-flat formal schemeGq such that (Gq)η = Gan

m . In other words, Gq is a flat formal model for Ganm .

3.3. A family of formal models for p-multiplication. As before choose q ∈ K× such that|q| < 1 and such that there exists a p-th root q1/p ∈ K. A closer look at the map (3) shows thatthe [p]-multiplication extends to a morphism of formal schemes

[p] : B(qi/p, q(i−1)/p)[p]−→ B(qi, qi−1).

The diagram (4) shows that these maps glue to a morphism

[p] : Gq1/p → Gq.

By construction, after tensoring − ⊗OK K all morphisms on algebras coincide with those definedin (1), (2), (3), respectively. We conclude:

Proposition 3.2. The map [p] : Gq1/p → Gq is an affine formal model of [p] : Ganm → Gan

m .


We moreover see directly from the construction:

Proposition 3.3. The map [p] : Gq1/p → Gq reduces mod π to the relative Frobenius map.

We now have everything together to finish our proof that (Ganm )∞ is perfectoid:

Proposition 3.4. The space Ganm has a [p]-F -formal tower. In particular, there exists a perfectoid

space (Ganm )∞ such that (Gan

m )∞ ∼ lim←−[p]Ganm .

Proof. Since K is perfectoid, we can find q ∈ K× such that |q| < 1 for which there exist arbitrarypn-th roots. We choose such a q and roots q1/pn for all n. Then the two Propositions above combineto show that

. . .[p]−→ Gq1/p2

[p]−→ Gq1/p[p]−→ Gq

is a [p]-F -formal tower. By Proposition 2.16 we obtain a perfectoid space (Ganm )∞ as desired. �

3.4. The action of T . The multiplication Ganm × Gan

m → Ganm restricts to give an action of the

annulus B(1, 1) on each of the affinoids in our rigid analytic cover as follows:

(5)

B(qi, qi−1)m←−B(1, 1)× B(qi, qi−1)

K〈qi/X,X/qi−1〉 → K〈Z, 1/Z〉⊗K〈qi/X,X/qi−1〉qi/X 7→ 1/Z ⊗ qi/X

X/qi−1 7→ Z ⊗X/qi−1

The same arguments as in the last section show that the map described in (5) has a flat formalmodel

B(qi, qi−1)m←− B(1, 1)×B(qi, qi−1).

We therefore conclude:

Proposition 3.5. For any q ∈ K× with |q| < 1, the formal torus T := B(1, 1) has a natural actionon Gq via a map

m : T ×Gq → Gq.

This map is a formal model of the action of the annulus B(1, 1) on Ganm . Furthermore, this action

is compatible with the models for [p] in the sense that if there is a p-th root q1/p ∈ K, then thefollowing diagram commutes.

T ×Gq1/p Gq1/p

T ×Gq Gq.

[p]×[p]

m

[p]

m

Proof. The existence of m follows from the above consideration concerning the map (5). The restis clear from the construction: All adic rings we have used in the construction are OK-subalgebrasof the affinoid K-algebras used to define Gan

m , so the equalities hold because they hold for Ganm . �


3.5. The case of general tori. By taking products everywhere, all of the statements in thissection immediately generalise to split tori:

Corollary 3.6. Let T be a split torus over K of the form T = (Ganm )d. Then for any q ∈ K× with

|q| < 1 the formal scheme Tq := (Gq)d is a formal model of T . Assume that there is a p-th root

q1/p ∈ K, then the [p]-multiplication map has an affine formal model [p] : Tq1/p → Tq that locally

on polyannuli is of the form [p] :∏dj=1 B(qij/p, q(ij−1)/p)→

∏dj=1 B(qij , qij−1). Moreover this map

reduces mod π to the relative Frobenius morphism.

Corollary 3.7. Let T be a split torus over K, considered as a rigid space. Then T has a [p]-F -formal tower. In particular, there exists a perfectoid space T∞ such that T∞ ∼ lim←−[p]

T .

Corollary 3.8. Let T be any split torus over K. For any q ∈ K× with |q| < 1, the formalcompletion T has a natural action on Tq via a map

m : T × Tq → Tq.

This map is a formal model of the action of the annulus T on T . Furthermore, this action iscompatible with the models for [p] in the sense that if there is a p-th root q1/p ∈ K, then the map

[p] : T1/pq → Tq is semi-linear with respect to [p] : T → T .

4. A [p]-F -formal tower for Raynaud extensions

In this section we study the p-multiplication tower of the Raynaud extensions associated toabeloid varieties over an algebraically closed perfectoid field K. The main result of this section isTheorem 4.11, namely the Raynaud extension E of an abeloid variety A over K admits a [p]-F -formal tower, showing that there exists a perfectoid tilde-limit E∞ ∼ lim←−[p]

E.

Remark 4.1. We are able to prove our main theorem over a general perfectoid field, but we optto work over an algebraically closed field to simplify the exposition. The essential ideas all appearin this case, and at the end of each section we sketch the changes needed to move to the generalcase. For a general perfectoid field K, the diagram 7 might only be defined after a finite extension;restricting to the algebraically closed case now allows us to ignore this issue. We sketch how to getaround this issue using Galois descent in Remark 4.12.

4.1. Raynaud extensions. We briefly sketch the theory of Raynaud extensions here, and referthe readers to [6] for more details on the setup.

Let A be an abelian variety over K. There exists a unique connected open rigid analytic subgroupA of A which extends to a formal smooth OK-group scheme E with semi-abelian reduction. ThisE fits into a short exact sequence of formal group schemes

(6) 0→ T → Eπ−→ B → 0

where B is the completion of an abelian variety B over K of good reduction (as usual we also denoteby B the rigid space associated to it), and T is the completion of a torus T of rank r over K. Therigid generic fibre T η of the torus T canonically embeds into the rigid torus T an which again wesimply denote by T . One can show that this induces a pushout exact sequence in the category ofrigid groups. More precisely, there exists a rigid group variety E such that the following diagramcommutes and the left square is a pushout.


(7)

0 T η Eη Bη 0

0 T E B 0

The abelian variety A we started with can then be uniformized in terms of E as follows:

Definition 4.2. A subset M of a rigid space G is called discrete if the intersection of M withany affinoid open subset of G is a finite set of points. Let G be a rigid group, then a lattice in Gof rank r is a discrete subgroup M of G which is isomorphic to the constant rigid group Zr.

Theorem 4.3. There exists a lattice M ⊆ E of rank equal to the rank r of the torus for whichthe quotient E/M exists as a rigid space and has a group structure such that E → E/M is a rigidgroup homomorphism. Moreover, there is a natural isomorphism

A = E/M.

The data of the extension (6) together with the lattice M ⊆ E is what we refer to as a Raynauduniformisation of A. This will in fact be the only input we need to construct the perfectoid tilde-limit A∞. Consequently, our method generalises to the class of rigid groups which admit Raynauduniformisation, namely to abeloid varieties:

Theorem 4.4 (Lutkebohmert, [6], Theorem 7.6.4). Let A be an abeloid variety, that is a connectedsmooth proper commutative rigid group over K. Then A admits a Raynaud uniformisation.

In the situation of Raynaud uniformisation, since M is discrete, the local geometry of A isdetermined by the local geometry of E. We will therefore first study the [p]-multiplication tower ofE and will then deduce properties of the [p]-multiplication tower of A.

Our strategy is to study the local geometry of E and E via T and B. An obstacle in doing thisis that the categories of formal or rigid groups are not abelian, which makes working with shortexact sequences a subtle issue. Another issue is that one cannot direcly study short exact sequenceslocally on T , E or B. An important tool is therefore the following Lemma:

Lemma 4.5. The short exact sequence (6) admits local sections, that is there is a cover of B byformal open subschemes Ui such that there exist local sections s : Ui → E of π. In particular, onecan cover E by formal open subschemes of the form T × Ui ↪→ E.

Proof. This is proved in Proposition A.2.5 in [6], where it is fomulated in terms of the groupExt(B, T ). Also see [3], §1. �

This Lemma suggests that instead of considering Raynaud extensions from the abelian categoryviewpoint, one should consider them as principal T -bundles of formal schemes.

Remark 4.6. This is the language we use in the rest of the paper: We will work with fibre bundlesof formal schemes, rigid spaces and schemes. The main technical tool we need is the associatedfibre construction in these settings. For a rigorous treatment of these we refer to the Appendix.

First of all, we note that the sequence (6) from the last section gives rise to a principal T -bundleE → B. The fact that E is obtained from Eη via push-out along T η → T can now conveniently

be expressed in terms of the associated fibre bundle by saying that E = T ×Tη Eη in the sense ofDefinition A.9. We then have the following description of [p]:


Lemma 4.7. The map [p] : E → E coincides with the morphism

[p]×[p] [p] : T ×Tη Eη → T ×Tη Eηinduced by the different [p]-multiplication maps by Proposition A.18.

Proof. Lemma A.19 in light of Remark A.20 applied to the maps g = [p] : T η → T η, h = [p] : T → T

and f = [p] : Eη → Eη says that [p]×[p] [p] is the unique morphism of fibre bundles E → E makingthe following diagram commute:

(8)

T E

T E

T η Eη

T η Eη

[p] ∃!

[p] [p]

Since [p] : E → E is such a map, the Lemma follows. �

4.2. A [p]-F -formal tower for E. In this subsection we prove that E admits a [p]-F -formal tower.The first step is to construct a family of formal models for E.

Lemma 4.8. Let q ∈ K× with |q| < 1. Let Tq be the formal model from Corollary 3.6. Then there

is a formal scheme Eq := Tq ×T E that is a flat formal model of the rigid space E. Furthermore,there exists a morphism

Eq := Tq ×T E → B

which is a fibre bundle and a formal model of E → B.

Proof. Recall from Corollary ?? that Tq has a T -action that is a model of the T η-action on T . In

particular, the associated fibre construction for the principal T -bundle E → B gives a fibre bundle

Eq := Tq ×T E → B. Since Tq is a formal model of T , one sees by Lemma A.21 that Eq is a formal

model of T ×Tη Eη which by definition is equal to E. Finally, since Tq and B are flat, so is Eq. �

Next we construct a model for the [p]-multiplication map. Here we can use again that [p] existson E and on Tq1/p .

Lemma 4.9. Let q ∈ K× be such that |q| < 1 and let q1/p ∈ K be a p-th root of q. Then there isan affine morphism

[p] : Eq1/p → Eq

which is a formal model of [p] : E → E.

Proof. Recall that the multiplication map [p] : T → T has a formal model [p] : Tq1/p → Tq byCorollary 3.6. This fits into a commutative diagram

Tq1/p Tq

T T .

[p]

[p]


Functoriality of the associated fibre construction in the general case, Proposition A.18, appliedto the diagram below then gives a natural map Eq1/p → E making the diagram commute:

(9)

Tq Eq

Tq1/p Tq1/p ×T E

T E

T E

[p] ∃

[p] [p]

By Lemma 4.7 this diagram equals diagram (8) on the generic fibre.To see that the morphism [p] : Eq1/p → Eq is affine, first note that [p] : B → B is an affine

morphism. The map [p] : Tq1/p → Tq is affine by construction, namely by Corollary 3.6 it is locally

on Tq of the form [p] :∏dj=1 B(qij/p, q(ij−1)/p) →

∏dj=1 B(qij , qij−1). Note that both of these

affine open subsets are fixed by the action of T . We conclude from the construction in the proof ofProposition A.18 that the morphism [p] : Eq1/p → Eq locally on the target is of the form

[p] :

d∏j=1

B(qij/p, q(ij−1)/p)× U ′ →d∏j=1

B(qij , qij−1)× U

for an affine open formal subscheme U ⊆ B with affine preimage U ′ under [p] : B → B. This showsthat the morphism is affine locally on the target, and therefore is affine. �

We have thus proved the first part of what we want to show about tilde-limits of E:

Lemma 4.10. With notation as above, the rigid group E admits a [p]-formal tower of the form

. . .[p]−→ Eq1/p2

[p]−→ Eq1/p[p]−→ Eq

for some q ∈ K×. In particular, there exists an adic space E∞ such that E∞ ∼ lim←−[p]E.

Proof. By Lemma 4.9, any choice of q ∈ K× with |q| < 1 and a compatible system of pn-th rootsq1/pn ∈ K× gives a tower

. . .[p]−→ Eq1/p2

[p]−→ Eq1/p[p]−→ Eq

that on the generic fibre equals . . .[p]−→ E

[p]−→ E. This is the desired [p]-formal tower. �

We are now ready to prove the main result of this section, namely that E∞ is perfectoid.

Theorem 4.11. Let K be perfectoid. Then the [p]-formal tower from Lemma 4.10

. . .[p]−→ Eq1/p2

[p]−→ Eq1/p[p]−→ Eq

is already a [p]-F -formal tower. In particular, the space E∞ is perfectoid.


Proof. It suffices to prove that the reduction mod π of the map [p] : Eq1/p −→ Eq factors throughrelative Frobenius.

In the following we denote reduction of a formal scheme by a ∼ over the respective symbol, forexample the reductions of T , E and T are denoted by T , E and T.

Recall that [p] : Eq1/p −→ Eq was constructed using the [p]-multiplication cube in diagram (9)and functoriality of the associated bundle. Also recall that all statements we have used about fibrebundles also hold when we replace formal schemes over OK by schemes over OK/π, and formationof the associated bundle commutes with this reduction. In particular,

Eq = Tq ×T E.

By Corollary 3.6, the model of the multiplication map [p] : Tq1/p → Tq reduces to relative Frobeniusover p. In particular, we have a natural isomorphism

T(p)

q1/p∼= Tq

and we can identify T(p)

q1/p= Tq in the following. The same is true for T (p) = T .

Since E and T are group schemes, the reduction of [p] on them factors through the relative Frobe-nius maps FE and FT respectively. By functoriality of relative Frobenius (”Frobenius commuteswith any map”) we have a commutative diagram

E E(p)

T T (p).

FE

FT

In other words, FE is an FT -linear morphism of fibre bundles. Again by functoriality of Frobeniuswe also have a commutative diagram

Tq1/p T(p)

q1/p

T T (p).

FT

FT

By Proposition A.18, we thus obtain a natural morphism

FT ×FT FE : Tq1/p ×T E → T

(p)

q1/p×T

(p)

E(p).

Using the explicit description of FT ×FT FE in the proof of Proposition A.18, we easily check that

this morphism is just the relative Frobenius of Eq1/p : This is a consequence of the fact that relative

Frobenius on the fibre product T × U for any U ⊆ B is just the product of the relative Frobeniusmorphisms of T and U , and thus the morphisms θi from Lemma A.8 are all trivial.

But this means that again by Proposition A.18, the reduction of the formal model of the p-multiplication cube in diagram 9 admits the following factorisation:


Tq Eq

T(p)q E

(p)

q1/p

Tq1/p Eq1/p

T E

T (p) E(p)

T E

F F

F F

Since the composed maps E → E on the bottom right, T → T on the bottom left and Tq1/p → Tqon the upper left by construction are the reductions of the respective p-multiplication maps [p], thefunctoriality of the associated bundle construction in Proposition A.18 implies that the two mapson the upper right compose to the reduction of [p]×[p] [p]. But [p]×[p] [p] is equal to [p] : Eq1/p −→ Eqby definition of the latter. This completes the proof that the reduction of [p] : Eq1/p −→ Eq factors

through the relative Frobenius on Eq1/p .The conclusion that E∞ is perfectoid then follows from Proposition 2.16. �

Remark 4.12. With some work, the arguments in this section can be extended to any perfectoidbase field. For instance, the Raynaud uniformisation of Theorem 4.3 might only be defined over afinite extension L of K. Our argument then gives a perfectoid space over the (necessarily perfectoid)field L. We can then use Galois descent to get a perfectoid space over our original field K. Thisuses that the quotient of a perfectoid space by a finite group often remains perfectoid: see Theorem1.4 of [5] for details. Finally, one checks that this Galois descent commutes with tilde-limits.

5. The case of abeloid varieties

We now want to prove the main result of this work, building on the preceding chapters:

Theorem 5.1. Let A be an abeloid variety over an algebraically closed perfectoid field K. Forinstance, we may take A to be an abelian variety. Then there is a perfectoid space A∞ such that

A∞ ∼ lim←−[p]

A.

Before we proceed with the proof of the main theorem which occupies the rest of the section, letus recall some notation:

(1) g is the dimension of A.(2) E is the Raynaud extension associated to A from Proposition 4.4, which is an extension of

a split rigid torus T of rank r by an abelian variety B of good reduction.(3) M ⊆ E is a lattice of rank r such that A = E/M .

5.1. Covering A by subspaces of E. The first step towards towards the proof is to construct acover of A = E/M by subspaces of E that behaves well under [p]-multiplication.


As a first step we recall how to relate the lattice M to a Euclidean lattice in Rr, cf §2.7 and §6.2in [6]. On the level of points, Gan

m has an absolute value map

| − | : Ganm (K) = K× → R×, x 7→ |x|

which induces the following group homomorphism from the torus T :

| − | : T (K) = (K×)r → (R×)r, (x1, . . . , xn) 7→ (|x1|, . . . , |xn|)

Since when working with lattices we prefer additive notation, we also consider the map

` : T (K) = (K×)r → Rr, x1, . . . , xn 7→ (− log |x1|, . . . ,− log |xn|).

Note that this map has dense image due to our assumptions on K.The formal torus T corresponds on K-points to T η(K) = (O×K)r and is thus in the kernel of

`. We can therefore extend ` to E(K) as follows: Locally over an open subspace U ⊆ B we have

E|U = T ×Tη Eη|U and we define ` by projection from the first factor. The different E|U are then

glued on intersections using the T η-action on T . But since ` on T is invariant under the T η-action,the maps glue together to a group homomorphism

` : E(K)→ Rr.

Since A = E/M is proper, the lattice M is sent by ` to a Euclidean lattice Λ ⊂ Rr of full rankr (see Proposition 6.1.4 in [6]). In particular, the map l restricts to an isomorphism of discretetorsion-free groups

` : M∼−→ Λ ⊆ Rr.

The idea is now that one can understand the quotient E/M in terms of the quotient Rr/Λ. Weare going to make this precise in the following:

For any d ∈ Rr>0, consider the cuboid with length 2d centered at the origin:

S(d) = {(s1, . . . , sr) ∈ Rr||si| ≤ di}

Choose d small enough such that S(d) intersects Λ only in 0 ∈ Λ and such that the translatesof S(d) by Λ are all disjoint. We may moreover assume that exp(−di) ∈ |K×| for all i. Chooseq1, . . . , qr ∈ K such that |qi| = exp(−di). We denote by B(q, q−1) the affinoid open rigid multi-annulus in T centered at 1 of radii |qi| < 1 < |qi|−1 in every direction.

Lemma 5.2. The inverse image `−1(S(d)) ⊆ E(K) is the underlying set of the admissible open

subspace E(q) := B(q, q−1)×Tη Eη of E considered as a rigid space.

Proof. One shows this first for the map T → Rr, where it is clear that the preimage is B(q, q−1).

This is also described in §6.4 of [4]. The statement for B(q, q−1)×Tη Eη follows by direct inspectionon local trivialisations B(q, q−1)× U for U ⊆ B. �

Note that by our choice of d, the map Rr → Rr/Λ maps S(d) bijectively onto its image.

Lemma 5.2 then says that we can use B(q, q−1)×Tη Eη as a chart for E/M around the origin.In order to obtain charts around other points of E/M , we simply need to consider translations:

Recall that for any c ∈ T (K), the translation c· : T → T is an isomorphism of rigid spaces thatsends 1 to c. We denote the image of any admissible open subspace U under translation by c · U .

For the next Lemma, recall thatK is algebraically closed and hence we can identify the underlyingset of E considered as a rigid space with the set of K-points E(K).


Ei

Ej

Figure 1. Given two charts Ei and Ej , the chart Ej is glued to Ei along inter-sections with all translates of Ei by q ∈M .

Lemma 5.3. With notation as before, let c ∈ T (K) be any point and let s = l(c). Then the inverseimage `−1(s+ S(d)) ⊆ E(K) of the translation of S(d) by s is the underlying set of the admissible

open subset E(c, q) := (c · B(q, q−1))×Tη Eη ⊆ E.

Proof. Since ` commutes with the translations

T (K) Rr

T (K) Rr,

`

·c +s

`

this is an immediate consequence of Lemma 5.2. �

Definition 5.4. We call the spaces E(c, q) ⊆ E cuboids centered at c. More precisely, they arelocally a cuboid c · B(q, q−1) ⊆ T times an admissible open subspace of the abelian variety B.

Lemma 5.5. There exist finitely many admissible open cuboids E1, . . . , Ek ⊆ E which map iso-morphically onto their images in A = E/M and which cover A admissibly.

One can reconstruct A from any such cover by glueing E1, . . . , Ek as follows: By construction,for any Ei the translates m · Ei by m ∈ M are pairwise disjoint and we thus have a canonicalprojection π from the union ∪m∈M (m ·Ei) ⊆ E to Ei. Let Eij := (

⋃m∈M m ·Ei) ∩Ej ⊆ E. Then

we glue Ej to Ei via the map

Eij →⋃m∈M

m · Eiπ−→ Ei.

Proof. Since Rr/Λ is compact, we can find finitely many s1, . . . , sk and d1, . . . , dk ∈ Rr>0 such thatRr/Λ is covered by the si +S(di). When we choose corresponding c1, . . . , ck ∈ T (K) and q1, . . . , qkas in Lemma 5.3, then the corresponding Ei := E(ci, qi) give an admissible cover of A = E/M byadmissible open subsets of E.

In order to reconstruct A, note that ∪m∈Mm · Ei is precisely the preimage of Ei under theprojection E → E/M . In particular, the subspace Eij is precisely the preimage of Ei under the


composition Ej ↪→ E → E/M . In other words, the subspace Eij ⊆ Ej is the intersection of Ei andEj when considered as subspaces of A. This shows that as charts of A, the spaces Ei and Ej areglued via Eij as described. �

Finally, we need some control about what happens to the cubes under [p]-multiplication. Recallfrom Lemma 5.3 that we can always assume that c admits p-th roots.

Lemma 5.6. Let c1/p be a p-th root of c and let q1/p be a p-th root of q in (K×)r. Then under

[p] : E → E, the admissible open E(ci, q) pulls back to the admissible open E(c1/pi , q1/p).

Proof. It is clear that under [p] : T → T , the admissible open cuboid c ·B(q, q−1) centered at c pullsback to c1/p · B(q1/p, q−1/p). Note that this is independent of the choices of c1/p and q1/p. Nowrecall that in terms of fibre bundles, multiplication [p] : E → E is

[p]×[p] [p] : T ×Tη Eη → T ×Tη Eη

by Lemma 4.7. Thus (c · B(q, q−1))×Tη Eη pulls back to (c1/p · B(q1/p, q−1/p))×Tη Eη. �

5.2. The two towers. In this section we want to separate the [p]-multiplication of A into twodifferent towers, which we think of as being a “ramified” tower and an “etale” tower. Of course incharacteristic 0 both towers will actually be etale, and these words are only meant to describe thebehaviour of the maps relative to the lattice M .

For the ramified tower, we first make an auxiliary choice of certain torsion subgroups of A: SinceK is algebraically closed, we can choose lattices M1/pn ⊆ E such that [p] : E → E restricts to

isomorphisms M1/pn+1 →M1/pn for all n.

Remark 5.7. Such a choice is equivalent to the choice of subgroups Dn ⊆ A[pn] of rank prn for all

n such that pDn+1 = Dn and Dn + E[pn] = A[pn]. Namely, given the lattices M1/pn+1

, we obtain

the desired torsion subgroups by setting Dn := M1/pn+1

/M . This is because any such lattice givesa splitting of the short exact sequence 0→ E[pn]→ A[pn]→M/Mpn → 0.

Conversely, given subgroups Dn ⊆ A[pn] with properties as above, we recover M1/pn as thekernel of E → A→ A/Dn.

One might call the choice of Dn for all n a partial anticanonical Γ0(p∞)-structure, because if Badmits a canonical subgroup (that is, if B has bad reduction or if it has good reduction and thereduction satisfies a condition on its Hasse invariant), the choice of a (full) anticanonical Γ0(p∞)-structure on A is equivalent to the choice of a partial anticanonical Γ0(p∞)-structure on A andan anticanonical Γ0(p∞)-structure on B. Note however that A always has a partial anticanonicalsubgroup even if B does not have a canonical subgroup.

Note that in the case of K perfectoid but not necessarily algebraically closed, one can still carryout the constructions in the following using partial anticanonical Γ0(p∞)-structures, whereas thelattices M1/pn might not be defined over K.

Following the remark, denote by Dn the torsion subgroup M1/pn/M ⊆ A. The quotient A/Dn =E/M1/pn is then another abeloid variety over K and the quotient map vn : E/M → E/M1/pn isan isogeny of degree prn through which [pn] : A→ A factors:


(10)

E/M1/pn

E/M E/M.

[pn]

[pn]

vn

We think of these maps as being an analogue of Frobenius and Verschiebung, which is why wedenote the left map by v. Putting everything together, the [p]-multiplication tower splits into twotowers

(11)

. . ....

...

E/M E/M1/p E/M1/p2

E/M E/M1/p

. E/M

v

[p]

v

[p] [p]

[p]

v

[p]

Since each quotient M1/pn/M is a finite etale group scheme, all horizontal maps are finite etale.The vertical tower on the other hand fits into a second commutative diagram of rigid groups whichcompares it to the [p]-tower of E:

(12)

......

...

0 M1/p2 E E/M1/p2 0

0 M1/p E E/M1/p 0

0 M E E/M 0

∼= [p] [p]

∼= [p] [p]

5.3. Constructing a limit of the vertical tower. Our first step is to show that the tower on theright has a perfectoid tilde-limit. Recall from Lemma 5.5 that E/M can be covered by admissibleopen subspaces E1, . . . , Ek ⊆ E which map isomorphically onto an admissible open via E → E/M .

Denote by E1/pn

i ⊆ E the pullback along [pn] : E → E. Also denote by E1/pn

ij ⊆ E the pullback of

the intersection Eij . We can then reconstruct the space E/M1/pn from the E1/pn

i as follows:

Lemma 5.8.


E E/M1/pn

E E/M

[pn]

Figure 2. Illustration of how [pn] : E/M1/pn → E/M can be glued from the maps

E1/pn

j → Ej . Here Ej on bottom and E1/pn

j on top are represented by the greycuboids in the middle. On the left they are embedded into E whereas on the rightthey are considered as charts for E/M and E/M1/p.

(1) The restriction to E1/pn

i ⊆ E of E → E/M1/pn is an isomorphism onto its image. In

particular, we can view E1/pn

i as a chart of E/M1/pn , and this is the preimage of Ei under

E/M1/pn → E/M .

(2) The collection of E1/pn

i is an atlas for E/M1/pn .

(3) We can reconstruct E/M1/pn from glueing the E1/pn

i along the E1/pn

ij .

(4) The map [pn] : E/M1/pn → E/M can be glued from the restrictions of [pn] : E → E to

E1/pn

i → Ei, that is these maps commute with the glueing maps on E1/pn

ij .

The situation is thus like in Figure 2.

Proof. The first part follows because the map on the left of diagram 12 is an isomorphism. Thesecond follows from the pullback of the Ei along E/M1/pn → E/M , using that the diagram com-

mutes. We thus obtain an admissible cover by cuboids E1/pn

1 , . . . , E1/pn

k of E/M1/pn . The second

part of Lemma 5.5 applied to E/M1/pn then shows that E/M1/pn can be reconstructed by glueing

along subspaces E1/pn

ij .

Finally, in order to see that one can glue together the map [pn] : E/M1/pn → E/M from

the E1/pn

i , use that intersection of cuboids are again cuboids, and so E1/pn

ij is a disjoint union

of cuboids. It then follows from Lemma 5.6 that Eij pulls back under [pn] to the intersection

E1/pn

ij ⊆ E/M1/pn . That [pn] commutes with the glueing maps is clear because we know from

diagram (10) that [pn] : E → E induces a morphism [pn] : E/M1/pn → E/M . �

We are now ready to prove:


Proposition 5.9. There is a perfectoid space E/M1/p∞ such that

E/M1/p∞ ∼ lim←−n

E/M1/pn .

Proof. Denote by E1/p∞

i the pullback of Ei ⊆ E to E∞. This is an open subspace of a perfectoidspace and hence perfectoid. Moreover, by Proposition 2.4 we have

E1/p∞

i ∼ lim←−E1/pn

i .

Given two different Ei, Ej , we know by Lemma 5.8 that at every step in the tower, the pullbacks

E1/pn

i and E1/pn

j to E/M1/pn intersect in E1/pn

ij . We can thus glue the E1/p∞

i along pullbacks

E1/p∞

ij of the intersections Eij = Ei ∩ Ej to E∞ and thus obtain a perfectoid space E/M1/p∞ .

This is a tilde-limit for lim←−[p]E/M1/pn because by construction it is so locally, and the definition

of tilde-limits in Definition 2.4.1 of [14] is local on the source. �

5.4. Constructing a limit of the horizontal tower. In order to construct a tilde-limit for lim←−A,

we can now use that the horizontal maps in diagram (11) are all finite etale. They are even finitecovering maps, in the following sense:

Lemma 5.10. For any 0 ≤ m ≤ n, the preimage of E1/pn

i from Lemma 5.8 under the horizontal

map vn−m : E/M1/pm → E/M1/pn is isomorphic to pr(n−m) disjoint copies of E1/pn

i . More canoni-

cally, it can be described as the isomorphic image of M1/pn/M1/pm×E1/pm

i under the multiplication

map E/M1/pm × E/M1/pm → E/M1/pm .

Proof. By the first part of Lemma 5.8, we know that the preimage of E1/pn

i under the projection

E → E/M1/pn is a disjoint union of translates of E1/pn

i by M1/pn . The result then follows because

M1/pn/M1/pm = (Z/pn−mZ)r. �

We also record the following immediate consequence:

Lemma 5.11. The preimage of Ei under [pn] : A → A is isomorphic to prn disjoint copies of

E1/pn

i . More canonically, we can describe the preimage as the isomorphic image of Dn × E1/pn

i

under the multiplication A×A→ A. The situation is thus as in figure 3.

Proof. This follows from the first part of Lemma 5.8 combined with Lemma 5.10 in the case ofm = 0. �

Lemma 5.12. The squares in diagram (11) are all pullback diagrams.

E/M1/pn E/M1/pn+1

E/M1/pn−1

E/M1/pn

[p]

v

[p]

v

Proof. This can for instance be checked locally: The admissible open subset E1/pn

i ⊆ E/M1/pn

from Lemma 5.8 is pulled back to E1/pn+1

i under the vertical map [p] : E/M1/pn+1 → E/M1/pn .

The preimage of E1/pn

i under the horizontal map E/M1/pn−1 → E/M1/pn is pr disjoint copies

of E1/pn

i by Lemma 5.10. The pullback of E1/pn

i to the upper right is thus pr disjoint copies of

E1/pn+1

i , which is clearly the fibre product. �


E/M E/M1/pn

E/M

[pn]

vn

[pn]

Figure 3. Illustration of how [p] : E/M → E/M factors in a part that is “ram-ified” (the vertical tower) and a part that is “etale” (the horizontal tower) withrespect to our cover.

Lemma 5.13. The horizontal maps in diagram (11) induce natural finite etale morphisms v :E/M1/p∞ → E/M1/p∞ that fit into Cartesian diagrams

E/M1/p∞ E/M1/p∞

E/M1/pn−m E/M1/pn

vm

vm

In particular, the preimage of E1/p∞

i under vm is isomorphic to prm copies of E1/p∞

i .

Proof. Since E/M → E/M1/p is finite etale, the fibre product with E/M1/p∞ → E/M1/p existsand is perfectoid by Proposition 7.10 of [8].

The universal property of the fibre product then gives a unique map

E/M1/p∞ → E/M ×E/M1/p E/M1/p∞

making the natural diagrams commute. On the other hand, using Lemma 5.12 we see that the fibreproduct has compatible maps into the vertical inverse system over E/M . Since by Proposition 2.5the perfectoid tilde-limit E/M1/p∞ is universal for maps from perfectoid spaces to the inversesystem, we obtain a unique map into the other direction. �

We thus obtain a pro-etale tower

(13) . . .v−→ E/M1/p∞ v−→ E/M1/p∞ v−→ E/M1/p∞

which we think of as being a kind of vertical “limit” of diagram 11.We now want to explicitly construct a tilde-limit of this tower. Recall from Remark 5.7 the finite

etale subgroups Dn = M1/pn/M ⊆ E/M = A. Since K is algebraically closed, they are constant


and uncanonically isomorphic to (Z/pnZ)r = Spa(Map((Z/pnZ)r,K),Map((Z/pnZ)r,OK)). The

[p]-multiplication on E maps M1/pn+1

onto M1/pn and therefore the [p]-multiplication tower ofE/M induces a tower

. . .[p]−→ Dn+1

[p]−→ Dn → . . . .

Lemma 5.14. There exists a perfectoid space D∞ such that D∞ ∼ lim←−[p]Dn. Any choice of

compatible isomorphisms Dn∼= Z/pnZr induces an isomorphism

D∞ = Spa(Mapcts(Zrp,K),Mapcts(Zrp,OK)).

Such ”profinite” perfectoid spaces like D∞ are mentioned in [13] before Remark 8.2.2.

Proof. The Dn are etale over (K,OK) and thus affinoid perfectoid. Proposition 2.7 then shows thatD∞ exists. Given isomorphisms Dn = (Z/pnZ)r = Spa(Map((Z/pnZ)r,K),Map((Z/pnZ)r,OK)),Proposition 2.7 moreover says that the tilde-limit is given by the π-adic completion of(

lim−→n

Map((Z/pnZ)r,K), lim−→n

Map((Z/pnZ)r,OK))

= (Maplc(Zrp,K),Maplc(Zrp,OK))

where Maplc denotes the locally constant morphisms, for the p-adic topology on Zp. Since theπ-adic completion of these are the continuous morphisms, this proves the Lemma. �

Proposition 5.15. There exists a perfectoid space (E/M1/p∞)∞ such that

(E/M1/p∞)∞ ∼ lim←−v

(E/M1/p∞).

Moreover, the projection map π : (E/M1/p∞)∞ → E/M1/p∞ is a profinite covering map, namelyevery point of E/M1/p∞ has an open neighbourhood U such that π−1(U) is isomorphic to D∞×U .

Proof. By Lemma 5.5, the preimage of E1/p∞

i under vm of E/M1/p∞ is isomorphic to Dm×E1/p∞

i .By Lemma 2.12 we then have

D∞ × E1/p∞

i ∼ lim←−m

Dm × E1/p∞

i .

As before, these local tilde-limits glue using Propositions 2.4 and 2.5. �

5.5. The diagonal tower: proof of the main theorem. We now want to show that (E/M1/p∞)∞is in fact a tilde-limit of the [p]-multiplication tower. In other words, this says that the horizontallimit of the vertical tilde-limits in diagram 11 is also a diagonal tilde-limit. This isn’t just a formalconsequence since tilde-limits aren’t limits. But using the local geometry of the maps in the towerin terms of cuboids, it is still easy to see:

Proposition 5.16. The perfectoid space A∞ := (E/M1/p∞)∞ is a tilde-limit of lim←−[p]A. It is

independent up to unique isomorphism of the choice of partial anticanonical Γ0(p∞)-structure, butit remembers the choice as a pro-finite etale closed subgroup D∞ ⊆ A∞.

The preimage of Ei ⊆ A under the projection A∞ → A is isomorphic to D∞ × E1/p∞

i .

Proof. It is clear from (E/M1/p∞)∞ ∼ lim←−v E/M1/p∞ and E/M1/p∞ ∼ lim←−[p]

E/M1/pn that the

underlying topological space of (E/M1/p∞)∞ is indeed isomorphic to lim←−[p]|E/M |.

In order to show that it is a tilde-limit of lim←−[p]E/M , it thus suffices to give an explicit cover of

(E/M1/p∞)∞ by open affinoids satisfying the tilde-limit property.


Recall that by construction of (E/M1/p∞) we have a cover of E/M by open subsets Ei that pull

back to perfectoid open subspaces E1/p∞

i for which E1/p∞

i ∼ lim←−E1/pn

i . Moreover, by the second

part of Proposition 5.15 we know that the pullback of E1/p∞

i to (E/M1/p∞)∞ is D∞ × E1/p∞

i .On the other hand, when we go along the diagonal tower, we obtain the inverse system

· · · → Dn+1 × E1/pn+1

i → Dn × E1/pn

i → . . . .

By Lemma 2.12 this inverse system has perfectoid tilde-limitD∞×E1/p∞

i . These local tilde-limitsglue together to give the desired tilde-limit A∞.

That A∞ is independent of the Γ0(p∞)-structure up to unique isomorphism is a consequence ofthe universal property of the perfectoid tilde-limit. To see that D∞ is a closed subgroup of A∞,

choose i such that the unit section of E/M lies in Ei. Then the unit section Spa(K,OK)→ E1/p∞

i

induces a closed immersion D∞ ↪→ D∞ × E1/p∞

i ↪→ A∞. �

This finishes the proof of Theorem 5.1.Note that while the approach via cuboids Ei may look a bit technical on first glance, it has the

advantage of giving an explicit description of A∞ as being glued from pieces of D∞×E∞ by glueingdata that is controlled by the lattices M1/pn . This might be interesting for applications.

Remark 5.17. To extend the result to a general perfectoid base field, some changes are necessaryas the lattices M1/pn may no longer be defined over K. One approach is to note that as inLemma 5.11, the preimage of a sufficiently small cuboid Ei under [pn] : A→ A is isomorphic to prn

disjoint copies of E1/pn

i . Roughly speaking, this is because the cuboid E(c, q) depends only on |c|and |q|, so we can change c and q to admit p−th roots. Then in the inverse limit, we get profinitely

many copies of E1/p∞

i above Ei. These glue as before to give the perfectoid tilde-limit A∞. Thisapproach skips over the two towers entirely and is a bit simpler geometrically.

Another approach is to define M1/pn as the pullback of M under [pn] : E → E. This may nolonger be a lattice, but we can still define Dn := M1/pn/M and check that it is the analytification ofa finite subgroup scheme of A[pn]. This allows us to construct the vertical and horizontal towers asbefore, but the arguments are a bit different as the morphisms E/M → E/M1/pn are no longer localcoverings. As this approach retains the two towers description, it makes it possible to generalizethe results of the following section to a general perfectoid base field.

Finally, we note that if K is any perfectoid field of characteristic p, then instead of the verticaland horizontal tower one can use the Frobenius and Verschiebung tower, respectively, and one canshow that lim←−V A

perf ∼ lim←−[p]A is the desired perfectoid tilde limit.

6. Limits of the covering maps

In this section we use the explicit constructions of the space A∞ to study its geometry moreclosely. We retain notation and assumptions from the last chapter.

Over the course of the proof of Theorem 5.1, we have used three different towers: The tower

· · · → E[p]−→ E, the tower · · · → E/M

[p]−→ E/M and the tower · · · → E/M1/p [p]−→ E/M . The threeare related by covering maps which fit into a commutative diagram of towers

E E/M E/M1/pn+1

E E/M E/M1/pn

[p] [p] [p]


As we have seen in the last sections, all three towers have perfectoid tilde-limits, that we havedenoted by E∞, A∞ and E/M1/p∞ .

By Proposition 2.19 the map π : E → A = E/M in the limit induces a natural group homo-morphism ι : E∞ → A∞. A similar universal property argument shows that we obtain a naturalgroup homomorphism A∞ → E/M1/p∞ . The composition of these two maps is the morphismE∞ → E/M1/p∞ , which is the limit of the maps E → E/M1/pn in the above diagram. We nowwant to look at these morphisms more closely one after the other.

We start with the morphism E∞ → E/M1/p∞ :

Proposition 6.1. Denote by M∞ ∼= M the perfectoid tilde-limit of the tower

. . . M1/p2 M1/p M.[p]

∼[p]

∼[p]

∼

There is a natural map M∞ → E∞ with respect to which we can interpret M∞ as a lattice of rankr in E∞. The map fits into a short exact sequence of perfectoid groups

0→M∞ → E∞ → E/M1/p∞ → 0

that is locally split. In particular, we can view E∞ as an M∞-torsor over E/M1/p∞ .

Proof. The map M∞ → E∞ is induced by the universal property of the perfectoid tilde-limit asusual. In order to see that the sequence is exact, we need to see that the first map is a kernel ofthe second, and the second map is a categorical quotient of the first. To this end, we first analysethe morphism locally: The projections to the inverse system fit into a commutative diagram

M∞ E∞ E/M1/p∞

......

...

M1/pn E E/M1/pn

M E E/M

[pn] [pn] [pn]

Let us consider the preimages of Ei ⊆ E/M under these morphisms: By Lemma 5.5 we see thatthe pullback to E is

⊔q∈M qEi. We can also see this as the isomorphic image of M ×Ei under the

multiplication map E × E → E. The pullback of Ei along [pn] : E/M1/pn → E/M is E1/pn

i as we

have seen in Lemma 5.8. The same Lemma shows that the pullback of this along E → E/M1/pn

is⊔q∈M1/pn qE

1/pn

i = M1/pn × E1/pn

i . We conclude that the pullback to E∞ is M∞ × E1/p∞

i . By

construction of E/M1/p∞ in the proof of Proposition 5.9, the pullback of Ei to E/M1/p∞ is E1/p∞

i .All in all, we obtain a pullback diagram

E∞ E/M1/p∞

M∞ × E1/p∞

i E1/p∞

i

E E/M

M × Ei Ei


We conclude that E∞ → E/M1/p∞ is a principal M∞-torsor of perfectoid groups. It is then clearthat M∞ is the preimage of 0 ∈ E/M1/p∞ , from which one easily verifies that M∞ ↪→ E∞ has theuniversal property of the kernel. It remains to see that E∞ → E/M1/p∞ has the universal propertyof the cokernel: Given any perfectoid group H and a group homomorphism E∞ → H that is trivial

on M∞, the restriction M∞×E1/p∞

i → H gives a natural map E1/p∞

i → H. Since by construction

of E/M1/p∞ the spaces E1/p∞

i and E1/p∞

j are glued on E1/p∞

ij using translation by M∞, these glue

together to the desired morphism of E/M1/p∞ . �

The case of ι : A∞ → E/M1/p∞ is similar:

Proposition 6.2. The subgroup D∞ ⊆ A∞ gives rise to a short exact sequence of perfectoid groups

0→ D∞ → A∞ → E/M1/p∞ → 0

that is locally split. In particular, we can view A∞ as a D∞-torsor over E/M1/p∞ .

Proof. By Proposition 5.15 the pullback of E1/p∞

i under A∞ → E/M1/p∞ is

D∞ × E1/p∞

i → E1/p∞

i

which shows that A∞ → E/M1/p∞ is a D∞-torsor. As in the last proof, this implies that thesequence in the Proposition is a short exact sequence. �

Finally, we consider the case of ι : E → A = E/M . While the limits of the last two towers werefibre bundles again, the map ι shows quite a different behaviour and on the opposite is an injectivegroup homomorphism. This may seem strange at first, but it is actually what one might expectfollowing the intuition of the following example:

Remark 6.3. Consider the following inverse system of abstract groups:

0 Z R R/Z 0

0 Z R R/Z 0

[p] [p] [p]

While at finite level the maps on the right are all covering maps, in the inverse limit the homologicalalgebra of lim←− produces a long exact sequence

0 0 R lim←−[p]R/Z lim←−

1

[p]Z = Zp/Z 0.

So in the limit the covering map becomes the kernel of a map to Zp/Z.

For perfectoid groups the homological algebra argument of course doesn’t apply. Nevertheless,we can again use the explicit covers of the last section to show that the situation is very similar asin the remark. In the following, we use the notion of injective morphism from [11], Definition 5.1.

Theorem 6.4. The map ι : E∞ → A∞ is an injective group homomorphism. It fits into thefollowing commutative diagram of locally split short exact sequences of perfectoid groups:

0 M∞ E∞ E/M1/p∞ 0

0 D∞ A∞ E/M1/p∞ 0


The morphism ι is compatible with the splittings: Locally on open subspaces U ⊆ E/M1/p∞ themorphism E∞ ↪→ A∞ is of the form M∞ × U → D∞ × U . In particular, one can describe A∞ asthe associated fibre bundle

A∞ = D∞ ×M∞ E∞.

Proof. Recall that the map ι : E∞ → A∞ arises by a universal property from the inverse system

......

...

E A E/M1/p

E A E/M.

[p] [p]

v

f

Using Lemmas 5.8 and 5.10 we see that the pullback of this diagram to Ei ⊆ E/M is

......

...⊔q∈M1/p

q · E1/pn

i

⊔q∈Dn

E1/pn

i E1/pn

i

⊔q∈M

q · Ei Ei Ei

We see from this description and from the last part of Proposition 5.16 that the pullback to infinitelevel is the sequence

(14) M∞ × E1/p∞

i → D∞ × E1/p∞

i → E1/p∞

i .

This shows that the diagram of short exact sequences commutes. Since the E1/p∞

i cover E/M1/p∞

by construction, it also shows the description of A∞ in terms of the associated fibre bundle.It remains to prove that ι is injective: Choose a basis of M , and thus a trivialisation of all Dn. We

then see that the map M∞ → D∞ on the level of the underlying topological spaces can be describedas the inclusion Zr ↪→ Zrp. This shows that M∞ ↪→ D∞ is an injective group homomorphism ofperfectoid spaces. Thus by equation (14), the morphism E∞ → A∞ is injective as well. �

Corollary 6.5. The injection E∞ → A∞ induces a short exact sequence of perfectoid groups

0→M∞ → D∞ × E∞ → A∞ → 0

where the map on the left is the diagonal embedding of M∞ into D∞ × E∞. In particular, we candescribe A∞ as the quotient (D∞ × E∞)/M∞ in the category of perfectoid groups.

Proof. The map D∞ × E∞ → A∞ is just the composition of D∞ × E∞ ↪→ A∞ × A∞ with themultiplication of A∞. It is then clear from the short exact sequences of Theorem 6.4 that M∞ isthe kernel of this map. That A∞ has the universal property of the cokernel is a consequence of theuniversal property of the associated fibre bundle construction: Explicitly, this follows from the factthat locally over U ⊆ E/M1/p∞ , the map on the right is the projection

D∞ ×M∞ × U → D∞ × U.This gives a local splitting, and thus the necessary map in the universal property of the cokernel. �


We can see the short exact sequence of 6.5 as the analogue at infinity of the short exact sequenceof rigid groups

0→M → E → A→ 0.

In particular, while as a rigid analytic space A is locally isomorphic to open subspaces of E, theperfectoid space A∞ is locally isomorphic to open subspaces of D∞ × E∞.

Appendix A. Fibre bundles of formal and rigid spaces

In this chapter we review the theory of fibre bundles with structure group T and in particularof principal T -bundles in the setting of formal and rigid geometry.

In this chapter we denote by T a commutative formal group scheme over OK . We denote themultiplication map by m : T × T → T . By a T -action on a formal scheme X we mean a morphismmX : T ×X → X such that the usual associativity diagram commutes.

Definition A.1. By a T -linear map of schemes X and Y with T -actions we mean a morphismφ : X → Y such that the following diagram commutes

T ×X T × Y

X Y

mX

idT ×φ

mY

φ

We denote by FormActT the category of formal schemes with action by T .

The definition of a principal T -bundle is just what we get when we take the definition of aprincipal G-bundle and replace the category of topological spaces by the category of formal schemes.

Notation A.2. In the following, if π : E → B is a morphism of formal schemes, then for a formalopen subscheme U ⊆ B we denote E|U := π−1(U) ⊆ E.

Definition A.3. Let T be a formal group scheme. Let F be a formal scheme with an actionm : T × F → F . A morphism π : E → B of formal schemes is called a fibre bundle with fibreF and structure group T if there is a cover U of B of open formal subschemes Ui ⊆ B withisomorphisms ϕi : F × Ui

∼−→ E|Ui which satisfy the following conditions:

(a) For every Ui ∈ U, the following diagram commutes:

F × Ui E|Ui

Ui

ϕi

p2π

(b) For every two Ui, Uj ∈ U with intersection Uij , the commutative diagram

F × Uij E|Uij F × Uij

Uij

ϕi

p2π

p2

ϕj

produces an isomorphism φij := ϕ−1j ◦ϕi : F ×Uij → F ×Uij with the following property:

There exists a morphism ψij : Uij → T such that

φij = F × Uijψij×id× id−−−−−−−→ T × F × Uij

m×id−−−→ F × Uij


Definition A.4. When we take F equal to the formal scheme T with the action on itself by leftmultiplication, then a fibre bundle π : E → B with fibre T and structure group T is called aprincipal T -bundle. This is also called a T -torsor.

Example. For the short exact sequence 0 → T → Eπ−→ B → 0 from Section 4, E

π−→ B definesa principal T -bundle by Lemma 4.5. Moreover, for any formal open subscheme U ⊆ B, the mapE|U → U is still a principal T -bundle. This is what we mean when we say that the notion ofprincipal T -bundles is better suited for studying E locally on B than the notion of short exactsequences.

The morphism φij from condition (b) is fully determined by the morphism ψij : Uij → T . By aglueing argument, one shows:

Lemma A.5. Suppose we are given formal schemes F and B and a formal group scheme T with anaction on F . Then fibre bundles π : E → B with fibre F and structure group T are equivalent to thedata (up to refinement) of a cover U of B by formal open subschemes and morphisms ψij : Uij → Tfor all Ui, Uj ∈ U that satisfy the cocycle condition ψjk · ψij = ψik, by which we mean that thefollowing diagram commutes:

(15)

Uijk T × T

Uijk T.

ψij×ψjk

m

ψik

In order to define the category of fibre bundles, we also need the following:

Lemma A.6. Let E → B be a fibre bundle with fibre F and structure group T . With notationslike in Definition A.3 we have a natural T -action on F × Ui for each i when we let T act triviallyon Ui. These actions glue together to a T -action on E.

Proof. This is immediate from condition (b). �

Definition A.7. Let π : E → B be a fibre bundle with fibre F and structure group T and letπ′ : E′ → B′ be a fibre bundle with fibre F ′ and structure group T . Then a morphism of fibrebundles f : (E′, B′, π′)→ (E,B, π) is a commutative diagram of formal schemes

E′ B′

E B

fE

π′

fB

π

in which the morphism fE is also T -linear (we often abbreviate this by writing f : E′ → E). Wethus obtain the category of fibre bundles over T that we denote by FormFibBunT and the fullsubcategory of principal T -bundles, that we denote by FormPrinBunT .

In the case of principal T -bundles, this data can be given as follows: Let U be of B a cover overwhich E is trivialised. Then we can always refine U in such a way that for all U ∈ U the fibrebundle E′ is trivial over U ′ := f−1

B (U). The induced map fE : T × U ′ → T × U is then T -linearand thus can be reconstructed from the induced map

θ : U ′ = 1× U ′ ↪→ T × U ′ fE−−→ T × U p1−→ T.


Lemma A.8. Fix a morphism fB : B′ → B and fix notation as above. Then the data of a morphismf = (fE , fB) of principal T -bundles is equivalent to the data of morphisms θi : U ′i = f−1

B (Ui)→ Tfor some cover U of B by formal open subschemes Ui such that each θi induces a map fE,i suchthat for all i, j the following diagram commutes:

T × U ′ij T × U ′ij

T × Uij T × Uij .

fE,i

φ′ij

fE,j

φij

Moreover, commutativity of the above diagram is equivalent to commutativity of

U ′ij T × T

T × T T.

ψ′ij×θj

(ψij◦f)×θi m

m

Or in short hand notation,

ψ′ij(u)θj(u) = ψij(f(u)) · θi(u)

Proof. One direction is clear. For the other, the first part follows from glueing. The second part isa consequence of all maps in the first diagram being T -linear. �

Definition A.9. Let π : E → B be a principial T -bundle. Let F be a formal scheme with an actionby T . Since the data in the equivalent characterisation of Lemma A.5 is completely independentof the fibre, the morphisms ψij : Uij → T by Lemma A.5 define a fibre bundle with fibre F andstructure group T that we denote by F ×T E. This is called the associated bundle or Borel-Weilconstruction.

Note that in many authors in differential geometry and topology denote the associated bundleby ”F ×T E” instead of F ×T E. In our setting, however, this is slightly confusing since we oftenhave natural maps from T to F and E, but F ×T E is usually not their fibre product. In fact itbehaves more like a pushout, for instance in the case that E comes from a short exact sequence.

Proposition A.10. The associated bundle construction is a bifunctor

−×T − : FormActT × FormPrinBunT → FormFibBunT

from the categories of formal schemes with T -action × the category of principal T -bundles to thecategory of fibre bundles with structure group T .

Proof. Let E and E′ be principal T -bundles and let f : E′ → E be a morphism of T -bundles.Let F and F ′ be formal schemes with T -action and let h : F ′ → F be a T -equivariant morphism.Then we can find compatible covers U′ of B′ and U of B such that locally we have diagrams like inLemma A.8. Then locally F ×T E and F ′ ×T E′ are of the form F × Ui and F ′ × U ′i such that weobtain a natural map

F ′ × U ′ih×T f−−−−→ F × Ui, (x, u) 7→ (h(x)θi(u), fB(u))

(of course this description is just a short hand for a diagram of maps, and not a description in termsof ”points”). These maps glue together over the cover, since on intersections Lemma A.8 impliesthat we have a commutative diagram


F ′ × U ′ij F × Uij

F ′ × U ′ij F × Uij .

h×T f

ψ′ij×id

h×T f

ψij×id

One easily checks that this is functorial in both components. �

Lemma A.11. Let S be another formal group scheme that receives an action of T from a grouphomomorphism g : T → S. Then for any principal T -bundle E, the Borel construction S ×T E isa principal S-bundle.

Proof. This follows from Lemma A.5. The only thing we need to check is that the cocycle conditionfrom diagram (15) also holds with respect to S. But g is a homomorphism and therefore the followingdiagram commutes:

T × T S × S

T S.

m

g×g

m

g

�

Lemma A.12. The Borel construction is a functor S ×T − from principal T -bundles to principalS-bundles.

Proof. This is a consequence of Lemma A.8. One obtains the necessary data by composing themorphisms θ′ : U ′i → T with the morphism T → S. These morphisms glue together because thesecond diagram of Lemma A.8 commutes, as one easily sees from the fact that T → S is a morphismof formal groups. �

The Borel construction satisfies the following universal property:

Lemma A.13. Let g : T → S be a homomorphism of formal group schemes and let E → B bea principal T -bundle. Let X be any principal S-bundle. Note that X receives a T -action fromg. Then there is a functorial one-to-one correspondence between T -linear morphisms E → X andmorphisms of principal S-bundles S ×T E → X.

A.1. The semi-linear case. We later want to consider morphisms of fibre bundles that are inducedfrom morphisms of short exact sequences. In this situation, in order to describe the morphism ofthe kernels, we need to incorporate morphisms of the structure group into the notion of morphismsof fibre bundles. For this we need semi-linear group actions.

Definition A.14. Let T and S be formal group schemes and let g : T → S be a homomorphism.Let X and Y be formal schemes with actions m : T × X → X and m : S × Y → Y respectively.Then by a g-linear morphism f : X → Y we mean a morphism of formal schemes such that thefollowing diagram commutes

T ×X S × Y

X Y.

g×f

m m

f


Definition A.15. We denote by FormAct the category of pairs (T,X) where T is a formal groupscheme and X is a formal scheme with T -action, and morphisms are pairs of (g, f) where g is agroup homomorphism and f is a g-linear morphism. It has a natural forgetful functor to FormGrp,the category of formal group schemes.

Definition A.16. Let g : T ′ → T be a homomorphism of formal group schemes. Let π : E → Bbe a fibre bundle with fibre F and structure group T and let π′ : E′ → B′ be a fibre bundle withfibre F ′ and structure group T ′ . Then a g-linear morphism of principal bundles is a diagram

E′ B′

E B

fE

π′

fB

π

such that fE is g-linear. We denote by FormFibBun the category of fibre bundles (E,B, π, T, F )with arrows being the morphisms of principal bundles. It has a natural forgetful functor

(E,B, π, T, F ) 7→ T

to the category FormGrp. We denote by FormPrinBun the full subcategory of principal bundles.

We get the natural analogue of Lemma A.8:

Lemma A.17. With the notations from Lemma A.8, a g-linear morphism of a principal T ′-bundleto a principal T -bundle is equivalent to the data of morphisms θ : U ′i → T such that the followingdiagram commutes on intersections:

U ′ij T ′ × T T × T

T × T T

ψ′ij×θj

(ψij◦f)×θi

g×id

m

m

Or in short hand notation,

(16) g(ψ′ij(u)) · θj(u) = ψij(f(u)) · θi(u).

Similarly as in Proposition A.10 one can conclude from this that change of fibre is functorial inthe following sense:

Proposition A.18. Given any homomorphism of group schemes g : T ′ → T and a g-linear homo-morphism h : F ′ → F of formal schemes with T ′ and T -actions respectively, and a homomorphismf : E′ → E of principal T ′ and T -bundles over g, one obtains a morphism

h×g f : F ′ ×T′E′ → F ×T E

of fibre bundles over g, in a way that is functorial in h, g, f . More precisely, the associated bundleconstruction is a fibered bifunctor

−×− − : FormAct×FormGrp FormPrinBun→ FormFibBun.

Proof. Let (E,B,π,T ) and (E′,B′,π′,T ′) be principal bundles. Let F and F ′ be formal schemes withT -action and T ′-action respectively. Let g : T → T ′ be a group homomorphism and let h : F ′ → Fbe a g-equivariant morphism. Let f : E′ → E be a morphism of principal fibre bundles over g.Then we can find compatible covers U′ of B′ and U of B such that locally we have diagrams like in


Lemma A.8. Then locally F ×T E and F ′ ×T E′ are of the form F × Ui and F ′ × U ′i such that weobtain a natural map

F ′ × U ′i(h×gf)−−−−−→ F × Ui, (x, u) 7→ (h(x)θi(u), fB(u))

(as before this description is just a short hand for a diagram of maps, and not a description in termsof ”points”). These maps glue together over the cover, since on intersection Lemma A.8 impliesthat we have a commutative diagram

F ′ × U ′ij F × Uij

F ′ × U ′ij F × Uij .

h×gf

ψ′ij×id

h×gf

ψij×id

More precisely, by g-linearity of h one has

h(x · ψ′ij(u)) · θj(u) = h(x) · g(ψ′ij(u)) · θj(u)(16)= h(x) · ψij(f(u)) · θi(u).

This shows that the maps glue to a morphism h×g f as desired.By refining covers, one easily checks that this is functorial in both components. �

We obtain a variant of Lemma A.13 in the semilinear case:

Lemma A.19. Let E′ be a principal T ′ bundle and let E be a principal T -bundle. Let H ′ and Hbe formal group schemes and assume there is a commutative diagram of group homomorphisms

H ′ H

T ′ T.

h

g

Let moreover f : E′ → E be a g-linear morphism of fibre bundles. Then the map h ×g f fromProposition A.18 is the unique h-linear morphism of fibre bundles making the following diagramcommute:

H ′ ×T ′ E′ H ×T E

E′ E.

h×fg

f

Proof. The morphism exists by Proposition A.18. The vertical maps in the commutative diagramexist by functoriality via E = T ×T E → H ×T E.

On the other hand, on any compatible trivialisation T ′ × U ′ → T × U of f : E′ → E there isclearly only one way to extend this to H ′ × U ′ → H × U in a h-linear way. �

Remark A.20. All that we have done in this chapter can be done in completely the same waywith formal schemes replaced by rigid spaces (covers being replaced by admissible covers) and alsofor schemes, or in fact for any site. We have preferred to use formal schemes to make things moreexplicit. The different categories of fibre bundles are well-behaved with respect to the usual functorsbetween the different categories: For instance, by functoriality of fibre products there are naturalrigidification and reduction functors from formal principal T -bundles over OK to rigid principal Tη-

bundles over K on the generic fibre, and to principal T -bundles on the reduction OK/π. Moreover,these generic fibre and reduction functors commute with the associated fibre construction:


Lemma A.21. Let T be a formal group scheme and let π : E → B be a principal T -bundle. Let Fbe a formal scheme with an action by T . Then

(F ×T E)η = Fη ×Tη EηProof. This can be checked locally on any trivialising cover, where it is clear. �

References

[1] Bhargav Bhatt, The Hodge-Tate decomposition via perfectoid spaces, Lecture notes from the Arizona Winter

School 2017, available at swc.math.arizona.edu/aws/2017/2017BhattNotes.pdf

[2] Siegfried Bosch, Lectures on Formal and Rigid Geometry, Lecture Notes in Mathematics, vol 2015. Springer,

Berlin/Heidelberg/New York (2014).

[3] Siegfried Bosch, Werner Lutkebohmert, Degenerating abelian varieties, Topology 30 (1991), 653-698.[4] Jean Fresnel and Marius van der Put, Rigid Analytic Geometry and its Applications, Progress in Mathematics,

vol. 218. Birkhauser Boston, Inc., Boston (2004).

[5] David Hansen, Quotients of adic spaces by finite groups, Math. Res. Letters, to appear.[6] Werner Lutkebohmert, Rigid Geometry of Curves and Their Jacobians, Springer, Ergeb. 3. Folge, vol. 61.

Springer International Publishing (2016).[7] Vincent Pilloni and Benoit Stroh, Cohomologie coherente et representations Galoisiennes Annales

mathematiques du Quebec, vol. 40. (2016), 167-202, Springer

[8] Peter Scholze, Perfectoid spaces, Publ. Math. de l’IHES 116 (2012), no. 1, 245-313.[9] Peter Scholze, p-adic Hodge theory for rigid analytic varieties, Forum Math. Pi 1 (2013), el, 77.

[10] Peter Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), no.

3, 945-1066.[11] Peter Scholze, etale cohomology of diamonds, preprint arXiv:1709.07343 (2017)

[12] Peter Scholze, Perfectoid spaces and their applications, Proceedings of the ICM (2014)

[13] Peter Scholze and Jared Weinstein, Berkeley lectures on p-adic geometry, revised version, 2017, available athttp://www.math.uni-bonn.de/people/scholze/Berkeley.pdf

[14] Peter Scholze and Jared Weinstein, Moduli of p-divisible groups, Camb. J. Math. 1 (2013), no. 2, 145-237.

swc.math.arizona.edu/aws/2017/2017BhattNotes.pdf

http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf

Documents

Introductionpeople.math.harvard.edu/~zyao/paper/Perfd_AV.pdfBhargav Bhatt for proposing the project, for his guidance and for his constant encouragement, and we would like to thank