Notes on diamonds∗

David Hansen†

August 25, 2016

Diamonds are a beautiful class of geometric objects introduced by PeterScholze in his course at Berkeley in the Fall of 2014. There is a fantastic set ofnotes available from this course (“the Berkeley notes”). The present documentgrew out of my attempts to understand various particular claims in the Berkeleynotes, and various other questions about diamonds. These notes undoubtedlycontain many mistakes, and are not intended as a substitute for the Berkeleynotes.

I’m grateful to Johan de Jong, Christian Johansson, Peter Scholze, and JaredWeinstein for some very helpful conversations.

Contents1 Preliminaries 2

1.1 Sheaves on sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Perfectoid spaces and tilting . . . . . . . . . . . . . . . . . . . . . 31.3 The big pro-étale site . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 A remark on quotients by group actions . . . . . . . . . . . . . . 81.5 A remark on closed immersions . . . . . . . . . . . . . . . . . . . 8

2 Diamonds: definition and first properties 102.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Diamonds as pro-étale equivalence relations . . . . . . . . . . . . 112.3 The underlying topological space, and open subdiamonds . . . . 122.4 Pro-étale torsors . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Fiber products and direct products . . . . . . . . . . . . . . . . . 172.6 Some topological properties . . . . . . . . . . . . . . . . . . . . . 192.7 The miracle theorems . . . . . . . . . . . . . . . . . . . . . . . . 242.8 Some sites associated with a diamond . . . . . . . . . . . . . . . 28∗Version of 8/25/2016. Changes from the previous version: i. Dramatically improved some

of the results in §2.4, cf. Theorem 2.17 in particular; ii. Added (a preliminary version of) §7.

†Department of Mathematics, Columbia University, 2990 Broadway, New York NY 10027;

hansen@math.columbia.edu

1

3 Diamonds associated with adic spaces 283.1 The affinoid case . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Some basic compatibilities . . . . . . . . . . . . . . . . . . . . . . 35

4 Examples 374.1 The diamond of SpaQp . . . . . . . . . . . . . . . . . . . . . . . 374.2 Diamonds over a base . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Self-products of the diamond of SpaQp . . . . . . . . . . . . . . 394.4 The sheaf SpdZp . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 The diamond B

+

dR/Fil

n . . . . . . . . . . . . . . . . . . . . . . . 41

5 Moduli of shtukas 425.1 The Fargues-Fontaine curve . . . . . . . . . . . . . . . . . . . . . 445.2 The de Rham affine Grassmannian . . . . . . . . . . . . . . . . . 485.3 Moduli of shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.4 The Newton stratification . . . . . . . . . . . . . . . . . . . . . . 545.5 Admissible and inadmissible loci . . . . . . . . . . . . . . . . . . 55

6 More on morphisms 566.1 Smooth morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2 Morphisms locally of finite type . . . . . . . . . . . . . . . . . . . 57

7 Into the abyss 597.1 Absolute diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2 Stacks: first definitions . . . . . . . . . . . . . . . . . . . . . . . . 607.3 Diamond stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

1 PreliminariesWe assume basic familiarity with adic spaces and perfectoid spaces, including(but not limited to) Lectures 1-7 of the Berkeley notes. Unless explicitly statedotherwise, all adic spaces are analytic adic spaces over SpaZp; we denote thiscategory by Adic. We also follow the convention that perfectoid spaces do notlive over a fixed perfectoid base field.

We reserve the notation ∼= for canonical isomorphisms.

1.1 Sheaves on sitesLet C be a site. Recall that by definition, a presheaf F on C is a contravari-ant functor C → Sets. Given X ∈ C, we write hX = HomC(−, X) for thecontravariant Yoneda embedding of X.

A sheaf is a presheaf such that for every covering {Ui → U}i∈I in C, theequalizer sequence

0 → F(U) →�

i

F(Ui) ⇒�

i,j

F(Ui ×U Uj)

2

is exact. We write Sh(C) for the category of sheaves on C. The followingdefinition is standard.

Definition 1.1. A morphism F → G of sheaves is surjective if for any U ∈ Cand any section s ∈ G(U), there exists a covering {Ui → U}i∈I in C such thats|Ui ∈ G(Ui) lies in the image of F(Ui) → G(Ui) for each i ∈ I.

We also adopt the following conventions.

Definition 1.2. Let C be a site. A morphism F → G of (pre)sheaves on C isrepresentable if for every X ∈ C and every morphism of (pre)sheaves hX → G,there is an isomorphism s : hY

∼→ F ×G hX for some Y ∈ C.

As usual, the pair (Y, s) in the previous definition is unique up to uniqueisomorphism. Usually we suppress s and just write “...an isomorphism hY �F ×G hX ...”. Note that we follow the Stacks Project in saying “representable”instead of “relatively representable”.

Definition 1.3. Let C be a site, and let “blah” be a property of morphisms in Cwhich is preserved under arbitrary pullback. A morphism F → G of (pre)sheaveson C is “blah” if it is representable and, for every X ∈ C and every morphismof (pre)sheaves hX → G, the morphism Y → X corresponding to the upperhorizontal arrow in the pullback diagram

F ×G hX � hY

��

�� hX

��F �� G

is “blah”.

Proposition 1.4. If F → G is “blah”, then for any H → G the pullback F ×GH→ H is “blah”.

Proof. Given hX → H, we have (F×GH)×HhX = F×GhX , so this is immediate.

Definition 1.5. If F is a sheaf with an action of a group G, we write F/G forthe sheafification of the presheaf sending U ∈ C to F(U)/G.

1.2 Perfectoid spaces and tilting

Let �Perf denote the category of all perfectoid spaces. Let Perf denote thecategory of perfectoid spaces in characteristic p, so tilting defines a functor

�Perf → Perf

X �→ X�.

As Jared Weinstein drolly remarks, tilting is an “idempotent functor”: (X�)�=

X�. We remind the reader that tilting has essentially every compatibility one

3

could dream of: there is a natural homeomorphism |X| ∼= |X�| compatible withopen affinoid perfectoid subspaces and with rational subsets thereof, there arenatural equivalences Xan

∼= X�

anand Xet

∼= X�

etwith associated equivalences of

topoi, etc. However, tilting is forgetful: to recover X from X�, one needs to

specify extra data.

Proposition 1.6. If X is a perfectoid space, then the assignment U �→ W (O+

X(U)

�)

on open affinoid perfectoid subsets U ⊂ X defines a sheaf of rings A+

Xon X

which depends only on the tilt of X. This sheaf of rings comes with a naturalsurjective ring map θX : A+

X→ O+

X.

Definition 1.7 (Fargues-Fontaine, Scholze, Kedlaya-Liu). Let X be a perfec-toid space. An ideal sheaf J ⊂ A+

Xis primitive of degree one if locally on a

covering X = ∪iUi by affinoid perfectoids, J |Ui = (ξi) is principal and gener-ated by an element of the form ξi ∈ W (O+

X(Ui)

�) of the form ξi = p+[�i]αi with

�i ∈ O+

X(Ui)

� some pseudouniformizer and with αi ∈ W (O+

X(Ui)

�) arbitrary.

Let Perf+ denote the category of pairs (X,J ) where X ∈ Perf and J ⊂ A+

X

is an ideal sheaf which is primitive of degree one, with morphisms (X,J ) →(Y, I) given by morphisms f : X → Y such that f

−1I · A+

X= J . Note that

Perf+ is a category fibered in sets over Perf. For any X ∈ �Perf, the ideal sheaf

ker θX ⊂ A+

X= A+

X� is primitive of degree one, so we get a functor

�Perf → Perf+

X �→�X

�, ker θX ⊂ A+

X�

�.

Proposition 1.8 (Scholze, Kedlaya-Liu). This functor defines a natural equiv-alence �Perf ∼= Perf

+. For any perfectoid space S, tilting defines a natural equiv-alence �Perf/S

∼= Perf/S� .

Explicitly, the “untilt” X� associated with a pair (X,J ) ∈ Perf

+ has integralstructure sheaf O+

X� = A+

X/J .

A word on notation: The notation “Perf” follows the Berkeley notes.We are introducing the notation “ �Perf” here on the grounds that tilting allowsone to regard arbitrary perfectoid spaces as liftings/deformations of perfectoidspaces in characteristic p; the tradition of denoting “lifts” or “deformations” ofsome mathematical object via a superscripted tilde seems to be well-established,hence our choice of notation.

1.3 The big pro-étale siteIn this section we recall some material from the Berkeley notes, and prove somebasic properties of the big pro-étale site.

Definition 1.9.

4

1. A morphism Spa(B,B+) → Spa(A, A

+) of affinoid perfectoid spaces is

affinoid pro-étale if (B,B+) admits a presentation as the completion of a

filtered direct limit(B,B

+) =

�limi→

(Ai, A+

i)

of perfectoid (A, A+)-algebras, with each Spa(Ai, A

+

i) → Spa(A, A

+) étale.

2. A morphism f : Y → X of perfectoid spaces is pro-étale if for every pointy ∈ Y , there is an open affinoid perfectoid subset V ⊂ Y containing y andan open affinoid perfectoid subset U ⊂ X containing f(V ) such that theinduced morphism V → U is affinoid pro-étale.

Proposition 1.10. Suppose f : Y → X is a pro-étale morphism of perfectoidspaces.

i. If Z → X is any morphism of perfectoid spaces, then Z ×X Y → Z is apro-étale morphism.

ii. If g : Z → Y is a morphism of perfectoid spaces such that f◦g is pro-étale,then g is pro-étale.

Next we define a notion of pro-étale cover.

Definition 1.11. A pro-étale morphism of perfectoid spaces f : Y → X is apro-étale cover(ing) if for every qc open subset U ⊂ X, there is some qc opensubset V ⊂ Y with f(V ) = U .

We record the following for later use (the proof follows as in Proposition15.5.1 of the Berkeley notes).

Proposition 1.12. If f : Y → X is a pro-étale covering, then |f | : |Y | → |X|is a quotient map.

In order to show that perfectoid spaces with pro-étale covers form a site, wealso need to verify the following proposition.

Proposition 1.13. Suppose f : Y → X is a pro-étale covering of perfectoidspaces. If g : Z → X is any morphism of perfectoid spaces, then Z ×X Y → Z

is a pro-étale covering.

Proof. Let W ⊂ Z be a quasicompact open subset, so W ⊆ g−1

(g(W )). Choosea quasicompact open U ⊂ X with g(W ) ⊆ U , and then choose a quasicompactopen V ⊂ Y with f(V ) = U . Then W ×U V ⊂ Z ×X Y is a quasicompact openwith image W in Z.

Proposition 1.14. Suppose

Z

g

�����

����

f◦g �� X

Y

f

���������

is a diagram of pro-étale morphisms of perfectoid spaces, and suppose f ◦ g is apro-étale cover. Then f is a pro-étale cover.

5

Proof. Let U ⊂ X be any qc open; we need to find a qc open V ⊂ Y with f(V ) =

U . By assumption, we may choose some qc open W ⊂ Z with (f ◦ g)(W ) = U .Choose a covering ∪i∈IVi of f

−1(U) by qc opens in Y , so ∪i∈Ig

−1(Vi) ∩W is

an open covering of W . Since W is qc, we can find a finite subset I� ⊂ I such

that ∪i∈I�g−1

(Vi) ∩W is an open covering of W . Then

g(W ) = ∪i∈I�g�g−1

(Vi) ∩W�⊆ ∪i∈I�Vi,

so V := ∪i∈I�Vi is a qc open subset of Y such that g(W ) ⊆ V ⊆ f−1

(U), andtherefore f(V ) = U as desired.

Definition 1.15. The big pro-étale site is the site Perfproet whose underly-

ing category is the category Perf of perfectoid spaces in characteristic p, withcoverings given by pro-étale coverings.

There is also a small pro-étale site Xproet for any perfectoid space X, withobjects given by perfectoid spaces pro-étale over X and covers given by pro-étalecovers.

We now turn to sheaves on Perfproet.

Proposition 1.16. For any X ∈ Perf, the presheaf hX = Hom(−, X) is a sheafon the big pro-étale site.

Proof sketch. One reduces to the case of X affinoid. Recall that for any affinoidadic space X = Spa(R,R

+) and any adic space Y , there’s a natural identification

Hom(Y, X) = Hom�(R,R

+), (O(Y ),O(Y )

+)�.

Let Y ∈ Perf be some perfectoid space with a given pro-étale cover Y → Y , sowe need to show exactness of the sequence

0 → Hom(Y,X) → Hom(Y , X) ⇒ Hom(Y ×Y Y , X).

But OY and O+

Yare sheaves on Yproet, so we get an exact sequence

0 → (O(Y ),O(Y )+) → (O(Y ),O(Y )

+) ⇒ (O(Y ×Y Y ),O(Y ×Y Y )

+),

and we’re done upon applying the left-exact functor Hom ((R,R+),−).

By our conventions, given any property of morphisms of perfectoid spacespreserved under arbitrary pullback, there is a corresponding notion for mor-phisms of sheaves on Perf

proet. In particular, we may speak of a morphism ofsheaves F → G on Perf

proet being an open immersion, Zariski closed immer-sion, finite étale, étale, pro-étale, a pro-étale cover, etc. (Note, however, that“surjective” will always mean “surjective as a morphism of sheaves.”)

Proposition 1.17. A morphism F → G of sheaves on Perfproet is a pro-étale

cover if and only if it is surjective and pro-étale.

6

Proof. Let F → G be a pro-étale morphism. By definition, the sheaf morphismF → G is surjective if, for any hX → G with associated hY � F ×G hX , we canfind some pro-étale cover X → X and a section h

X→ F such that the diagram

F �� G

hY��

��

hX

��

hX

�����������������

�����������������

commutes. If F → G is a pro-étale cover, then X = Y does the job, so F → Gis surjective.

Suppose conversely that F → G is a surjective pro-étale morphism, andconsider a map hX → G, so we get a diagram

F �� G

hY��

��

hX

��

hX

�����������������

�����������������

with Y as above. We need to show that the associated morphism Y → X is apro-étale cover. Since the square is cartesian, we can fill in another arrow fromh

Xto hX , vis.

F �� G

hY��

��

hX

��

hX

�����������������

�����������������

����������

Since Y → X and X → X are both pro-étale, the morphism X → Y is pro-étaleby Proposition 1.10.ii, so the diagram

X

�����

����

�� X

Y

����������

satisfies the hypotheses of Proposition 1.14 and thus Y → X is a pro-étalecover.

7

1.4 A remark on quotients by group actionsLet X be an adic space with a (right) action of a group G. There is a naturalcandidate for a quotient space X/G, given by the following construction. LetX =

�|X|,OX , {| · |x}x∈|X|

�be the data underlying X. Then we define an

object X/G =�|X/G|,OX/G, {| · |y}y∈|X/G|

�in Huber’s ambient category V as

follows:

• |X/G| := |X|/G with the quotient topology. Let q : |X| → |X/G| be theevident map.

• OX/G := (q∗OX)G. This is clearly a sheaf (since (−)

G is left-exact) ofcomplete topological rings.

• | · |y is the valuation on the stalk OX/G,y induced by OX/G,y → OX,x

|·|x→Γ|·|x ∪ {0} where x ∈ q

−1(y) is any preimage of y.

By construction X/G defines an object in V with a canonical G-equivariantmorphism q : X → X/G in V (for the trivial action on X/G), and it’s easy tocheck that X/G is categorical in the category V : if Y ∈ V is arbitrary and G

acts on HomV (X,Y ) by precomposition, then

HomV (X,Y )G

= HomV (X/G, Y ).

When X/G is an adic space, we call it the categorical quotient of X by G. Ofcourse, if X lives in some full subcategory C ⊂ Adic (e.g. perfectoid spaces,or rigid analytic spaces over a fixed nonarchimedean field K) we might requirethat X/G also lie in C, in which case the quotient X/G is also categorical in C.

1.5 A remark on closed immersionsRecall that two valuations v, w on a ring A are equivalent iff v(a) ≤ v(b) ⇔w(a) ≤ w(b) for all (a, b) ∈ A

2.

Lemma 1.18. Let A be a Tate ring, and let S ⊂ A be any dense subset. Then S

separates distinct points in Cont(A), in the sense that if v and w are inequivalentcontinuous valuations on A, we can find elements a, b ∈ S such that v(a) ≤ v(b)

and w(a) > w(b).

Proof. Assume v and w are inequivalent, and choose a, b ∈ A such that v(a) ≤v(b) and w(a) > w(b). Let � be a pseudouniformizer of A. Clearly w(a) > 0; ifv(b) = 0 then we can replace b by b + �

N for some large N while maintainingthe inequality w(a) > w(b), and then v(b + �

N) = v(�)

N �= 0, so without lossof generality we can assume that v(b) > 0 and w(a) > 0.

Now, by the density of S in A we can choose a� ∈ S such that

v(a− a�) ≤ v(b) andw(a− a

�) < w(a)

and b� ∈ S such that

v(b− b�) < v(b) andw(b− b

�) < w(a).

8

But thenv(a

�) ≤ max(v(a− a

�), v(a)) ≤ v(b) = v(b

�)

andw(a

�) = w(a) > max(w(b− b

�), w(b)) ≥ w(b

�),

so we’re done after replacing a and b by a� and b

�.

Lemma 1.19. Let f : R → S be a morphism of Tate rings with dense image.Then Cont(S) → Cont(R) is injective.

Proof. Immediate from the previous lemma.

Definition 1.20. A morphism Y → X of adic spaces is a closed immersion ifit is a closed immersion of locally ringed spaces (cf. Tag 01HK in the StacksProject).

We’d like to introduce the following more general notion; the goal here is tosimultaneously capture the usual notion of a closed immersion, and the notionof a “Zariski-closed embedding” of perfectoid spaces from Peter’s torsion paper.

Definition 1.21. A morphism Z = Spa(S, S+) → X = Spa(R,R

+) of affinoid

adic spaces is a weak closed immersion if the map f : R → S has dense imageand the injective map |Z|→ |X| identifies |Z| homeomorphically with

V (ker f) := {x ∈ |X| | |r|x = 0∀r ∈ ker f} .

A morphism of general adic spaces Z → X is a weak closed immersion ifX admits a covering by open affinoids Xi such that f

−1(Xi) is affinoid and

f−1

(Xi) → Xi is a weak closed immersion for all i.

Of course a closed immersion is a weak closed immersion. Note that ingeneral, the notion of a global weak closed immersion may not be well-behaved;for example, if f : Z → X is a weak closed immersion and U ⊂ X is an opensubset, is f

−1(U) → U a weak closed immersion? This is already unclear (to

me) when Z and X are affinoids.The next lemma shows, however, that when Z → X is a weak closed im-

mersion of affinoids with Z uniform, things are better behaved. Note that ifSpa(B,B

+) → Spa(A, A

+) is any map of affinoid adic spaces such that A → B

has dense image, then the map

Hom(T, Spa(B,B+)) → Hom(T, Spa(A, A

+))

is injective for any adic space T . This implies the uniqueness statements in thefollowing lemma.

Lemma 1.22. Let Z = Spa(S, S+) → X = Spa(R,R

+) be a weak closed

immersion such that S is uniform. Then for any map

g : W = Spa(A, A+) → X

such that A is uniform, g factors (uniquely) through a map h : W → Z if andonly if |g|(|W |) ⊆ |Z|. In particular, Z is determined uniquely up to uniqueisomorphism by the closed subset |Z| ⊂ |X|.

9

Proof. Let S be the filtered set of rational subsets U ⊂ X with |Z| ⊂ |U |,so |Z| = ∩U∈S |U |. By the universal property of rational subsets, the mapsZ → X and W → X factor uniquely through maps Z → U and W → U forany U ∈ S. Writing U = Spa(RU , R

+

U), these correspond to continuous maps

(RU , R+

U) → (S, S

+) and (RU , R

+

U) → (A, A

+). Taking the direct limit gives a

maplim→S

(RU , R+

U) → (S, S

+)

with dense image and uniform target, which moreover identifies (S, S+) isomor-

phically with the uniform completion of lim→S (RU , R

+

U). But then since (A, A

+)

is uniform by assumption, the map

lim→S

(RU , R+

U) → (A, A

+)

extends uniquely to a map (S, S+) → (A, A

+), as desired.

Corollary 1.23. Let Z = Spa(S, S+) → X = Spa(R,R

+) be a weak closed

immersion such that S is either strongly Noetherian or stably uniform. Thenfor any rational subset U ⊂ X, the morphism Z ×X U → U is a weak closedimmersion with strongly Noetherian or stably uniform source mapping |Z×X U |homeomorphically onto |U | ∩ |Z|.

In particular, weak closed immersions Z → X of affinoid adic spaces with Z

strongly Noetherian or stably uniform can be glued without any issues.

2 Diamonds: definition and first propertiesUntil further notice, all sheaves are sheaves of sets on Perf

proet.

2.1 The definitionWe now come to the key definition.

Definition 2.1. A diamond is a sheaf D on Perfproet which admits a surjective

and pro-étale morphism hX → D from a representable sheaf. A morphism ofdiamonds is a morphism of sheaves on Perf

proet. We write Dia for the categoryof diamonds.

If D is a diamond, we refer to any choice of a surjective pro-étale morphismhX → D as a presentation of D.

Proposition 2.2. Let F → G be a pro-étale morphism of sheaves on Perfproet.

If G is a diamond, then so is F . If F → G is surjective and pro-étale, then Fis a diamond if and only if G is a diamond.

Proof. Easy and left to the reader.

Here is a first sanity check.

10

Proposition 2.3. The functor

Perf → Dia

X �→ hX

is fully faithful.

Proof. This is an easy consequence of Proposition 1.16.

Given X ∈ Perf, we also denote hX interchangeably by X♦. Later we will

associate a diamond X♦ with any analytic adic space X over SpaZp.

Definition 2.4. If D is a diamond, a subdiamond of D is a subfunctor E ⊂ Dwhich is naturally isomorphic to a diamond.

Note that if E is a subdiamond of a diamond D, the natural monomorphismE → D need not be a representable morphism.

2.2 Diamonds as pro-étale equivalence relationsLet D be a diamond with a given presentation hX → D. We see by representabil-ity that the fiber product hX×D hX is representable, say by hZ , and we get twomorphisms s, t : Z ⇒ X corresponding to the two projections of hX×DhX � hZ

to hX . Since hX → D is pro-étale, s and t are both pro-étale. Note that hZ

is naturally a subfunctor of hX × hX , and that hZ(T ) ⊂ hX(T )× hX(T ) givesan equivalence relation on hX(T ) for any T ∈ Perf. In short, s, t : Z ⇒ X is apro-étale equivalence relation on X.

Proposition 2.5. The diagram

hZ ⇒ hX → D

is a coequalizer diagram in Sh(Perfproet

).

Proof. This is immediate from the sheaf-theoretic surjectivity of hX → D, cf.Lemma 7.12.3 (Tag 00WL) in the Stacks Project.

In analogy with the situation for algebraic spaces, it’s natural to ask if theconverse of this proposition is true: given a pro-étale equivalence relation inperfectoid spaces s, t : Z ⇒ X, is D = Coeq (hZ ⇒ hX) a diamond? Theanswer is unclear, at least to this author.1 The key difficulty seems to beproving that hX → D is representable (and pro-étale): trying to naively imitatethe arguments in Section 52.10 of the Stacks Project, one quickly runs into somedescent-theoretic questions whose answers seem negative in general.

1Scholze has informed us (7/14/2016) that the answer to this question is “no”, and that the

“official” definition of a diamond in the final version of the Berkeley notes will be a sheaf on

Perfproet

isomorphic to Coeq (hZ ⇒ hX) for some pro-étale equivalence relation s, t : Z ⇒ X

in perfectoid spaces. By Proposition 2.5, this new definition is strictly more general than the

old definition. Due to lack of energy on my part, and the fact that I don’t know any examples

of diamonds for this new definition which don’t fall under the framework of the old definition,

I’m not planning (at present) to rewrite these notes to take the “new” definition into account.

Sorry. —DH

11

2.3 The underlying topological space, and open subdia-monds

Given a presentation of a diamond hX → D, let hX ×D hX � hZ as above, andlet Z ⇒ X be the associated pro-étale equivalence relation. It’s not hard to showthat the induced map |Z| → |X| × |X| is injective and defines an equivalencerelation ∼|Z| on the topological space |X|. Let |D| = |X|/ ∼|Z| be the set ofequivalence classes for this relation, with the quotient topology relative to themap |X| � |D|.

Proposition 2.6. The topological space |D| is well-defined independently ofall choices, and the association D �→ |D| defines a functor from diamonds totopological spaces.

Proof. A refinement of a given presentation hX → D is a commutative diagram

hX�

������

����

�� D

hX

����������

with all morphisms surjective and pro-étale. Note in particular that X� → X

is a pro-étale cover, so |X �| → |X| is a quotient map. Given such a diagram,define Z

� by hX� ×D hX� � hZ� analogously with Z. To prove the proposition,it suffices to construct a natural homeomorphism r : |X|/ ∼|Z|∼= |X �|/ ∼|Z�| forany such diagram; indeed, any two presentations hX1

→ D and hX2→ D admit

a common refinement, since one can simply choose hX� with hX� � hX1×D hX2

.To construct r, note that since

hZ� = hX� ×D hX� ,

= hX ×D hX ×hX×hX (hX� × hX�)

= hZ ×hX×hX (hX� × hX�),

we get a natural surjective2 map

f : |Z �|→ |Z|×|X|×|X| (|X �|× |X �|).

Since the composition of this map with the inclusion

|Z|×|X|×|X| (|X �|× |X �|) ⊂ |X �|× |X �|

recovers the natural inclusion |Z �| ⊂ |X �|× |X �|, we deduce that f is bijective.In particular, we find that the equivalence relation ∼|Z�| on |X �| is exactly thepreimage of ∼|Z| under |X �| � |X|, so |X �|/ ∼|Z�|∼= |X|/ ∼|Z| as sets. To

2e.g. by playing with (K, K

+)-valued points for affinoid fields (K, K

+)

12

upgrade this to a homeomorphism, note that we have a natural commutativediagram of sets

|X �|

π�

��

q �� |X|

π

��|X �|/ ∼|Z�| r

�� |X|/ ∼|Z|

where r is the set-theoretic bijection we’ve just constructed. By definition,the targets of π and π

� have the quotient topologies induced by the naturaltopologies on their sources. Given U ⊂ |X|/ ∼|Z| open, then

π�−1

(r−1

(U)) = q−1

(π−1

(U))

is open in |X �|, so r−1

(U) is open and therefore r is continuous. On the otherhand, given V ⊂ |X �|/ ∼|Z�| open, then

π�−1

(V ) = q−1

(π−1

(r(V )))

is open in |X �|. Since q and π are quotient maps, we deduce that π−1

(r(V ))

and r(V ) are open, so r−1 is continuous as desired.

Functoriality of |D| is left to the reader.

Definition 2.7. Given a diamondD, an open subdiamond E ⊂ D is a subfunctorof D such that for some presentation hX → D, E ×D hX � hY is representableand the associated map Y → X is an open immersion.

This definition seems a priori more general than simply requiring that E →D be an open immersion, but it turns out not to be:

Proposition 2.8. Open subdiamonds E ⊂ D coincide with open immersionsE → D.

Proof. Left to the reader; in any case, the next proposition supersedes thisclaim.

It turns out that open subdiamonds are “purely topological” relative to theirambient diamond:

Proposition 2.9. Given a diamond D, the association E �→ |E| defines aninclusion-preserving bijection from open subdiamonds of D to open subsets of|D|.Proof. We sketch the construction of the inverse association. Given an opensubset U ⊂ |D|, we define EU ⊂ D as the subfunctor characterized by theproperty that hX → D factors through EU → D iff the associated map |X|→ |D|has image contained in U . One easily checks that EU → D is an open immersion:given any m : hX → D with associated |m| : |X| → |D|, the functor EU ×D hX

is represented by hY , where Y ⊂ X is the open immersion of perfectoid spacesdefined by setting

|Y | = |m|−1(U) ⊆ |X|.

13

2.4 Pro-étale torsorsThis section is partly based on Sections 4.2-4.3 of [Wei15].

Definition 2.10. Let G be a locally profinite group. A morphism f : Y → X

of perfectoid spaces is a pro-étale G-torsor if there is a G-action on Y lying overthe trivial action on X such that Y ×X Y ∼= G × Y and such that f admits asection pro-étale-locally on X.

Likewise, a morphism F → G of sheaves on Perfproet is a pro-étale G-torsor if

there is a G-action on F lying over the trivial G-action on G such that, pro-étalelocally on G, we have F � G ×G.

Remark. If F → G is a pro-étale G-torsor, the induced map F/G → G isan isomorphism (since it becomes an isomorphism pro-étale-locally on G, andeverything is a pro-étale sheaf).

Proposition 2.11. A map of sheaves F → G is a pro-étale G-torsor if andonly if there exists a surjective pro-etale map G� → G together with a sections : G� → F such that the natural map

G� ×G → G� ×G F(x, g) �→ (x, s(x) · g)

is an isomorphism.

Proof. This is just a rewording of the definition.

Proposition 2.12. If F → G is any pro-étale G-torsor, then G acts freely ongeometric points of F .

Proof. Given a geometric point x ∈ G(C,OC) lying under a given G-orbit S of(C,OC)-points in F , choose a cover G� → G and a section s : G� → F as in theprevious proposition. We may lift x to a (C,OC) point x of G�; but then themap G� × G → G� ×G F postcomposed with projection to F identifies x × G

bijectively with s(x) ·G = S ⊂ F(C,OC).

Proposition 2.13. If F → G is any pro-étale G-torsor and K ⊂ G is any opensubgroup of G, then the induced map F/K → G is (representable and) étale.

Proof. Choose G� and s as in Proposition 2.11, so we have an isomorphism

G� ×G∼→ G� ×G F

(x, g) �→ (x, s(x) · g)

as before. Applying (−)/K to this gives an isomorphism

G� ×G/K∼→ G� ×G F/K.

14

In particular, we get a pullback diagram

G� ×G/K

��

�� G�

��F/K �� G

where the vertical arrows are surjective and pro-étale. But G/K is a discreteset, so the upper horizontal arrow is separated and étale. Now we’re done afteran application of the following key lemma.

Lemma 2.14. Let H → G be a morphism of sheaves, and suppose there existsa surjective pro-etale sheaf map G� → G such that

H×G G� → G�

is (representable and) separated and étale. Then H→ G is separated and etale.

Proof. This reduces formally to the following statement: separated étale mapsform a stack for the pro-étale topology on any perfectoid space X. More pre-cisely, the fibered category p : Et → Xproet defined by

p−1

(U) = {V → U separated etale}

for any U ∈ Xproet is a stack. This is true because Peter told me it is. (Butseriously, I’ll try to figure out a proof and add it here.)

Proposition 2.15 (Weinstein). If F → G is a pro-étale G-torsor with G profi-nite, then F → G is surjective and pro-finite étale. In particular, G is a diamondif and only if F is a diamond.

This is slightly tricky, since we don’t know a priori that F → G is repre-sentable.

Proof. Surjectivity is clear. For the pro-étale claim, write G = lim←i G/Hi withHi open normal in G. Then F = lim←i F/Hi as sheaves, so it suffices to showthat each F/Hi → G is finite étale. This is true on a pro-étale cover of G, bydefinition, so we’re reduced to showing that whether or not a morphism F → Gis finite étale can be checked pro-étale-locally on G. We may clearly assume Gis representable.

So let F → hX be any sheaf map, and let hX→ hX be a surjective pro-étale

map such that F ×hX hX

is representable, say by hY

, and such that hY→ h

X

is finite étale. Since

Y ×X,pri

(X ×X X) ∼= F ×X X ×X,pri

(X ×X X)

∼= F ×X (X ×X X),

where pri : X×X X → X denotes either of the two projections, the two pullbacksof Y under pr

iare canonically isomorphic, and one easily checks that these

15

isomorphisms satisfy the usual cocycle condition. In particular, the finite étalemap Y → X inherits a descent datum relative to the pro-étale cover X → X,so (again by Lemma 4.2.4 of [Wei15]) Y descends uniquely to a finite étale mapY → X, and one easily checks that F ∼= hY .

The final claim follows immediately from Proposition 2.2.

Theorem 2.16 (Weinstein). Let G be a profinite group, and let F be a sheafon Perf

proet equipped with an action of G. Suppose G acts freely on the geo-metric points F(C,OC) for any algebraically closed nonarchimedean field C incharacteristic p. Then F → F/G is a pro-étale G-torsor, and is surjective andpro-étale.

Here F(C,OC) is of course shorthand for the set of sheaf maps hSpa(C,OC) →F .

Proof. This is exactly Proposition 4.3.2 of [Wei15]. Note that Proposition 4.3.2of loc. cit. is only stated in the case where F is a representable sheaf, butWeinstein’s proof doesn’t use this assumption in any way.

We’ve now developed enough material to prove the following basic structuralstatement.

Theorem 2.17. Let F → G be a pro-étale G-torsor for G an arbitrary locallyprofinite group. Then F → G is (representable and) pro-étale. In particular Fis a diamond if and only if G is.

Proof. Choose an open profinite subgroup K ⊂ G. By Proposition 2.13, the mapF/K → G is surjective and étale. On the other hand, G acts freely on geometricpoints of F , so a fortiori K does as well, and then F → F/K is surjective andpro-finite étale by Theorem 2.15. But now we’re done after factoring our originalmap as the composite

F → F/K → G,

since both morphisms here are representable and pro-étale.

Considering the circuitious route we took to get this theorem, maybe Ishouldn’t feel bad that it took me a year to prove.

The special case of Theorem 2.16 where F is assumed representable givesour first real mechanism for constructing diamonds. Due to its importance, westate it separately:

Theorem 2.18. Let X be a perfectoid space in characteristic p, and let G

be a profinite group acting on X. Suppose G acts freely on the set X(C,OC)

for any algebraically closed nonarchimedean field C in characteristic p. ThenhX → hX/G is a pro-étale G-torsor, hX/G is a diamond and hX → hX/G is apresentation. Furthermore, there is a natural homeomorphism |hX/G| ∼= |X|/G.

Proof. By Theorem 2.16, hX → hX/G is a pro-étale G-torsor, so the sheaf maphX → hX/G is surjective and pro-étale by Proposition 2.15. The statement ontopological spaces is clear.

16

2.5 Fiber products and direct productsTheorem 2.19. Given any morphisms of diamonds E → D, F → D, the fiberproduct of sheaves E ×D F is a diamond.

Proof. Choose presentations hX → D, hY → E , hZ → F . One easily checksthat each of the morphisms

(hY ×D hZ)×D hX → hY ×D hZ → hY ×D F → E ×D F

is surjective and pro-étale (e.g., hY ×D F → E ×D F is the pullback of thesurjective pro-étale map hY → E along E×DF → E), so the composite morphismis surjective and pro-étale. On the other hand, we claim that (hY ×D hZ)×D hX

is a representable sheaf. To see this, write

(hY ×D hZ)×D hX∼= (hY ×D hX)×hX (hZ ×D hX).

Since hX → D is representable, we deduce that hY ×D hX and hZ ×D hX areboth representable, say with hY ×D hX = hU and hZ ×D hX = hV . Therefore

(hY ×D hZ)×D hX∼= hU ×hX hV

is the fiber product of sheaves associated with representable objects. SinceU ×X V exists and hU ×hX hV

∼= hU×XV , we’re done.

Next we show that Dia admits direct products.

Theorem 2.20. For any two diamonds D, E, the product D × E is canonicallya diamond, where the product is taken in the category of sheaves of sets onPerf

proet.

Proof. Choose presentations hX → D, hY → E . One easily checks that themorphism

hX × hY → hX × E → D × Eis a composition of surjective pro-étale morphisms, so itself is surjective pro-étale. It then suffices to show that hX×hY is representable. This is the contentof the following proposition.

Proposition. Let X, Y be two perfectoid spaces in characteristic p. ThenhX × hY

∼= hX×Y for a certain canonical perfectoid space “X × Y ” whose for-mation is functorial in X and Y .

Proof. It suffices to prove the result when X and Y are affinoid perfectoid;the general case then follows by gluing. So let (A, A

+), (B,B

+) be two per-

fectoid Tate-Huber pairs in characteristic p, with associated affinoid perfectoidspaces X = Spa(A, A

+), Y = Spa(B,B

+). We begin by defining X × Y . Set

C = A⊗Fp B, and let C+ be the integral closure of A

+ ⊗Fp B+ in C. Choose

pseudouniformizers �A ∈ A, �B ∈ B, and set I = (�A, �B) ⊂ C+ (here we

abbreviate �A ⊗ 1 ∈ C by �A, and likewise for �B ; we continue to use this

17

abbreviation in what follows). Let D+ be the I-adic completion of C

+. Thenwe define

X × Y = Spa(D+, D

+) � {x | |�A�B |x = 0} .

To see that X × Y is perfectoid, consider the open subsets

Un = {x | |�A|nx ≤ |�B |x �= 0, |�B |nx ≤ |�A|x �= 0}

of Spa(D+, D

+). It’s easy to see that Un ⊂ Un+1 and X × Y =

�n≥1

Un.

Furthermore, Un = U

�{�n+1

A ,�n+1

B }�A�B

�is a rational subset of Spa(D

+, D

+), so we

may describe it explicitly as an affinoid adic space. Precisely, let Cn = A⊗Fp B

and let C+

nbe the integral closure of (A

+⊗Fp B+)[

�nA

�B,

�nB

�A] in Cn; give C

+

nthe

I · C+

n-adic topology, and give Cn the topology making C

+

nan open subring.

(Note that the I ·C+

n-adic, �A ·C+

n-adic, and �B ·C+

n-adic topologies on C

+

nall

coincide.) Let Dn and D+

nbe the completions of Cn and C

+

nfor these topologies.

Then �O(Un),O(Un)

+�

= (Dn, D+

n).

One checks directly that Dn is a complete perfect uniform Tate ring, whichexactly characterizes the perfectoid rings in characteristic p. Therefore each Un

is perfectoid, so X × Y is perfectoid.Next we verify that X × Y has the claimed property at the level of sheaves.

Note that we have natural continuous ring maps A+ → D

+, B

+ → D+ inducing

morphisms Spa(D+, D

+) → Spa(A

+, A

+) and Spa(D

+, D

+) → Spa(B

+, B

+),

and the latter morphisms restricted to X×Y factor through natural morphismspr

X: X × Y → X and pr

Y: X × Y → Y , respectively.

Choose any Z ∈ Perf, and suppose we’re given a morphism Z → X × Y ,i.e. an element of hX×Y (Z). Composing this morphism with the morphismspr

X,pr

Y, we get morphisms Z → X, Z → Y , i.e. an element of hX(Z)×hY (Z).

This association is clearly functorial in Z, and so defines a map of sheaves

F : hX×Y → hX × hY .

To go the other way, assume that Z is affinoid perfectoid, say Z = Spa(R,R+)

for (R,R+) some perfectoid Tate-Huber pair in characteristic p. Suppose we’re

given an element of hX(Z)× hY (Z). This is equivalent to the data of a pair ofcontinuous ring maps f : A → R, g : B → R such that f(A

+), g(B

+) ⊂ R

+.Consider the evident ring map f ⊗ g : C → R. One checks directly that(f ⊗ g)(C

+) ⊂ R

+, using the obvious inclusion

(f ⊗ g)(A+ ⊗Fp B

+) ⊂ R

+

together with the fact that R+ is integrally closed in R. Choose a pseudouni-

formizer �R ∈ R; replacing �R by �1/p

j

Rif necessary, we may assume that �R

divides both f(�A) and g(�B) in R+. This immediately implies that the ring

map f ⊗ g : C+ → R

+ induces compatible ring maps C+/I

n → R+/�

n

R, so

18

passing to the inverse limit we get a continuous ring map f⊗g : D+ → R

+.Passing to Spa’s, we may consider the composite

i : Spa(R,R+) → Spa(R

+, R

+)

Spa(f⊗g)−→ Spa(D+, D

+)

where the lefthand arrow is the evident open immersion. Since f⊗g carries�A�B to a unit in R, i factors through the open subset X×Y ⊂ Spa(D

+, D

+),

so we get a map i : Spa(R,R+) → X × Y , i.e. an element of hX×Y (Z). Sum-

marizing our efforts in this paragraph so far, we’ve described a natural map ofsets

G : hX(Z)× hY (Z) → hX×Y (Z)

for any affinoid perfectoid Z. One checks directly that for any map Z → Z� of

affinoid perfectoids, the associated diagram

hX(Z�)× hY (Z

�)

��

�� hX×Y (Z�)

��hX(Z)× hY (Z) �� hX×Y (Z)

commutes, so G extends to a morphism of sheaves. Finally, one checks that F

and G are naturally inverse.

Though it’s not strictly necessary for the above proof, let me note that D+

is a perfectoid Huber ring in Gabber-Ramero’s sense, and hence is sheafy.Remark. If k is any characteristic p field with the discrete topology, there is anevident category Perfk of perfectoid spaces over Spa k, and the above proof alsoshows (by replacing −⊗Fp − with −⊗k− everywhere) that for two such spacesthe product “X ×k Y ” exists. Likewise, for two diamonds D, E over Spa k, theproduct D ×k E exists.

Corollary 2.21. Let D be a diamond. Then the diagonal D → D × D isrepresentable if and only if any morphism hU → D from a representable sheafis representable.

Proof. This is general nonsense (cf. Tag 0024 in the Stacks Project).

2.6 Some topological propertiesIn this section we define quasicompact and quasiseparated objects and mor-phisms in Dia.

Definition 2.22. Let D be a diamond.

1. D is quasicompact (qc) if there is a presentation hX → D with X quasi-compact.

2. D is quasiseparated (qs) if the diamond hX×D hY is quasicompact for anymorphisms hX → D, hY → D with X and Y quasicompact.

19

Remark. If X is any quasicompact perfectoid space, we can find a surjectiveétale map Y → X with Y affinoid perfectoid. In particular, we could equallywell define a diamond D to be quasicompact if and only if it admits a presen-tation hX → D with X affinoid perfectoid. We’ll use this equivalence withoutparticular comment.

Proposition 2.23. If D is quasiseparated, then so is any open subdiamondE ⊂ D.

Proof. Given morphisms hX → E , hY → E , we have

hX ×E hY∼= hX ×D hY

since E is a subfunctor of D. If D is quasiseparated, this product is quasicompactby assumption.

Proposition 2.24. If D is a quasicompact diamond, then |D| is quasicompact.

Proof. Choose a presentation hX → D with X quasicompact, so we get a quo-tient map on topological spaces |X| � |D|. Then a collection of open subsets of|D| give a covering if and only if they pull back to an open covering of |X|; butthe latter space is quasicompact by assumption, so any such covering admits afinite refinement.

Definition 2.25. Let f : D → E be a morphism of diamonds.

1. f is quasicompact (qc) if the diamond D ×E hX is quasicompact for anymorphism hX → E with X affinoid perfectoid.

2. f is quasiseparated (qs) if the diagonal morphism ∆ : D → D ×E D isquasicompact. (Note that ∆ may not be representable.)

Proposition 2.26. Let f : D → E be a morphism of diamonds. Then f isquasicompact if and only if, for every morphism of diamonds F → E with Fquasicompact, the diamond D ×E F is quasicompact.

Proof. “If” is clear. For “only if”, suppose f is quasicompact, and choose anymap F → E with F quasicompact as in the statement of the proposition. Choosea presentation hX → F with X affinoid perfectoid. Then

D ×E hX → D ×E F

is a surjective and pro-étale map of diamonds with quasicompact source. Choosea presentation hU → D ×E hX with U affinoid perfectoid. The composite map

hU → D ×E hX → D ×E F

is then surjective and pro-étale by construction, so it gives a presentation ofD×E F . Since U is affinoid perfectoid, we see that D×E F is quasicompact, asdesired.

20

Using this proposition, one easily checks that pullbacks and compositions ofquasicompact morphisms are quasicompact.Remark. If f : Y → X is a morphism of schemes, one can check that f is quasi-compact in the usual sense (i.e., X admits a covering by open affine subschemesXi such that each f

−1(Xi) is quasicompact) if and only if, for every morphism

V → X with V affine, the scheme Y ×X V is quasicompact.Following Scholze, we also make the following definitions.

Definition 2.27. A diamond D is spatial if D is quasicompact and quasisepa-rated and the subsets

{|E| ⊂ |D|, E ⊂ D a qc open subdiamond}

give a neighborhood basis of |D|. A diamond D is locally spatial if it has anopen covering by spatial diamonds.

Remark. A diamond D is “spatial” in the sense of the Berkeley notes if and onlyif it is “locally spatial and quasiseparated” in the sense of the definitions here.Likewise, a diamond is “quasicompact and spatial” in the sense of the Berkeleynotes if and only if it is “spatial” in the sense of the definition here. Thesenew definitions of spatial and locally spatial were indicated to me by Peter (inearly July 2016); part of the motivation is the verbal analogy exhibited in thefollowing proposition:

Proposition 2.28. If D is spatial, then |D| is a spectral space. If D is locallyspatial, then |D| is locally spectral.

Proof. Any topological space which admits an open covering by spectral spacesis locally spectral, so it suffices to prove the first claim. The first claim followsfrom the arguments in §16.2 of the Berkeley notes.

Proposition 2.29. i. If D is spatial, then so is any qc open subdiamond.ii. If D is spatial and E ⊂ D is any open subdiamond, then E is locally

spatial.

Proof. i. First observe that if D is spatial and D� ⊂ D is any open subdiamond,then D� is quasiseparated. Furthermore, if D� is quasicompact and Ei ⊂ D runsover a set of qc open subdiamonds such that the |Ei|’s give a neighborhood basisin |D| of some point x ∈ |D�|, then the diamonds Ei ∩ D� = Ei ×D D� are stillquasicompact and the open subsets |Ei ∩ D�| = |Ei| ∩ |D�| give a neighborhoodbasis of x in |D�|. This shows that any qc open subdiamond of a spatial diamondis spatial, and that such subdiamonds are stable under finite intersection.

ii. We can write E as the filtered colimit

E = lim→U⊂D qc open, |U|⊆|E|

U .

But each U is qc by assumption, hence spatial by part i., so we get an opencovering of E by spatial diamonds.

21

Corollary 2.30. If D is locally spatial, then so is any open subdiamond E ⊆ D.

Proof. Choose an open covering of D by spatial subdiamonds Ui, i ∈ I. ThenE = ∪i∈I(E ∩Ui) gives an open covering of E , and each E ∩Ui is open in Ui andhence locally spatial by the previous proposition. We can therefore choose anopen cover of each E ∩ Ui by spatial subdiamonds Vij ⊆ E ∩ Ui, j ∈ Ji. Butthen E = ∪i∈I,j∈JiVij gives an open covering of E by spatial subdiamonds, asdesired.

Proposition 2.31. If D is a locally spatial diamond, then |D| is quasicompactif and only if D is quasicompact.

Proof. We already proved the “if” direction for any diamond.For the “only if” direction, choose an open covering of D by spatial subdia-

monds Di ⊂ D, i ∈ I. Then the open subsets |Di| give a covering of |D|; sincethe latter is quasicompact by assumption, we can choose a finite subset I

� ⊂ I

such that |D| = ∪i∈I� |Di|. Since open coverings of a diamond coincide withopen coverings of its topological space, the diamonds Di, i ∈ I

� give an opencovering of D. In particular, the map

�

i∈I�

Di → D

is surjective and étale. For each i ∈ I�, choose a presentation hXi → Di with

Xi affinoid perfectoid, and let S =�

i∈I� Xi. Then the composite map

hS∼=

�

i∈I�

hXi →�

i∈I�

Di → D

is surjective and pro-étale, and S is quasicompact (and in fact affinoid perfec-toid), so D is quasicompact.

Corollary 2.32. Let f : D → E be a morphism of diamonds. If f is quasicom-pact and E is locally spatial, then |f | : |D|→ |E| is quasicompact.

Proof. Let U ⊂ |E| be any quasicompact open subset, with U ⊂ E the associatedopen subdiamond. Let V = |f |−1

(U) ⊂ |D| be the preimage of U , so V =

|D ×E U|. We need to check that V is quasicompact.By Corollary 2.30, U is locally spatial. Since |U| = U is quasicompact, U is

then quasicompact by Proposition 2.31. But then D ×E U is quasicompact byProposition 2.26, so |D ×E U| is quasicompact as desired.

Corollary 2.33. Let D be a quasiseparated locally spatial diamond, and letE ⊆ D be an open subdiamond. Then the following are equivalent:

i. The map E → D is quasicompact.ii. |E| is a retrocompact open subset of |D|.

22

Proof. i. implies ii. Immediate (without any quasiseparatedness hypthesis)by the previous corollary.

ii. implies i. The key point is that since D is quasiseparated and locallyspatial, there are natural inclusion-preseving bijections between

a) quasicompact open subsets of |D|,b) quasicompact open subdiamonds of D, andc) spatial open subdiamonds of D.Indeed, the bijection of a) and b) follows from Proposition 2.31, and the

bijection of b) and c) follows because an open subdiamond of a quasiseparatedlocally spatial diamond is spatial if and only if it is quasicompact. Moreover,open subsets as in a) give a neighborhood basis of |D| stable under finite inter-section and finite union. We exploit these observations as follows.

Suppose |E| ⊂ |D| is a retrocompact open subset. Let f : X♦ → D be any

map from an affinoid perfectoid space. It suffices to show that E ×D X♦ is

quasicompact. Since the subset |f |(|X|) ⊂ |D| is quasicompact, we may choose(by our observations above regarding qc open subsets of |D|) a quasicompactopen subdiamond U ⊂ D such that |f |(|X|) ⊆ |U|, i.e. such that f factorsthrough a map X

♦ → U . Let E � = E ×D U be the open subdiamond of Dassociated with the open subset |E|∩ |U|. By assumption, |E| is a retrocompactopen, and |U| is a quasicompact open, so |E|∩ |U| is a quasicompact open subsetof |D|. Therefore, E � is a quasicompact open subdiamond of D. We have anidentification

E ×D X♦

= E � ×U X♦.

But E � and X♦ are both quasicompact, and U is spatial so in particular qua-

siseparated, and therefore E×D X♦

= E �×U X♦ is quasicompact as desired.

Definition 2.34. A morphism f : D → E of diamonds is separated if it isquasiseparated and the map |∆| : |D|→ |D ×E D| is a closed embedding.

Proposition 2.35. A quasiseparated morphism f : D → E of locally spatialdiamonds is separated if and only if it satisfies the valuative criterion of sepa-ratedness.

Proof. Omitted.

Definition 2.36. A morphism f : D → E of locally spatial diamonds is spatialif the associated map |D|→ |E| of locally spectral spaces is a spectral map.

We now turn to the notion of a taut diamond. Recall the following definition(cf. [Hub96, Def. 5.1.2]).

Definition 2.37. i. A locally spectral space X is taut if it is quasiseparated(i.e. the intersection of any two quasicompact open subsets is quasicompact)and the closure U of any quasicompact open subset U is quasicompact.

Note that if X is a taut locally spectral space, then any quasicompact opensubset U ⊂ X is spectral.

Definition 2.38. A diamond D is taut if D is locally spatial and the locallyspectral space |D| is taut.

23

2.7 The miracle theoremsGiven an adic space X, there is typically a huge difference between locally closedsubsets of |X| and locally closed immersions into X; the latter notion is muchmore restrictive. In this section we prove a “miracle theorem” showing that thisdifference largely disappears when X is a w-local affinoid perfectoid space. Wethen bootstrap this up to an equally miraculous construction which allows oneto “diamondize” a very general class of locally closed subsets E ⊂ |D|, where Dis an arbitrary diamond. I learned the statements of Theorems 2.39 and 2.42below from Peter Scholze.

Recall that a spectral space X is w-local if every connected component ofX has a unique closed point, and the subset X

c of closed points is closed inX. This implies that the composition X

c�→ X

γ

� π0(X) is a homeomorphism,where we give π0(X) its natural profinite topology.

In the statement and proof of the following theorem only, we will denote adicspaces by calligraphic letters, to distinguish them from the (many) topologicalspaces appearing.

Theorem 2.39. Let X = Spa(R,R+) be an affinoid perfectoid space whose

underlying topological space X = |X | is w-local. Then for any locally closedgeneralizing subset T ⊂ X, there exists a locally closed immersion of perfectoidspaces f : T → X identifying |T | homeomorphically with T . The pair (T , f) isunique up to unique isomorphism and is universal for morphisms g : Z → Xwith g(|Z|) ⊆ T .

Here a locally closed immersion is a morphism of perfectoid spaces f : Y → Xwhich can be factored as Y → U → X where U → X is an open immersion andY → U is a Zariski-closed embedding in the sense of Scholze’s torsion paper.

We make some preliminary reductions. Write T = U ∩Z where Z (resp. U)is closed (resp. open) in X. Let U

� ⊂ U be any quasicompact open, and setT�= T ∩ U

�= Z ∩ U

�. Note that T� is again locally closed and generalizing,

and is closed in the constructible topology on X. Varying U� over all quasi-

compact opens in U , an easy gluing argument together with the uniqueness of(T , f) shows that it suffices to prove the theorem with T replaced by T

� in theobvious sense (i.e. it suffices to prove the theorem when T = U ∩ Z with U

quasicompact).So, let T = U ∩ Z ⊂ X be locally closed and generalizing where Z (resp.

U) is closed (resp. quasicompact open) in X. As we’ve already noted, T ⊂ X

is closed in the constructible topology; this implies that T with its inducedtopology is a spectral space by Tag 0902 in the Stacks Project.

Lemma 2.40. Notation and assumptions as above, we have:i. T ⊂ X with its induced topology is a spectral space, and T = T ∩ U is a

quasicompact open in T .ii. T ⊂ X is generalizing.iii. Under the natural quotient map γ : X → π0(X), we have T = γ

−1(γ(T ))

with γ(T ) ⊂ π0(X) closed.

24

Proof. i. The space T is closed in the spectral space X, and thus is spectralwith the induced topology by Tag 0902 in the Stacks Project. For the secondclaim, note that the inclusion T ⊆ T ∩ U is obvious, so it suffices to show theopposite inclusion. But T = U ∩ Z ⊆ Z = Z, so T ∩ U ⊆ Z ∩ U = T .

ii. Let x ≺ y ∈ X be any points with x ∈ T . Since X is spectral and T ⊂ X

is closed in the constructible topology, Tag 0903 implies that there exists a pointz ∈ T such that x ≺ z. Since T is generalizing, the maximal generalization z0

of z ∈ X lies in T . But the maximal generalization x0 of x coincides with z0,so then

y ∈ G(x) ⊆ {x0} = {z0} ⊆ T

as desired.iii. Here we use the w-local structure of X in a more serious way. More

precisely, let X = |Spa(A, A+)| be the topological space of some affinoid analytic

adic space, and suppose X is w-local. Let γ : X � π0(X) ∼= Xc be the

continuous map as above. Then we claim the association

W ⊂ X �→ γ(W ) ⊂ π0(X)

defines a bijection from closed generalizing subsets of X to closed subsets ofπ0(X), with inverse given by γ

−1(−).

To see this, note that any point x ∈ X has a unique rank one generalizationx0, as well as a unique (by w-locality) closed specialization x

c; since generaliza-tions in analytic adic spaces form a totally ordered chain, we have x

c ≺ x ≺ x0

and γ−1

(xc) = {x0}. This implies that the generalizations of x

c form a totallyordered chain3 coinciding with the set of specializations of x0.4 Suppose nowthat W ⊂ X is closed and generalizing, and let x ∈ X be any point. Since W

is closed and thus specializing as well, we have x ∈ W if and only if xc ∈ W , if

and only if γ−1

(xc) = {x0} ⊆ W . This shows that W = γ

−1(γ(W )), and we get

that γ(W ) is closed in π0(X) since γ(W ) ∼= W ∩ Xc is closed in X

c ∼= π0(X)

(cf. also the proof of Tag 096C). Taking W = T , which is allowed by part ii.,the claim follows.

Proof of the theorem. Notation as above, part iii. of the Lemma lets us writeγ(T ) as a cofiltered limit (i.e. intersection) of open-closed subsets Vi ⊂ π0(X)

over some index set i ∈ I. The preimages γ−1

(Vi) ⊂ |X | are quasicompactopen-closed subspaces, and thus correspond to unique open-closed immersionsof perfectoid spaces Vi → X which again form a cofiltered system. Each spaceVi is affinoid: choosing any pseudouniformizer � ∈ R

◦◦ ∩ R× and writing ei ∈

O(X ) ∼= R for the idempotent cutting out Vi, the formula

Vi = {x ∈ X | |1− ei|x ≤ |�|x, |�|x ≤ |�|x}3Whose order type, however, could be some enormous ordinal.

4This last coincidence is extremely special to the w-local situation, and also relies crucially

on the fact that X arises as |Spa(A, A+

)|.

25

exhibits Vi as a rational subset of X , with Vj a rational subset of Vi if j ≥ i.Now we get an affinoid perfectoid space

“T ” = lim←i∈I

Vi → X

such that |T | ∼= T , and one easily checks that this map is a Zariski-closedembedding in the sense of Scholze’s torsion paper (the associated ideal in R isthe ideal generated by 1− ei for all i ∈ I). But then

T = T ∩ U ∼= |T ×X U|

where of course U → X is the open immersion of perfectoid spaces correspondingto the open subset U ⊂ X, so T = T ×X U is the space we seek. One easilychecks the universal property and uniqueness. Note that the map T → Xfactors as T → U → X , i.e. as a Zariski-closed embedding followed by an openimmersion, so this really is a locally closed immersion.

Now we bootstrap this theorem up to a much more general result, using thefact that diamonds always have coverings made of w-local affinoid perfectoids.Before doing so, we need a little lemma.

Lemma 2.41. If f : Y → X is a continuous map of topological spaces andS ⊂ X is generalizing, then f

−1(S) ⊂ Y is generalizing.

Proof. Let y ≺ z ∈ Y be points with y ∈ f−1

(S). We need to check thatf(z) ∈ S. Since y ∈ {z}, we get

f(y) ∈ f({z}) ⊆ {f(z)},

with the second inclusion following by continuity, so f(y) ≺ f(z) in X. Sincef(y) ∈ S and S is generalizing, we get f(z) ∈ S as desired.

Theorem 2.42. Let D be a diamond with associated topological space |D|. Thenfor any locally closed generalizing subset E ⊂ |D|, there is a canonical subdi-amond E ⊂ D such that |E| ∼= E homeomorphically inside |D|, characterizedby the following universal property: a map X

♦ → D (for X ∈ Perf arbitrary)factors through a map X

♦ → E if and only if the associated map |X|→ |D| hasimage contained in E.

Remark. It’s easy to check that E has the same universal property with respectto maps G → D from an arbitrary diamond G.

Proof. Observe that if E ⊂ |D| is any subset whatsoever, we can use the samerecipe as in the theorem to define a subfunctor E ⊂ D of “maps to D factoringthrough E on topological spaces”. We’re going to show that when E is locallyclosed and generalizing, E is a diamond and |E| = E.

So, fix E ⊂ |D| locally closed and generalizing, with associated subfunctorE ⊂ D. Let

U♦

=

�

i∈I

X♦i

f→ D

26

be a representable surjective pro-étale map, where U =�

i∈IXi is perfectoid

with each Xi affinoid perfectoid. After possibly replacing each Xi by an affinoidpro-étale cover, we may assume the |Xi|’s are all w-local. Then |f | : |U |→ |D|is a quotient map, and the subset |f |−1

(E) is locally closed and generalizing in|U | =

�i∈I

|X♦i| ∼=

�i∈I

|Xi| by the previous lemma. Applying Theorem 2.39“one i ∈ I at a time” now gives a locally closed immersion of perfectoid spacesV → U such that |V | ∼= |U | ×|D| E = |f |−1

(E). Note that V♦ is naturally a

subfunctor of U♦.

By the universal property of E , the map V♦ → D factors uniquely through

a map V♦ → E . We claim that:

i. V♦

= U♦ ×D E as subfunctors of U

♦, andii. under this identification, the map V

♦ → E identifies with the natural pro-jection U

♦ ×D E → E .To check i., we simply observe that if g : Y

♦ → U♦ is any map, then g factors

via a map Y♦ → V

♦ if and only if |g|(|Y |) ⊆ |V | = |U | ×|D| E = |f |−1(E), by

the universal property of V → U ; but this holds if and only if f ◦ g : Y♦ → D

factors via a map Y♦ → E . Checking ii. is now an easy trace through the

definitions, keeping in mind that the diagram

V♦

= U♦ ×D E ��

��

U♦

f

��E �� D

is cartesion and that the horizontal arrows therein are monomorphisms. Butnow, since this diagram is cartesian, we get that the map V

♦ → E is repre-sentable, surjective and pro-étale, since it’s the pullback of the representablesurjective pro-étale map U

♦ → D along the map E → D. Therefore E is adiamond, and

|E| = im(|V ♦|→ |D|) = |f |�|f |−1

(E)�

= E

as desired.

Definition 2.43. A monomorphism of diamonds E → D is a locally closed em-bedding if E arises by applying Theorem 2.42 to some locally closed generalizingsubset E ⊂ |D|. We’ll also say (equivalently) that E ⊂ D is a locally closedsubdiamond of D.

We emphasize that a locally closed embedding E → D in this sense need notbe relatively representable! However, it is true that E ×D X

♦ is representableby a perfectoid space whenever X is a w-local perfectoid space.

Corollary 2.44. Given a diamond D, the association E �→ |E| defines aninclusion-preserving bijection from locally closed subdiamonds of D to locallyclosed generalizing subsets of |D|.

27

Proposition 2.45. Let D be a locally spatial diamond, and let E → D be alocally closed embedding. Then E is locally spatial.

Proof. Write |E| = E as an intersection U ∩Z with U open and Z closed. ThenU ⊂ |D| corresponds to an open immersion U → D, so U is locally spatial byCorollary 2.30. Now U ∩ Z ⊂ U is closed and generalizing in U , and we getE → U as the associated locally closed embedding. Arguing locally on an opencovering of U by spatial subdiamonds Ui, we get that each Ei = E ×U Ui → Ui

is the locally closed embedding associated with the closed generalizing subsetE ∩ |Ui| ⊂ |Ui|, and it suffices to show that each Ei is spatial. So now we’rereduced to showing that if D is spatial and E ⊂ D is a locally closed embeddingsuch that |E| ⊂ |D| is closed and generalizing, then E is spatial too; this is aneasy exercise.

Proposition 2.46. Let D be a quasiseparated locally spatial diamond, and letE ⊂ D be a locally closed subdiamond. Then the following are equivalent:

i. The morphism E → D is quasicompact.ii. |E| is a retrocompact subset of |D|.

Sketch. i. again implies ii. by Corollary 2.32.To see that ii. implies i., argue as in the previous proposition.

2.8 Some sites associated with a diamondHere we content ourselves with a general nonsense definition.

Definition 2.47. Let D be a diamond, and let • be one of the decorations• ∈ {an, fet, et,proet}, respectively. Then there is a site “D•” with objectsgiven by diamonds E over D such that the map E → D is (respectively) anopen immersion, a finite étale map, an étale map, or a pro-étale map, and withcovers given by collections {Ei → E} such that

�Ei → E is surjective as a map

of sheaves.

This “uniform” covering condition is indeed reasonable, since the pro-étaletopology refines the other three.

3 Diamonds associated with adic spacesDefinition 3.1. Given a perfectoid space Y ∈ Perf, an untilt of Y is a pair(Y

�, ι) where Y

� is a perfectoid space and ι : Y�� ∼→ Y is an isomorphism. A

morphism m : (Y�, ι) → (Y

��, ι�) of untilts of Y is a morphism of perfectoid

spaces m : Y� → Y

�� such that ι� ◦m

�= ι.

Note that for any morphism m : (Y�, ι) → (Y

��, ι�) of untilts of Y , m

�=

ι�−1 ◦ ι is an isomorphism, so m is necessarily an isomorphism as well. Thus

the category of untilts of Y naturally forms a groupoid. Note also that a given(Y

�, ι) has no automorphisms, since if m : Y

� → Y� tilts to the identity map,

28

then m must be the identity. Let SpdZp : Perf → Sets be the presheaf sendingY ∈ Perf to the set of isomorphism classes of untilts of Y . We will see laterthat SpdZp is actually a sheaf.

Now let X be an analytic adic space over SpaZp.

Definition 3.2. Given any perfectoid space Y ∈ Perf, an untilt of Y over X isa triple (Y

�, ι, f) where (Y

�, ι) is an untilt of Y and f : Y

� → X is a morphismof adic spaces. A morphism m : (Y

�, ι, f) → (Y

��, ι�, f�) of untilts of Y over

X is a morphism of perfectoid spaces m : Y� → Y

�� such that ι� ◦m

�= ι and

f� ◦m = f .

Again, any morphism of untilts of Y over X is necessarily an isomorphism,and a given (Y

�, ι, f) has no automorphisms.

LetX

♦: Perf

proet → Sets

be the presheaf sending Y ∈ Perf to the set of isomorphism classes of untilts ofY over X. The association X �→ X

♦ is clearly a functor.

Proposition 3.3. If X is a perfectoid space, then X♦ ∼= hX� . (So, in particular

X♦ ∼= (X

�)♦.)

Proof. This is an immediate consequence of the equivalence �Perf/X∼= Perf/X�

induced by tilting.

In light of this proposition, one might think of X♦ as some kind of “tilt” of

X even when X isn’t a perfectoid space. The main result in this section is thefollowing theorem.

Theorem 3.4. For X any analytic adic space over SpaZp, X

♦ is a diamond.Moreover, X

♦ is locally spatial.

The functor X �→ X♦ has many compatibilities, some of which will be

discussed below. Let us note in particular that, although the structure mapX → SpaZp does induce a natural transformation X

♦ → SpdZp, our basicpoint of view is to “forget” this map from X

♦ to SpdZp. We shall return to thispoint in detail later on. We also note that X

♦ has a canonical Frobenius FX♦ ,sending (Y

�, ι, f) ∈ X

♦(Y ) to (Y

�, FY ◦ ι, f).

3.1 The affinoid caseThe crucial ingredient in the proof of Theorem 3.4 is the following lemma, whichtreats the case of X affinoid.

Lemma 3.5. Let X = Spa(R,R+) be an affinoid adic space. Choose a directed

system (Ri, R+

i), i ∈ I of finite étale Galois (R,R

+)-algebras such that

(R, R+) =

�limi→

(Ri, R+

i)

29

is perfectoid, where the completion is for the topology making limi→R+

iopen and

bounded. (By an argument of Colmez and Faltings, such a direct system alwaysexists.) Let Gi be the Galois group of Ri over R, so Gi acts on Xi = Spa(Ri, R

+

i)

and by continuity G = lim←i G acts on X = Spa(R, R+). Then:

1. The space X� with its induced action of G satisfies the hypotheses of Theo-

rem 2.18. Consequently, hX� → h

X�/G is a pro-étale G-torsor and hX�/G

is a diamond.

2. There is a natural isomorphism

X♦ ∼= h

X�/G.

In particular, X♦ is a diamond.

It’s essentially trivial to check that X� with its action of G satisfies the

hypotheses of Theorem 2.18, so we’re reduced to showing the isomorphism X♦ ∼=

hX�/G.

Lemma 3.6. Let X, X and G be as in Lemma 3.5. For any perfectoid spaceT , there is a natural bijection

Hom(T,X) ∼= HomG(T /T, X)/ �

where the right-hand side denotes isomorphism classes of G-equivariant mapsT → X from pro-étale G-torsors T /T .

Proof. Given f : T → X, set Ti = Xi ×X,f T . By the almost purity theorem,this is perfectoid and is a finite étale Gi-torsor over T with an evident map toXi. Let T = lim←i Ti; this exists as a perfectoid space since the transition mapsare all finite étale, and the maps Ti → Xi compile into a map f : T → X by aneasy continuity argument.

In the other direction, recall that for any affinoid adic space X = Spa(R,R+)

and any adic space Y , there’s a natural identification

Hom(Y, X) = Hom�(R,R

+), (O(Y ),O(Y )

+)�.

In particular, in our setup, giving a G-equivariant map f : T → X from apro-étale G-torsor T /T is equivalent to giving a continuous G-equivariant map

f : (R, R+) → (O(T ),O(T )

+).

By pro-étale descent for perfectoid spaces, the map

(O(T ),O(T )+) → (O(T )

G,O(T )

+G)

is an isomorphism, so restricting f to G-invariant elements gives a map

(RG

, R+G

) → (O(T ),O(T )+)

30

which clearly only depends on the isomorphism class of T /T and f . Since theimage of the map (R,R

+) → (R, R

+) is contained in (R

G, R

+G), composing the

restriction of f with this inclusion induces a map

f ∈ Hom�(R,R

+), (O(T ),O(T )

+)�

= Hom(T,X).

Finally, one checks that these associations f �→ f and f �→ f are mutuallyinverse.

Returning now to the setup of Lemma 3.5, we’re ready to construct the mapX

♦ → hX�/G. Given some Y ∈ Perf and some (Y

�, ι, f) ∈ X

♦(Y ), we need to

describe an element of (hX�/G)(Y ). By the construction of Lemma 3.6, giving f

is the same as giving a pro-étale G-torsor Y�/Y

� together with a G-equivariantmap f : Y

� → X (up to isomorphism). Symbolically,

X♦(Y ) ∼=

�(Y

�, ι, f)

�/ �

∼=�

(Y�, ι, Y

�/Y

�, f)

�/ �

(with the obvious meaning in the second line). Tilting this data (and using ι)gives a pro-étale G-torsor Y /Y together with a G-equivariant map f

�: Y → X

�.But giving this latter data is equivalent to giving a Y -point of (h

X�/G)! Indeed,since h

X� → hX�/G is a pro-étale G-torsor, we get a natural identification

(hX�/G)(Y ) = HomG(Y /Y, X

�)/ � .

This gives the desired map. Symbolically,

X♦(Y ) ∼=

�(Y

�, ι, f)

�/ �

∼=�

(Y�, ι, Y

�/Y

�, f)

�/ �

tilt→ HomG(Y /Y, X�)/ �

=: (hX�/G)(Y ).

To finish the proof of Lemma 3.5, we need to invert the map labelled “tilt→”in the previous diagram. In other words, given a pro-étale G-torsor Y /Y anda G-equivariant map a : Y → ˜

X�, we need to (canonically and functorially)produce an untilt (Y

�, ι), a pro-étale G-torsor Y

�/Y

� and a G-equivariant mapa

�: Y

� → X. This is the least formal part of the proof, and we do it in threesteps as follows:

1. By the equivalence �Perf/X

∼= Perf/X� , we immediately get (Y

�, ι) and a

�

for free.

2. By Propositions 3.7 and 3.9 below, the G-action on Y lifts to a uniqueG-action on Y

� compatible with a� and ι.

31

3. Finally, in Proposition 3.10 below, we construct (Y�, ι) as the categorical

quotient of (Y�, ι) by its natural G-action.

Proposition 3.7. Let Y ∈ Perf be a perfectoid space with an action of somegroup G, and let (Y

�, ι) be a given untilt of Y . Then there is at most one G-

action on Y� for which ι is G-equivariant. If this action exists, we say (Y

�, ι)

is a G-equivariant untilt of Y .

Proof. Let ψ : G → Aut(Y ) be the action map, and let φ, φ�: G → Aut(Y

�) be

two action maps. If ι is G-equivariant for φ and φ�, then

ι ◦ φ�

g= ψg ◦ ι = ι ◦ φ

�g

�

for all g ∈ G, so φ�g

� ◦ (φ�

g)−1

= (φ�g◦ φ

−1

g)�

= id for all g ∈ G, and thereforeφ�g◦ φ

−1

g= id for all g ∈ G.

Proposition 3.8. Let Y ∈ Perf be a perfectoid space with an action of somegroup G, and let (Y

�, ι) be a given untilt of Y . Then (Y

�, ι) is a G-equivariant

untilt if and only if the natural pullback action of G on A+

Ypreserves the ideal

sheaf ker θY � .

Proof. Easy and left to the reader.

Proposition 3.9. Let X,Y ∈ Perf be perfectoid spaces with an action of agroup G, and let f : Y → X be a G-equivariant morphism. If (X

�, ι) is a G-

equivariant untilt of X, then the untilt (Y�, ι�) of Y induced by the equivalence

Perf/X∼= �Perf/X� is a G-equivariant untilt, and f

�: Y

� → X� is G-equivariant.

Proof. Let ker θX� ⊂ A+

Xbe the ideal sheaf describing the untilt X

�. By theprevious proposition, ker θX� is preserved by the G-action on X. Since ker θY � =

f−1

ker θX� ·A+

Yand f is G-equivariant, the ideal sheaf ker θY � is preserved by the

G-action which therefore descends to the quotient O+

Y � , so Y� is G-equivariant.

The G-equivariance of f� is then obvious.

Let Y → Y be a pro-étale G-torsor, and let (Y�, ι) be an untilt of Y with the

trivial G-action. Applying Proposition 3.9, we get a G-equivariant untilt (Y�, ι)

of Y over (Y�, ι). The next proposition reverses this construction.

Proposition 3.10. Let Y ∈ Perf be a perfectoid space, and let Y → Y be apro-étale G-torsor for some profinite group G. Let (Y

�, ι) be a G-equivariant

untilt of Y . Then the categorical quotient Y�

= Y�/G exists as a perfectoid

space, and ι induces a canonical isomorphism

ι : Y��

= (Y�/G)

� ∼= Y��

/G∼→ Y /G ∼= Y,

so (Y, ι) is an untilt of Y . This association induces an equivalence of categories

{untilts of Y } ∼= {G− equivariant untilts of Y }.

32

Note: Although Proposition 3.10 doesn’t appear explicitly in the Berkeleynotes, I don’t claim any originality in formulating or proving it: a special caseis very strongly implicit in the proofs of Theorem 9.4.5 and Lemma 9.4.6 in theBerkeley notes. Cf. also the remarks following Proposition 4.2 below.

Proof. We may assume Y and Y are affinoid, say with Y = Spa(A, A+) and

Y = Spa(A, A+). Let Y

�= Spa(A

�, A

�+) be our given untilt (so we also have

an isomorphism ι : A+ ∼→ A

��+) with its action of G. Set A�

= (A�)G and

likewise for A�+. We need to show that (A

�, A

�+) is perfectoid with tilt (A, A

+).

Let � ∈ A+ be a good pseudouniformizer, i.e. a pseudouniformizer such that

��

= (��)� ∈ A

�+ satisfies ��|p in A

�+; let us say � is very good if there isanother good �

� such that ��� is good. Note that �

� is a G-invariant elementof A

�+, since it’s the image of the G-invariant element [�] ∈ W (A+) under the

G-eqivariant map W (A+) → A

�+. Note also that for any good �, ι induces aG-equivariant isomorphism A

�+/�

� ∼= A+/�. The key claim (whose proof we

momentarily defer) is as follows:Claim: The isomorphism A

�+/�

� ∼= A+/� induces an isomorphism A

�+/�

� ∼=A

+/� when � is very good.Granted this claim, we first observe that A

� is perfectoid. Since A� is clearly

a uniform Tate ring (it’s a closed subring of A�), so we only need to verify the

Frobenius condition. Applying the claim twice, with � and �p such that �

p isvery good, we get a commutative diagram

A�+

/��

���

Φ �� A�+/(�

�)p

���

A+/�

Φ �� A+/�

p

where Φ is the pth power map as usual; since the lower Φ is an isomorphism,the upper Φ is too.

Now we have the map A�+ → A

�+, so tilting this gives an injective mapA

��+ → A��+ ∼= A

+ whose image factors through (A+)G

= A+; thus we have

get an injective map A��+ → A

+. But reducing mod �, this becomes theisomorphism A

��+/� ∼= A

+/� ∼= A

�+/�

�, so A+

= A��+

+ �A+ and thus

A+ ∼= A

��+ by topological Nakayama.It remains to prove the claim. We have short exact sequences

0 → A�+ ·��

→ A�+ → A

�+/�

� → 0

and0 → A

+ ·��

→ A+ → A

+/� → 0

whose final terms are canonically identified, so passing to continuous groupcohomology gives exact sequences

0 → A�+

/�� → H

0(G, A

�+/�

�) → H

1

cts(G, A

�+)

33

and0 → A

+/� → H

0(G, A

+/�) → H

1

cts(G, A

+)

in which the middle terms are canonically identified. Suppose now that � isvery good, and choose some �

� with ��� good. We claim then that

A�+

/��= im

�H

0(G, A

�+/(��

�)�) → H

0(G, A

�+/�

�)

�

andA

+/� = im

�H

0(G, A

+/(��

�)) → H

0(G, A

+/�)

�.

This gives our desired isomorphism, since the right-hand terms in these identitiesare canonically identified.

To see this, consider the commutative diagram

0 �� A�+/(��

�)� ��

��

H0(G, A

�+/(��

�)�) ��

��

H1

cts(G, A

�+)

·(��)�

��

0 �� A�+/�

� �� H0(G, A

�+/�

�) �� H1

cts(G, A

�+)

in which the first two downwards maps are the obvious reduction maps. Clearly

A�+

/�� ⊆ im

�H

0(G, A

�+/(��

�)�) → H

0(G, A

�+/�

�)

�

by going around the lefthand square. For the inclusion in the other direction,the key point is that the H

1’s are almost zero (in the almost setting defined byp-power roots of �

� for any good �)5, so the image of the middle downwardsmap dies in the lower H

1. The proof for A+/� is exactly analogous.

3.2 The general caseIn this subsection we finish the proof of Theorem 3.4. For now we simply sketchthe material in this section; a full proof will appear in the next version of thesenotes. In any case, the idea is clear: given X, choose a covering X = ∪i∈IXi

by affinoid adic spaces Xi. By Theorem 3.5, each X♦i

is a diamond, and we’dlike to glue up the X

♦i

’s to form X♦.

First possible proof: Use Theorem 3.5 together with the following two propo-sitions.

Proposition 3.11. The presheaf X♦ is a sheaf.

This is nontrivial (the proof uses Lemma 3.5).

Proposition 3.12. Suppose we are given sheaves D and (Di)i∈I on Perfproet,

together with sheaf maps Di → D. Suppose furthermore that1. Each map Di → D is an open immersion.2. The sheaf map

�Di → D is surjective.

3. The sheaves Di are diamonds.Then D is a diamond.

5(use pro-étale descent plus Cartan-Leray for the covering Y → Y )

34

Proof sketch. Choose presentations hXi → Di, and then show that the inducedsheaf map

�hXi → D is surjective and pro-étale.

Second possible proof. Formalize the notion of “gluing along open subdia-monds”:

Definition 3.13. A gluing datum in diamonds is a quadruple (I, (Di), (Eij), (ϕij))

where I is a set, Di, i ∈ I is a collection of diamonds, Eij ⊂ Di is an opensubdiamond for each i, j ∈ I, and ϕij : Eij

∼→ Eji is an isomorphism. We fur-ther require that Eii = Di and ϕii = id, and that for any i, j, k ∈ I we haveϕ−1

ij(Eji ∩ Ejk) = Eij ∩ Eik and furthermore the diagram

Eij ∩ Eik

ϕij

������������

ϕik �� Eki ∩ Ekj

Eji ∩ Ejk

ϕjk

������������

commutes.

Proposition 3.14. Given a gluing datum in diamonds (I, (Di), (Eij), (ϕij)) asabove, there exists a diamond D and open subdiamonds Ei ⊂ D together withisomorphisms ϕi : Di

∼→ Ei such that ϕi(Eij) = Ei ∩ Ej and ϕij = ϕ−1

j|Ei∩Ej ◦

ϕi|Eij . Furthermore, morphisms to D are described as follows:

Hom(hX ,D) =

�open cover X = ∪i∈IXi and (gi : hXi → Di) such that

g−1

i(Eij) = hXi∩Xj and gj |hXi∩Xj

= ϕij ◦ gi|hXi∩Xj

�.

The second of these proofs is arguably the better way to go, since it givesmore refined information.

3.3 Some basic compatibilitiesLet X be an analytic adic space over SpaZp. In this section we discuss some ofthe compatibilities between X and X

♦.

Proposition 3.15. There is a natural homeomorphism |X♦| ∼= |X|.Proof. One reduces (via Proposition 3.14) to the case of X affinoid. Followingthe proof in the Berkeley notes, we may then take X = Spa(R,R

+) and X =

Spa(R, R+), Xi, G, etc. to be chosen as in the statement of Lemma 3.5, and

then we trace through the following chain of natural isomorphisms:

|X♦| ∼= |hX�/G|

∼= |X�|/G

∼= |X|/G

∼=�lim←i

|Xi|�

/G

∼= lim←i

|X|∼= |X|.

35

Here the first line follows from Lemma 3.5 and Theorem 2.18, and the third linefollows from naturality of tilting. Writing (R∞, R

+

∞) = limi→(Ri, R+

i) for the

uncompleted direct limit, the fourth line follows from the identifications

|X| = |Spa(R, R+)| = |Spa(R∞, R∞)| ∼= lim

←i

|Xi|,

i.e. because the topological space |Spa(A, A+)| = |Spa �(A, A+)| only depends on

the completion of a given Huber pair (A, A+).

Proposition 3.16. The functor (−)♦ induces a natural equivalence Xfet

∼=X

♦fet

.

Note: The objects of the site Xfet are “generalized adic spaces” in the senseof Scholze-Weinstein which are finite étale over X. This slight complicationis necessary at the present time: if X = Spa(R,R

+) is some sheafy affinoid,

then one wants an equivalence Xfet∼= (R)fet, but it’s not clear in general that

arbitrary finite étale R-algebras are sheafy! Allowing generalized adic spaces,everything is okay: given any S ∈ (R)fet, the generalized adic space Spa(S, S

+)

(where S+ is the integral closure of R

+ in S) is the associated object of Xfet.In any case, when X is locally Noetherian or perfectoid, the objects of Xfet arehonest adic spaces.

Proof. One reduces to the affinoid case, and then proceeds as in Lemma 17.3.8of the Berkeley notes.

Proposition 3.17. If X is locally Noetherian or perfectoid (or more generally ifX has a well-behaved étale site), the functor (−)

♦ induces a natural equivalenceXet

∼= X♦et

.

Proof. The étale site is “generated by” open embeddings and finite étale maps,so this follows from the previous results.

Next we record a compatibility of (−)♦ with weak closed immersions (recall

we defined the latter in §1.5).

Lemma 3.18. Let Z = Spa(S, S+) → X = Spa(R,R

+) be a weak closed

immersion of affinoid adic spaces. Then |Z| ⊂ |X| is closed and generalizing.

Proof. Any V (I) ⊂ X is closed, and any morphism of analytic adic spaces isgeneralizing.

In particular, if Z = Spa(S, S+) → X = Spa(R,R

+) is a weak closed immer-

sion, this lemma shows that we can produce a subdiamond Z ⊂ X♦ by applying

Theorem 2.42 to the closed generalizing subset |Z| ⊂ |X| = |X♦|.

Proposition 3.19. If Z = Spa(S, S+) → X = Spa(R,R

+) and Z ⊂ X

♦ areas in the previous sentence, and S is uniform,6 then Z = Z

♦ as subfunctors ofX

♦.6This condition can be dropped if one is willing to work with pre-adic spaces.

36

Proof. By the universal property of Z, the inclusion Z♦ ⊂ X

♦ factors througha map Z

♦ → Z. Therefore, it’s enough to check that if T = Spa(A, A+) ∈ Perf

and f : T♦ → X

♦ are arbitrary, and |f | carries |T | into |Z| ⊂ |X| = |X♦| (sof factors through a map T

♦ → Z), then f factors through a map T♦ → Z

♦.Let f be any morphism of this type. Since the map f corresponds to a mapf

�: T

�= Spa(A

�, A

�+) → X for T

� some untilt of T with the property that|f �| carries |T �| ∼= |T | into |Z|, it’s enough to see that f

� factors through a mapT

� → Z. But this is exactly Lemma 1.22.

4 ExamplesIf (R,R

+) is a sheafy Tate-Huber pair over Zp, we write Spd(R,R

+) for the

diamond Spa(R,R+)♦, and we further abbreviate Spa R = Spa(R,R

◦) and

SpdR = Spd(R,R◦).

4.1 The diamond of SpaQp

By the general theory of Section 3, the adic space SpaQp has an associated dia-mond SpdQ

p. We explain here its canonical presentation, and its interpretation

as a functor.Let ζpn , n ≥ 1 be a compatible sequence of primitive p

nth roots of unity, andlet Qcyc

p= �Qp(ζp∞). Let us describe the tilt of Qcyc

p. Setting On = Zp[ζpn ], we

have an isomorphism of Fp-algebras

On/(ζp − 1) = Fp[t]/(tp

n−1(p−1)

)

ζpn �→ 1 + t,

or equivalently

On/(ζp − 1) = Fp[tp1−n

]/(tp−1

)

ζpn �→ 1 + tp1−n

.

Taking the inductive limit over n, we get

Zcyc

p/(ζp − 1) = Fp[t

1/p∞

]/(tp−1

),

and then applying (−)�

= lim ←x�→x

p(−) gives Zcyc,�

p∼= Fp[[t

1/p∞

]] and Qcyc,�

p∼=

Fp((t1/p

∞)). Note that the sharp map Qcyc,�

p→ Qcyc

psends t to

� := limn→∞

(ζpn − 1)p

n−1

.

Since Qcyc

p= �Qp(ζp∞) is a perfectoid pro-étale Z×

p-torsor over Qp with tilt

Qcyc,�

p∼= Fp((t

1/p∞

)), Lemma 3.5 implies the following result.

37

Proposition 4.1. There is a natural isomorphism SpdQp∼= hSpaFp((t1/p∞ ))/Z×p ,

where a ∈ Z×p

acts by sending t1/p

nto (1 + t

1/pn)a − 1.

A similar result holds for any finite extension K/Qp. Precisely, let Fq bethe residue field of K, and let G ∈ OK [[X,Y ]] be a Lubin-Tate formal OK-module law. For a ∈ OK , let [a](T ) = expG(a logG(T )) ∈ OK [[T ]] be the seriesrepresenting multiplication by a on G. Then SpdK ∼= hSpaFq((t1/q∞ ))/O×K wherea ∈ O×

Kacts by sending t

1/qn

to [a](t1/q

n).

Next we show that perfectoid spaces over Qp are equivalent to perfectoidspaces in characteristic p equipped with a “structure morphism” to SpdQp. Tomake the latter notion precise, note that Perf is a full subcategory of Dia, sowe may speak of the slice category Perf/D for any diamond D.

Proposition 4.2. The functor

�Perf/SpaQp→ Perf/SpdQp

Y �→ Y♦

is an equivalence of categories.

This is Theorem 9.4.5 in the Berkeley notes. The proof we give below mightseem suspiciously short; this is because, in our interpretation, the key Proposi-tion 3.10 does all of the heavy lifting, but in fact we reverse-engineered Propo-sition 3.10 through a close reading of Peter’s proof of Theorem 9.4.5 sketchedin the Berkeley notes.

Proof. Since we’ll need it later, we prove a more general result: for any analyticadic space X over SpaZp, passage to diamonds induces an equivalence �Perf/X

∼=Perf/X♦ .

We construct the essential inverse. Assume for simplicity that X = Spa(R,R+)

is affinoid, so we may choose X = Spa(R, R+), Xi, G, as in the statement of

Lemma 3.5, with X♦ ∼= (X

�)♦/G, etc. Suppose we are given some Y ∈ Perf

together with a morphism Y♦ → X

♦; we need to produce (canonically andfunctorially) an untilt Y

� over X. Since (X�)♦ → X

♦ is a pro-étale G-torsor,we get a pullback diagram

Y♦ ��

��

Y♦

��(X

�)♦ ��

X♦

for some Y ∈ Perf, where the horizontal arrows are pro-étale G-torsors and allmaps are G-equivariant. In particular, we get a G-equivariant map Y → X

�

and thus a G-equivariant untilt Y� → X. By Proposition 3.10, Y

�= Y

�/G is

perfectoid, and is the desired untilt of Y .

38

4.2 Diamonds over a baseLet X be any analytic adic space over SpaZp. There are two natural ways ofdefining “diamonds fibered over X”:

1. We could define them by redoing the definition of diamonds with Perfproet

replaced by the site �Perfproet

/X.

2. We could define them as diamonds equipped with a morphism to X♦.

Proposition 4.3. The two preceding definitions are canonically equivalent.

Proof. Let D be a diamond equipped with a morphism to X♦. There is clearly

an identificationSh(Perf

proet)/X♦ ∼= Sh(Perf

proet

/X♦ ),

so we can regard D as a sheaf on Perfproet

/X♦ . Now the key point is that we have(by the proof of 4.2) an equivalence of sites

Perfproet

/X♦ ∼= �Perfproet

/X,

so a sheaf equipped with a morphism to X♦ “untilts” to a sheaf on �Perf

proet

/X.

We leave the remaining details to the interested reader.

The upshot is that for K/Qp a nonarchimedean field, diamonds over SpdK canbe interpreted as functors on the category of perfectoid spaces over Spa K, oreven as covariant functors from perfectoid (K,OK)-algebras to Sets, and con-versely, it makes sense to ask whether a given functor from perfectoid (K,OK)-algebras to Sets or a given pro-étale sheaf on the category of perfectoid spacesover Spa K is a diamond over SpdK. In any case, we’ll use all these equivalenceswithout further comment.

4.3 Self-products of the diamond of SpaQp

By Proposition 2.20, the diamond (SpdQp)n is well-defined. This is an intricate

object for n > 1, no longer in the essential image of the functor (−)♦. We note,

among other things, that |(SpdQp)n| has Krull dimension n− 1. This seems to

suggest that the (nonexistent) structure map “SpaQp → SpaF1” has relativedimension one.

4.4 The sheaf SpdZp

Recall the presheaf SpdZp on Perfproet. The diamond SpdQ

pis naturally a

subfunctor of SpdZp, and there is also a subfunctor “SpdFp” corresponding to

characteristic p untilts (of which there aren’t very many). The object SpdZp isnot a diamond, but it comes close:

39

Proposition 4.4. For any S ∈ Perf, there is a canonical analytic adic space“S × SpaZ

p” over SpaZ

psuch that

(S × SpaZp)♦ ∼= S

♦ × SpdZp.

In particular, S♦ × SpdZp is a diamond and SpdZp is a sheaf.

Proof. The idea is as follows: Given S = Spa(A, A+) ∈ Perf with � ∈ A any

choice of pseudouniformizer, set

S × SpaZp

= SpaW (A+) � {x | |[�]|x = 0}.

Thi turns out to be an honest adic space, independent of the choice of �, andthe analogous space for general S is then constructed by gluing. See Sections11.2-11.3 in the Berkeley notes.

Now we sketch the identification (S × SpaZp)♦ ∼= S

♦ × SpdZp. We mayassume S = Spa(A, A

+) is affinoid. For any Y = Spa(R,R

+) ∈ Perf, the

Y -points of (S × SpaZp)♦ are given by triples (Y

�, ι, f) where the data of

f : Y�= Spa(R

�, R

�+) → S × SpaZp

is equivalent to the data of a ring map f : W (A+) → R

�+ such that f([�])

is invertible in R�. Reducing this ring map mod (p, [�]) gives a ring map

A+/� → R

�+/(p, [�]) which after applying (−)

� gives a map A+ → R

��+, whichinduces a corresponding map f

�: Y

�� → S on Spa’s. Composing with ι−1 we

get a morphism f ◦ ι−1

: Y → S, and the data (Y�, ι, (f ◦ ι

−1)♦) corresponds

exactly to a Y -point of S♦ × SpdZp.

We can bootstrap this a bit:

Proposition 4.5. For any diamond D, the sheaf D × SpdZp is a diamond.

Proof. Choose a surjective and pro-étale map X♦ → D for some X ∈ Perf;

then X♦ × SpdZp → D × SpdZp is surjective and pro-étale, and the source is

a diamond, so we conclude by Proposition 2.2.

We also note the following:

Proposition 4.6. Given any S ∈ Perf, there is a natural bijection betweenuntilts (S

�, ι) and sections s of the projection S

♦×SpdZp → S♦, and any untilt

induces a closed immersion i : S� → S × SpaZp such that

s : S♦ (ι

♦)−1

→ S��♦ ∼= S

�♦ i♦→ S

♦ × SpdZp

gives the corresponding section. This also yields a natural bijection betweenuntilts of S and closed immersions S

� → S × SpaZp for which the compositemap S

�♦ → S♦ × SpdZp → S

♦ is an isomorphism.

40

Let (SpdZp)n

+⊂ (SpdZ

p)n be the subfunctor whose Y -points parametrize

n-tuples of untilts (Y�

i, ιi) (1 ≤ i ≤ n) with at least one of the Y

�

i’s living over

SpaQp. We leave it to the reader to show that (SpdZp)n

+is a diamond, covered

by the open subdiamonds

Di = SpdZp × · · ·× SpdZp × SpdQp × SpdZp × · · ·× SpdZp

(with SpdQp in the ith position) for 1 ≤ i ≤ n. Is this the largest subfunctorof (SpdZp)

n which is naturally a diamond?

4.5 The diamond B+

dR/Filn

For any perfectoid Tate ring R/Qp, we have the de Rham period ring B+

dR(R),

defined as the completion of W (R�◦

)[1

p] along the kernel of the natural surjection

θ : W (R�◦

)[1

p] � R. When R = Cp, this is the usual Fontaine ring B

+

dR. In

general ker θ is principal and generated by some non-zerodivisor ξ, so B+

dR(R) is

filtered by the ideals Fili= (ker θ)

i with associated gradeds griB+

dR(R) � ξ

iR.

Let B+

dR/Fil

n → SpdQp be the functor whose sections over a given Spa(R,R+)♦ →

SpdQp are given by the set B+

dR(R

�)/Fil

n, where R� is the untilt of R deter-

mined by the given map to SpdQp.

Theorem 4.7. The functor B+

dR/Fil

n is a diamond.

This is Proposition 18.2.3 in the Berkeley notes, where a “qpf” proof is given.We complement this with a “pro-étale” proof here.

Proof. Induction on n. The case n = 1 is clear, since B+

dR/Fil

1 ∼= A1,♦ whereA1

= ∪n≥1SpaQp �pnx� is the affine line over SpaQ

p.

Let B = (SpaZcyc

p[[T ]])η, and let B = lim←ϕ B where ϕ is the endomorphism

of B given by the map T �→ (1 + T )p − 1. Explicitly, B = (Spa A)η where A is

the (p, T )-adic completion of

Zcyc

p[[T ]][T1, T2, . . . ]/(ϕ(T1)− T,ϕ(T2)− T1, . . . ,ϕ(Ti+1)− Ti, . . . ).

Then B is perfectoid, and the map π : B → A1 characterized by π∗x = log(1+T )

is a perfectoid pro-étale covering of A1. (More precisely, B♦ → A1,♦ is a pro-étale G-torsor, with G = Qp(1) � Z×

p.)

Claim: There is an isomorphism of functors

B+

dR/Fil

n ×A1,♦ B♦ � B+

dR/Fil

n−1 ×SpdQp B♦

for any n ≥ 2.Granted this claim, we deduce the theorem as follows: Since B♦ → A1,♦ is

surjective and pro-étale, the map

B+

dR/Fil

n ×A1,♦ B♦ → B+

dR/Fil

n

41

is surjective and pro-étale, so by Proposition 2.2 it’s enough to show thatB

+

dR/Fil

n ×A1,♦ B♦ is a diamond. But the Claim together with Proposition2.19 reduce this to the fact that B

+

dR/Fil

n−1 is a diamond, which is exactly ourinduction hypothesis.

It remains to prove the claim. In general, the map θ induces a natural trans-formation B

+

dR/Fil

n → A1,♦ of functors over SpdQp. Let s : B → B+

dR/Fil

n bethe natural transformation sending r = (r0, r1, r2, . . . ) ∈ B(R

�) to log[1 + r

�]

where r�

= (r0, r1, . . . ) ∈ (R�◦

/p)�. Then (θ ◦ s)(r) = log(1 + r0) = π(r). In

particular, the diagramB♦

π

��s

�����������

B+

dR/Fil

n θ �� A1,♦

commutes, and π is surjective and pro-étale. The claim is then deduced asfollows: given (a, b) ∈ B

+

dR/Fil

n × B♦ with θ(a) = π(b), the element a − s(b)

lives in ker θ ⊂ B+

dR/Fil

n, so we get a natural isomorphism

B+

dR/Fil

n ×A1,♦ B♦ ∼= ker θ ×SpdQp B♦

(a, b) → (a− s(b), b)

(x + s(y), y) ← (x, y).

Furthermore, after base change to SpdQcyc

pwe have an exact sequence

0 → B+

dR/Fil

n−1 ×SpdQp SpdQcyc

p

t→ B+

dR/Fil

n ×SpdQp SpdQcyc

p→ A1,♦

Qcyc

p→ 0

where t is the usual log[ε] of p-adic Hodge theory, so we get an isomorphism

ker θ ×SpdQp SpdQcyc

p� B

+

dR/Fil

n−1 ×SpdQp SpdQcyc

p

after making a choice of t. But B naturally lives over Qcyc

p, so then

ker θ ×SpdQp B♦ ∼= (ker θ ×SpdQp SpdQcyc

p)×SpdQcyc

pB♦

��B

+

dR/Fil

n−1 ×SpdQp SpdQcyc

p

�×SpdQcyc

pB♦

� B+

dR/Fil

n−1 ×SpdQp B♦,

and the claim follows.

5 Moduli of shtukasIn this section, we construct moduli spaces of mixed-characteristic local shtukasand explain their relation with Rapoport-Zink spaces and local Shimura vari-eties. The material in this section follows closely the ideas of Caraiani-Scholze,Fargues-Fontaine, Kedlaya-Liu, Scholze and Scholze-Weinstein.

42

Let E/Qp be a finite extension with residue field Fq; set E = �Eunr . LetG/Spec E be a connected reductive group, which we assume for simplicity isquasisplit. Fix a Borel subgroup B and maximal torus T ⊂ B defined over E,and let µ ∈ X∗(T)dom be a dominant cocharacter. A local shtuka datum is atriple D = (G, µ, b) where G and µ are as specified, and b ∈ G(E) is an elementwhose σ-conjugacy class [b] lies in B(G, µ

−1). 7

For any perfectoid space S over Fq, the datum of b gives rise to a G-bundleEb,S on the relative Fargues-Fontaine curve XS,E , functorially in S, whose iso-morphism class depends only on the σ-conjugacy class of b. Our goal in thissection is to explain the following definition and theorem, with all ingredientscarefully laid out.

Definition 5.1. The moduli space of shtukas associated with the datum D isthe functor ShtD,∞ : PerfFq

→ Sets sending S ∈ PerfFqto the set of isomor-

phism classes of tuples(S

�, ι,F , u,α)

where (S�, ι) is an untilt of S over E, F is a G-bundle over XS , u : F|XS�S�

∼→Eb,S |XS�S� is a µ-positioned modification of Eb,S , and α : Etriv,S

∼→ F is aG-bundle isomorphism.

The space ShtD,∞ is something like an infinite-level Rapoport-Zink space.In particular, the following heuristic might be helpful in parsing the definitionof ShtD,∞(S):

• the data of b is “like” the p-divisible group H = Hb ×FqS over S, with

Hb/Fq the p-divisible group whose Dieudonne module is determined by b,

• the data of (S�, ι), F and u “like” a quasideformation H of H to S

�,

• the data of α is “like” a trivialization of the rational Tate module VpH.

When G = GLn and µ is minuscule this heuristic can be made into literal truth,as we’ll see below, but in general the space ShtD,∞ is unrelated to p-divisiblegroups.

In any case, this is a very rich object. First of all, it’s obviously fiberedover Spd E, and tuples of this form have no automorphisms. We will see thatShtD,∞ defines a sheaf on Perf

proet

/Spd E. Next, the locally profinite group G :=

G(E) = Aut(Etriv) acts on ShtD,∞ by sending α to α ◦ g−1, and the locally

profinite group Jb := Jb(E) < Aut(Eb) acts by sending u to j ◦ u; these twogroup actions obviously commute. There are also two period maps on ShtD,∞,the Grothendieck-Messing period map

πGM : ShtD,∞ → GrD,GM � GrG,µ ×Spd E Spd E

7For clarity, we do not take a maximally general setup here: one can instead consider a

tuple of dominant cocharacters µ1, . . . , µn, and then b should satisfy [b] ∈ B(G,−P

1≤i≤n µi).

43

and the Hodge-Tate period map

πHT : ShtD,∞ → GrD,HT∼= GrG,−µ ×Spd E Spd E.

(Here GrG,µ is an open Schubert cell in the BdR-affine Grassmannian GrG de-fined below.) The map πGM interprets u as a µ-positioned modification of Eb

along S�, while the map πHT interprets α

−1 ◦ u−1 as a −µ-positioned modi-

fication of Etriv along S�. There are natural actions of Jb and G on GrD,GM

and GrD,HT, respectively, and πGM and πHT are then G × Jb-equivariant for(respectively) the trivial actions of G and Jb on their targets.

Theorem 5.2. Let the notation and assumptions be as above. Then:i. The image of the period morphism πGM is a non-empty, open and partially

proper subdiamond Gradm

D,GMof the diamond GrD,GM, stable under the action of

Jb.ii. The induced morphism πGM : ShtD,∞ → Gr

adm

D,GMis representable and

pro-étale, and makes ShtD,∞ into a pro-étale G-torsor over Gradm

D,GM. In partic-

ular, ShtD,∞ is a diamond over Spd E.iii. For any open compact subgroup K ⊂ G, the quotient ShtD,K = ShtD,∞/K

is a diamond étale over GrD,GM, and

ShtD,∞ → ShtD,K

is a pro-étale K-torsor. The fibers of the induced map ShtD,K → Gradm

D,GMover

geometric points are identified with the discrete set G/K.iv. When µ is minuscule, the diamonds GrD,GM, Gr

adm

D,GMand ShtD,K are

in the essential image of the functor (−)♦ from smooth rigid analytic spaces

over Spa E. In particular, the rigid space MD,K over Spa E such that M♦D,K

∼=ShtD,K is the local Shimura variety with K-level structure associated with Dsought by Rapoport and Viehmann.

v. When µ is minuscule and G = GLn, there is a natural isomorphismShtD,∞ ∼= M♦

Hb,∞ compatible with all structures, where MHb,∞ is an infinite-level Rapoport-Zink space.

Let us note right away that parts i.-iii. of the previous theorem are anessentially immediate consequence of the ideas developed in the Berkeley notes(although the definition of ShtD,∞ is only implicit there). The key result inthe proof of part iv. is a theorem of Caraiani-Scholze asserting that whenµ is minuscule, GrG,µ is canonically identified with the diamond F�

♦G,µ

of acertain rigid analytic flag variety. Finally, we deduce part v. from an “explicit”description of MHb,∞ in terms of p-adic Hodge theory due to Scholze-Weinstein.

5.1 The Fargues-Fontaine curveThroughout the rest of Section 5, E/Qp denotes a finite extension with uni-formizer π and residue field Fq = Fpf . For any perfect Fq-algebra R, set

44

WOE (R) = W (R) ⊗W (Fq) OE , so R �→ WOE (R) gives a functor from per-fect Fq-algebras to π-torsion-free π-adically complete strict OE-algebras. Letϕ = ϕq = ϕ

f ⊗ id be the natural q-Frobenius on WOE (R). Let E = WOE (Fq)[1

p]

be the completion of the maximal unramfied extension of E, so σ = ϕ generatesGal(E/E).

Let S ∈ PerfFq be given. In this section we construct analytic adic spacesYS and XS over Spa E such that Y♦

S∼= S

♦×Fq SpdE and X♦S∼= S

♦/Frob

Zq×Fq

SpdE, functorially in S. (If we need to emphasize E, we write XS,E etc.) WhenS = Spa(F,OF ) for F/Fq a perfectoid field, XS = XF,E is the adic Fargues-Fontaine curve.

Suppose first that S = Spa(A, A+) is affinoid perfectoid, and let � ∈ A

+ beany choice of pseudouniformizer. Then we set

YS = SpaWOE (A+) � {x | |[�]p|x = 0}.

Here WOE (A+) is given the (p, [�])-adic topology. The space YS is in fact an

honest adic space: the affinoid adic spaces

YS,n = SpaWOE (A+)

�[�]

n

p,

pn

[�]

�[1

p] ⊂ YS

sit nestedly as YS,1 ⊂ YS,2 ⊂ · · · ⊂ YS,n ⊂ · · · with YS = ∪n≥1YS,n, and adirect calculation verifies that each YS,n is preperfectoid and hence honest. Bygluing we obtain YS for a general S ∈ PerfFq . The Frobenius ϕq on YS turnsout to be properly discontinuous, and we then set XS = YS/ϕ

Zq. This is the

relative Fargues-Fontaine curve over S.The adic spaces XS and YS don’t admit a morphism to S, but they behave

as if they do. For example, the identification X♦S∼= S

♦/Frob

Zq×Fq SpdE gives

a natural continuous projection |XS |→ |S| defined as the composition

|XS | = |X♦S|→ |S♦

/FrobZq| ∼= |S♦| = |S|

(using that Frobq acts trivially on |S|). Any morphism T → S in PerfFq inducesa canonical morphism XT → XS compatible with the previously defined mapon topological spaces, and if T → S is “blah” where “blah” is any one of theconditions “closed immersion”, “open immersion”, “finite étale”, “étale”, etc., thenXT → XS is also “blah”. We also note that, for any untilt (S

�, ι) of S over E,

we get a canonical closed immersion i : S��→ XS of E-adic spaces such that the

composite|S| ∼= |S�| |i|→ |XS |→ |S|

is the identity.Let Bun(XS) denote the category of vector bundles on XS , with morphisms

given by isomorphisms. Note that for any morphism f : T → S we get a canon-ical pullback f

∗: Bun(XS) → Bun(XT ) compatible with all bundle operations.

Now fix a reductive algebraic group G/Spec E, and let Rep(G) = {(ρ, V )}denote the tensor category of algebraic representations ρ of G on finite-dimensionalE-vector spaces V .

45

Definition 5.3. A G-bundle on XS is an exact additive tensor functor

E : Rep(G) → Bun(XS)

(ρ, V ) �→ ρ ◦ E .

We write BunG(XS) for the category of G-bundles on XS , with morphisms givenby natural isomorphisms of tensor functors. When G = GLn/E, we identifyBunG(XS) with rank n vector bundles on XS in the obvious way.

Remark. One might hope that G-bundles on XS could also be defined suitablyas étale or pro-étale Gan-torsors8 over XS , but this seems technically difficult:the trivial G-bundle would then correspond to the adic space XS ×Spa E Gad,but it’s not clear (to DH) that this fiber product exists as an honest adic space.

We define BunG as the groupoid over PerfFq for which BunG(S) is thecategory of G-bundles on XS .

Theorem 5.4 (Fargues, Scholze). BunG is a stack for the pro-étale topology.

Proof sketch. This amounts to showing that the fibered category

U/S �→ {vector bundles onYU}

is a stack on Sproet; since YS is preperfectoid, this can be reduced to showingthat the fibered category Y/X �→ {vector bundles on Y } is a stack on Xproet forany X ∈ �Perf, which can be checked directly.

When S = Spa(C,OC) is a geometric point, there is a complete classificationof G-bundles on XS due to Fargues-Fontaine and Fargues. Precisely, given anyS ∈ PerfFq

and any b ∈ G(E), a recipe of Fargues recalled below produces aG-bundle Eb,S on XS whose isomorphism class depends only on the σ-conjugacyclass [b] of b: In particular we get a map

B(G) → BunG(XS)

[b] �→ [Eb,S ].

Theorem 5.5 (Fargues-Fontaine for G = GLn, Fargues for general G). WhenS = Spa(C, C

+) is a geometric point, the map B(G) → BunG(XS) is essentially

surjective on objects, and Eb,S � Eb�,S iff [b] = [b�].

As one consequence of this theorem, given any S ∈ PerfFq and any G-bundleE over XS , we get a function |S| → B(G) sending any s = Spa(K, K

+) ∈ S to

the unique conjugacy class [b(s)] such that s∗E � Eb(s),s; here s = Spa(C, C

+)

is any geometric point over s.We recall Fargues’s definition of Eb,S . Suppose first that S = Spa(A, A

+) ∈

PerfFqis affinoid perfectoid. A G-bundle on XS is the same as a ϕ-equivariant

G-bundle on YS . On the other hand, there is a natural functor from ϕ-modules8(Here Gan

denotes the adic group over Spa E associated with G.)

46

with G-structure over WOE (A+)[

1

p] to the category of ϕ-equivariant G-bundles

on YS . Given b, we define a ϕ-module with G-structure Mb over WOE (A+)[

1

p]

as follows: Mb is the tensor functor on Rep(G) sending (ρ, V ) to

WOE (A+)[

1

p]⊗

E(V ⊗E E) ∼= WOE (A

+)[

1

p]⊗E V

with the Frobenius action given by ϕ ⊗ ρ(b)(id ⊗ σ). We then define Eb,S asthe associated G-bundle on XS . This construction is clearly functorial in S, sowe obtain the bundle Eb,S for general S by gluing. Note that for any morphismf : T → S of perfectoid spaces over Fq, the pullback map f

∗ on G-bundlesinduces a canonical isomorphism f

∗Eb,S∼= Eb,T . In other words, S �→ Eb,S

behaves “as if” it were pulled back from a bundle Eb on the (nonexistent) XSpaFq

.Note also that if we take b = id, then Eid,S recovers the trivial G-bundle (andmakes sense for any S ∈ PerfFq ).

The automorphism group of the bundle Eb turns out to be quite rich:

Proposition 5.6. Let Jb denote the group sheaf on PerfFqgiven by S �→

Aut(Eb,S). Then Jb is the sheafification of the presheaf sending any S = Spa(A, A+) ∈

PerfFqto the group

Jb(S) =

�g ∈ G(B+

crys,E(A

+/�)) | g = bϕq(g)b

−1

�.

Furthermore, Jb is canonically a semidirect product Jb∼= Ub � Jb, where Jb =

Jb(E) denotes the E-points of the algebraic group

Jb(A) =

�g ∈ G(A⊗E E) | g = bσ(g)b

−1

�

and where Ub is defined as the subgroup which acts trivially on the graded piecesof the Harder-Narasimhan filtration of the vector bundle ρ◦Eb,S for any (ρ, V ) ∈Rep(G).

This object deserves further study. We note that Jb = Jb exactly when b

is basic (and in particular, if b = id, we have Jid = G(E)); in general, Ub is anontrivial “unipotent group” of “dimension �νb, 2ρ�.” The sheaf Jb is an absolutediamond in the sense of §7; in particular, the group sheaf J

b,E= Jb ×Fq

Spd E

(i.e., the restriction of Jb to Perf/Spd E

) is provably a group diamond over Spd E.Finally, we briefly study “trivial” G-bundles on XS .

Definition 5.7. Given S ∈ PerfFq , a G-bundle E on XS is pointwise-trivialif for all geometric points s = Spa(C, C

+) → S, the pullback of E to Xs is

(non-canonically) isomorphic to Eid,s.

Theorem 5.8. Given any S ∈ PerfFq together with a pointwise-trivial G-bundleE on XS, the functor

T rivE/S : Perf/S → Sets

{f : T → S} �→ IsomBunG(XT ) (Eid,T , f∗E)

is representable by a perfectoid space pro-étale over S, and the map T rivE/S → S

a pro-étale G-torsor.

47

Proof. A more general result is proved in [Han16].

5.2 The de Rham affine GrassmannianRecall that for any perfectoid Tate ring R/Qp, we have the de Rham periodrings B+

dR(R) and BdR(R).

Fix a connected reductive quasisplit group G/Spec E. Choose a maximaltorus T and a Borel B containing it, both defined over E, and let X∗(T)dom ⊂X∗(T) be the associated set of dominant cocharacters. The functor of interestin this section is the following:

Definition 5.9. The de Rham affine Grassmannian GrG is the functor onperfectoid (E,OE)-algebras sending (R,R

+) to the set of (isomorphism classes

of) G-torsors over Spec B+

dR(R) equipped with a trivialization over Spec BdR(R).

We regard GrG as a functor fibered over SpdE in the obvious way.For any cocharacter µ ∈ X∗(T)dom, let GrG,≤µ ⊂ GrG be the functor

on perfectoid (E,OE)-algebras sending (R,R+) to the set of G-torsors over

Spec B+

dR(R) equipped with a µ-bounded trivialization over Spec BdR(R).

One checks directly that GrG,µ is a sheaf on Perfproet

/Spd E. We remark that

GrG,≤µ contains a subfunctor GrG,µ where the trivialization has relative po-sition given exactly by µ; this coincides with GrG,µ when µ is minuscule, butin general there is a nontrivial stratification GrG,≤µ“ = ”

�ν≤µ

GrG,ν . Forcompleteness, we also state a Tannakian interpretation of GrG,µ and GrG,≤µ:

Proposition 5.10. GrG,≤µ (resp. GrG,µ) is the functor on perfectoid (E,OE)-algebras sending (R,R

+) to the set of associations

Λ : (ρ, V ) ∈ Rep(G) → {ΛV ⊂ V ⊗E BdR(R) a B+

dR(R)−lattice}

compatible with tensor products and short exact sequences such that for all(ρ, V ) ∈ Rep(G) and all geometric points s = Spa(C, C

+) → Spa(R,R

+), there

is an E-basis v1, . . . , vdimV of V and a B+

dR(C)-basis e1, . . . , edimV of s

∗ΛV such

that

e1

e2

...edimV

= (ρ ◦ ν)(ξ) ·

v1

v2

...vdimV

for some generator ξ of ker�θ : B+

dR(C) → C

�and some ν ∈ X∗(T)dom with

ν ≤ µ (resp. with ν = µ).

Theorem 5.11 (Scholze). The sheaf GrG,≤µ is a qc spatial diamond overSpdE, and GrG,µ is an open subdiamond of GrG,≤µ.

This is one of the main theorems from Peter’s Berkeley course; the proof isdifficult.

We can also interpret GrG,≤µ and GrG,µ in terms of vector bundles onthe Fargues-Fontaine curve. More precisely, we have the following proposition,which follows easily from Theorem 8.9.6 in [KL15]:

48

Proposition 5.12. The sheaf GrG,≤µ (resp. GrG,µ) is the functor on perfectoid(E,OE)-algebras sending (R,R

+) (with associated S = Spa(R,R

+), X = XS�

and i : S �→ X ) to the set of isomorphism classes of pairs (E , u) where E is aG-bundle on X and

u : E|X�S

∼→ Etriv|X�S

is a µ-bounded (resp. µ-positioned) modification of the trivial G-bundle alongS.

Remark. Our convention on the meaning of “µ-bounded” is that when G = GLn

and µ = (k1 ≥ k2 ≥ · · · ≥ kn) with kn ≥ 0, then E as in the previous propositionextends to a subsheaf of Etriv.

We also need a “twisted” form GrEbG,µ

of GrG,µ defined over Spd E:

Proposition 5.13. For any b ∈ G(E), let GrEbG,µ

be the functor on perfectoid(E,O

E)-algebras sending (R,R

+) (with associated S = Spa(R,R

+), X = XS�

and i : S �→ X ) to the set of isomorphism classes of pairs (F , u) where F is aG-bundle on X and

u : F|X�S

∼→ Eb,S� |X�S

is a µ-bounded modification of the bundle Eb,S� along S. Then GrEbG,µ

is a dia-mond, canonically isomorphic to GrG,µ ×Spd E Spd E, and the group diamondJ

b,E= Jb ×Fq

Spd E acts naturally on GrEbG,µ

.

Remark 5.14. Suppose b ∈ B(G, µ−1

) is the unique µ−1-ordinary element; then

the dimension of Jb,E

coincides with the dimension of GrEbG,µ

. It seems reason-able to conjecture that in this case,

i. The action of Jb,E

on GrEbG,µ

has a unique open orbit, andii. This open orbit coincides with the admissible locus Gr

Eb,adm

G,µdefined

below.Is there an “orbit-theoretic description” of Gr

Eb,adm

G,µfor more general b?

5.3 Moduli of shtukasProof of Theorem 5.2,i.-iv. Given a local shtuka datum D = (G, µ, b), we setGrD,GM := Gr

EbG,µ

. Recall this is the diamond over Spd E whose S-points overa given (S

�, ι) ∈ (Spd E)(S) correspond to µ-positioned modifications (F , u) of

Eb,S along the graph of S�. We then get a morphism

πGM : ShtD,∞ → GrD,GM

given by forgetting α. We define Gradm

D,GMas the subfunctor of GrD,GM parametriz-

ing F ’s which are pointwise-trivial: precisely, a given S → GrD,GM factors overGr

adm

D,GMiff for all geometric points s = Spa(C, C

+) → S the specialization s

∗Fof the associated F is isomorphic to the trivial G-bundle on Xs. An easy calcu-lation on geometric points shows that the image of πGM is exactly Gr

adm

G,GM, and

that the fibers of πGM over any geometric point of Gradm

D,GMform a G-torsor.

49

We need to prove that Gradm

D,GMis open and partially proper in GrD,GM. Let

S ∈ PerfFqbe any perfectoid space with a map f : S

♦ → GrD,GM over Spd E.By definition, this map gives rise to a µ-bounded modification FS of the G-bundle Eb,S over XS . We need to show that the locus in S where FS becomespointwise-trivial is open and partially proper. Let (ρ, V ) be any algebraic rep-resentation of G, so we get vector bundles ρ ◦ FS and ρ ◦ Fb,S with the first asa ρ ◦ µ-bounded modification of the second. An easy calculation (using the factthat b ∈ B(G, µ

−1)) shows that ρ ◦ FS has degree zero for any (ρ, V ), so the

locus where FS is trivial9 coincides with the locus where adG◦FS is trivial. Thisin turn coincides with the locus where the associated ϕ-module M(adG ◦ FS)

over RS is pointwise étale, so Theorem 9.3.6 of [KL15] finishes the job. By atheorem of Fontaine and Rapoport, Gr

adm

D,GMhas points over finite extensions

of E, so in particular is nonempty. (Here again we’re using the containmentb ∈ B(G, µ

−1).) This proves part i.

Summarizing, we have a surjective G-equivariant map πGM : ShtD,∞ →Gr

adm

D,GMwhose target is a diamond over Spd E and whose fibers over any ge-

ometric point form a G-torsor. A direct calculation shows the natural actionmap

G× ShtD,∞ → ShtD,∞ ×Gr

adm

D,GM

ShtD,∞

is an isomorphism. To prove part ii. of the theorem, we need to show that πGM

is representable and pro-étale. To show this, let f : S♦ → Gr

adm

D,GMbe any map,

so we get a pointwise-trivial G-bundle FS = ”f∗Funiv

” on XS . Then we get apullback diagram

T riv♦FS/S

��

��

S♦

f

��ShtD,∞

πGM �� Gradm

D,GM

with T rivFS/S defined as in Theorem 5.8. Theorem 5.8 shows that the upperhorizontal arrow here is pro-étale with source a representable sheaf, so πGM isrepresentable and pro-étale. This proves parts i. and ii. of the theorem. Partiii. follows immediately from Proposition 2.13.

For part iv., the essential point is the following result, proved by Caraiani-Scholze:

Proposition. There is a flag variety F�G,µ over Spa E together with anatural morphism

π : GrG,µ → F�♦G,µ

which is an isomorphism when µ is minuscule.In particular, since GrD,GM � GrG,µ ×Spd E Spd E, the diamond GrD,GM

for minuscule µ is in the essential image of the functor (−)♦ from smooth rigid

analytic spaces over Spd E. Since ShtD,K is étale over GrD,GM, Proposition3.17 implies that ShtD,K also comes this same category via (−)

♦.9This is not quite true; one still needs to take the Kottwitz class into account.

50

In the rest of this section, we prove part v. of the theorem. The key ingre-dient here is Theorem D from Scholze-Weinstein, which we recall below.

Set G = GLn/E, so a G-bundle on XS is just a rank n vector bundle. Letµ be as in the theorem, so µ = (1, . . . , 1, 0, . . . , 0) with d 1’s and n − d 0’s forsome 0 ≤ d ≤ n. Let b ∈ GLn(E) be an element with [b] ∈ B(G, µ

−1), so we

get a local shtuka datum D. Let Hb be the associated π-divisible OE-moduleover Fq, and let M(Hb) be the (covariant) rational OE-Dieudonne module ofHb, i.e. M(Hb) = E

n with the endomorphism F given by F = πb ·σ. Note thatHb is well-defined up quasi-isogeny by our specification of M(Hb).

Now we explain how to construct a natural transformation ShtD,∞ →M♦Hb,∞

inducing the desired isomorphism.

Proposition 5.15. Let S = Spa(R,R+) with (R,R

+) any perfectoid (E,O

E)-

algebra, so we have our associated i : S → XS� . Then

H0(XS� , Eb,S�) ∼=

�M(Hb)⊗E

B+

crys,E(R

+/π)

�F⊗ϕ=π

andH

0(S, i

∗Eb,S�) ∼= M(Hb)⊗ER,

functorially in (R,R+). Under these isomorphisms, the natural map

θ :

�M(Hb)⊗E

B+

crys,E(R

+/π)

�F⊗ϕ=π

→ M(Hb)⊗ER

of vector spaces over E identifies with the map

H0(XS� , Eb,S�) → H

0(XS� , i∗i

∗Eb,S�) = H0(S, i

∗Eb,S�).

Proof. Unwind it all.

Next we recall Theorem D of [SW13], in the context of π-divisible OE-modules. Let MHb,∞ be the infinite level Rapoport-Zink space over Spa E

associated with Hb.

Theorem 5.16 (Scholze-Weinstein). The space MHb,∞ is isomorphic to thefunctor on perfectoid (E,O

E)-algebras sending (R,R

+) to the set of ordered

tuples

s1, . . . , sn ∈�M(Hb)⊗E

B+

crys,E(R

+/π)

�F⊗ϕ=π

satisfying the following two conditions:i. The cokernel W of the map

Rn

θ(si)−→ M(Hb)⊗ER

is a finite projective R-module of rank d.ii. For all geometric points x : Spa(C, C

+) → Spa(R,R

+), the induced

sequence of vector spaces

0 → En x

∗si−→

�M(Hb)⊗E

B+

crys,E(C

+/π)

�F⊗ϕ=π

→ W ⊗R C → 0

is exact.

51

On the other hand, we have the following easy reduction regarding ShtD,∞.

Proposition 5.17. As a functor on perfectoid (E,OE

)-algebras, ShtD,∞ sends(R,R

+) (with associated S = Spa(R,R

+), X = XS� and closed immersion

i : S �→ X ) to the set of injective OX -module morphisms u : On

X → Eb,S� suchthat coker u � i∗W for some rank d vector bundle W on S.

Maintain the setup of this proposition; going to global sections and usingProposition 5.15 plus the identification H

0(X ,OX ) ∼= E, we get

H0(u) : E

n → H0(XS� , Eb,S�) ∼=

�M(Hb)⊗E

B+

crys,E(R

+/π)

�F⊗ϕ=π

.

Let s1(u), . . . , sn(u) be the images of the standard basis vectors under this map.We’re going to show that this tuple satisfies the requirements of Theorem 5.16;this implies that the association u �→ si(u) defines a morphism

ξ : ShtD,∞ →M♦Hb,∞

of diamonds over Spd E.Pulling back the exact sequence

On

X → Eb,S� → i∗W → 0

of OX -modules along the closed immersion i gives an exact sequence

On

S→ i

∗Eb,S� →W → 0

of vector bundles on S. By [KL15], passage to global sections gives an equiv-alence from vector bundles on S to finitely generated projective R-modules10.Setting W = H

0(S,W), taking global sections in the previous exact sequence,

and using Proposition 5.15, we then get an exact sequence

Rn → M(Hb)⊗E

R → W → 0,

with the first arrow identified with θ(s1), . . . , θ(sn). This verifies the first con-dition in Theorem 5.16.

For the second condition, let η = Spa(C, C+)

x→ Spa(R,R+) be any geomet-

ric point, so we get a morphism fx : Xη� → XS� as described previously, and apoint iη : η �→ Xη� sitting in a pullback square

η

x

��

iη �� Xη�

fx

��S

i

�� XS�

10With bundle morphisms corresponding to R-module morphisms M → N having projective

cokernel

52

where the horizontal maps are closed immersions. Clearly f∗xEb,S� = Eb,η�

Pulling back the exact sequence

0 → On

XS�→ Eb,S� → i∗W → 0

along fx and applying H0(Xη� ,−) gives an exact sequence

0 → En → H

0(Xη� , Eb,η�) → H

0(Xη� , f

∗xi∗W) → 0,

exact on the right because H1(Xη� ,OX ) = 0. Using Proposition 5.15 again, the

middle term identifies with�M(Hb)⊗E

B+

crys,E(C

+/π)

�F⊗ϕ=π

, and the secondarrow then identifies with x

∗si. Finally, f

∗xi∗W = iη∗x

∗W, so

H0(Xη� , f

∗xi∗W) = H

0(Xη� , iη∗x

∗W) = H0(η, x

∗W) = W ⊗R C

and the exact sequence in question identifies with the sequence

0 → En x

∗si−→

�M(Hb)⊗E

B+

crys,E(C

+/π)

�F⊗ϕ=π

→ W ⊗R C → 0,

and thus the latter sequence is exact. This verifies the second condition inTheorem 5.16.

Proposition 5.18. The morphism ξ : ShtD,∞ → M♦Hb,∞ defined above is an

isomorphism.

Sketch. There are two possible ways of doing this. The first is to explicitlyconstruct ξ

−1 by carefully reversing all the steps in the definition of ξ.The second is to use the Grothendieck-Messing period maps. These sit in a

GLn(E)-equivariant diagram

ShtD,∞ξ ��

π�GM �����������

M♦Hb,∞

πGM

������

����

�

F�♦

where F� is the rigid analytic flag variety over Spa E whose (R,R+)-points

parametrize rank d projective R-module quotients of M(Hb) ⊗ER. Clearly

imπ�GM

⊆ imπGM; by Theorem C of Scholze-Weinstein, the opposite inclu-sion holds. Let F�

♦,adm denote the common image of these maps in F�♦,

so F�♦,adm ⊂ F� is open and partially proper. We already know that π

�GM

is apro-étale GLn(E)-torsor over F�

♦,adm; by Theorem D in Scholze-Weinstein, it’seasy to check that πGM is also a pro-étale GLn(E)-torsor over F�

♦,adm. Thefollowing easy proposition then implies ξ is an isomorphism.

Proposition 5.19. Let G be a sheaf on Perfproet, and let G be a locally profinite

group. Then any G-equivariant morphism f : F → F � of pro-étale G-torsorsover G is an isomorphism.

53

Proof. Pro-étale-locally on G, we have G-equivariant isomorphisms F � G ×G � F � by the definition of a pro-étale G-torsor, so our given f becomes anisomorphism pro-étale-locally on G, and then f must be an isomorphism sinceeverything is a sheaf.

5.4 The Newton stratificationMaintain the setup of Definition 5.9 and the discussion thereafter. FollowingCaraiani-Scholze, we use Proposition 5.12 and Fargues’s classification of G-bundles on the curve to define a G-stable Newton stratification of GrG,µ indexedby elements of B(G):

Definition 5.20. For any [b] ∈ B(G), the Newton stratum Grb

G,µis the fol-

lowing subfunctor of GrG,µ: given S ∈ PerfFq and a map f : S♦ → GrG,µ,

with induced G-bundle E over XS and µ-positioned modification u : E|XS�S�∼→

Etriv,S |XS�S� as in Proposition 5.12, then f factors over Grb

G,µif and only if for

all geometric points s : Spa(C, C+) → S we have s

∗E � Eb,s as G-bundles onXs.

In general, these strata are only defined as subfunctors, a priori; they alsogive a stratification of |GrG,µ| by locally closed subsets.

Proposition 5.21. The stratum Grb

G,µis nonempty if and only if b ∈ B(G, µ).

Remark 5.22. In the case of minuscule µ, this was proved by Rapoport in hisappendix to Scholze’s Lubin-Tate tower article.

Proof. “Only if” was proved by Caraiani-Scholze. “If” follows from the nexttheorem and the non-emptyness of the space ShtDb,∞ therein.

When b is the basic (resp. µ-ordinary) element of B(G, µ), the associatedstratum is an open (resp. closed) subdiamond of GrG,µ.

Theorem 5.23. For any b ∈ B(G, µ), consider the local shtuka datum Db =

(G, µ−1

, b). Then the stratum Grb

G,µcoincides with the image of the Hodge-Tate

period morphism

ShtDb,∞πHT→ GrDb,HT

∼= GrG,µ ×Spd E Spd E → GrG,µ.

Furthermore, letting Jb,E

denote the group sheaf on Perf/Spd E

defined as before,there is a natural action of J

b,Eon ShtDb,∞ extending the action of Jb and

equivariant for the trivial action on the target of πHT such that the action map

ShtDb,∞ × Jb,E

→ ShtDb,∞ ×GrDb,HTShtDb,∞

is an isomorphism and on geometric points we have Grb

G,µ(C) ∼= ShtDb,∞(C)/J

b,E(C).

54

This strongly suggests that we should regard Grb

G,µas the coarse moduli

space of the stack [ShtDb,∞/Jb,E

].However, something better is true. Precisely, note that Gr

b

G,µ⊂ GrG,µ is

defined by applying the recipe “F ⊂ |D| � F ⊂ D” from the proof of Theorem2.42 to a suitable subspace |Gr

b

G,µ| ⊂ |GrG,µ|. Caraiani-Scholze show using

results of Kedlaya-Liu that |Grb

G,µ| is locally closed in |GrG,µ|, and it’s clearly

generalizing: x : Spa(C, C+) → GrG,µ factors through Gr

b

G,µif and only if the

rank one generization Spa(C,OC) → GrG,µ so factors, because the category ofG-bundles on XSpa(R,R+) is totally insensitive to changing R

+! So now we mayapply Theorem 2.42 to deduce the following result.

Theorem 5.24. Each Newton stratum Grb

G,µis a diamond.

5.5 Admissible and inadmissible lociHere we consider some examples, in the setting of GLn over Qp. So, fix E = Qp,G = GLn, B the upper-triangular Borel, µ a B-dominant cocharacter withweights (k1 ≥ · · · ≥ kn). After twisting, we may assume that all weights of µ

are ≥ 0. Fix b ∈ GLn(Qp) with [b] ∈ B(G, µ−1

) and Newton cocharacter νb.WLOG we may replace b by a σ-conjugate lying in GLn(Qpd) for some (minimal)d ≥ 1, according to whether b satisfies an associated decency equation. We againunwind the definition of GrD,GM (with slightly lighter notation):

Definition 5.25. Gr = Grµ,b is the functor on perfectoid spaces over Qpd

sending S = Spa(A, A+) to the set of isomorphism classes of pairs (E , u), where

E is a rank n vector bundle on XS� and u : E �→ Eb,S� is an injection of vectorbundles which is an isomorphism away from i : S �→ XS� and which induces aµ-positioned modification

0 → E → Eb,S� → Q → 0

along the graph of S.

The last clause here means concretely that Q is an OX -module sheaf onXS� whose stalks are zero away from points lying in the subset |S| ⊂ |XS� |such that for any point x = Spa(K, K

+) → S, we have H

0(Xx� , x

∗Q) �

⊕n

i=1B+

dR(K)/Fil

ki .Inside Grµ,b we have the admissible locus Gr

adm

µ,b, and its complement Gr

inadm

µ,b.

To be clear: we’ve already seen that Gradm

µ,bis an open and partially proper sub-

diamond, so in particular |Gradm

µ,b|c ⊂ |Grµ,b| is open and specializing. Thus its

complement |Gradm

µ,b|c ⊂ |Grµ,b| is closed and generalizing; we define Gr

inadm

µ,bas

the subdiamond associated with |Gradm

µ,b|c ⊂ |Grµ,b| via Theorem 2.42.

What do these spaces look like? Note that in some sense, Grinadm

µ,bshould

usually be kind of a strange object: for example, when End(Eb) is a divisionalgebra, one can check that Grµ,b and Gr

adm

µ,bhave the same K-points for any

finite extension K/Qp. We’ll see examples like this where Grinadm

µ,b, which cannot

have points over finite extensions K/Qp, is nevertheless nonempty.

55

Example 5.26. Take n = 2, µ = (3, 0), b =

�1

p−3

�(so d = 2), so

Eb = O(3

2). The space Gr = Grµ,b is, morally speaking, a B+

dR/Fil

2-bundle overP1; in any case, it has dimension three. For any point x : Spa(C, C

+) → Gr we

get a modification0 → Ex → O(

3

2) → Q → 0

on XC where Q is a cyclic B+

dR(C)-module of length three. Since Ex has degree

0 and rank 2, and all slopes ≤ 1, the classification theorem of Fargues-Fontaineshows that E � O2 or E � O(1) ⊕ O(−1). These two possibilities correspondexactly to the admissible and inadmissible loci. Pushing this a bit further, onegets the following nice description of Gr

inadm:

Proposition (H.-Weinstein). The inadmissible locus here is canonically andJb-equivariantly isomorphic to the pro-étale sheaf over SpdQp2 which sends aperfectoid Qp2-algebras (A, A

+) to

�B+

crys(A)

ϕ2=p � 0

�/Q×

p.

Sketch. Let S = Spa(A, A+) be as in the statement, and let f : S → Gr

inadm

be an S-point. Let0 → E u→ Eb,S� → Q → 0

be the corresponding modification. Then Ex � O(1)⊕O(−1) at any geometricpoint x = Spa(C, C

+) → S, so by a theorem of Kedlaya-Liu we get a filtration

of E by line bundles0 → E>0 → E → E<0 → 0

where E>0 has slope one at all points. In particular, we have an isomor-phism E>0 � O(1) pro-étale-locally on S, which is moreover unique up to Q×

p-

ambiguity. Composing this isomorphism with u gives a nowhere-vanishing mapO(1) → O(3/2) up to Q×

p-ambiguity, or equivalently a nowhere-vanishing map

O → O(1/2) up to Q×p

-ambiguity. But this is just a nowhere-vanishing sectionof O(1/2) over XS� , i.e. an element of B+,ϕ

2=p

crys(A) � 0, again (pro-étale-locally)

up to Q×p

-ambiguity.

6 More on morphisms

6.1 Smooth morphismsClearly a smooth morphism of relative dimension zero should just be an étalemorphism; for nonzero relative dimension, Scholze has suggested the followingdefinition:

Definition 6.1 (Scholze). A morphism f : X → Y of diamonds is smooth ofpure dimension d > 0 if, pro-étale-locally on Y and analytic-locally on X, f canbe factored (if d > 0) into a composite

Xg→ W

h→ Y

56

where g is smooth of pure dimension d− 1 and h can be fit into a commutativediagram

Wh ��

q

��

B1

Y

pr

��W

h �� Y

whereB1

Y= “Y ×SpdFp SpdFp �T � ”

is the relative one-dimensional closed unit disk over Y11 (with pr the evident

projection), h is étale, and q realizes W as a pro-étale Γ-torsor over W for somepro-p group Γ.

A morphism f : X → Y of diamonds is smooth if X admits a coveringby open subdiamonds Xi such that each Xi → Y is smooth of some relativedimension di. If moreover sup

idi < ∞, we say f is finite-dimensional.

6.2 Morphisms locally of finite typeLemma 6.2. Let k be a perfect field, and let X be a reduced k-scheme locally offinite type which is equidimensional of dimension d. Then there exists a denseopen subscheme U ⊆ X which is smooth and equidimensional of dimension d.

Proof. One can just take U = Reg(X) to be the locus of points where OX,x isregular: this is open by a classical result, and dense since by the reducednessassumption it contains the generic point of any irreducible component of X. Cf.Tag 0B8X in the Stacks Project.

Lemma 6.3. Let k be a perfect field, and let X be a reduced finite-dimensionalk-scheme locally of finite type. Then we can find a finite chain of reduced closedsubschemes

X = Z0 ⊃ Z1 ⊃ Z2 · · · ⊃ Zn = ∅such that dimZi ≤ dimX − i and Zi � Zi+1 with its induced reduced structureis smooth over Spec k.

Proof. Easy downward induction on dimX, using the previous lemma to getstarted: let X0 (resp. X1) be the union of all dimX-dimensional (resp. < dimX-dimensional) irreducible components of X, and take

Z1 = X1 ∪ (X0 � Reg(X0))

with its induced reduced structure.11

I.e., B1Y has functor of points given by Hom(T

♦,B1

Y ) = Hom(T♦

, Y ) × H0(T,O

+T ) for

arbitrary T ∈ Perf.

57

Amusingly, the definition of a locally finite type diamond seems less acces-sible than the definition of a smooth diamond. Granted a workable notion ofsmoothness, though, the previous lemma suggests the following inductive defi-nition of a diamond being locally of finite type over an analytic field.12

Definition 6.4. Let F be a nonarchimedean field in which p is topologicallynilpotent, and let D be a locally spatial diamond over SpdF . Then D is locallyfinite type of dimension ≤ d if there exists an open subdiamond U ⊆ D withthe following properties:

i. The diamond U is smooth of dimension ≤ d.ii. |U| ⊆ |D| is specializing.iii. The locally closed subdiamond Z ⊂ D associated with the closed gener-

alizing subset |D| � |U| ⊆ |D| by Theorem 2.42 is locally finite type of relativedimension ≤ d− 1, and the morphism Z → D is quasicompact.

More generally, we say a locally spatial diamond D/SpdF is locally of finitetype if we can find a covering of D by open subdiamonds (Di ⊂ D)i∈I such thateach Di is locally finite type of dimension ≤ di for some unspecified di < ∞.

Here is the first sanity check.

Proposition 6.5. Let F be a perfect nonarchimedean field in which p is topo-logically nilpotent, and let X be any rigid analytic space over Spa F . Then X

♦

is locally finite type over SpdF . Furthermore, if X is quasicompact, then X♦

is locally finite type of dimension ≤ dimX.

Proof. Since X and Xred have the same diamond, we can assume that X is

reduced. Covering X by admissible open affinoid subspaces and using the com-patibility of (−)

♦ with open immersions, it clearly suffices to prove the secondclaim in the case where X = Sp(A) is a reduced affinoid.

Let A be any reduced affinoid F -algebra; since any affinoid algebra is excel-lent (cf. Conrad’s paper on irreducible components of rigid spaces), the subsetReg(A) ⊆ SpecA is Zariski-open, and then dense as well since by assump-tion it contains the generic points of all irreducible components. Feeding thisobservation into an easy variant of the proof of Lemma 6.2, we get (for anyfinite-dimensional reduced rigid space X) a chain

X = Z0 ⊃ Z1 ⊃ Z2 · · · ⊃ Zn ⊃ ∅

of reduced Zariski-closed rigid subspaces with exactly the same properties as inLemma 6.3.

Now, observe that if we take D = X♦ and U = (X � Z1)

♦ as in Definition6.4, then U is smooth of dimension ≤ dimX and |U| = |X|� |Z1| is indeed openand specializing inside |D| = |X|, so |Z1| ⊂ |D| is closed and generalizing andTheorem 2.42 then produces a subdiamond Z ⊂ D associated with |Z1|. Theproposition then follows by downward induction on dimX if we can identify Zwith the subdiamond Z

♦1⊂ X

♦. But this is just Proposition 3.19.12

DH: I am writing this out solely for my own amusement; there is no guarantee that this

is the “correct” definition. Presumably matters will be clarified in Peter’s forthcoming book.

58

Definition 6.6. Let F be as above. A locally spatial diamond D over SpdF isproper if it is locally finite type, separated, quasicompact, and partially properas a sheaf.

Here we say a sheaf F on Perf is partially proper if the natural restric-tion map F(Spa(R,R

+)) → F(Spa(R,R

◦)) is bijective for any characteristic p

perfectoid ring R and any subring R+ of integral elements.

Proposition 6.7. With this definition, X♦ is proper if X is any proper rigid

space over Spa F . Moreover, the closed Schubert cell GrG,≤µ is proper overSpdE.

7 Into the abyssIn this section we define categories of diamond stacks and smooth diamondstacks.

7.1 Absolute diamondsDefinition 7.1. An absolute diamond is a sheaf F on Perf such that F ×D isa diamond for any diamond D.

Note that unlike the category of diamonds, the category of absolute dia-monds has a final object, namely the "point" SpdFp. This is also the final objectin Sh(Perf). Note also the following handy “domino principle”: if D1 × · · ·×Di

is any finite product of absolute diamonds, then the whole product is a diamondas soon as one of the Di’s is a diamond.

Proposition 7.2. If E1 → D ← E2 is a diagram of absolute diamonds, thenE1×DE2 is an absolute diamond. In particular, the category of absolute diamondshas all finite limits.

Proof. Let T be any diamond; then

(E1 ×D E2)× T ∼= (E1 × T )×D×T (E2 × T ).

But each • × T on the right-hand side here is a diamond by assumption, andthe category of diamonds admits fiber products, so (E1×D E2)×T is a diamondas desired.

Proposition 7.3. If E1

f1→ D f2← E2 is a diagram of absolute diamonds and oneof the Ei’s is a diamond, then E1 ×D E2 is a diamond.

Proof. Write

E1 ×D E2∼= (E1 × E2)×f1×f2,(D×D),∆ D∼= (E1 × E2)×id×id×f1×f2,(E1×E2×D×D),id×id×∆ (E1 × E2 ×D).

Then by the domino principle, each of the three grouped direct products in thelower line here is a diamond, and diamonds admit fiber products.

59

According to Proposition 4.5, the sheaf SpdZp is an absolute diamond. Sofar, this is really the only (non-diamond) example we’ve seen in these notes.However, there are plenty of other interesting examples. Here’s just one ofthem:

Example 7.4. Fix coprime positive integers h, d, and set

b =

1

1

. . .1

p−d

∈ GLh(Qp).

Then for any S ∈ Perf, the recipe described in §5.1 produces a rank h vectorbundle Eb,S on the relative curve XS . (This is the bundle which is often denotedby “O(

d

h)”.) Then the “absolute Banach-Colmez space”

Bϕh=p

d

: Perf → Sets

S �→ H0(XS , Eb,S)

is an absolute diamond which is never a diamond. For example, when d = 1,this functor is (noncanonically) isomorphic to SpdFp[[T

1/p∞

]], which visiblyhas a “non-analytic point” corresponding to T = 0; but diamonds only have“analytic points”.

7.2 Stacks: first definitionsWe start with some useful general-nonsense observations. First, note that ifF is any sheaf on Perf, we can talk uniformly about perfectoid spaces over F ,(absolute) diamonds over F , and sheaves over F . This is just because Perf

and Dia both embed fully faithfully into Sh(Perf), and one can form the sliceSh(Perf)/F . This is of course compatible with the usual notions when F = hX

is representable. We also note that “sheaves G on Perf/F ” are trivially the sameas “morphisms G → F of sheaves on Perf”, i.e. one can “commute the slice”:Sh(Perf)/F = Sh(Perf/F ); I’ll use this freely without any further comment.

Definition 7.5. If F is any sheaf on Perf, a stack over F is a pro-étale stackin groupoids over Perf/F . If F = hSpdFp we speak of a stack over Perf.

Note that we have the chain of relations “perfectoid spaces over F ⊂ dia-monds over F ⊂ absolute diamonds over F ⊂ sheaves over F ⊂ stacks over F”for any F . Here (of course) “⊂” means fully faithful embedding of categories,and the final inclusion sends a sheaf G on Perf/F to the associated “categoryfibered in setoids” over Perf/F .

Definition 7.6. A 1-morphism Y → X of stacks over Perf is representable indiamonds if for any diamond D and any map D → X , the pullback Y ×X D

60

is (2-equivalent to the category fibered in setoids associated with)13 a diamondover D.

Example 7.7. i. Any morphism of absolute diamonds is representable in dia-monds. This is an immediate consequence of Proposition 7.3.

ii. A sheaf F is an absolute diamond if and only if the “structure morphism”F → SpdFp is representable in diamonds. This is trivial from the definition ofan absolute diamond.

iii. Any pro-étale G-torsor F → G is representable in diamonds. Thisimmediately follows from Theorem 2.17.

Proposition 7.8. Let X be a stack on Perf. Then the following are equivalent:i. The diagonal X → X × X is representable in diamonds.ii. Any morphism D → X with D an absolute diamond is representable in

diamonds.

Proof. Omitted.

Thanks to this proposition, the following definition is well-posed:

Definition 7.9. Let f : Y → X be a morphism of stacks over Perf, and let P bea property of morphisms of diamonds which is preserved under arbitrary basechange and which is pro-étale-local on the target. Then we say f has propertyP if f is representable in diamonds, and if for any diamond D and any mapD → X , the morphism of diamonds

Y ×X D → D

has property P .

The list of admissible P includes the properties “open immersion”, “locallyclosed embedding”, “finite étale”, “étale”, “smooth”, “surjective”, “v-cover”,... WhenY → X is a morphism of diamonds, this definition agrees with the usual meaningof these properties.

Example 7.10. The map SpdQp → SpdFp is smooth in this sense!

Warning: The structure morphism BunG → SpdFp is not representable indiamonds, so it’s not smooth in this sense! Therefore we need to do somethinga little more elaborate in order to declare BunG “smooth”.

7.3 Diamond stacksDefinition 7.11. A diamond stack is a stack X on Perf such that:

i. The diagonal X → X × X is representable in diamonds, andii. There exists an absolute diamond S and a morphism f : S → X which is

surjective and smooth.If moreover S in condition ii. can be chosen smooth, we say X is a smooth

diamond stack. We refer to a pair (S, f) as a chart for X .13

As usual, we’ll commit the abuse of never bringing up this parenthetical thing again.

61

There’s an obvious relative version of this definition, which we omit. Anyway,this is supposed to capture the stack BunG, which of course is the main dragonwe’re trying to slay.

In general it’s not obvious that BunG or anything like it is a diamond stack,let alone a smooth one.

Example 7.12. To shorten things I’ll write pt for SpdFp. Let D/Qp be theusual quaternion division algebra. Then I claim the gerbe [pt/D

×] is a smooth

diamond stack. This is not obvious, because the map pt → [pt/D×

] is notsmooth - a more exotic chart is required. We’ll obtain a suitable chart byjudiciously applying the following proposition.

Proposition 7.13. Let G be a locally profinite group, and let E → E � be asurjective G-equivariant morphism of absolute diamonds which is smooth as amorphism of stacks. Suppose moreover that E lives as a pro-étale G-torsor oversome absolute diamond D, so D ∼= E/G. Then the natural map

f : D = E/G → [E �/G]

is surjective and smooth as a morphism of stacks, and [E �/G] is a diamond stackwith (D, f) a chart.

First we check that [E �/G] satisfies condition i. of Definition 7.11. Thisamounts to the following lemma.

Lemma 7.14. Let S be an absolute diamond, and let D be an absolute diamondover S with an action of a locally profinite group G lying over the trivial actionon S. Then the (relative) diagonal of [D/G] is representable in diamonds.

Proof. Let us spell out the fibered category [D/G]. An object of [D/G] is aquadruple (U, P, π,ϕ) where U ∈ Perf/S , π : P → U is a pro-étale G-torsor, andϕ : P

♦ → D is a G-equivariant map compatible with the structure maps to S.A morphism

(U�, P

�, π�, ϕ�) → (U, P, π, ϕ)

in [D/G] is a morphism f : U� → U of perfectoid spaces over S together with

a G-equivariant morphism h : P� → P over f which induces a G-equivariant

isomorphism P� ∼= P ×π,U,f U

� and such that ϕ = ϕ� ◦ h. The functor p is given

by(U, P, π, ϕ) �→ U ∈ Perf/S .

In other words, the fiber category of [D/G] over a given U ∈ Perf/S is thegroupoid of pro-étale G-torsors over U equipped with a G-equivariant map toD.

Returning to the problem at hand, we need to show that for any U ∈ Perf/S

and any morphism U(x1,x2)−→ X × X , the sheaf

IsomU (x1, x2) = U ×X×X ,∆ X

62

is representable by a diamond. Specifying x1 and x2 amounts to specifyingquadruples (U, Pi, πi, ϕi) as above, and by definition IsomU (x1, x2) is the sheafon Perf/U sending f : T → U to the set of isomorphisms i : f

∗P

♦1

∼→ f∗P

♦2

fitting into a G-equivariant commutative diagram

D

f∗P

♦1

f∗π1

��

i

∼ ��

ϕ1

�����������

f∗P

♦2

ϕ2

��

f∗π1����

����

���

T♦

of diamonds / absolute diamonds over S. In particular, IsomU (x1, x2) is nat-urally a subfunctor of the sheaf IsomGTor(U)(P1, P2) on Perf/U which sendsf : T → U to the set of G-equivariant isomorphisms i : f

∗P1

∼→ f∗P2 over

T . We claim that these sheaves naturally sit in a G-equivariant commutativediagram

D ∆ �� D ×S D

IsomU (x1, x2)×U♦ P♦1

a

��

��

ω

��

IsomGTor(U)(P1, P2)×U♦ P♦1

b

��

��IsomU (x1, x2)

�� IsomGTor(U)(P1, P2)

of sheaves on Perf/S where both squares are pullback squares. For the lowersquare this is true by construction. For the upper square, we need to definethe morphism b. Giving a T -point of IsomGTor(U)(P1, P2) ×U♦ P

♦1

is the sameas specifying a triple (f, i, s) consisting morphism f : T → U together witha G-equivariant isomorphism i : f

∗P1

∼→ f∗P2 and a section s : T → f

∗P1

of the map f∗P1 → T . Then b sends such a triple (f, i, s) to the T point of

D ×S D given by T(ϕ1◦s,ϕ2◦i◦s)−→ D × D, which factors through D ×S D by the

commutativity of the diagrams

P♦i

πi

��

ϕi �� D

��U

♦ �� S

for i = 1, 2. Since i and the ϕ’s are G-equivariant, b is G-equivariant. Further-more, one sees by direct inspection that b(f, i, s) factors through ∆(D) if andonly if ϕ1 ◦s = ϕ2 ◦ i◦s as morphisms T → D; varying s using the G-action, the

63

G-equivariance of b shows that ϕ1 ◦s = ϕ2 ◦ i◦s for one s if and only if the sameequality holds for all s : T → f

∗P1, if and only if (f, i) comes from a T -point of

IsomU (x1, x2). This shows that the upper square is a pullback square.Supposing momentarily that IsomGTor(U)(P1, P2)×U♦ P

♦1

is a diamond, wefinish the argument as follows. Going back to the diagram, observe that D andD×S D are absolute diamonds; since moreover IsomGTor(U)(P1, P2)×U♦ P

♦1

is adiamond and the upper square is a pullback square, Proposition 7.3 now showsthat the sheaf IsomU (x1, x2)×U♦ P

♦1

is a diamond as well. But then

ω : IsomU (x1, x2)×U♦ P♦1→ IsomU (x1, x2)

is surjective and pro-étale, since it’s the pullback of the surjective pro-étale mapP

♦1→ U

♦ along IsomU (x1, x2) → U♦, and we’ve just shown that the source of

ω is a diamond, so its target is a diamond as desired.Finally, to see that IsomGTor(U)(P1, P2) ×U♦ P

♦1

is a diamond, we simplyobserve that the natural map

IsomGTor(U)(P1, P2)×U♦ P♦1→ P

♦2×U♦ P

♦1

is an isomorphism. This can be checked pro-étale-locally on U , so one easilyverifies it by going to a pro-étale cover where both the Pi’s become trivializable.

References[Han16] David Hansen, Period maps and variations of p-adic Hodge structure,

preprint.

[Hub96] Roland Huber, Etale cohomology of rigid analytic varieties and adicspaces, book.

[KL15] Kiran Kedlaya and Ruochuan Liu, Relative p-adic Hodge theory I.

[SW13] Peter Scholze and Jared Weinstein, Moduli of p-divisible groups, Cam-bridge J. Math.

[Wei15] Jared Weinstein, Gal(Qp/Qp) as a geometric fundamental group.

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