Notes on diamonds∗
David Hansen†
August 25, 2016
Diamonds are a beautiful class of geometric objects introduced by Peter
Scholze in his course at Berkeley in the Fall of 2014. There is a fantastic set of
notes available from this course (“the Berkeley notes”). The present document
grew out of my attempts to understand various particular claims in the Berkeley
notes, and various other questions about diamonds. These notes undoubtedly
contain many mistakes, and are not intended as a substitute for the Berkeley
notes.
I’m grateful to Johan de Jong, Christian Johansson, Peter Scholze, and Jared
Weinstein for some very helpful conversations.
Contents
1 Preliminaries
1.1 Sheaves on sites . . . . . . . . . . . . . .
1.2 Perfectoid spaces and tilting . . . . . . .
1.3 The big pro-étale site . . . . . . . . . . .
1.4 A remark on quotients by group actions
1.5 A remark on closed immersions . . . . .
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2
2
3
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8
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2 Diamonds: definition and first properties
2.1 The definition . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Diamonds as pro-étale equivalence relations . . . . . . . .
2.3 The underlying topological space, and open subdiamonds
2.4 Pro-étale torsors . . . . . . . . . . . . . . . . . . . . . . .
2.5 Fiber products and direct products . . . . . . . . . . . . .
2.6 Some topological properties . . . . . . . . . . . . . . . . .
2.7 The miracle theorems . . . . . . . . . . . . . . . . . . . .
2.8 Some sites associated with a diamond . . . . . . . . . . .
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∗ 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;
[email protected]
1
3 Diamonds associated with adic
3.1 The affinoid case . . . . . . .
3.2 The general case . . . . . . .
3.3 Some basic compatibilities . .
spaces
28
. . . . . . . . . . . . . . . . . . . . 29
. . . . . . . . . . . . . . . . . . . . 34
. . . . . . . . . . . . . . . . . . . . 35
4 Examples
4.1 The diamond of Spa Qp . . . .
4.2 Diamonds over a base . . . . .
4.3 Self-products of the diamond of
4.4 The sheaf Spd Zp . . . . . . . .
+
4.5 The diamond BdR
/Filn . . . .
. . . . .
. . . . .
Spa Qp
. . . . .
. . . . .
5 Moduli of shtukas
5.1 The Fargues-Fontaine curve . . . .
5.2 The de Rham affine Grassmannian
5.3 Moduli of shtukas . . . . . . . . . .
5.4 The Newton stratification . . . . .
5.5 Admissible and inadmissible loci .
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37
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42
44
48
49
54
55
6 More on morphisms
56
6.1 Smooth morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.2 Morphisms locally of finite type . . . . . . . . . . . . . . . . . . . 57
7 Into the abyss
59
7.1 Absolute diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.2 Stacks: first definitions . . . . . . . . . . . . . . . . . . . . . . . . 60
7.3 Diamond stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1
Preliminaries
We assume basic familiarity with adic spaces and perfectoid spaces, including
(but not limited to) Lectures 1-7 of the Berkeley notes. Unless explicitly stated
otherwise, all adic spaces are analytic adic spaces over Spa Zp ; we denote this
category by Adic. We also follow the convention that perfectoid spaces do not
live over a fixed perfectoid base field.
We reserve the notation ∼
= for canonical isomorphisms.
1.1
Sheaves on sites
Let C be a site. Recall that by definition, a presheaf F on C is a contravariant functor C → Sets. Given X ∈ C, we write hX = HomC (−, X) for the
contravariant Yoneda embedding of X.
A sheaf is a presheaf such that for every covering {Ui → U }i∈I in C, the
equalizer sequence
�
�
0 → F(U ) →
F(Ui ) ⇒
F(Ui ×U Uj )
i
i,j
2
is exact. We write Sh(C) for the category of sheaves on C. The following
definition is standard.
Definition 1.1. A morphism F → G of sheaves is surjective if for any U ∈ C
and any section s ∈ G(U ), there exists a covering {Ui → U }i∈I in C such that
s|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 is
representable 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 unique
isomorphism. 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 C
which is preserved under arbitrary pullback. A morphism F → G of (pre)sheaves
on C is “blah” if it is representable and, for every X ∈ C and every morphism
of (pre)sheaves hX → G, the morphism Y → X corresponding to the upper
horizontal 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 ×G
H → H is “blah”.
Proof. Given hX → H, we have (F ×G H)×H hX = F ×G hX , so this is immediate.
Definition 1.5. If F is a sheaf with an action of a group G, we write F/G for
the sheafification of the presheaf sending U ∈ C to F(U )/G.
1.2
Perfectoid spaces and tilting
� denote the category of all perfectoid spaces. Let Perf denote the
Let Perf
category of perfectoid spaces in characteristic p, so tilting defines a functor
�
Perf
X
→ Perf
�→ 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
∼ |X � | compatible with
could dream of: there is a natural homeomorphism |X| =
open affinoid perfectoid subspaces and with rational subsets thereof, there are
�
�
natural equivalences Xan ∼
and Xét ∼
with associated equivalences of
= Xan
= Xét
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 (OX
(U )� )
+
on open affinoid perfectoid subsets U ⊂ X defines a sheaf of rings ÃX on X
which depends only on the tilt of X. This sheaf of rings comes with a natural
+
surjective ring map θX : Ã+
X → OX .
Definition 1.7 (Fargues-Fontaine, Scholze, Kedlaya-Liu). Let X be a perfectoid space. An ideal sheaf J ⊂ Ã+
X is primitive of degree one if locally on a
covering X = ∪i Ui by affinoid perfectoids, J |Ui = (ξi ) is principal and gener+
ated by an element of the form ξi ∈ W (OX
(Ui )� ) of the form ξi = p+[�i ]αi with
+
+
�
�i ∈ OX (Ui ) some pseudouniformizer and with αi ∈ W (OX
(Ui )� ) arbitrary.
Let Perf + denote the category of pairs (X, J ) where X ∈ Perf and J ⊂ Ã+
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 −1 I · Ã+
X = J . Note that
� the ideal sheaf
Perf + is a category fibered in sets over Perf. For any X ∈ Perf,
+
ker θX ⊂ Ã+
X = ÃX � is primitive of degree one, so we get a functor
�
Perf
X
+
→ Perf
�
�
�→
X � , ker θX ⊂ Ã+
.
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 integral
+
+
structure sheaf OX
� = ÃX /J .
A word on notation: The notation “Perf” follows the Berkeley notes.
� here on the grounds that tilting allows
We are introducing the notation “ Perf”
one to regard arbitrary perfectoid spaces as liftings/deformations of perfectoid
spaces in characteristic p; the tradition of denoting “lifts” or “deformations” of
some mathematical object via a superscripted tilde seems to be well-established,
hence our choice of notation.
1.3
The big pro-étale site
In this section we recall some material from the Berkeley notes, and prove some
basic 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 + ) = lim�
(Ai , A+
i )
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 point
y ∈ Y , there is an open affinoid perfectoid subset V ⊂ Y containing y and
an open affinoid perfectoid subset U ⊂ X containing f (V ) such that the
induced morphism V → U is affinoid pro-étale.
Proposition 1.10. Suppose f : Y → X is a pro-étale morphism of perfectoid
spaces.
i. If Z → X is any morphism of perfectoid spaces, then Z ×X Y → Z is a
pro-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 a
pro-étale cover(ing) if for every qc open subset U ⊂ X, there is some qc open
subset V ⊂ Y with f (V ) = U .
We record the following for later use (the proof follows as in Proposition
15.5.1 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, we
also need to verify the following proposition.
Proposition 1.13. Suppose f : Y → X is a pro-étale covering of perfectoid
spaces. 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 )). Choose
a quasicompact open U ⊂ X with g(W ) ⊆ U , and then choose a quasicompact
open V ⊂ Y with f (V ) = U . Then W ×U V ⊂ Z ×X Y is a quasicompact open
with image W in Z.
Proposition 1.14. Suppose
f ◦g
Z�
�� g
��
��
�
Y
�X
�
�
f ��
�
�
��
is a diagram of pro-étale morphisms of perfectoid spaces, and suppose f ◦ g is a
pro-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∈I Vi of f −1 (U ) by qc opens in Y , so ∪i∈I g −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 ), and
therefore f (V ) = U as desired.
Definition 1.15. The big pro-étale site is the site Perf proet whose underlying category is the category Perf of perfectoid spaces in characteristic p, with
coverings given by pro-étale coverings.
There is also a small pro-étale site Xproét for any perfectoid space X, with
objects given by perfectoid spaces pro-étale over X and covers given by pro-étale
covers.
We now turn to sheaves on Perf proet .
Proposition 1.16. For any X ∈ Perf, the presheaf hX = Hom(−, X) is a sheaf
on the big pro-étale site.
Proof sketch. One reduces to the case of X affinoid. 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 )+ ) .
Let Y ∈ Perf be some perfectoid space with a given pro-étale cover Ỹ → Y , so
we need to show exactness of the sequence
0 → Hom(Y, X) → Hom(Ỹ , X) ⇒ Hom(Ỹ ×Y Ỹ , X).
But OY and OY+ are sheaves on Yproet , so we get an exact sequence
0 → (O(Y ), O(Y )+ ) → (O(Ỹ ), O(Ỹ )+ ) ⇒ (O(Ỹ ×Y Ỹ ), O(Ỹ ×Y Ỹ )+ ),
and we’re done upon applying the left-exact functor Hom ((R, R+ ), −).
By our conventions, given any property of morphisms of perfectoid spaces
preserved under arbitrary pullback, there is a corresponding notion for morphisms of sheaves on Perf proet . In particular, we may speak of a morphism of
sheaves F → G on Perf proet being an open immersion, Zariski closed immersion, 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 Perf proet 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 morphism
F → G is surjective if, for any hX → G with associated hY � F ×G hX , we can
find some pro-étale cover X̃ → X and a section hX̃ → F such that the diagram
�G
F
��
�
�
�
�
�
��
�
�
�� hY
�� hX
�
�
�
�
���
�� �����
�
� ����
hX̃
commutes. If F → G is a pro-étale cover, then X̃ = Y does the job, so F → G
is surjective.
Suppose conversely that F → G is a surjective pro-étale morphism, and
consider a map hX → G, so we get a diagram
�G
F
��
�
�
�
�
�
��
�
�� hX
� hY
��
���
�
�
�
��
��
���
�������
hX̃
with Y as above. We need to show that the associated morphism Y → X is a
pro-étale cover. Since the square is cartesian, we can fill in another arrow from
hX̃ to hX , vis.
�G
F
��
�
�
�
�
�
��
�
�� hX
� hY
�� ���
���
�
�
�
�
�
�� ��� ����
���������
hX̃
Since Y → X and X̃ → X are both pro-étale, the morphism X̃ → Y is pro-étale
by Proposition 1.10.ii, so the diagram
X̃ �
��
��
��
�
Y
�X
��
�
��
��
�
�
satisfies the hypotheses of Proposition 1.14 and thus Y → X is a pro-étale
cover.
7
1.4
A remark on quotients by group actions
Let X be an adic space with a (right) action of a group G. There is a natural
candidate
� for a quotient space
� X/G, given by the following construction. Let
X = |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 the
evident map.
• OX/G := (q∗ OX )G . This is clearly a sheaf (since (−)G is left-exact) of
complete topological rings.
|·|x
• | · |y is the valuation on the stalk OX/G,y induced by OX/G,y → OX,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-equivariant
morphism q : X → X/G in V (for the trivial action on X/G), and it’s easy to
check 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. Of
course, 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 require
that X/G also lie in C, in which case the quotient X/G is also categorical in C.
1.5
A remark on closed immersions
Recall that two valuations v, w on a ring A are equivalent iff v(a) ≤ v(b) ⇔
w(a) ≤ w(b) for all (a, b) ∈ A2 .
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 inequivalent
continuous 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; if
v(b) = 0 then we can replace b by b + �N for some large N while maintaining
the inequality w(a) > w(b), and then v(b + �N ) = v(�)N �= 0, so without loss
of 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) and w(a − a� ) < w(a)
and b� ∈ S such that
v(b − b� ) < v(b) and w(b − b� ) < w(a).
8
But then
and
v(a� ) ≤ max(v(a − a� ), v(a)) ≤ v(b) = v(b� )
w(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 if
it is a closed immersion of locally ringed spaces (cf. Tag 01HK in the Stacks
Project).
We’d like to introduce the following more general notion; the goal here is to
simultaneously capture the usual notion of a closed immersion, and the notion
of 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 image
and 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 if
X 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 in
general, 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 open
subset, 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 immersion of affinoids with Z uniform, things are better behaved. Note that if
Spa(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 the
following 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 and
only if |g|(|W |) ⊆ |Z|. In particular, Z is determined uniquely up to unique
isomorphism 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 maps
Z → X and W → X factor uniquely through maps Z → U and W → U for
+
any U ∈ S. Writing U = Spa(RU , RU
), these correspond to continuous maps
+
+
+
(RU , RU ) → (S, S ) and (RU , RU ) → (A, A+ ). Taking the direct limit gives a
map
+
lim
(RU , RU
) → (S, S + )
→
S
with dense image and uniform target, which moreover identifies (S, S + ) isomor+
phically with the uniform completion of lim→ (RU , RU
). But then since (A, A+ )
S
is uniform by assumption, the map
+
lim
(RU , RU
) → (A, A+ )
→
S
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. Then
for any rational subset U ⊂ X, the morphism Z ×X U → U is a weak closed
immersion 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 properties
Until further notice, all sheaves are sheaves of sets on Perf proet .
2.1
The definition
We now come to the key definition.
Definition 2.1. A diamond is a sheaf D on Perf proet which admits a surjective
and pro-étale morphism hX → D from a representable sheaf. A morphism of
diamonds is a morphism of sheaves on Perf proet . We write Dia for the category
of diamonds.
If D is a diamond, we refer to any choice of a surjective pro-étale morphism
hX → D as a presentation of D.
Proposition 2.2. Let F → G be a pro-étale morphism of sheaves on Perf proet .
If G is a diamond, then so is F. If F → G is surjective and pro-étale, then F
is 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
X
→ Dia
�→ 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 Spa Zp .
Definition 2.4. If D is a diamond, a subdiamond of D is a subfunctor E ⊂ D
which is naturally isomorphic to a diamond.
Note that if E is a subdiamond of a diamond D, the natural monomorphism
E → D need not be a representable morphism.
2.2
Diamonds as pro-étale equivalence relations
Let D be a diamond with a given presentation hX → D. We see by representability that the fiber product hX ×D hX is representable, say by hZ , and we get two
morphisms s, t : Z ⇒ X corresponding to the two projections of hX ×D hX � 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 ) gives
an equivalence relation on hX (T ) for any T ∈ Perf. In short, s, t : Z ⇒ X is a
pro-étale equivalence relation on X.
Proposition 2.5. The diagram
hZ ⇒ hX → D
is a coequalizer diagram in Sh(Perf proet ).
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 the
converse of this proposition is true: given a pro-étale equivalence relation in
perfectoid spaces s, t : Z ⇒ X, is D = Coeq (hZ ⇒ hX ) a diamond? The
answer is unclear, at least to this author.1 The key difficulty seems to be
proving that hX → D is representable (and pro-étale): trying to naively imitate
the arguments in Section 52.10 of the Stacks Project, one quickly runs into some
descent-theoretic questions whose answers seem negative in general.
1 Scholze 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
Perf proet 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 subdiamonds
Given a presentation of a diamond hX → D, let hX ×D hX � hZ as above, and
let Z ⇒ X be the associated pro-étale equivalence relation. It’s not hard to show
that the induced map |Z| → |X| × |X| is injective and defines an equivalence
relation ∼|Z| on the topological space |X|. Let |D| = |X|/ ∼|Z| be the set of
equivalence classes for this relation, with the quotient topology relative to the
map |X| � |D|.
Proposition 2.6. The topological space |D| is well-defined independently of
all choices, and the association D �→ |D| defines a functor from diamonds to
topological spaces.
Proof. A refinement of a given presentation hX → D is a commutative diagram
hX � �
��
��
��
��
hX
�D
��
�
��
��
�
�
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 � | for
any 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 the
preimage of ∼|Z| under |X � | � |X|, so |X � |/ ∼|Z � | ∼
= |X|/ ∼|Z| as sets. To
2 e.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 commutative
diagram of sets
q
� |X|
|X � |
π�
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 natural
topologies 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 other
hand, 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 diamond D, an open subdiamond E ⊂ D is a subfunctor
of D such that for some presentation hX → D, E ×D hX � hY is representable
and 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 immersions
E → D.
Proof. Left to the reader; in any case, the next proposition supersedes this
claim.
It turns out that open subdiamonds are “purely topological” relative to their
ambient diamond:
Proposition 2.9. Given a diamond D, the association E →
�
|E| defines an
inclusion-preserving bijection from open subdiamonds of D to open subsets of
|D|.
Proof. We sketch the construction of the inverse association. Given an open
subset U ⊂ |D|, we define EU ⊂ D as the subfunctor characterized by the
property 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 spaces
defined by setting
|Y | = |m|−1 (U ) ⊆ |X|.
13
2.4
Pro-étale torsors
This 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 over
the trivial action on X such that Y ×X Y ∼
= G × Y and such that f admits a
section pro-étale-locally on X.
Likewise, a morphism F → G of sheaves on Perf proet 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-étale
locally on G, we have F � G × G.
Remark. If F → G is a pro-étale G-torsor, the induced map F/G → G is
an isomorphism (since it becomes an isomorphism pro-étale-locally on G, and
everything is a pro-étale sheaf).
Proposition 2.11. A map of sheaves F → G is a pro-étale G-torsor if and
only if there exists a surjective pro-etale map G � → G together with a section
s : 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 on
geometric 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 the
previous proposition. We may lift x to a (C, OC ) point x̃ of G � ; but then the
map G � × G → G � ×G F postcomposed with projection to F identifies x̃ × G
bijectively with s(x̃) · G = S ⊂ F(C, OC ).
Proposition 2.13. If F → G is any pro-étale G-torsor and K ⊂ G is any open
subgroup 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 discrete
set, so the upper horizontal arrow is separated and étale. Now we’re done after
an application of the following key lemma.
Lemma 2.14. Let H → G be a morphism of sheaves, and suppose there exists
a 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 maps
form a stack for the pro-étale topology on any perfectoid space X. More precisely, the fibered category p : Ét → Xproet defined by
p−1 (U ) = {V → U separated étale}
for any U ∈ Xproet is a stack. This is true because Peter told me it is. (But
seriously, 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 profinite, then F → G is surjective and pro-finite étale. In particular, G is a diamond
if and only if F is a diamond.
This is slightly tricky, since we don’t know a priori that F → G is representable.
Proof. Surjectivity is clear. For the pro-étale claim, write G = lim←i G/Hi with
Hi open normal in G. Then F = lim←i F/Hi as sheaves, so it suffices to show
that each F/Hi → G is finite étale. This is true on a pro-étale cover of G, by
definition, so we’re reduced to showing that whether or not a morphism F → G
is finite étale can be checked pro-étale-locally on G. We may clearly assume G
is representable.
So let F → hX be any sheaf map, and let hX̃ → hX be a surjective pro-étale
map such that F ×hX hX̃ is representable, say by hỸ , and such that hỸ → hX̃
is finite étale. Since
Ỹ ×X̃,pri (X̃ ×X X̃) ∼
= F ×X X̃ ×X̃,pri (X̃ ×X X̃)
∼
= F ×X (X̃ ×X X̃),
where pri : X̃ ×X X̃ → X̃ denotes either of the two projections, the two pullbacks
of Ỹ under pri are canonically isomorphic, and one easily checks that these
15
isomorphisms satisfy the usual cocycle condition. In particular, the finite étale
map Ỹ → X̃ inherits a descent datum relative to the pro-étale cover X̃ → X,
so (again by Lemma 4.2.4 of [Wei15]) Ỹ descends uniquely to a finite étale map
Y → 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 sheaf
on Perf proet equipped with an action of G. Suppose G acts freely on the geometric points F(C, OC ) for any algebraically closed nonarchimedean field C in
characteristic p. Then F → F/G is a pro-étale G-torsor, and is surjective and
pro-étale.
F.
Here F(C, OC ) is of course shorthand for the set of sheaf maps hSpa(C,OC ) →
Proof. This is exactly Proposition 4.3.2 of [Wei15]. Note that Proposition 4.3.2
of loc. cit. is only stated in the case where F is a representable sheaf, but
Weinstein’s proof doesn’t use this assumption in any way.
We’ve now developed enough material to prove the following basic structural
statement.
Theorem 2.17. Let F → G be a pro-étale G-torsor for G an arbitrary locally
profinite group. Then F → G is (representable and) pro-étale. In particular F
is a diamond if and only if G is.
Proof. Choose an open profinite subgroup K ⊂ G. By Proposition 2.13, the map
F/K → G is surjective and étale. On the other hand, G acts freely on geometric
points of F, so a fortiori K does as well, and then F → F/K is surjective and
pro-finite étale by Theorem 2.15. But now we’re done after factoring our original
map 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 I
shouldn’t feel bad that it took me a year to prove.
The special case of Theorem 2.16 where F is assumed representable gives
our first real mechanism for constructing diamonds. Due to its importance, we
state 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. Then
hX → hX /G is a pro-étale G-torsor, hX /G is a diamond and hX → hX /G is a
presentation. 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 map
hX → hX /G is surjective and pro-étale by Proposition 2.15. The statement on
topological spaces is clear.
16
2.5
Fiber products and direct products
Theorem 2.19. Given any morphisms of diamonds E → D, F → D, the fiber
product of sheaves E ×D F is a diamond.
Proof. Choose presentations hX → D, hY → E, hZ → F. One easily checks
that 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 the
surjective pro-étale map hY → E along E ×D F → E), so the composite morphism
is 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 are
both 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. Since
U ×X V exists and hU ×hX hV ∼
= hU ×X V , 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 canonically
a diamond, where the product is taken in the category of sheaves of sets on
Perf proet .
Proof. Choose presentations hX → D, hY → E. One easily checks that the
morphism
hX × hY → hX × E → D × E
is a composition of surjective pro-étale morphisms, so itself is surjective proétale. It then suffices to show that hX × hY is representable. This is the content
of the following proposition.
Proposition. Let X, Y be two perfectoid spaces in characteristic p. Then
hX × hY ∼
= hX×Y for a certain canonical perfectoid space “X × Y ” whose formation 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 perfectoid Tate-Huber pairs in characteristic p, with associated affinoid perfectoid
spaces 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 + . Then
we 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 .
� n+1 n+1 �
{�A ,�B }
Furthermore, Un = U
is a rational subset of Spa(D+ , D+ ), so we
�A �B
may describe it explicitly as an affinoid adic space. Precisely, let Cn = A ⊗Fp B
n
�A
�n
and let Cn+ be the integral closure of (A+ ⊗Fp B + )[ �B
, �B
] in Cn ; give Cn+ the
A
I · Cn+ -adic topology, and give Cn the topology making Cn+ an open subring.
(Note that the I · Cn+ -adic, �A · Cn+ -adic, and �B · Cn+ -adic topologies on Cn+ all
coincide.) Let Dn and Dn+ be the completions of Cn and Cn+ for these topologies.
Then
�
�
O(Un ), O(Un )+ = (Dn , Dn+ ).
One checks directly that Dn is a complete perfect uniform Tate ring, which
exactly 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 morphisms
prX : X × Y → X and prY : 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 morphisms
prX , prY , 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 of
continuous 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 pseudouni1/pj
formizer �R ∈ R; replacing �R by �R if necessary, we may assume that �R
divides both f (�A ) and g(�B ) in R+ . This immediately implies that the ring
n
map f ⊗ g : C + → R+ induces compatible ring maps C + /I n → R+ /�R
, so
18
ˆ : D + → R+ .
passing to the inverse limit we get a continuous ring map f ⊗g
Passing to Spa’s, we may consider the composite
i : Spa(R, R+ ) → Spa(R+ , R+ )
ˆ
Spa(f ⊗g)
−→
Spa(D+ , D+ )
ˆ carries
where the lefthand arrow is the evident open immersion. Since f ⊗g
�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). Summarizing our efforts in this paragraph so far, we’ve described a natural map of
sets
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 an
evident category Perf k of perfectoid spaces over Spa k, and the above proof also
shows (by replacing − ⊗Fp − with − ⊗k − everywhere) that for two such spaces
the product “X ×k Y ” exists. Likewise, for two diamonds D, E over Spa k, the
product D ×k E exists.
Corollary 2.21. Let D be a diamond. Then the diagonal D → D × D is
representable if and only if any morphism hU → D from a representable sheaf
is representable.
Proof. This is general nonsense (cf. Tag 0024 in the Stacks Project).
2.6
Some topological properties
In this section we define quasicompact and quasiseparated objects and morphisms in Dia.
Definition 2.22. Let D be a diamond.
1. D is quasicompact (qc) if there is a presentation hX → D with X quasicompact.
2. D is quasiseparated (qs) if the diamond hX ×D hY is quasicompact for any
morphisms hX → D, hY → D with X and Y quasicompact.
19
Remark. If X is any quasicompact perfectoid space, we can find a surjective
étale map Y → X with Y affinoid perfectoid. In particular, we could equally
well define a diamond D to be quasicompact if and only if it admits a presentation hX → D with X affinoid perfectoid. We’ll use this equivalence without
particular comment.
Proposition 2.23. If D is quasiseparated, then so is any open subdiamond
E ⊂ 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 quasicompact
by 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 quotient 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|; but
the latter space is quasicompact by assumption, so any such covering admits a
finite 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 any
morphism hX → E with X affinoid perfectoid.
2. f is quasiseparated (qs) if the diagonal morphism ∆ : D → D ×E D is
quasicompact. (Note that ∆ may not be representable.)
Proposition 2.26. Let f : D → E be a morphism of diamonds. Then f is
quasicompact if and only if, for every morphism of diamonds F → E with F
quasicompact, the diamond D ×E F is quasicompact.
Proof. “If” is clear. For “only if”, suppose f is quasicompact, and choose any
map F → E with F quasicompact as in the statement of the proposition. Choose
a 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. Choose
a 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 of
D ×E F. Since U is affinoid perfectoid, we see that D ×E F is quasicompact, as
desired.
20
Using this proposition, one easily checks that pullbacks and compositions of
quasicompact morphisms are quasicompact.
Remark. If f : Y → X is a morphism of schemes, one can check that f is quasicompact in the usual sense (i.e., X admits a covering by open affine subschemes
Xi 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 quasiseparated 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 an
open covering by spatial diamonds.
Remark. A diamond D is “spatial” in the sense of the Berkeley notes if and only
if 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 Berkeley
notes if and only if it is “spatial” in the sense of the definition here. These
new definitions of spatial and locally spatial were indicated to me by Peter (in
early July 2016); part of the motivation is the verbal analogy exhibited in the
following proposition:
Proposition 2.28. If D is spatial, then |D| is a spectral space. If D is locally
spatial, then |D| is locally spectral.
Proof. Any topological space which admits an open covering by spectral spaces
is locally spectral, so it suffices to prove the first claim. The first claim follows
from 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 runs
over a set of qc open subdiamonds such that the |Ei |’s give a neighborhood basis
in |D| of some point x ∈ |D� |, then the diamonds Ei ∩ D� = Ei ×D D� are still
quasicompact and the open subsets |Ei ∩ D� | = |Ei | ∩ |D� | give a neighborhood
basis of x in |D� |. This shows that any qc open subdiamond of a spatial diamond
is 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 open
covering 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. Then
E = ∪i∈I (E ∩ Ui ) gives an open covering of E, and each E ∩ Ui is open in Ui and
hence locally spatial by the previous proposition. We can therefore choose an
open cover of each E ∩ Ui by spatial subdiamonds Vij ⊆ E ∩ Ui , j ∈ Ji . But
then E = ∪i∈I,j∈Ji Vij gives an open covering of E by spatial subdiamonds, as
desired.
Proposition 2.31. If D is a locally spatial diamond, then |D| is quasicompact
if 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 subdiamonds Di ⊂ D, i ∈ I. Then the open subsets |Di | give a covering of |D|; since
the 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 with
open coverings of its topological space, the diamonds Di , i ∈ I � give an open
covering of D. In particular, the map
�
Di → D
i∈I �
�
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 perfectoid), so D is quasicompact.
Corollary 2.32. Let f : D → E be a morphism of diamonds. If f is quasicompact and E is locally spatial, then |f | : |D| → |E| is quasicompact.
Proof. Let U ⊂ |E| be any quasicompact open subset, with U ⊂ E the associated
open 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 by
Proposition 2.26, so |D ×E U| is quasicompact as desired.
Corollary 2.33. Let D be a quasiseparated locally spatial diamond, and let
E ⊆ 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 locally
spatial, there are natural inclusion-preseving bijections between
a) quasicompact open subsets of |D|,
b) quasicompact open subdiamonds of D, and
c) 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 quasiseparated
locally 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 intersection 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 quasicompact
open subdiamond U ⊂ D such that |f |(|X|) ⊆ |U|, i.e. such that f factors
through a map X ♦ → U. Let E � = E ×D U be the open subdiamond of D
associated with the open subset |E| ∩ |U|. By assumption, |E| is a retrocompact
open, and |U| is a quasicompact open, so |E| ∩ |U| is a quasicompact open subset
of |D|. Therefore, E � is a quasicompact open subdiamond of D. We have an
identification
E ×D X ♦ = E � ×U X ♦ .
But E � and X ♦ are both quasicompact, and U is spatial so in particular quasiseparated, 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 is
quasiseparated and the map |∆| : |D| → |D ×E D| is a closed embedding.
Proposition 2.35. A quasiseparated morphism f : D → E of locally spatial
diamonds is separated if and only if it satisfies the valuative criterion of separatedness.
Proof. Omitted.
Definition 2.36. A morphism f : D → E of locally spatial diamonds is spatial
if 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 open
subset U ⊂ X is spectral.
Definition 2.38. A diamond D is taut if D is locally spatial and the locally
spectral space |D| is taut.
23
2.7
The miracle theorems
Given an adic space X, there is typically a huge difference between locally closed
subsets of |X| and locally closed immersions into X; the latter notion is much
more restrictive. In this section we prove a “miracle theorem” showing that this
difference largely disappears when X is a w-local affinoid perfectoid space. We
then bootstrap this up to an equally miraculous construction which allows one
to “diamondize” a very general class of locally closed subsets E ⊂ |D|, where D
is an arbitrary diamond. I learned the statements of Theorems 2.39 and 2.42
below from Peter Scholze.
Recall that a spectral space X is w-local if every connected component of
X has a unique closed point, and the subset X c of closed points is closed in
γ
X. This implies that the composition X c �→ X � 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 adic
spaces by calligraphic letters, to distinguish them from the (many) topological
spaces 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 closed
generalizing subset T ⊂ X, there exists a locally closed immersion of perfectoid
spaces f : T → X identifying |T | homeomorphically with T . The pair (T , f ) is
unique up to unique isomorphism and is universal for morphisms g : Z → X
with g(|Z|) ⊆ T .
Here a locally closed immersion is a morphism of perfectoid spaces f : Y → X
which can be factored as Y → U → X where U → X is an open immersion and
Y → 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 set
T � = 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 quasicompact 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 the
obvious 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 induced
topology 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 spectral
with the induced topology by Tag 0902 in the Stacks Project. For the second
claim, note that the inclusion T ⊆ T ∩ U is obvious, so it suffices to show the
opposite 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 point
z ∈ 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) ∼
= X c 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 of
π0 (X), with inverse given by γ −1 (−).
To see this, note that any point x ∈ X has a unique rank one generalization
x0 , as well as a unique (by w-locality) closed specialization xc ; since generalizations in analytic adic spaces form a totally ordered chain, we have xc ≺ x ≺ x0
and γ −1 (xc ) = {x0 }. This implies that the generalizations of xc form a totally
ordered chain3 coinciding with the set of specializations of x0 .4 Suppose now
that 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 ∩ X c 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 write
γ(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 quasicompact
open-closed subspaces, and thus correspond to unique open-closed immersions
of perfectoid spaces Vi → X which again form a cofiltered system. Each space
Vi 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 }
3 Whose
order type, however, could be some enormous ordinal.
last coincidence is extremely special to the w-local situation, and also relies crucially
on the fact that X arises as |Spa(A, A+ )|.
4 This
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
Vi → X
←
i∈I
such that |T | ∼
= T , and one easily checks that this map is a Zariski-closed
embedding in the sense of Scholze’s torsion paper (the associated ideal in R is
the 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 corresponding
to the open subset U ⊂ X, so T = T ×X U is the space we seek. One easily
checks the universal property and uniqueness. Note that the map T → X
factors as T → U → X , i.e. as a Zariski-closed embedding followed by an open
immersion, so this really is a locally closed immersion.
Now we bootstrap this theorem up to a much more general result, using the
fact 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 and
S ⊂ 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 that
f (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. Since
f (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|. Then
for any locally closed generalizing subset E ⊂ |D|, there is a canonical subdiamond E ⊂ D such that |E| ∼
= E homeomorphically inside |D|, characterized
by 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| has
image contained in E.
Remark. It’s easy to check that E has the same universal property with respect
to maps G → D from an arbitrary diamond G.
Proof. Observe that if E ⊂ |D| is any subset whatsoever, we can use the same
recipe as in the theorem to define a subfunctor E ⊂ D of “maps to D factoring
through E on topological spaces”. We’re going to show that when E is locally
closed and generalizing, E is a diamond and |E| = E.
So, fix E ⊂ |D| locally closed and generalizing, with associated subfunctor
E ⊂ D. Let
�
f
U♦ =
Xi♦ → D
i∈I
26
�
be a representable surjective pro-étale map, where U = i∈I Xi is perfectoid
with each Xi affinoid perfectoid. After possibly replacing each Xi by an affinoid
pro-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 |Xi | = 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 spaces
V → 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 ♦ , and
ii. under this identification, the map V ♦ → E identifies with the natural projection 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♦
�
E
�
�D
f
is cartesion and that the horizontal arrows therein are monomorphisms. But
now, since this diagram is cartesian, we get that the map V ♦ → E is representable, surjective and pro-étale, since it’s the pullback of the representable
surjective pro-étale map U ♦ → D along the map E → D. Therefore E is a
diamond, 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 embedding if E arises by applying Theorem 2.42 to some locally closed generalizing
subset E ⊂ |D|. We’ll also say (equivalently) that E ⊂ D is a locally closed
subdiamond of D.
We emphasize that a locally closed embedding E → D in this sense need not
be relatively representable! However, it is true that E ×D X ♦ is representable
by a perfectoid space whenever X is a w-local perfectoid space.
Corollary 2.44. Given a diamond D, the association E �→ |E| defines an
inclusion-preserving bijection from locally closed subdiamonds of D to locally
closed generalizing subsets of |D|.
27
Proposition 2.45. Let D be a locally spatial diamond, and let E → D be a
locally closed embedding. Then E is locally spatial.
Proof. Write |E| = E as an intersection U ∩ Z with U open and Z closed. Then
U ⊂ |D| corresponds to an open immersion U → D, so U is locally spatial by
Corollary 2.30. Now U ∩ Z ⊂ U is closed and generalizing in U , and we get
E → U as the associated locally closed embedding. Arguing locally on an open
covering 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 subset
E ∩ |Ui | ⊂ |Ui |, and it suffices to show that each Ei is spatial. So now we’re
reduced to showing that if D is spatial and E ⊂ D is a locally closed embedding
such that |E| ⊂ |D| is closed and generalizing, then E is spatial too; this is an
easy exercise.
Proposition 2.46. Let D be a quasiseparated locally spatial diamond, and let
E ⊂ 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 diamond
Here we content ourselves with a general nonsense definition.
Definition 2.47. Let D be a diamond, and let • be one of the decorations
• ∈ {an, fét, ét, proét}, respectively. Then there is a site “D• ” with objects
given by diamonds E over D such that the map E → D is (respectively) an
open immersion, a finite étale map, an étale map,
� or a pro-étale map, and with
covers 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-étale
topology refines the other three.
3
Diamonds associated with adic spaces
Definition 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
◦ 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,
�−1
28
then m must be the identity. Let Spd Zp : Perf → Sets be the presheaf sending
Y ∈ Perf to the set of isomorphism classes of untilts of Y . We will see later
that Spd Zp is actually a sheaf.
Now let X be an analytic adic space over Spa Zp .
Definition 3.2. Given any perfectoid space Y ∈ Perf, an untilt of Y over X is
a triple (Y � , ι, f ) where (Y � , ι) is an untilt of Y and f : Y � → X is a morphism
of 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.
Let
X ♦ : Perf proet → Sets
be the presheaf sending Y ∈ Perf to the set of isomorphism classes of untilts of
Y 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
)
.)
=
� /X ∼
Proof. This is an immediate consequence of the equivalence Perf
= 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 the
following theorem.
Theorem 3.4. For X any analytic adic space over Spa Zp , 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 map
X → Spa Zp does induce a natural transformation X ♦ → Spd Zp , our basic
point of view is to “forget” this map from X ♦ to Spd Zp . We shall return to this
point 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 case
The crucial ingredient in the proof of Theorem 3.4 is the following lemma, which
treats the case of X affinoid.
Lemma 3.5. Let X = Spa(R, R+ ) be an affinoid adic space. Choose a directed
system (Ri , Ri+ ), i ∈ I of finite étale Galois (R, R+ )-algebras such that
+
�
(R̃, R̃+ ) = lim(R
i , Ri )
i→
29
is perfectoid, where the completion is for the topology making limi→ Ri+ open and
bounded. (By an argument of Colmez and Faltings, such a direct system always
exists.) Let Gi be the Galois group of Ri over R, so Gi acts on Xi = Spa(Ri , Ri+ )
and by continuity G = lim←i G acts on X̃ = Spa(R̃, R̃+ ). Then:
1. The space X̃ � with its induced action of G satisfies the hypotheses of Theorem 2.18. Consequently, hX̃ � → hX̃ � /G is a pro-étale G-torsor and hX̃ � /G
is a diamond.
2. There is a natural isomorphism
X♦ ∼
= hX̃ � /G.
In particular, X ♦ is a diamond.
It’s essentially trivial to check that X̃ � with its action of G satisfies the
hypotheses of Theorem 2.18, so we’re reduced to showing the isomorphism X ♦ ∼
=
hX̃ � /G.
Lemma 3.6. Let X, X̃ and G be as in Lemma 3.5. For any perfectoid space
T , there is a natural bijection
Hom(T, X) ∼
= HomG (T̃ /T, X̃)/ �
where the right-hand side denotes isomorphism classes of G-equivariant maps
T̃ → X̃ from pro-étale G-torsors T̃ /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 to
Xi . Let T̃ = lim←i Ti ; this exists as a perfectoid space since the transition maps
are all finite étale, and the maps Ti → Xi compile into a map f˜ : T̃ → X̃ by an
easy 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˜ : T̃ → X̃ from a
pro-étale G-torsor T̃ /T is equivalent to giving a continuous G-equivariant map
f˜ : (R̃, R̃+ ) → (O(T̃ ), O(T̃ )+ ).
By pro-étale descent for perfectoid spaces, the map
(O(T ), O(T )+ ) → (O(T̃ )G , O(T̃ )+G )
is an isomorphism, so restricting f˜ to G-invariant elements gives a map
(R̃G , R̃+G ) → (O(T ), O(T )+ )
30
which clearly only depends on the isomorphism class of T̃ /T and f˜. Since the
image of the map (R, R+ ) → (R̃, R̃+ ) is contained in (R̃G , R̃+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 mutually
inverse.
Returning now to the setup of Lemma 3.5, we’re ready to construct the map
X ♦ → hX̃ � /G. Given some Y ∈ Perf and some (Y � , ι, f ) ∈ X ♦ (Y ), we need to
describe an element of (hX̃ � /G)(Y ). By the construction of Lemma 3.6, giving f
is the same as giving a pro-étale G-torsor Ỹ � /Y � together with a G-equivariant
map f˜ : Ỹ � → X̃ (up to isomorphism). Symbolically,
�
(Y � , ι, f ) / �
�
�
∼
(Y � , ι, Ỹ � /Y � , f˜) / �
=
X ♦ (Y ) ∼
=
�
(with the obvious meaning in the second line). Tilting this data (and using ι)
gives a pro-étale G-torsor Ỹ /Y together with a G-equivariant map f˜� : Ỹ → X̃ � .
But giving this latter data is equivalent to giving a Y -point of (hX̃ � /G)! Indeed,
since hX̃ � → hX̃ � /G is a pro-étale G-torsor, we get a natural identification
(hX̃ � /G)(Y ) = HomG (Ỹ /Y, X̃ � )/ � .
This gives the desired map. Symbolically,
� �
�
X ♦ (Y ) ∼
(Y , ι, f ) / �
=
�
�
∼
(Y � , ι, Ỹ � /Y � , f˜) / �
=
tilt
→
=:
HomG (Ỹ /Y, X̃ � )/ �
(hX̃ � /G)(Y ).
tilt
To finish the proof of Lemma 3.5, we need to invert the map labelled “ →”
in the previous diagram. In other words, given a pro-étale G-torsor Ỹ /Y and
a G-equivariant map a : Ỹ → X̃ � , we need to (canonically and functorially)
produce an untilt (Y � , ι), a pro-étale G-torsor Ỹ � /Y � and a G-equivariant map
a� : Ỹ � → X̃. This is the least formal part of the proof, and we do it in three
steps as follows:
�
�
∼
�
1. By the equivalence Perf
/X̃ = Perf /X̃ � , we immediately get (Ỹ , ι̃) and a
for free.
2. By Propositions 3.7 and 3.9 below, the G-action on Ỹ lifts to a unique
G-action on Ỹ � compatible with a� and ι̃.
31
3. Finally, in Proposition 3.10 below, we construct (Y � , ι) as the categorical
quotient of (Ỹ � , ι̃) by its natural G-action.
Proposition 3.7. Let Y ∈ Perf be a perfectoid space with an action of some
group G, and let (Y � , ι) be a given untilt of Y . Then there is at most one Gaction 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 therefore
φ�g ◦ φ−1
=
id
for
all
g
∈
G.
g
Proposition 3.8. Let Y ∈ Perf be a perfectoid space with an action of some
group 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 Ã+
Y preserves 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 a
group G, and let f : Y → X be a G-equivariant morphism. If (X � , ι) is a Gequivariant untilt of X, then the untilt (Y � , ι� ) of Y induced by the equivalence
� /X � is a G-equivariant untilt, and f � : Y � → X � is G-equivariant.
Perf /X ∼
= Perf
�
Proof. Let ker θX � ⊂ Ã+
X be the ideal sheaf describing the untilt X . By the
previous proposition, ker θX � is preserved by the G-action on X. Since ker θY � =
f −1 ker θX � ·Ã+
Y and f is G-equivariant, the ideal sheaf ker θY � is preserved by the
G-action which therefore descends to the quotient OY+� , so Y � is G-equivariant.
The G-equivariance of f � is then obvious.
Let Ỹ → 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 (Ỹ � , ι̃)
of Ỹ over (Y � , ι). The next proposition reverses this construction.
Proposition 3.10. Let Y ∈ Perf be a perfectoid space, and let Ỹ → Y be a
pro-étale G-torsor for some profinite group G. Let (Ỹ � , ι̃) be a G-equivariant
untilt of Ỹ . Then the categorical quotient Y � = Ỹ � /G exists as a perfectoid
space, and ι̃ induces a canonical isomorphism
∼
ι : Y �� = (Ỹ � /G)� ∼
= Ỹ �� /G → Ỹ /G ∼
= Y,
so (Y, ι) is an untilt of Y . This association induces an equivalence of categories
{untilts of Y } ∼
= {G − equivariant untilts of Ỹ }.
32
Note: Although Proposition 3.10 doesn’t appear explicitly in the Berkeley
notes, I don’t claim any originality in formulating or proving it: a special case
is very strongly implicit in the proofs of Theorem 9.4.5 and Lemma 9.4.6 in the
Berkeley notes. Cf. also the remarks following Proposition 4.2 below.
Proof. We may assume Y and Ỹ are affinoid, say with Y = Spa(A, A+ ) and
Ỹ = Spa(Ã, Ã+ ). Let Ỹ � = Spa(Ã� , Ã�+ ) be our given untilt (so we also have
∼
an isomorphism ι̃ : Ã+ → Ã��+ ) with its action of G. Set A� = (Ã� )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
�� = (�� )� ∈ Ã�+ satisfies �� |p in Ã�+ ; let us say � is very good if there is
another good �� such that ��� is good. Note that �� is a G-invariant element
of Ã�+ , since it’s the image of the G-invariant element [�] ∈ W (Ã+ ) under the
G-eqivariant map W (Ã+ ) → Ã�+ . Note also that for any good �, ι̃ induces a
G-equivariant isomorphism Ã�+ /�� ∼
= Ã+ /�. The key claim (whose proof we
momentarily defer) is as follows:
Claim: The isomorphism Ã�+ /�� ∼
= Ã+ /� 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 Ã� ), so we only need to verify the
Frobenius condition. Applying the claim twice, with � and �p such that �p is
very 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�+ → Ã�+ , so tilting this gives an injective map
A��+ → Ã��+ ∼
= Ã+ whose image factors through (Ã+ )G = A+ ; thus we have
get an injective map A��+ → A+ . But reducing mod �, this becomes the
isomorphism 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 → Ã�+ → Ã�+ → Ã�+ /�� → 0
and
·� �
0 → Ã+ → Ã+ → Ã+ /� → 0
whose final terms are canonically identified, so passing to continuous group
cohomology gives exact sequences
1
0 → A�+ /�� → H 0 (G, Ã�+ /�� ) → Hcts
(G, �+ )
33
and
1
0 → A+ /� → H 0 (G, Ã+ /�) → Hcts
(G, Ã+ )
in which the middle terms are canonically identified. Suppose now that � is
very good, and choose some �� with ��� good. We claim then that
�
�
A�+ /�� = im H 0 (G, Ã�+ /(��� )� ) → H 0 (G, Ã�+ /�� )
and
�
�
A+ /� = im H 0 (G, Ã+ /(��� )) → H 0 (G, Ã+ /�) .
This gives our desired isomorphism, since the right-hand terms in these identities
are canonically identified.
To see this, consider the commutative diagram
0
0
� A�+ /(��� )�
� H 0 (G, �+ /(��� )� )
� H 1 (G, �+ )
cts
�
�
� H 0 (G, �+ /�� )
�
� H 1 (G, �+ )
cts
� A�+ /��
·(� � )�
in which the first two downwards maps are the obvious reduction maps. Clearly
�
�
A�+ /�� ⊆ im H 0 (G, Ã�+ /(��� )� ) → H 0 (G, Ã�+ /�� )
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 by
p-power roots of �� for any good �)5 , so the image of the middle downwards
map dies in the lower H 1 . The proof for A+ /� is exactly analogous.
3.2
The general case
In this subsection we finish the proof of Theorem 3.4. For now we simply sketch
the material in this section; a full proof will appear in the next version of these
notes. In any case, the idea is clear: given X, choose a covering X = ∪i∈I Xi
by affinoid adic spaces Xi . By Theorem 3.5, each Xi♦ is a diamond, and we’d
like to glue up the Xi♦ ’s to form X ♦ .
First possible proof: Use Theorem 3.5 together with the following two propositions.
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 Perf proet ,
together with sheaf maps Di → D. Suppose furthermore that
1. 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 Ỹ → Y )
34
Proof sketch.
� Choose presentations hXi → Di , and then show that the induced
sheaf map hXi → D is surjective and pro-étale.
Second possible proof. Formalize the notion of “gluing along open subdiamonds”:
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 open
∼
subdiamond for each i, j ∈ I, and ϕij : Eij → Eji is an isomorphism. We further require that Eii = Di and ϕii = id, and that for any i, j, k ∈ I we have
ϕ−1
ij (Eji ∩ Ejk ) = Eij ∩ Eik and furthermore the diagram
ϕik
� Eki ∩ Ekj
Eij ∩ Eik
���
��
ϕjk ���
��ϕ�ij
�
���
�
�
��
���
Eji ∩ Ejk
commutes.
Proposition 3.14. Given a gluing datum in diamonds (I, (Di ), (Eij ), (ϕij )) as
above, there exists a diamond D and open subdiamonds Ei ⊂ D together with
∼
isomorphisms 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:
�
�
open cover X = ∪i∈I Xi and (gi : hXi → Di ) such that
Hom(hX , D) =
.
gi−1 (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 gives
more refined information.
3.3
Some basic compatibilities
Let X be an analytic adic space over Spa Zp . In this section we discuss some of
the compatibilities between X and X ♦ .
∼ |X|.
Proposition 3.15. There is a natural homeomorphism |X ♦ | =
Proof. One reduces (via Proposition 3.14) to the case of X affinoid. Following
the proof in the Berkeley notes, we may then take X = Spa(R, R+ ) and X̃ =
Spa(R̃, R̃+ ), 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 ♦ | ∼
= |hX̃ � /G|
∼
= |X̃ � |/G
∼
= |X̃|/G
�
�
∼
lim |Xi | /G
=
←i
∼
= lim |X|
←i
∼
= |X|.
35
Here the first line follows from Lemma 3.5 and Theorem 2.18, and the third line
+
follows from naturality of tilting. Writing (R∞ , R∞
) = limi→ (Ri , Ri+ ) for the
uncompleted direct limit, the fourth line follows from the identifications
|X̃| = |Spa(R̃, R̃+ )| = |Spa(R∞ , R∞ )| ∼
= lim |Xi |,
←i
�
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 Xfét ∼
=
♦
Xfét
.
Note: The objects of the site Xfét are “generalized adic spaces” in the sense
of Scholze-Weinstein which are finite étale over X. This slight complication
is necessary at the present time: if X = Spa(R, R+ ) is some sheafy affinoid,
then one wants an equivalence Xfét ∼
= (R)fét , 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)fét , the generalized adic space Spa(S, S + )
(where S + is the integral closure of R+ in S) is the associated object of Xfét .
In any case, when X is locally Noetherian or perfectoid, the objects of Xfét are
honest adic spaces.
Proof. One reduces to the affinoid case, and then proceeds as in Lemma 17.3.8
of the Berkeley notes.
Proposition 3.17. If X is locally Noetherian or perfectoid (or more generally if
X has a well-behaved étale site), the functor (−)♦ induces a natural equivalence
♦
Xét ∼
.
= Xét
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 is
generalizing.
In particular, if Z = Spa(S, S + ) → X = Spa(R, R+ ) is a weak closed immersion, 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 ♦ are
as in the previous sentence, and S is uniform,6 then Z = Z ♦ as subfunctors of
X ♦.
6 This
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 through
a 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 ♦ | (so
f 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 map
f � : 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 map
�
T → Z. But this is exactly Lemma 1.22.
4
Examples
If (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
Spd R = Spd(R, R◦ ).
4.1
The diamond of Spa Qp
By the general theory of Section 3, the adic space Spa Qp has an associated diamond Spd Qp . We explain here its canonical presentation, and its interpretation
as a functor.
Let ζpn , n ≥ 1 be a compatible sequence of primitive pn th roots of unity, and
cyc
�
let Qcyc
p = Qp (ζp∞ ). Let us describe the tilt of Qp . Setting On = Zp [ζpn ], we
have an isomorphism of Fp -algebras
On /(ζp − 1)
ζpn
n−1
=
Fp [t]/(tp
(p−1)
)
�→ 1 + t,
or equivalently
On /(ζp − 1)
ζpn
1−n
=
Fp [tp
�→
1 + tp
]/(tp−1 )
1−n
.
Taking the inductive limit over n, we get
∞
1/p
Zcyc
]/(tp−1 ),
p /(ζp − 1) = Fp [t
∞
∼
∼
gives Zcyc,�
= Fp [[t1/p ]] and Qcyc,�
=
p
p
cyc,�
cyc
)). Note that the sharp map Qp
→ Qp sends t to
and then applying (−)� = lim
∞
Fp ((t1/p
← (−)
x�→xp
n−1
� := lim (ζpn − 1)p
n→∞
.
×
Since Qcyc
= Q�
p (ζp∞ ) is a perfectoid pro-étale Zp -torsor over Qp with tilt
p
cyc,� ∼
1/p∞
Qp
)), Lemma 3.5 implies the following result.
= Fp ((t
37
Proposition 4.1. There is a natural isomorphism Spd Qp ∼
= hSpa Fp ((t1/p∞ )) /Z×
p,
n
n
1/p
where a ∈ Z×
to (1 + t1/p )a − 1.
p acts by sending t
A similar result holds for any finite extension K/Qp . Precisely, let Fq be
the 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 series
×
representing multiplication by a on G. Then Spd K ∼
where
= hSpa Fq ((t1/q∞ )) /OK
n
n
×
1/q
1/q
a ∈ OK acts by sending t
to [a](t
).
Next we show that perfectoid spaces over Qp are equivalent to perfectoid
spaces in characteristic p equipped with a “structure morphism” to Spd Qp . To
make the latter notion precise, note that Perf is a full subcategory of Dia, so
we may speak of the slice category Perf /D for any diamond D.
Proposition 4.2. The functor
� /Spa Q
Perf
p
Y
→ Perf /Spd Qp
�→ Y ♦
is an equivalence of categories.
This is Theorem 9.4.5 in the Berkeley notes. The proof we give below might
seem suspiciously short; this is because, in our interpretation, the key Proposition 3.10 does all of the heavy lifting, but in fact we reverse-engineered Proposition 3.10 through a close reading of Peter’s proof of Theorem 9.4.5 sketched
in the Berkeley notes.
Proof. Since we’ll need it later, we prove a more general result: for any analytic
� /X ∼
adic space X over Spa Zp , passage to diamonds induces an equivalence Perf
=
Perf /X ♦ .
We construct the essential inverse. Assume for simplicity that X = Spa(R, R+ )
is affinoid, so we may choose X̃ = Spa(R̃, R̃+ ), Xi , G, as in the statement of
Lemma 3.5, with X ♦ ∼
= (X̃ � )♦ /G, etc. Suppose we are given some Y ∈ Perf
together with a morphism Y ♦ → X ♦ ; we need to produce (canonically and
functorially) an untilt Y � over X. Since (X̃ � )♦ → X ♦ is a pro-étale G-torsor,
we get a pullback diagram
Ỹ ♦
� Y♦
�
(X̃ � )♦
�
� X♦
for some Ỹ ∈ Perf, where the horizontal arrows are pro-étale G-torsors and all
maps are G-equivariant. In particular, we get a G-equivariant map Ỹ → X̃ �
and thus a G-equivariant untilt Ỹ � → X̃. By Proposition 3.10, Y � = Ỹ � /G is
perfectoid, and is the desired untilt of Y .
38
4.2
Diamonds over a base
Let X be any analytic adic space over Spa Zp . There are two natural ways of
defining “diamonds fibered over X”:
1. We could define them by redoing the definition of diamonds with Perf proet
proet
�
replaced by the site Perf
/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 identification
proet
Sh(Perf proet )/X ♦ ∼
= Sh(Perf /X ♦ ),
so we can regard D as a sheaf on Perf proet
. Now the key point is that we have
/X ♦
(by the proof of 4.2) an equivalence of sites
proet
∼ Perf
�
Perf proet
/X ,
/X ♦ =
proet
�
so a sheaf equipped with a morphism to X ♦ “untilts” to a sheaf on Perf
/X .
We leave the remaining details to the interested reader.
The upshot is that for K/Qp a nonarchimedean field, diamonds over Spd K can
be interpreted as functors on the category of perfectoid spaces over Spa K, or
even as covariant functors from perfectoid (K, OK )-algebras to Sets, and conversely, 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 spaces
over Spa K is a diamond over Spd K. In any case, we’ll use all these equivalences
without further comment.
4.3
Self-products of the diamond of Spa Qp
By Proposition 2.20, the diamond (Spd Qp )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 |(Spd Qp )n | has Krull dimension n − 1. This seems to
suggest that the (nonexistent) structure map “Spa Qp → Spa F1 ” has relative
dimension one.
4.4
The sheaf Spd Zp
Recall the presheaf Spd Zp on Perf proet . The diamond Spd Qp is naturally a
subfunctor of Spd Zp , and there is also a subfunctor “Spd Fp ” corresponding to
characteristic p untilts (of which there aren’t very many). The object Spd Zp is
not a diamond, but it comes close:
39
Proposition 4.4. For any S ∈ Perf, there is a canonical analytic adic space
“S × Spa Zp ” over Spa Zp such that
(S × Spa Zp )♦ ∼
= S ♦ × Spd Zp .
In particular, S ♦ × Spd Zp is a diamond and Spd Zp is a sheaf.
Proof. The idea is as follows: Given S = Spa(A, A+ ) ∈ Perf with � ∈ A any
choice of pseudouniformizer, set
S × Spa Zp = Spa W (A+ ) � {x | |[�]|x = 0}.
Thi turns out to be an honest adic space, independent of the choice of �, and
the analogous space for general S is then constructed by gluing. See Sections
11.2-11.3 in the Berkeley notes.
Now we sketch the identification (S × Spa Zp )♦ ∼
= S ♦ × Spd Zp . We may
assume S = Spa(A, A+ ) is affinoid. For any Y = Spa(R, R+ ) ∈ Perf, the
Y -points of (S × Spa Zp )♦ are given by triples (Y � , ι, f ) where the data of
f : Y � = Spa(R� , R�+ ) → S × Spa Zp
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��+ , which
induces 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 ♦ × Spd Zp .
We can bootstrap this a bit:
Proposition 4.5. For any diamond D, the sheaf D × Spd Zp is a diamond.
Proof. Choose a surjective and pro-étale map X ♦ → D for some X ∈ Perf;
then X ♦ × Spd Zp → D × Spd Zp 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 between
untilts (S � , ι) and sections s of the projection S ♦ × Spd Zp → S ♦ , and any untilt
induces a closed immersion i : S � → S × Spa Zp such that
s : S♦
(ι♦ )−1
♦
i
→ S ��♦ ∼
= S �♦ → S ♦ × Spd Zp
gives the corresponding section. This also yields a natural bijection between
untilts of S and closed immersions S � → S × Spa Zp for which the composite
map S �♦ → S ♦ × Spd Zp → S ♦ is an isomorphism.
40
Let (Spd Zp )n+ ⊂ (Spd Zp )n be the subfunctor whose Y -points parametrize
n-tuples of untilts (Yi� , ιi ) (1 ≤ i ≤ n) with at least one of the Yi� ’s living over
Spa Qp . We leave it to the reader to show that (Spd Zp )n+ is a diamond, covered
by the open subdiamonds
Di = Spd Zp × · · · × Spd Zp × Spd Qp × Spd Zp × · · · × Spd Zp
(with Spd Qp in the ith position) for 1 ≤ i ≤ n. Is this the largest subfunctor
of (Spd Zp )n which is naturally a diamond?
4.5
+
The diamond BdR
/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�◦ )[ p1 ] along the kernel of the natural surjection
+
θ : W (R�◦ )[ p1 ] � R. When R = Cp , this is the usual Fontaine ring BdR
. In
+
general ker θ is principal and generated by some non-zerodivisor ξ, so BdR (R) is
i
filtered by the ideals Fili = (ker θ)i with associated gradeds gri B+
dR (R) � ξ R.
n
+
Let BdR /Fil → Spd Qp be the functor whose sections over a given Spa(R, R+ )♦ →
n
�
�
Spd Qp are given by the set B+
dR (R )/Fil , where R is the untilt of R determined by the given map to Spd Qp .
+
Theorem 4.7. The functor BdR
/Filn 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 BdR
/Fil1 ∼
= A1,♦ where
1
n
A = ∪n≥1 Spa Qp �p x� is the affine line over Spa Qp .
Let B = (Spa Zcyc
p [[T ]])η , and let B̃ = lim←ϕ B where ϕ is the endomorphism
of B given by the map T �→ (1 + T )p − 1. Explicitly, B̃ = (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 B̃ is perfectoid, and the map π : B̃ → A1 characterized by π ∗ x = log(1+T )
is a perfectoid pro-étale covering of A1 . (More precisely, B̃♦ → A1,♦ is a proétale G-torsor, with G = Qp (1) � Z×
p .)
Claim: There is an isomorphism of functors
+
BdR
/Filn ×A1,♦ B̃♦
+
� BdR
/Filn−1 ×Spd Qp B̃♦
for any n ≥ 2.
Granted this claim, we deduce the theorem as follows: Since B̃♦ → A1,♦ is
surjective and pro-étale, the map
+
+
BdR
/Filn ×A1,♦ B̃♦ → BdR
/Filn
41
is surjective and pro-étale, so by Proposition 2.2 it’s enough to show that
+
BdR
/Filn ×A1,♦ B̃♦ is a diamond. But the Claim together with Proposition
+
2.19 reduce this to the fact that BdR
/Filn−1 is a diamond, which is exactly our
induction hypothesis.
It remains to prove the claim. In general, the map θ induces a natural trans+
+
formation BdR
/Filn → A1,♦ of functors over Spd Qp . Let s : B̃ → BdR
/Filn be
the natural transformation sending r = (r0 , r1 , r2 , . . . ) ∈ B̃(R� ) to log[1 + r� ]
where r� = (r0 , r1 , . . . ) ∈ (R�◦ /p)� . Then (θ ◦ s)(r) = log(1 + r0 ) = π(r). In
particular, the diagram
♦
+
BdR
� B̃
��
�
�
π
�� s
���
�
θ
� A1,♦
/Filn
commutes, and π is surjective and pro-étale. The claim is then deduced as
+
follows: given (a, b) ∈ BdR
/Filn × B̃♦ with θ(a) = π(b), the element a − s(b)
n
+
lives in ker θ ⊂ BdR /Fil , so we get a natural isomorphism
+
BdR
/Filn ×A1,♦ B̃♦
∼
= ker θ ×Spd Qp B̃♦
(a, b) → (a − s(b), b)
(x + s(y), y) ← (x, y).
Furthermore, after base change to Spd Qcyc
we have an exact sequence
p
t
1,♦
n
+
+
cyc
0 → BdR
/Filn−1 ×Spd Qp Spd Qcyc
→0
p → BdR /Fil ×Spd Qp Spd Qp → AQcyc
p
where t is the usual log[ε] of p-adic Hodge theory, so we get an isomorphism
n−1
+
ker θ ×Spd Qp Spd Qcyc
×Spd Qp Spd Qcyc
p � BdR /Fil
p
after making a choice of t. But B̃ naturally lives over Qcyc
p , so then
ker θ ×Spd Qp B̃♦
♦
∼
cyc
= (ker θ ×Spd Qp Spd Qcyc
p ) ×Spd Qp B̃
� +
�
� BdR
/Filn−1 ×Spd Qp Spd Qcyc
×Spd Qcyc
B̃♦
p
p
+
� BdR
/Filn−1 ×Spd Qp B̃♦ ,
and the claim follows.
5
Moduli of shtukas
In this section, we construct moduli spaces of mixed-characteristic local shtukas
and explain their relation with Rapoport-Zink spaces and local Shimura varieties. The material in this section follows closely the ideas of Caraiani-Scholze,
Fargues-Fontaine, Kedlaya-Liu, Scholze and Scholze-Weinstein.
42
unr . Let
�
Let E/Qp be a finite extension with residue field Fq ; set Ĕ = E
G/Spec E be a connected reductive group, which we assume for simplicity is
quasisplit. 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 a
triple D = (G, µ, b) where G and µ are as specified, and b ∈ G(Ĕ) is an element
whose 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-bundle
Eb,S on the relative Fargues-Fontaine curve XS,E , functorially in S, whose isomorphism class depends only on the σ-conjugacy class of b. Our goal in this
section is to explain the following definition and theorem, with all ingredients
carefully laid out.
Definition 5.1. The moduli space of shtukas associated with the datum D is
the functor ShtD,∞ : Perf Fq → Sets sending S ∈ Perf Fq to the set of isomorphism classes of tuples
(S � , ι, F, u, α)
∼
where (S � , ι) is an untilt of S over Ĕ, 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 a
G-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 definition
of ShtD,∞ (S):
• the data of b is “like” the p-divisible group H = Hb ×Fq S 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 H̃ of H to S � ,
• the data of α is “like” a trivialization of the rational Tate module Vp H̃.
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-divisible
groups.
In any case, this is a very rich object. First of all, it’s obviously fibered
over Spd Ĕ, and tuples of this form have no automorphisms. We will see that
ShtD,∞ defines a sheaf on Perf proet
. Next, the locally profinite group G :=
/Spd Ĕ
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 two
group 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 Ĕ
7 For clarity, we do not take a maximally general setup here: one can instead consider a
P
tuple of dominant cocharacters µ1 , . . . , µn , and then b should satisfy [b] ∈ B(G, − 1≤i≤n µi ).
43
and the Hodge-Tate period map
πHT : ShtD,∞ → GrD,HT ∼
= GrG,−µ ×Spd E Spd Ĕ.
(Here GrG,µ is an open Schubert cell in the BdR -affine Grassmannian GrG defined 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 modification 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,GM of the diamond GrD,GM , stable under the action of
Jb .
ii. The induced morphism πGM : ShtD,∞ → Gradm
D,GM is representable and
pro-étale, and makes ShtD,∞ into a pro-étale G-torsor over Gradm
D,GM . In particular, ShtD,∞ is a diamond over Spd Ĕ.
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,GM over
geometric points are identified with the discrete set G/K.
iv. When µ is minuscule, the diamonds GrD,GM , Gradm
D,GM and ShtD,K are
♦
in the essential image of the functor (−) from smooth rigid analytic spaces
∼
over Spa Ĕ. In particular, the rigid space MD,K over Spa Ĕ such that M♦
D,K =
ShtD,K is the local Shimura variety with K-level structure associated with D
sought by Rapoport and Viehmann.
v. When µ is minuscule and G = GLn , there is a natural isomorphism
ShtD,∞ ∼
= M♦
Hb ,∞ compatible with all structures, where MHb ,∞ is an infinitelevel Rapoport-Zink space.
Let us note right away that parts i.-iii. of the previous theorem are an
essentially immediate consequence of the ideas developed in the Berkeley notes
(although the definition of ShtD,∞ is only implicit there). The key result in
the proof of part iv. is a theorem of Caraiani-Scholze asserting that when
µ is minuscule, GrG,µ is canonically identified with the diamond F�♦
G,µ of a
certain 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 curve
Throughout the rest of Section 5, E/Qp denotes a finite extension with uniformizer 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 perfect Fq -algebras to π-torsion-free π-adically complete strict OE -algebras. Let
ϕ = ϕq = ϕf ⊗ id be the natural q-Frobenius on WOE (R). Let Ĕ = WOE (Fq )[ p1 ]
be the completion of the maximal unramfied extension of E, so σ = ϕ generates
Gal(Ĕ/E).
Let S ∈ Perf Fq be given. In this section we construct analytic adic spaces
YS and XS over Spa E such that YS♦ ∼
= S ♦ ×Fq Spd E and XS♦ ∼
= S ♦ /FrobZ
q ×Fq
Spd E, functorially in S. (If we need to emphasize E, we write XS,E etc.) When
S = Spa(F, OF ) for F/Fq a perfectoid field, XS = XF,E is the adic FarguesFontaine curve.
Suppose first that S = Spa(A, A+ ) is affinoid perfectoid, and let � ∈ A+ be
any choice of pseudouniformizer. Then we set
YS = Spa WOE (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
� n n�
p
1
YS,n = Spa WOE (A+ ) [�]
p , [�] [ p ] ⊂ YS
sit nestedly as YS,1 ⊂ YS,2 ⊂ · · · ⊂ YS,n ⊂ · · · with YS = ∪n≥1 YS,n , and a
direct calculation verifies that each YS,n is preperfectoid and hence honest. By
gluing we obtain YS for a general S ∈ Perf Fq . The Frobenius ϕq on YS turns
out to be properly discontinuous, and we then set XS = YS /ϕZ
q . 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 XS♦ ∼
= S ♦ /FrobZ
q ×Fq Spd E gives
a natural continuous projection |XS | → |S| defined as the composition
∼ ♦
|XS | = |XS♦ | → |S ♦ /FrobZ
q | = |S | = |S|
(using that Frobq acts trivially on |S|). Any morphism T → S in Perf Fq induces
a canonical morphism XT → XS compatible with the previously defined map
on topological spaces, and if T → S is “blah” where “blah” is any one of the
conditions “closed immersion”, “open immersion”, “finite étale”, “étale”, etc., then
XT → 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
|i|
|S| ∼
= |S � | → |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 canonical 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-dimensional
E-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 given
by natural isomorphisms of tensor functors. When G = GLn /E, we identify
BunG (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 suitably
as 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 Perf Fq for which BunG (S) is the
category 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 on YU }
is a stack on Sproet ; since YS is preperfectoid, this can be reduced to showing
that the fibered category Y /X �→ {vector bundles on Y } is a stack on Xproet for
� which can be checked directly.
any X ∈ Perf,
When S = Spa(C, OC ) is a geometric point, there is a complete classification
of G-bundles on XS due to Fargues-Fontaine and Fargues. Precisely, given any
S ∈ Perf Fq and any b ∈ G(Ĕ), a recipe of Fargues recalled below produces a
G-bundle Eb,S on XS whose isomorphism class depends only on the σ-conjugacy
class [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). When
S = 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 ∈ Perf Fq and any G-bundle
E 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+ ) ∈
Perf Fq is 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-modules
8 (Here
Gan denotes the adic group over Spa E associated with G.)
46
with G-structure over WOE (A+ )[ p1 ] to the category of ϕ-equivariant G-bundles
on YS . Given b, we define a ϕ-module with G-structure Mb over WOE (A+ )[ p1 ]
as follows: Mb is the tensor functor on Rep(G) sending (ρ, V ) to
WOE (A+ )[ p1 ] ⊗Ĕ (V ⊗E Ĕ) ∼
= WOE (A+ )[ p1 ] ⊗E V
with the Frobenius action given by ϕ ⊗ ρ(b)(id ⊗ σ). We then define Eb,S as
the associated G-bundle on XS . This construction is clearly functorial in S, so
we obtain the bundle Eb,S for general S by gluing. Note that for any morphism
f : T → S of perfectoid spaces over Fq , the pullback map f ∗ on G-bundles
induces 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) XSpa Fq .
Note also that if we take b = id, then Eid,S recovers the trivial G-bundle (and
makes sense for any S ∈ Perf Fq ).
The automorphism group of the bundle Eb turns out to be quite rich:
Proposition 5.6. Let Jb denote the group sheaf on Perf Fq given by S �→
Aut(Eb,S ). Then Jb is the sheafification of the presheaf sending any S = Spa(A, A+ ) ∈
Perf Fq to the group
�
�
+
−1
Jb (S) = g ∈ G(B+
(A
/�))
|
g
=
bϕ
(g)b
.
q
crys,E
∼ Ub � Jb , where Jb =
Furthermore, Jb is canonically a semidirect product Jb =
Jb (E) denotes the E-points of the algebraic group
�
�
Jb (A) = g ∈ G(A ⊗E Ĕ) | g = bσ(g)b−1
and where Ub is defined as the subgroup which acts trivially on the graded pieces
of 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 a
nontrivial “unipotent group” of “dimension �νb , 2ρ�.” The sheaf Jb is an absolute
diamond in the sense of §7; in particular, the group sheaf Jb,Ĕ = Jb ×Fq Spd Ĕ
(i.e., the restriction of Jb to Perf /Spd Ĕ ) is provably a group diamond over Spd Ĕ.
Finally, we briefly study “trivial” G-bundles on XS .
Definition 5.7. Given S ∈ Perf Fq , a G-bundle E on XS is pointwise-trivial
if 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 ∈ Perf Fq together with a pointwise-trivial G-bundle
E on XS , the functor
T rivE/S : Perf /S
{f : T → S}
→
�→
Sets
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 Grassmannian
Recall that for any perfectoid Tate ring R/Qp , we have the de Rham period
rings B+
dR (R) and BdR (R).
Fix a connected reductive quasisplit group G/Spec E. Choose a maximal
torus 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 interest
in this section is the following:
Definition 5.9. The de Rham affine Grassmannian GrG is the functor on
perfectoid (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 Spd E 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 Perf proet
/Spd E . We remark that
GrG,≤µ contains a subfunctor GrG,µ where the trivialization has relative position given exactly by µ; this coincides with GrG,µ when µ�
is minuscule, but
in general there is a nontrivial stratification GrG,≤µ “ = ” ν≤µ GrG,ν . For
completeness, 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
v1
 e2 
 v2 





 = (ρ ◦ ν)(ξ) · 

..
..




.
.
edimV
vdimV
�
�
+
for some generator ξ of ker θ : BdR (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 over
Spd E, and GrG,µ is an open subdiamond of GrG,≤µ .
This is one of the main theorems from Peter’s Berkeley course; the proof is
difficult.
We can also interpret GrG,≤µ and GrG,µ in terms of vector bundles on
the 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 a
G-bundle on X and
∼
u : E|X �S → Etriv |X �S
is a µ-bounded (resp. µ-positioned) modification of the trivial G-bundle along
S.
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 proposition
extends to a subsheaf of Etriv .
b
We also need a “twisted” form GrEG,µ
of GrG,µ defined over Spd Ĕ:
b
Proposition 5.13. For any b ∈ G(Ĕ), let GrEG,µ
be the functor on perfectoid
+
(Ĕ, OĔ )-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 a
G-bundle on X and
∼
u : F|X �S → Eb,S � |X �S
b
is a µ-bounded modification of the bundle Eb,S � along S. Then GrEG,µ
is a diamond, canonically isomorphic to GrG,µ ×Spd E Spd Ĕ, and the group diamond
b
Jb,Ĕ = Jb ×Fq Spd Ĕ acts naturally on GrEG,µ
.
Remark 5.14. Suppose b ∈ B(G, µ−1 ) is the unique µ−1 -ordinary element; then
b
the dimension of Jb,Ĕ coincides with the dimension of GrEG,µ
. It seems reasonable to conjecture that in this case,
b
i. The action of Jb,Ĕ on GrEG,µ
has a unique open orbit, and
b ,adm
ii. This open orbit coincides with the admissible locus GrEG,µ
defined
below.
b ,adm
Is there an “orbit-theoretic description” of GrEG,µ
for more general b?
5.3
Moduli of shtukas
Proof of Theorem 5.2,i.-iv. Given a local shtuka datum D = (G, µ, b), we set
b
GrD,GM := GrEG,µ
. Recall this is the diamond over Spd Ĕ whose S-points over
�
a given (S , ι) ∈ (Spd Ĕ)(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,GM as the subfunctor of GrD,GM parametrizing F’s which are pointwise-trivial: precisely, a given S → GrD,GM factors over
∗
+
Gradm
D,GM iff for all geometric points s = Spa(C, C ) → S the specialization s F
of the associated F is isomorphic to the trivial G-bundle on Xs . An easy calculation on geometric points shows that the image of πGM is exactly Gradm
G,GM , and
adm
that the fibers of πGM over any geometric point of GrD,GM form a G-torsor.
49
We need to prove that Gradm
D,GM is open and partially proper in GrD,GM . Let
S ∈ Perf Fq be any perfectoid space with a map f : S ♦ → GrD,GM over Spd Ĕ.
By definition, this map gives rise to a µ-bounded modification FS of the Gbundle Eb,S over XS . We need to show that the locus in S where FS becomes
pointwise-trivial is open and partially proper. Let (ρ, V ) be any algebraic representation of G, so we get vector bundles ρ ◦ FS and ρ ◦ Fb,S with the first as
a ρ ◦ µ-bounded modification of the second. An easy calculation (using the fact
that 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. This
in 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 a
theorem of Fontaine and Rapoport, Gradm
D,GM has points over finite extensions
of Ĕ, so in particular is nonempty. (Here again we’re using the containment
b ∈ B(G, µ−1 ).) This proves part i.
Summarizing, we have a surjective G-equivariant map πGM : ShtD,∞ →
adm
GrD,GM whose target is a diamond over Spd Ĕ and whose fibers over any geometric point form a G-torsor. A direct calculation shows the natural action
map
G × ShtD,∞ → ShtD,∞ ×Gradm
ShtD,∞
D,GM
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 ♦ → Gradm
D,GM be any map,
∗ univ
so we get a pointwise-trivial G-bundle FS = ”f F
” on XS . Then we get a
pullback diagram
� S♦
T riv♦
FS /S
f
�
ShtD,∞
πGM
�
� Gradm
D,GM
with T rivFS /S defined as in Theorem 5.8. Theorem 5.8 shows that the upper
horizontal arrow here is pro-étale with source a representable sheaf, so πGM is
representable and pro-étale. This proves parts i. and ii. of the theorem. Part
iii. follows immediately from Proposition 2.13.
For part iv., the essential point is the following result, proved by CaraianiScholze:
Proposition. There is a flag variety F�G,µ over Spa E together with a
natural 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 Ĕ, the diamond GrD,GM
for minuscule µ is in the essential image of the functor (−)♦ from smooth rigid
analytic spaces over Spd Ĕ. Since ShtD,K is étale over GrD,GM , Proposition
3.17 implies that ShtD,K also comes this same category via (−)♦ .
9 This
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 ingredient 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. Let
µ be as in the theorem, so µ = (1, . . . , 1, 0, . . . , 0) with d 1’s and n − d 0’s for
some 0 ≤ d ≤ n. Let b ∈ GLn (Ĕ) 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 -module
over Fq , and let M (Hb ) be the (covariant) rational OE -Dieudonne module of
Hb , i.e. M (Hb ) = Ĕ n with the endomorphism F given by F = πb · σ. Note that
Hb 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 (Ĕ, OĔ )algebra, so we have our associated i : S → XS � . Then
�
�F ⊗ϕ=π
+
H 0 (XS � , Eb,S � ) ∼
= M (Hb ) ⊗Ĕ B+
crys,E (R /π)
and
H 0 (S, i∗ Eb,S � ) ∼
= M (Hb ) ⊗Ĕ R,
functorially in (R, R+ ). Under these isomorphisms, the natural map
�
�F ⊗ϕ=π
+
θ : M (Hb ) ⊗Ĕ B+
(R
/π)
→ M (Hb ) ⊗Ĕ R
crys,E
of vector spaces over E identifies with the map
H 0 (XS � , Eb,S � ) → H 0 (XS � , i∗ i∗ Eb,S � ) = H 0 (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 Ĕ
associated with Hb .
Theorem 5.16 (Scholze-Weinstein). The space MHb ,∞ is isomorphic to the
functor on perfectoid (Ĕ, OĔ )-algebras sending (R, R+ ) to the set of ordered
tuples
�
�F ⊗ϕ=π
+
s1 , . . . , sn ∈ M (Hb ) ⊗Ĕ B+
crys,E (R /π)
satisfying the following two conditions:
i. The cokernel W of the map
θ(si )
Rn −→ M (Hb ) ⊗Ĕ R
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
�
�F ⊗ϕ=π
x∗ s
+
0 → E n −→i M (Hb ) ⊗Ĕ B+
→ W ⊗R C → 0
crys,E (C /π)
is exact.
51
On the other hand, we have the following easy reduction regarding ShtD,∞ .
Proposition 5.17. As a functor on perfectoid (Ĕ, OĔ )-algebras, ShtD,∞ sends
(R, R+ ) (with associated S = Spa(R, R+ ), X = XS � and closed immersion
n
i : S �→ X ) to the set of injective OX -module morphisms u : OX
→ Eb,S � such
that coker u � i∗ W for some rank d vector bundle W on S.
Maintain the setup of this proposition; going to global sections and using
Proposition 5.15 plus the identification H 0 (X , OX ) ∼
= E, we get
�
�F ⊗ϕ=π
+
H 0 (u) : E n → H 0 (XS � , Eb,S � ) ∼
(R
/π)
.
= M (Hb ) ⊗Ĕ B+
crys,E
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 Ĕ.
Pulling back the exact sequence
n
OX
→ Eb,S � → i∗ W → 0
of OX -modules along the closed immersion i gives an exact sequence
OSn → i∗ Eb,S � → W → 0
of vector bundles on S. By [KL15], passage to global sections gives an equivalence 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 ) ⊗Ĕ R → W → 0,
with the first arrow identified with θ(s1 ), . . . , θ(sn ). This verifies the first condition in Theorem 5.16.
x
For the second condition, let η = Spa(C, C + ) → Spa(R, R+ ) be any geometric point, so we get a morphism fx : Xη� → XS � as described previously, and a
point iη : η �→ Xη� sitting in a pullback square
η
iη
� Xη �
x
�
S
fx
�
i
� XS �
10 With bundle morphisms corresponding to R-module morphisms M → N having projective
cokernel
52
where the horizontal maps are closed immersions.
Pulling back the exact sequence
n
0 → OX
S�
Clearly fx∗ Eb,S � = Eb,η�
→ Eb,S � → i∗ W → 0
along fx and applying H 0 (Xη� , −) gives an exact sequence
0 → E n → H 0 (Xη� , Eb,η� ) → H 0 (Xη� , fx∗ i∗ W) → 0,
exact on the right because H 1 (Xη� , OX ) = 0. Using Proposition 5.15 again, the
�
�F ⊗ϕ=π
+
middle term identifies with M (Hb ) ⊗Ĕ Bcrys,E
(C + /π)
, and the second
∗
∗
∗
arrow then identifies with x si . Finally, fx i∗ W = iη∗ x W, so
H 0 (Xη� , fx∗ i∗ W) = H 0 (Xη� , iη∗ x∗ W) = H 0 (η, x∗ W) = W ⊗R C
and the exact sequence in question identifies with the sequence
�
�F ⊗ϕ=π
x∗ s
+
0 → E n −→i M (Hb ) ⊗Ĕ B+
→ W ⊗R C → 0,
crys,E (C /π)
and thus the latter sequence is exact. This verifies the second condition in
Theorem 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 explicitly
construct 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
ξ
� M♦
ShtD,∞
Hb ,∞
��
�
��
�
�
��
��
��
�
�� πGM
πGM
��
���
F�♦
where F� is the rigid analytic flag variety over Spa Ĕ whose (R, R+ )-points
parametrize rank d projective R-module quotients of M (Hb ) ⊗Ĕ R. Clearly
�
imπGM
⊆ imπGM ; by Theorem C of Scholze-Weinstein, the opposite inclusion 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 πGM
is a
♦,adm
pro-étale GLn (E)-torsor over F�
; by Theorem D in Scholze-Weinstein, it’s
easy to check that πGM is also a pro-étale GLn (E)-torsor over F�♦,adm . The
following easy proposition then implies ξ is an isomorphism.
Proposition 5.19. Let G be a sheaf on Perf proet , and let G be a locally profinite
group. Then any G-equivariant morphism f : F → F � of pro-étale G-torsors
over 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 an
isomorphism pro-étale-locally on G, and then f must be an isomorphism since
everything is a sheaf.
5.4
The Newton stratification
Maintain the setup of Definition 5.9 and the discussion thereafter. Following
Caraiani-Scholze, we use Proposition 5.12 and Fargues’s classification of Gbundles on the curve to define a G-stable Newton stratification of GrG,µ indexed
by elements of B(G):
Definition 5.20. For any [b] ∈ B(G), the Newton stratum GrbG,µ is the following subfunctor of GrG,µ : given S ∈ Perf Fq 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 GrbG,µ if and only if for
all geometric points s : Spa(C, C + ) → S we have s∗ E � Eb,s as G-bundles on
Xs .
In general, these strata are only defined as subfunctors, a priori; they also
give a stratification of |GrG,µ | by locally closed subsets.
Proposition 5.21. The stratum GrbG,µ is nonempty if and only if b ∈ B(G, µ).
Remark 5.22. In the case of minuscule µ, this was proved by Rapoport in his
appendix to Scholze’s Lubin-Tate tower article.
Proof. “Only if” was proved by Caraiani-Scholze. “If” follows from the next
theorem and the non-emptyness of the space ShtDb ,∞ therein.
When b is the basic (resp. µ-ordinary) element of B(G, µ), the associated
stratum 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 GrbG,µ coincides with the image of the Hodge-Tate
period morphism
πHT
ShtDb ,∞ →
GrDb ,HT ∼
= GrG,µ ×Spd E Spd Ĕ → GrG,µ .
Furthermore, letting Jb,Ĕ denote the group sheaf on Perf /Spd Ĕ defined as before,
there is a natural action of Jb,Ĕ on ShtDb ,∞ extending the action of Jb and
equivariant for the trivial action on the target of πHT such that the action map
ShtDb ,∞ × Jb,Ĕ → ShtDb ,∞ ×GrDb ,HT ShtDb ,∞
is an isomorphism and on geometric points we have GrbG,µ (C) ∼
= ShtDb ,∞ (C)/Jb,Ĕ (C).
54
This strongly suggests that we should regard GrbG,µ as the coarse moduli
space of the stack [ShtDb ,∞ /Jb,Ĕ ].
However, something better is true. Precisely, note that GrbG,µ ⊂ GrG,µ is
defined by applying the recipe “F ⊂ |D| � F ⊂ D” from the proof of Theorem
2.42 to a suitable subspace |GrbG,µ | ⊂ |GrG,µ |. Caraiani-Scholze show using
results of Kedlaya-Liu that |GrbG,µ | is locally closed in |GrG,µ |, and it’s clearly
generalizing: x : Spa(C, C + ) → GrG,µ factors through GrbG,µ if and only if the
rank one generization Spa(C, OC ) → GrG,µ so factors, because the category of
G-bundles on XSpa(R,R+ ) is totally insensitive to changing R+ ! So now we may
apply Theorem 2.42 to deduce the following result.
Theorem 5.24. Each Newton stratum GrbG,µ is a diamond.
5.5
Admissible and inadmissible loci
Here 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 with
weights (k1 ≥ · · · ≥ kn ). After twisting, we may assume that all weights of µ
are ≥ 0. Fix b ∈ GLn (Q̆p ) 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 again
unwind 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 vector
bundles which is an isomorphism away from i : S �→ XS � and which induces a
µ-positioned modification
along the graph of S.
0 → E → Eb,S � → Q → 0
The last clause here means concretely that Q is an OX -module sheaf on
XS � 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) �
ki
⊕ni=1 B+
dR (K)/Fil .
inadm
Inside Grµ,b we have the admissible locus Gradm
.
µ,b , and its complement Grµ,b
adm
To be clear: we’ve already seen that Grµ,b is an open and partially proper subc
diamond, so in particular |Gradm
µ,b | ⊂ |Grµ,b | is open and specializing. Thus its
adm c
complement |Grµ,b | ⊂ |Grµ,b | is closed and generalizing; we define Grinadm
as
µ,b
adm c
the subdiamond associated with |Grµ,b | ⊂ |Grµ,b | via Theorem 2.42.
What do these spaces look like? Note that in some sense, Grinadm
should
µ,b
usually be kind of a strange object: for example, when End(Eb ) is a division
algebra, one can check that Grµ,b and Gradm
µ,b have the same K-points for any
finite extension K/Qp . We’ll see examples like this where Grinadm
, which cannot
µ,b
have points over finite extensions K/Qp , is nevertheless nonempty.
55
Example 5.26. Take n = 2, µ = (3, 0), b =
�
1
�
(so d = 2), so
p−3
2
Eb = O( 32 ). The space Gr = Grµ,b is, morally speaking, a B+
dR /Fil -bundle over
1
P ; in any case, it has dimension three. For any point x : Spa(C, C + ) → Gr we
get a modification
0 → Ex → O( 32 ) → 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-Fontaine
shows that E � O2 or E � O(1) ⊕ O(−1). These two possibilities correspond
exactly to the admissible and inadmissible loci. Pushing this a bit further, one
gets the following nice description of Grinadm :
Proposition (H.-Weinstein). The inadmissible locus here is canonically and
Jb -equivariantly isomorphic to the pro-étale sheaf over Spd Qp2 which sends a
perfectoid Qp2 -algebras (A, A+ ) to
�
�
ϕ2 =p
B+
(A)
�
0
/Q×
crys
p.
Sketch. Let S = Spa(A, A+ ) be as in the statement, and let f : S → Grinadm
be an S-point. Let
u
0 → E → Eb,S � → Q → 0
be the corresponding modification. Then Ex � O(1) ⊕ O(−1) at any geometric
point x = Spa(C, C + ) → S, so by a theorem of Kedlaya-Liu we get a filtration
of E by line bundles
0 → E >0 → E → E <0 → 0
where E >0 has slope one at all points. In particular, we have an isomorphism E >0 � O(1) pro-étale-locally on S, which is moreover unique up to Q×
pambiguity. Composing this isomorphism with u gives a nowhere-vanishing map
O(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 section
2
=p
of O(1/2) over XS � , i.e. an element of B+,ϕ
(A) � 0, again (pro-étale-locally)
crys
up to Q×
-ambiguity.
p
6
6.1
More on morphisms
Smooth morphisms
Clearly a smooth morphism of relative dimension zero should just be an étale
morphism; for nonzero relative dimension, Scholze has suggested the following
definition:
Definition 6.1 (Scholze). A morphism f : X → Y of diamonds is smooth of
pure dimension d > 0 if, pro-étale-locally on Y and analytic-locally on X, f can
be factored (if d > 0) into a composite
g
h
X→W →Y
56
where g is smooth of pure dimension d − 1 and h can be fit into a commutative
diagram
W̃
h̃
q
�
W
where
� B1
Y
pr
h
�
�Y
B1Y = “Y ×Spd Fp Spd Fp �T � ”
is the relative one-dimensional closed unit disk over Y 11 (with pr the evident
projection), h̃ is étale, and q realizes W̃ as a pro-étale Γ-torsor over W for some
pro-p group Γ.
A morphism f : X → Y of diamonds is smooth if X admits a covering
by open subdiamonds Xi such that each Xi → Y is smooth of some relative
dimension di . If moreover supi di < ∞, we say f is finite-dimensional.
6.2
Morphisms locally of finite type
Lemma 6.2. Let k be a perfect field, and let X be a reduced k-scheme locally of
finite type which is equidimensional of dimension d. Then there exists a dense
open 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 is
regular: this is open by a classical result, and dense since by the reducedness
assumption 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-dimensional
k-scheme locally of finite type. Then we can find a finite chain of reduced closed
subschemes
X = Z0 ⊃ Z1 ⊃ Z2 · · · ⊃ Zn = ∅
such that dimZi ≤ dimX − i and Zi � Zi+1 with its induced reduced structure
is smooth over Spec k.
Proof. Easy downward induction on dimX, using the previous lemma to get
started: let X0 (resp. X1 ) be the union of all dimX-dimensional (resp. < dimXdimensional) irreducible components of X, and take
Z1 = X1 ∪ (X0 � Reg(X0 ))
with its induced reduced structure.
11 I.e., B1 has functor of points given by Hom(T ♦ , B1 ) = Hom(T ♦ , Y ) × H 0 (T, O + ) for
Y
Y
T
arbitrary T ∈ Perf.
57
Amusingly, the definition of a locally finite type diamond seems less accessible than the definition of a smooth diamond. Granted a workable notion of
smoothness, though, the previous lemma suggests the following inductive definition 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 topologically
nilpotent, and let D be a locally spatial diamond over Spd F . Then D is locally
finite type of dimension ≤ d if there exists an open subdiamond U ⊆ D with
the 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 generalizing subset |D| � |U| ⊆ |D| by Theorem 2.42 is locally finite type of relative
dimension ≤ d − 1, and the morphism Z → D is quasicompact.
More generally, we say a locally spatial diamond D/Spd F is locally of finite
type if we can find a covering of D by open subdiamonds (Di ⊂ D)i∈I such that
each 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 topologically nilpotent, and let X be any rigid analytic space over Spa F . Then X ♦
is locally finite type over Spd F . Furthermore, if X is quasicompact, then X ♦
is locally finite type of dimension ≤ dimX.
Proof. Since X and X red have the same diamond, we can assume that X is
reduced. Covering X by admissible open affinoid subspaces and using the compatibility of (−)♦ with open immersions, it clearly suffices to prove the second
claim in the case where X = Sp(A) is a reduced affinoid.
Let A be any reduced affinoid F -algebra; since any affinoid algebra is excellent (cf. Conrad’s paper on irreducible components of rigid spaces), the subset
Reg(A) ⊆ SpecA is Zariski-open, and then dense as well since by assumption it contains the generic points of all irreducible components. Feeding this
observation into an easy variant of the proof of Lemma 6.2, we get (for any
finite-dimensional reduced rigid space X) a chain
X = Z0 ⊃ Z1 ⊃ Z2 · · · ⊃ Zn ⊃ ∅
of reduced Zariski-closed rigid subspaces with exactly the same properties as in
Lemma 6.3.
Now, observe that if we take D = X ♦ and U = (X � Z1 )♦ as in Definition
6.4, then U is smooth of dimension ≤ dimX and |U| = |X| � |Z1 | is indeed open
and specializing inside |D| = |X|, so |Z1 | ⊂ |D| is closed and generalizing and
Theorem 2.42 then produces a subdiamond Z ⊂ D associated with |Z1 |. The
proposition then follows by downward induction on dimX if we can identify Z
with the subdiamond Z1♦ ⊂ 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 Spd F is
proper if it is locally finite type, separated, quasicompact, and partially proper
as a sheaf.
Here we say a sheaf F on Perf is partially proper if the natural restriction 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 over
Spd E.
7
Into the abyss
In this section we define categories of diamond stacks and smooth diamond
stacks.
7.1
Absolute diamonds
Definition 7.1. An absolute diamond is a sheaf F on Perf such that F × D is
a diamond for any diamond D.
Note that unlike the category of diamonds, the category of absolute diamonds has a final object, namely the "point" Spd Fp . This is also the final object
in 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 diamond
as soon as one of the Di ’s is a diamond.
Proposition 7.2. If E1 → D ← E2 is a diagram of absolute diamonds, then
E1 ×D E2 is an absolute diamond. In particular, the category of absolute diamonds
has 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, and
the category of diamonds admits fiber products, so (E1 ×D E2 ) × T is a diamond
as desired.
f1
f2
Proposition 7.3. If E1 → D ← E2 is a diagram of absolute diamonds and one
of 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 the
lower line here is a diamond, and diamonds admit fiber products.
59
According to Proposition 4.5, the sheaf Spd Zp is an absolute diamond. So
far, 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 of
them:
Example 7.4. Fix coprime positive integers h, d, and set


1


1




.
..
b=
 ∈ GLh (Qp ).



1 
p−d
Then for any S ∈ Perf, the recipe described in §5.1 produces a rank h vector
bundle Eb,S on the relative curve XS . (This is the bundle which is often denoted
by “O( hd )”.) Then the “absolute Banach-Colmez space”
Bϕ
h
=pd
: Perf
S
→ Sets
�→ H 0 (XS , Eb,S )
is an absolute diamond which is never a diamond. For example, when d = 1,
∞
this functor is (noncanonically) isomorphic to Spd Fp [[T 1/p ]], which visibly
has a “non-analytic point” corresponding to T = 0; but diamonds only have
“analytic points”.
7.2
Stacks: first definitions
We start with some useful general-nonsense observations. First, note that if
F 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 slice
Sh(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 same
as “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 stack
in groupoids over Perf /F . If F = hSpd Fp we speak of a stack over Perf.
Note that we have the chain of relations “perfectoid spaces over F ⊂ diamonds 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 “category
fibered in setoids” over Perf /F .
Definition 7.6. A 1-morphism Y → X of stacks over Perf is representable in
diamonds 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 diamond
over D.
Example 7.7. i. Any morphism of absolute diamonds is representable in diamonds. 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 → Spd Fp is representable in diamonds. This is trivial from the definition of
an absolute diamond.
iii. Any pro-étale G-torsor F → G is representable in diamonds. This
immediately 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 be
a property of morphisms of diamonds which is preserved under arbitrary base
change and which is pro-étale-local on the target. Then we say f has property
P if f is representable in diamonds, and if for any diamond D and any map
D → X , the morphism of diamonds
Y ×X D → D
has property P .
The list of admissible P includes the properties “open immersion”, “locally
closed embedding”, “finite étale”, “étale”, “smooth”, “surjective”, “v-cover”,... When
Y → X is a morphism of diamonds, this definition agrees with the usual meaning
of these properties.
Example 7.10. The map Spd Qp → Spd Fp is smooth in this sense!
Warning: The structure morphism BunG → Spd Fp is not representable in
diamonds, so it’s not smooth in this sense! Therefore we need to do something
a little more elaborate in order to declare BunG “smooth”.
7.3
Diamond stacks
Definition 7.11. A diamond stack is a stack X on Perf such that:
i. The diagonal X → X × X is representable in diamonds, and
ii. 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 dragon
we’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 Spd Fp . Let D/Qp be the
usual 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 not
smooth - a more exotic chart is required. We’ll obtain a suitable chart by
judiciously applying the following proposition.
Proposition 7.13. Let G be a locally profinite group, and let E → E � be a
surjective G-equivariant morphism of absolute diamonds which is smooth as a
morphism of stacks. Suppose moreover that E lives as a pro-étale G-torsor over
some 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 stack
with (D, f ) a chart.
First we check that [E � /G] satisfies condition i. of Definition 7.11. This
amounts to the following lemma.
Lemma 7.14. Let S be an absolute diamond, and let D be an absolute diamond
over S with an action of a locally profinite group G lying over the trivial action
on 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 a
quadruple (U, P, π, ϕ) where U ∈ Perf /S , π : P → U is a pro-étale G-torsor, and
ϕ : 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 the
groupoid of pro-étale G-torsors over U equipped with a G-equivariant map to
D.
Returning to the problem at hand, we need to show that for any U ∈ Perf /S
(x1 ,x2 )
and any morphism U −→ X × X , the sheaf
IsomU (x1 , x2 ) = U ×X ×X ,∆ X
62
is representable by a diamond. Specifying x1 and x2 amounts to specifying
quadruples (U, Pi , πi , ϕi ) as above, and by definition IsomU (x1 , x2 ) is the sheaf
∼
on Perf /U sending f : T → U to the set of isomorphisms i : f ∗ P1♦ → f ∗ P2♦
fitting into a G-equivariant commutative diagram
�� D�
��
�
�
ϕ2
��
��
∼
f ∗ P1♦ i � f ∗ P2♦
�
��
f ∗ π1
��∗
�
�
� ��� f π1
♦
T
ϕ1
of diamonds / absolute diamonds over S. In particular, IsomU (x1 , x2 ) is naturally a subfunctor of the sheaf IsomGTor(U ) (P1 , P2 ) on Perf /U which sends
∼
f : 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 commutative
diagram
∆
D�
� D ×S D
�
a
b
� IsomGTor(U ) (P1 , P2 ) ×U ♦ P ♦
1
IsomU (x1 , x2 ) ×U ♦ P1♦
ω
�
� IsomGTor(U ) (P1 , P2 )
�
IsomU (x1 , x2 )
of sheaves on Perf /S where both squares are pullback squares. For the lower
square this is true by construction. For the upper square, we need to define
the morphism b. Giving a T -point of IsomGTor(U ) (P1 , P2 ) ×U ♦ P1♦ is the same
as specifying a triple (f, i, s) consisting morphism f : T → U together with
∼
a 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
(ϕ1 ◦s,ϕ2 ◦i◦s)
D ×S D given by T
−→
D × D, which factors through D ×S D by the
commutativity of the diagrams
Pi♦
ϕi
�D
πi
�
U♦
�
�S
for i = 1, 2. Since i and the ϕ’s are G-equivariant, b is G-equivariant. Furthermore, one sees by direct inspection that b(f, i, s) factors through ∆(D) if and
only 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 same
equality 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 ♦ P1♦ is a diamond, we
finish the argument as follows. Going back to the diagram, observe that D and
D ×S D are absolute diamonds; since moreover IsomGTor(U ) (P1 , P2 ) ×U ♦ P1♦ is a
diamond and the upper square is a pullback square, Proposition 7.3 now shows
that the sheaf IsomU (x1 , x2 ) ×U ♦ P1♦ is a diamond as well. But then
ω : IsomU (x1 , x2 ) ×U ♦ P1♦ → IsomU (x1 , x2 )
is surjective and pro-étale, since it’s the pullback of the surjective pro-étale map
P1♦ → 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 ♦ P1♦ is a diamond, we simply
observe that the natural map
IsomGTor(U ) (P1 , P2 ) ×U ♦ P1♦ → P2♦ ×U ♦ P1♦
is an isomorphism. This can be checked pro-étale-locally on U , so one easily
verifies 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 adic
spaces, book.
[KL15] Kiran Kedlaya and Ruochuan Liu, Relative p-adic Hodge theory I.
[SW13] Peter Scholze and Jared Weinstein, Moduli of p-divisible groups, Cambridge J. Math.
[Wei15] Jared Weinstein, Gal(Qp /Qp ) as a geometric fundamental group.
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