Perfectoid Space
Perfectoid Space
Perfectoid Space
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakultät
der
Rheinischen Friedrich-Wilhelms-Universität Bonn
vorgelegt von
Peter Scholze
aus
Dresden
Bonn 2011
Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen
Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn
Erscheinungsjahr: 2012
PERFECTOID SPACES
PETER SCHOLZE
Abstract. We introduce a certain class of so-called perfectoid rings and spaces, which
give a natural framework for Faltings’ almost purity theorem, and for which there
is a natural tilting operation which exchanges characteristic 0 and characteristic p.
We deduce the weight-monodromy conjecture in certain cases by reduction to equal
characteristic.
Contents
1. Introduction 2
2. Adic spaces 7
3. Perfectoid fields 15
4. Almost mathematics 18
5. Perfectoid algebras 21
6. Perfectoid spaces: Analytic topology 30
7. Perfectoid spaces: Etale topology 39
8. An example: Toric varieties 44
9. The weight-monodromy conjecture 47
References 50
1. Introduction
In commutative algebra and algebraic geometry, some of the most subtle problems
arise in the context of mixed characteristic, i.e. over local fields such as Qp which are
of characteristic 0, but whose residue field Fp is of characteristic p. The aim of this
paper is to establish a general framework for reducing certain problems about mixed
characteristic rings to problems about rings in characteristic p. We will use this frame-
work to establish a generalization of Faltings’s almost purity theorem, and new results
on Deligne’s weight-monodromy conjecture.
The basic result which we want to put into a larger context is the following canonical
isomorphism of Galois groups, due to Fontaine and Wintenberger, [13]. A special case
is the following result.
∞
Theorem 1.1. The absolute Galois groups of Qp (p1/p ) and Fp ((t)) are canonically
isomorphic.
In other words, after adjoining all p-power roots of p to a mixed characteristic field,
it looks like an equal characteristic ring in some way. Let us first explain how one can
∞
prove this theorem. Let K be the completion of Qp (p1/p ) and let K [ be the completion
∞
of Fp ((t))(t1/p ); it is enough to prove that the absolute Galois groups of K and K [ are
isomorphic. Let us first explain the relation between K and K [ , which in vague terms
consists in replacing the prime number p by a formal variable t. Let K ◦ and K [◦ be the
subrings of integral elements. Then
∞
∼ Fp [t1/p∞ ]/t = K [◦ /t ,
K ◦ /p = Zp [p1/p ]/p =
n n
where the middle isomorphism sends p1/p to t1/p . Using it, one can define a continuous
multiplicative, but nonadditive, map K [ → K, x 7→ x] , which sends t to p. On K [◦ , it
n n
is given by sending x to limn→∞ ynp , where yn ∈ K ◦ is any lift of the image of x1/p in
K [◦ /t = K ◦ /p. Then one has an identification
K [ = lim K , x 7→ (x] , (x1/p )] , . . .) .
←−p
x7→x
In order to prove the theorem, one has to construct a canonical finite extension L] of
K for any finite extension L of K [ . There is the following description. Say L is the
splitting field of a polynomial X d + ad−1 X d−1 + . . . + a0 , which is also the splitting field
1/pn 1/pn
of X d + ad−1 X d−1 + . . . + a0 for all n ≥ 0. Then L] can be defined as the splitting
1/pn 1/pn
field of X d + (ad−1 )] X d−1 + . . . + (a0 )] for n large enough: these fields stabilize as
n → ∞.
In fact, the same ideas work in greater generality.
Definition 1.2. A perfectoid field is a complete topological field K whose topology is
induced by a nondiscrete valuation of rank 1, such that the Frobenius Φ is surjective on
K ◦ /p.
Here K ◦ ⊂ K denotes the set of powerbounded elements. Generalizing the example
above, a construction of Fontaine associates to any perfectoid field K another perfectoid
field K [ of characteristic p, whose underlying multiplicative monoid can be described as
K [ = lim K .
←−p
x7→x
The theorem above generalizes to the following result.
Theorem 1.3. The absolute Galois groups of K and K [ are canonically isomorphic.
Our aim is to generalize this to a comparison of geometric objects over K with geo-
metric objects over K [ . The basic claim is the following.
PERFECTOID SPACES 3
Claim 1.4. The affine line A1K [ ‘is equal to’ the inverse limit limT 7→T p A1K , where T is
←−
the coordinate on A1 .
One way in which this is correct is the observation that it is true on K [ -, resp. K-,
valued points. Moreover, for any finite extension L of K corresponding to an extension
L[ of K [ , we have the same relation
L[ = lim L .
←−p
x7→x
Looking at the example above, we see that the explicit description of the map between
A1K [ and limT 7→T p A1K involves a limit procedure. For this reason, a formalization of this
←−
isomorphism has to be of an analytic nature, and we have to use some kind of rigid-
analytic geometry over K. We choose to work with Huber’s language of adic spaces,
which reinterprets rigid-analytic varieties as certain locally ringed topological spaces. In
particular, any variety X over K has an associated adic space X ad over K, which in
turn has an underlying topological space |X ad |.
Theorem 1.5. There is a homeomorphism of topological spaces
|(A1K [ )ad | ∼
= lim |(A1K )ad | .
←− p
T 7→T
Note that both sides of this isomorphism can be regarded as locally ringed topological
spaces. It is natural to ask whether one can compare the structure sheaves on both sides.
There is the obvious obstacle that the left-hand side has a sheaf of characteristic p rings,
whereas the right-hand side has a sheaf of characteristic 0 rings. Fontaine’s functors
make it possible to translate between the two worlds. There is the following result.
Definition 1.6. Let K be a perfectoid field. A perfectoid K-algebra is a Banach K-
algebra R such that the set of powerbounded elements R◦ ⊂ R is bounded, and such that
the Frobenius Φ is surjective on R◦ /p.
Theorem 1.7. There is natural equivalence of categories, called the tilting equivalence,
between the category of perfectoid K-algebras and the category of perfectoid K [ -algebras.
Here a perfectoid K-algebra R is sent to the perfectoid K [ -algebra
R[ = lim R .
←−p
x7→x
+
(ii) For any rational subset U ⊂ X with tilt U [ ⊂ X [ , the pair (OX (U ), OX (U )) is a
[ + [
perfectoid affinoid K-algebra with tilt (OX [ (U ), OX [ (U )).
+
(iii) The presheaves OX , OX are sheaves.
+
(iv) The cohomology group H i (X, OX ) is m-torsion for i > 0.
Here m ⊂ K ◦ is the subset of topologically nilpotent elements. Part (iv) implies
that H i (X, OX ) = 0 for i > 0, which gives Tate’s acyclicity theorem in the context of
perfectoid spaces. However, it says that this statement about the generic fibre extends
almost to the integral level, in the language of Faltings’s so-called almost mathematics.
In fact, this is a general property of perfectoid objects: Many statements that are true
on the generic fibre are automatically almost true on the integral level.
Using the theorem, one can define general perfectoid spaces by gluing affinoid perfec-
toid spaces X = Spa(R, R+ ). We arrive at the following theorem.
Theorem 1.9. The category of perfectoid spaces over K and the category of perfectoid
spaces over K [ are equivalent.
We denote the tilting functor by X 7→ X [ . Our next aim is to define an étale topos of
perfectoid spaces. This necessitates a generalization of Faltings’s almost purity theorem,
cf. [11], [12].
Theorem 1.10. Let R be a perfectoid K-algebra. Let S/R be finite étale. Then S is a
perfectoid K-algebra, and S ◦ is almost finite étale over R◦ .
In fact, as for perfectoid fields, it is easy to construct a fully faithful functor from the
category of finite étale R[ -algebras to finite étale R-algebras, and the problem becomes
to show that this functor is essentially surjective. But locally on X = Spa(R, R+ ), the
functor is essentially surjective by the result for perfectoid fields; one deduces the general
case by a gluing argument.
Using this theorem, one proves the following theorem. Here, Xét denotes the étale
site of a perfectoid space X, and we denote by Xét ∼ the associated topos.
Theorem 1.11. Let X be a perfectoid space over K with tilt X [ over K [ . Then tilting
induces an equivalence of sites Xét ∼ [ .
= Xét
As a concrete application of this theorem, we have the following result. Here, we use
the étale topoi of adic spaces, which are the same as the étale topoi of the corresponding
rigid-analytic variety. In particular, the same theorem holds for rigid-analytic varieties.
Theorem 1.12. The étale topos (Pn,ad )∼ is equivalent to the inverse limit limϕ (Pn,ad ∼
K )ét .
K [ ét ←−
Here, one has to interpret the latter as the inverse limit of a fibred topos in an obvious
way, and ϕ is the map given on coordinates by ϕ(x0 : . . . : xn ) = (xp0 : . . . : xpn ). The
same theorem stays true for proper toric varieties without change. We note that the
theorem gives rise to a projection map
π : PnK [ → PnK
defined on topological spaces and étale topoi of adic spaces, and which is given on
coordinates by π(x0 : . . . : xn ) = (x]0 : . . . : x]n ). In particular, we see again that this
isomorphism is of a deeply analytic and transcendental nature.
We note that (PnK [ )ad is itself not a perfectoid space, but limϕ (PnK [ )ad is, where ϕ :
←−
PnK [ → PnK [ denotes again the p-th power map on coordinates. However, ϕ is purely
inseparable and hence induces an isomorphism on topological spaces and étale topoi,
which is the reason that we have not written this inverse limit in Theorem 1.5 and
Theorem 1.12.
PERFECTOID SPACES 5
Finally, we apply these results to the weight-monodromy conjecture. Let us recall its
formulation. Let k be a local field whose residue field is of characteristic p, let Gk =
Gal(k̄/k), and let q be the cardinality of the residue field of k. For any finite-dimensional
Q̄` -representation V of Gk , we have the monodromy operator N : V → V (−1) induced
from the action of the `-adic inertia subgroup. It induces the monodromy filtration
FilN N N
i V ⊂ V , i ∈ Z, characterized by the property that N (Fili V ) ⊂ Fili−2 V (−1) for all
i ∈ Z and gri V ∼
N N i
= gr−i V (−i) via N for all i ≥ 0.
Conjecture 1.13 (Deligne, [9]). Let X be a proper smooth variety over k, and let
V = H i (Xk̄ , Q̄` ). Then for all j ∈ Z and for any geometric Frobenius Φ ∈ Gk , all
eigenvalues of Φ on grN j V are Weil numbers of weight i + j, i.e. algebraic numbers α
such that |α| = q (i+j)/2 for all complex absolute values.
Deligne, [10], proved this conjecture if k is of characteristic p, and the situation is
already defined over a curve. The general weight-monodromy conjecture over fields k of
characteristic p can be deduced from this case, as done by Terasoma, [33], and by Ito,
[26].
In mixed characteristic, the conjecture is wide open. Introducing what is now called
the Rapoport-Zink spectral sequence, Rapoport and Zink, [29], have proved the conjec-
ture when X has dimension at most 2 and X has semistable reduction. They also show
that in general it would follow from a suitable form of the standard conjectures, the
main point being that a certain linear pairing on cohomology groups should be nonde-
generate. Using de Jong’s alterations, [8], one can reduce the general case to the case of
semistable reduction, and in particular the case of dimension at most 2 follows. Apart
from that, other special cases are known. Notably, the case of varieties which admit
p-adic uniformization by Drinfeld’s upper half-space is proved by Ito, [25], by pushing
through the argument of Rapoport-Zink in this special case, making use of the special
nature of the components of the special fibre, which are explicit rational varieties.
On the other hand, there is a large amount of activity that uses automorphic ar-
guments to prove results in cases of certain Shimura varieties, notably those of type
U (1, n − 1) used in the book of Harris-Taylor [15]. Let us only mention the work of
Taylor and Yoshida, [32], later completed by Shin, [30], and Caraiani, [6], as well as the
independent work of Boyer, [4], [5]. Boyer’s results were used by Dat, [7], to handle the
case of varieties which admit uniformization by a covering of Drinfeld’s upper half-space,
thereby generalizing Ito’s result.
Our last main theorem is the following.
Theorem 1.14. Let k be a local field of characteristic 0. Let X be a geometrically con-
nected proper smooth variety over k such that X is a set-theoretic complete intersection
in a projective smooth toric variety. Then the weight-monodromy conjecture is true for
X.
Let us give a short sketch of the proof for a smooth hypersurface in X ⊂ Pn , which
is already a new result. We have the projection
π : PnK [ → PnK ,
and we can look at the preimage π −1 (X). One has an injective map H i (X) → H i (π −1 (X)),
and if π −1 (X) were an algebraic variety, then one could deduce the result from Deligne’s
theorem in equal characteristic. However, the map π is highly transcendental, and
π −1 (X) will not be given by equations. In general, it will look like some sort of fractal,
have infinite-dimensional cohomology, and will have infinite degree in the sense that it
will meet a line in infinitely many points. As an easy example, let
X = {x0 + x1 + x2 = 0} ⊂ (P2K )ad .
6 PETER SCHOLZE
Lemma 1.15. Let X̃ ⊂ (PnK )ad be a small open neighborhood of the hypersurface X.
Then there is a hypersurface Y ⊂ π −1 (X̃).
The proof of this lemma is by an explicit approximation algorithm for the homo-
geneous polynomial defining X, and is the reason that we have to restrict to complete
intersections. Using a result of Huber, one finds some X̃ such that H i (X) = H i (X̃), and
hence gets a map H i (X) = H i (X̃) → H i (Y ). As before, one checks that it is injective
and concludes.
After the results of this paper were first announced, Kiran Kedlaya informed us that
he had obtained related results in joint work with Ruochuan Liu, [27]. In particular,
in our terminology, they prove that for any perfectoid K-algebra R with tilt R[ , there
is an equivalence between the finite étale R-algebras and the finite étale R[ -algebras.
However, the tilting equivalence, the generalization of Faltings’s almost purity theorem
and the application to the weight-monodromy conjecture were not observed by them.
This led to an exchange of ideas, with the following two influences on this paper. In the
first version of this work, Theorem 1.3 was proved using a version of Faltings’s almost
purity theorem for fields, proved in [14], Chapter 6, using ramification theory. Kedlaya
observed that one could instead reduce to the case where K [ is algebraically closed,
which gives a more elementary proof of the theorem. We include both arguments here.
Secondly, a certain finiteness condition on the perfectoid K-algebra was imposed at
some places in the first version, a condition close to the notion of p-finiteness introduced
below; in most applications known to the author, this condition is satisfied. Kedlaya
made us aware of the possibility to deduce the general case by a simple limit argument.
Acknowledgments. First, I want to express my deep gratitude to my advisor M.
Rapoport, who suggested that I should think about the weight-monodromy conjecture,
and in particular suggested that it might be possible to reduce it to the case of equal
characteristic after a highly ramified base change. Next, I want to thank Gerd Faltings
for a crucial remark on a first version of this paper. Moreover, I wish to thank all
participants of the ARGOS seminar on perfectoid spaces at the University of Bonn in
the summer term 2011, for working through an early version of this manuscript and the
large number of suggestions for improvements. The same applies to Lorenzo Ramero,
whom I also want to thank for his very careful reading of the manuscript. Moreover, I
thank Roland Huber for answering my questions on adic spaces. Further thanks go to
Ahmed Abbes, Bhargav Bhatt, Pierre Colmez, Laurent Fargues, Jean-Marc Fontaine,
Ofer Gabber, Luc Illusie, Adrian Iovita, Kiran Kedlaya, Gerard Laumon, Ruochuan Liu,
Wieslawa Niziol, Arthur Ogus, Martin Olsson, Bernd Sturmfels and Jared Weinstein
for helpful discussions. Finally, I want to heartily thank the organizers of the CAGA
lecture series at the IHES for their invitation. This work is the author’s PhD thesis at
the University of Bonn, which was supported by the Hausdorff Center for Mathematics,
and the thesis was finished while the author was a Clay Research Fellow. He wants
to thank both institutions for their support. Moreover, parts of it were written while
visiting Harvard University, the Université Paris-Sud at Orsay, and the IHES, and the
author wants to thank these institutions for their hospitality.
PERFECTOID SPACES 7
2. Adic spaces
Throughout this paper, we make use of Huber’s theory of adic spaces. For this reason,
we recall some basic definitions and statements about adic spaces over nonarchimedean
local fields. We also compare Huber’s theory to the more classical language of rigid-
analytic geometry, and to the theory of Berkovich’s analytic spaces. The material of
this section can be found in [20], [19] and [18].
Definition 2.1. A nonarchimedean field is a topological field k whose topology is induced
by a nontrivial valuation of rank 1.
In particular, k admits a norm | · | : k → R≥0 , and it is easy to see that | · | is unique
up to automorphisms x 7→ xα , 0 < α < ∞, of R≥0 .
Throughout, we fix a nonarchimedean field k. Replacing k by its completion will not
change the theory, so we may and do assume that k is complete.
The idea of rigid-analytic geometry, and the closely related theories of Berkovich’s
analytic spaces and Huber’s adic spaces, is to have a nonarchimedean analogue of the
notion of complex analytic spaces over C. In particular, there should be a functor
{varieties/k} → {adic spaces/k} : X 7→ X ad ,
sending any variety over k to its analytification X ad . Moreover, it should be possible to
define subspaces of X ad by inequalities: For any f ∈ Γ(X, OX ), the subset
{x ∈ X ad | |f (x)| ≤ 1}
should make sense. In particular, any point x ∈ X ad should give rise to a valuation
function f 7→ |f (x)|. In classical rigid-analytic geometry, one considers only the max-
imal points of the scheme X. Each of them gives a map Γ(X, OX ) → k 0 for some
finite extension k 0 of k; composing with the unique extension of the absolute value of k
to k 0 gives a valuation on Γ(X, OX ). In Berkovich’s theory, one considers norm maps
Γ(X, OX ) → R≥0 inducing a fixed norm map on k. Equivalently, one considers valu-
ations of rank 1 on Γ(X, OX ). In Huber’s theory, one allows also valuations of higher
rank.
Definition 2.2. Let R be some ring. A valuation on R is given by a multiplicative map
| · | : R → Γ ∪ {0}, where Γ is some totally ordered abelian group, written multiplicatively,
such that |0| = 0, |1| = 1 and |x + y| ≤ max(|x|, |y|) for all x, y ∈ R.
If R is a topological ring, then a valuation | · | on R is said to be continuous if for all
γ ∈ Γ, the subset {x ∈ R | |x| < γ} ⊂ R is open.
Remark 2.3. The term valuation is somewhat unfortunate: If Γ = R>0 , then this would
usually be called a seminorm, and the term valuation would be used for (a constant
multiple of) the map x 7→ − log |x|. On the other hand, the term higher-rank norm is
much less commonly used than the term higher-rank valuation. For this reason, we stick
with Huber’s terminology.
Remark 2.4. Recall that a valuation ring is an integral domain R such that for any x 6= 0
in the fraction field K of R, at least one of x and x−1 is in R. Any valuation | · | on a
field K gives rise to the valuation subring R = {x | |x| ≤ 1}. Conversely, a valuation
ring R gives rise to a valuation on K with values in Γ = K × /R× , ordered by saying
that x ≤ y if x = yz for some z ∈ R. With respect to a suitable notion of equivalence
of valuations defined below, this induces a bijective correspondence between valuation
subrings of K and valuations on K.
If | · | : R → Γ ∪ {0} is a valuation on R, let Γ|·| ⊂ Γ denote the subgroup generated by
all |x|, x ∈ R, which are nonzero. The set supp(| · |) = {x ∈ R | |x| = 0} is a prime ideal
of R called the support of | · |. Let K be the quotient field of R/ supp(| · |). Then the
8 PETER SCHOLZE
is the ring of convergent power series on the ball given by |T1 |, . . . , |Tn | ≤ 1. Often, only
affinoid k-algebras of tft are considered; however, this paper will show that other classes
of affinoid k-algebras are of interest as well. We also note that any Tate k-algebra R,
resp. affinoid k-algebra (R, R+ ), admits the completion R̂, resp. (R̂, R̂+ ), which is again
a Tate, resp. affinoid, k-algebra. Everything depends only on the completion, so one
may assume that (R, R+ ) is complete in the following.
Definition 2.7. Let (R, R+ ) be an affinoid k-algebra. Let
X = Spa(R, R+ ) = {| · | : R → Γ ∪ {0} continuous valuation | ∀f ∈ R+ : |f | ≤ 1}/ ∼
= .
For any x ∈ X, write f 7→ |f (x)| for the corresponding valuation on R. We equip X
with the topology which has the open subsets
f1 , . . . , fn
U( ) = {x ∈ X | ∀i : |fi (x)| ≤ |g(x)|} ,
g
called rational subsets, as basis for the topology, where f1 , . . . , fn ∈ R generate R as an
ideal and g ∈ R.
Remark 2.8. Let $ ∈ k be topologically nilpotent, i.e. |$| < 1. Then to f1 , . . . , fn one
can add fn+1 = $N for some big integer N withoutP changing the rational subspace.
Indeed, there are elements h1 , . . . , hn ∈ R such that hi fi = 1. Multiplying by $N
N + +
for N sufficiently large, we have $ hi ∈ R , as R ⊂ R is open. Now for any x ∈
U ( f1 ,...,f
g
n
), we have
X
|$N (x)| = | ($N hi )(x)fi (x)| ≤ max |($N hi )(x)||fi (x)| ≤ |g(x)| ,
as desired. In particular, we see that on rational subsets, |g(x)| is nonzero, and bounded
from below.
PERFECTOID SPACES 9
The topological spaces Spa(R, R+ ) have some special properties reminiscent of the
properties of Spec(A) for a ring A. In fact, let us recall the following result of Hochster,
[17].
Definition/Proposition 2.9. A topological space X is called spectral if it satisfies the
following equivalent properties.
(i) There is some ring A such that X ∼ = Spec(A).
(ii) One can write X as an inverse limit of finite T0 spaces.
(iii) The space X is quasicompact, has a basis of quasicompact open subsets stable under
finite intersections, and every irreducible closed subset has a unique generic point.
In particular, spectral spaces are quasicompact, quasiseparated and T0 . Recall that
a topological space X is called quasiseparated if the intersection of any two quasicom-
pact open subsets is again quasicompact. In the following we will often abbreviate
quasicompact, resp. quasiseparated, as qc, resp. qs.
Proposition 2.10 ([18, Theorem 3.5]). For any affinoid k-algebra (R, R+ ), the space
Spa(R, R+ ) is spectral. The rational subsets form a basis of quasicompact open subsets
stable under finite intersections.
Proposition 2.11 ([18, Proposition 3.9]). Let (R, R+ ) be an affinoid k-algebra with
completion (R̂, R̂+ ). Then Spa(R, R+ ) ∼
= Spa(R̂, R̂+ ), identifying rational subsets.
Moreover, the space Spa(R, R+ ) is large enough to capture important properties.
Proposition 2.12. Let (R, R+ ) be an affinoid k-algebra, X = Spa(R, R+ ).
(i) If X = ∅, then R̂ = 0.
(ii) Let f ∈ R be such that |f (x)| =
6 0 for all x ∈ X. If R is complete, then f is invertible.
(iii) Let f ∈ R be such that |f (x)| ≤ 1 for all x ∈ X. Then f ∈ R+ .
Proof. Part (i) is [18], Proposition 3.6 (i). Part (ii) is [19], Lemma 1.4, and part (iii)
follows from [18], Lemma 3.3 (i).
We want to endow X = Spa(R, R+ ) with a structure sheaf OX . The construction is
as follows.
Definition 2.13. Let (R, R+ ) be an affinoid k-algebra, and let U = U ( f1 ,...,f g
n
)⊂X=
+ ×
Spa(R, R ) be a rational subset. Choose some R0 ⊂ R such that aR0 , a ∈ k , is a basis
of open neighborhoods of 0 in R. Consider the subalgebra R[ fg1 , . . . , fgn ] of R[g −1 ], and
equip it with the topology making aR0 [ fg1 , . . . , fgn ], a ∈ k × , a basis of open neighborhoods
of 0. Let B ⊂ R[ fg1 , . . . , fgn ] be the integral closure of R+ [ fg1 , . . . , fgn ] in R[ fg1 , . . . , fgn ].
Then (R[ fg1 , . . . , fgn ], B) is an affinoid k-algebra. Let (Rh fg1 , . . . , fgn i, B̂) be its completion.
Obviously,
f1 fn
, . . . , i, B̂) → Spa(R, R+ )
Spa(Rh
g g
factors over the open subset U ⊂ X.
Proposition 2.14 ([19, Proposition 1.3]). In the situation of the definition, the follow-
ing universal property is satisfied. For every complete affinoid k-algebra (S, S + ) with a
map (R, R+ ) → (S, S + ) such that the induced map Spa(S, S + ) → Spa(R, R+ ) factors
over U , there is a unique map
f1 fn
(Rh , . . . , i, B̂) → (S, S + )
g g
making the obvious diagram commute.
10 PETER SCHOLZE
We recall that for abstract reasons, there is up to equivalence at most one sober
topological space with the last property in the theorem. This gives the topological
space underlying the adic space a natural interpretation.
In the example X = Sp(khT i) discussed above, a typical example of a non-admissible
cover is the cover of {x | |x| ≤ 1} as
[
{x | |x| ≤ 1} = {x | |x| = 1} ∪ {x | |x| ≤ r} .
r<1
In the adic world, one can see this non-admissibility as being caused by the point of
type (5) which gives |x| a value γ < 1 bigger than any r < 1.
Moreover, adic spaces behave well with respect to formal models. In fact, one can
define adic spaces in greater generality so as to include locally noetherian formal schemes
as a full subcategory, but we will not need this more general theory here.
Theorem 2.22. Let X be some admissible formal scheme over k ◦ , let X be its generic
fibre in the sense of Raynaud, and let X ad be the associated adic space. Then there is a
continuous specialization map
sp : X ad → X ,
+
extending to a morphism of locally ringed topological spaces (X ad , OX ad ) → (X, OX ).
Now assume that X is a fixed quasicompact quasiseparated adic space locally of finite
type over k. By Raynaud, there exist formal models X for X, unique up to admissible
blowup. Then there is a homeomorphism
X∼ = lim X,
←−
X
where X runs over formal models of X, extending to an isomorphism of locally ringed
+ ∼
topological spaces (X, OX ) = limX (X, OX ), where the right-hand side is the inverse limit
←−
in the category of locally ringed topological spaces.
Proof. It is an easy exercise to deduce this from the previous theorem and the results
of [3], Section 4, and [19], Section 4.
In the example, one can start with X = Spf(k ◦ hT i) as a formal model; this gives A1κ
as underlying topological space. After that, one can perform iterated blowups at closed
points. This introduces additional P1κ ’s; the strict transform of each component survives
in the inverse limit and gives the closure of a point of type (2). Note that the point of
type (2) is given as the preimage of the generic point of the component in the formal
model.
We note that in order to get continuity of sp, it is necessary to use nonstrict equalities
in the definition of open subsets.
Now, let us state the following theorem about the comparison of Berkovich’s analytic
spaces and Huber’s adic spaces. For this, we need to recall the following definition:
Definition 2.23. An adic space X over k is called taut if it is quasiseparated and for
every quasicompact open subset U ⊂ X, the closure U of U in X is still quasicompact.
14 PETER SCHOLZE
Most natural adic spaces are taut, e.g. all affinoid adic spaces, or more generally all
qcqs adic spaces, and also all adic spaces associated to separated schemes of finite type
over k. However, in recent work of Hellmann, [16], studying an analogue of Rapoport-
Zink period domains in the context of deformations of Galois representations, it was
found that the weakly admissible locus is in general a nontaut adic space.
We note that one can also define taut rigid-analytic varieties, and that one gets an
equivalence of categories between the category of taut rigid-analytic varieties over k and
taut adic spaces locally of finite type over k. Hence the first equivalence in the following
theorem could be stated without reference to adic spaces.
Theorem 2.24 ([20, Proposition 8.3.1, Lemma 8.1.8]). There is an equivalence of cat-
egories
{hausdorff strictly k−analytic Berkovich spaces}
∼
= {taut adic spaces locally of finite type/k} ,
+
Proof. It is enough to note that kernel of the map OX,x → k(x)+ , which is also the
kernel of the map OX,x → k(x), is $-divisible.
Definition 2.26. An affinoid field is pair (K, K + ) consisting of a nonarchimedean field
K and an open valuation subring K + ⊂ K ◦ .
In other words, an affinoid field is given by a nonarchimedean field K equipped with
a continuous valuation (up to equivalence). In the situation above, (k(x), k(x)+ ) is an
affinoid field. The completion of an affinoid field is again an affinoid field. Also note
that affinoid fields for which k ⊂ K are affinoid k-algebras. The following description of
points is immediate.
Proposition 2.27. Let (R, R+ ) be an affinoid k-algebra. The points of Spa(R, R+ ) are
in bijection with maps (R, R+ ) → (K, K + ) to complete affinoid fields (K, K + ) such that
the quotient field of the image of R in K is dense.
PERFECTOID SPACES 15
Definition 2.28. For two points x, y in some topological space X, we say that x spe-
cializes to y (or y generalizes to x), written x y (or y ≺ x), if y lies in the closure
{x} of x.
Proposition 2.29 ([20, (1.1.6) - (1.1.10)]). Let (R, R+ ) be an affinoid k-algebra, and
let x, y ∈ X = Spa(R, R+ ) correspond to maps (R, R+ ) → (K, K + ), resp. (R, R+ ) →
(L, L+ ). Then x y if and only if K ∼= L as topological R-algebras and L+ ⊂ K + .
For any point y ∈ X, the set {x | x y} of generalizations of y is a totally ordered
chain of length exactly the rank of the valuation corresponding to y.
Note that in particular, for a given complete nonarchimedean field K with a map
R → K, there is the point x0 corresponding to (K, K ◦ ). This corresponds to the unique
continuous rank-1-valuation on K. The point x0 specializes to any other point for the
same K.
3. Perfectoid fields
Definition 3.1. A perfectoid field is a complete nonarchimedean field K of residue
characteristic p > 0 whose associated rank-1-valuation is nondiscrete, such that the
Frobenius is surjective on K ◦ /p.
We note that the requirement that the valuation is nondiscrete is needed to exclude
unramified extensions of Qp . It has the following consequence.
Lemma 3.2. Let |·| : K → Γ∪{0} be the unique rank-1-valuation on K, where Γ = |K × |
is chosen minimal. Then Γ is p-divisible.
Proof. As Γ 6= |p|Z , the group Γ is generated by the set of all |x| for x ∈ K with
|p| < |x| ≤ 1. For such x, choose some y such that |x − y p | ≤ |p|. Then |y|p = |y p | = |x|,
as desired.
The class of perfectoid fields naturally separates into the fields of characteristic 0 and
those of characteristic p. In characteristic p, a perfectoid field is the same as a complete
perfect nonarchimedean field.
Remark 3.3. The notion of a perfectoid field is closely related to the notion of a deeply
ramified field. Taking the definition of deeply ramified fields given in [14], we remark
that Proposition 6.6.6 of [14] says that a perfectoid field K is deeply ramified, and
conversely, a complete deeply ramified field with valuation of rank 1 is a perfectoid field.
Now we describe the process of tilting for perfectoid fields, which is a functor from
the category of all perfectoid fields to the category of perfectoid fields in characteristic
p.
For its first description, choose some element $ ∈ K × such that |p| ≤ |$| < 1. Now
consider
lim K ◦ /$ ,
←−
Φ
where Φ denotes the Frobenius morphism x 7→ xp . This gives a perfect ring of charac-
teristic p. We equip it with the inverse limit topology; note that each K ◦ /$ naturally
has the discrete topology.
Lemma 3.4. (i) There is a multiplicative homeomorphism
∼
=
lim K ◦ → lim K ◦ /$ ,
←−p ←−
x7→x Φ
given by projection. In particular, the right-hand side is independent of $. Moreover,
we get a map
lim K ◦ /$ → K ◦ : x 7→ x] .
←−
Φ
16 PETER SCHOLZE
this is possible by surjectivity of Φ on K ◦ /$. By the proof of part (i), we have |($[ )] −
$1p | ≤ |$|2 . This gives |($[ )] | = |$|, as desired.
PERFECTOID SPACES 17
4. Almost mathematics
We will use the book of Gabber-Ramero, [14], as our basic reference.
Fix a perfectoid field K. Let m = K ◦◦ ⊂ K ◦ be the subset of topologically nilpotent
elements; it is also the set {x ∈ K | |x| < 1}, and the unique maximal ideal of K ◦ . The
basic idea of almost mathematics is that one neglects m-torsion everywhere.
Definition 4.1. Let M be a K ◦ -module. An element x ∈ M is almost zero if mx = 0.
The module M is almost zero if all of its elements are almost zero; equivalently, mM = 0.
Lemma 4.2. The full subcategory of almost zero objects in K ◦ − mod is thick.
Proof. The only nontrivial part is to show that it is stable under extensions, so let
0 → M 0 → M → M 00 → 0
be a short exact sequence of K ◦ -modules, with mM 0 = mM 00 = 0. In general, one gets
that m2 M = 0. But in our situation, m2 = m, so M is almost zero.
We note that there is a sequence of localization functors
K ◦ − mod → K ◦ − mod/(m − torsion) → K − mod .
Their composite is the functor of passing from an integral structure to its generic fibre.
In this sense, the category in the middle can be seen as a slightly generic fibre, or as
an almost integral structure. It will turn out that in perfectoid situations, properties
and objects over the generic fibre will extend automatically to the slightly generic fibre,
in other words the generic fibre almost determines the integral level. It will be easy
to justify this philosophy if K has characteristic p, by using the following argument.
Assume that some statement is true over K. By using some finiteness property, it
follows that there is some big N such that it is true up to $N -torsion. But Frobenius is
bijective, hence the property stays true up to $N/p -torsion. Now iterate this argument
m
to see that it is true up to $N/p -torsion for all m, i.e. almost true.
Following these ideas, our proof of Theorem 3.7 will proceed as follows, using the
subscript fét to denote categories of finite étale (almost) algebras.
Kfét ∼
= K ◦a ∼
= (K ◦a /$)fét = (K [◦a /$[ )fét ∼
fét = K [◦a ∼
fét = K[ .
fét
Our principal aim in this section is to define all intermediate categories.
Definition 4.3. Define the category of almost K ◦ -modules as
K ◦a − mod = K ◦ − mod/(m − torsion) .
In particular, there is a localization functor M 7→ M a from K ◦ − mod to K ◦a − mod,
whose kernel is exactly the thick subcategory of almost zero modules.
Proposition 4.4 ([14, §2.2.2]). Let M , N be two K ◦ -modules. Then
HomK ◦a (M a , N a ) = HomK ◦ (m ⊗ M, N ) .
In particular, HomK ◦a (X, Y ) has a natural structure of K ◦ -module for any two K ◦a -
modules X and Y . The module HomK ◦a (X, Y ) has no almost zero elements.
For two K ◦a -modules M , N , we define alHom(X, Y ) = Hom(X, Y )a .
PERFECTOID SPACES 19
Proposition 4.5 ([14, §2.2.6, §2.2.12]). The category K ◦a − mod is an abelian ten-
sor category, where we define kernels, cokernels and tensor products in the unique way
compatible with their definition in K ◦ − mod, e.g.
M a ⊗ N a = (M ⊗ N )a
for any two K ◦ -modules M , N . For any three K ◦a -modules L, M, N , there is a functorial
isomorphism
Hom(L, alHom(M, N )) = Hom(L ⊗ M, N ) .
This means that K ◦a − mod has all abstract properties of the category of modules
over a ring. In particular, one can define in the usual abstract way the notion of a
K ◦a -algebra. For any K ◦a -algebra A, one also has the notion of an A-module. Any K ◦ -
algebra R defines a K ◦a -algebra Ra , as the tensor products are compatible. Moreover,
localization also gives a functor from R-modules to Ra -modules. For example, K ◦
gives the K ◦a -algebra A = K ◦a , and then A-modules are K ◦a -modules, so that the
terminology is consistent.
Proposition 4.6 ([14, Proposition 2.2.14]). There is a right adjoint
K ◦a − mod → K ◦ − mod : M 7→ M∗
to the localization functor M 7→ M a , given by the functor of almost elements
M∗ = HomK ◦a (K ◦a , M ) .
The adjunction morphism (M∗ )a → M is an isomorphism. If M is a K ◦ -module, then
(M a )∗ = Hom(m, M ).
If A is a K ◦a -algebra, then A∗ has a natural structure as K ◦ -algebra and Aa∗ = A.
In particular, any K ◦a -algebra comes via localization from a K ◦ -algebra. Moreover, the
functor M 7→ M∗ induces a functor from A-modules to A∗ -modules, and one sees that
also all A-modules come via localization from A∗ -modules. We note that the category
of A-modules is again an abelian tensor category, and all properties about the category
of K ◦a -modules stay true for the category of A-modules. We also note that one can
equivalently define A-algebras as algebras over the category of A-modules, or as K ◦a -
algebras B with an algebra morphism A → B.
Finally, we need to extend some notions from commutative algebra to the almost
context.
Definition/Proposition 4.7. Let A be any K ◦a -algebra.
(i) An A-module M is flat if the functor X 7→ M ⊗A X on A-modules is exact. If R is a
K ◦ -algebra and N is an R-module, then the Ra -module N a is flat if and only if for all
R-modules X and all i > 0, the module TorR i (N, X) is almost zero.
(ii) An A-module M is almost projective if the functor X 7→ alHomA (M, X) on A-
modules is exact. If R is a K ◦ -algebra and N is an R-module, then N a is almost pro-
jective over Ra if and only if for all R-modules X and all i > 0, the module ExtiR (N, X)
is almost zero.
(iii) If R is a K ◦ -algebra and N is an R-module, then M = N a is said to be an almost
finitely generated (resp. almost finitely presented) Ra -module if and only if for all ∈
m, there is some finitely generated (resp. finitely presented) R-module N with a map
f : N → N such that the kernel and cokernel of f are annihilated by . We say that
M is uniformly almost finitely generated if there is some integer n such that N can be
chosen to be generated by n elements, for all .
Proof. For parts (i) and (ii), cf. [14], Definition 2.4.4, §2.4.10 and Remark 2.4.12 (i).
For part (iii), cf. [14], Definition 2.3.8, Remark 2.3.9 (i) and Corollary 2.3.13.
20 PETER SCHOLZE
Remark 4.8. In (iii), we make the implicit statement that this property depends only
on the Ra -module N a . There is also the categorical notion of projectivity saying that
the functor X 7→ Hom(M, X) is exact, but not even K ◦a itself is projective in general:
One can check that the map
K ◦ = Hom(K ◦a , K ◦a ) → Hom(K ◦a , K ◦a /$) = Hom(m, K ◦ /$)
is in general not surjective, as the latter group contains sums of the form
i
X
$1−1/p xi
i≥0
5. Perfectoid algebras
Fix a perfectoid field K.
Definition 5.1. (i) A perfectoid K-algebra is a Banach K-algebra R such that the subset
R◦ ⊂ R of powerbounded elements is open and bounded, and the Frobenius morphism
Φ : R◦ /$ → R◦ /$ is surjective. Morphisms between perfectoid K-algebras are the
continuous morphisms of K-algebras.
(ii) A perfectoid K ◦a -algebra is a $-adically complete flat K ◦a -algebra A on which Frobe-
nius induces an isomorphism
1
Φ : A/$ p ∼= A/$ .
Morphisms between perfectoid K ◦a -algebras are the morphisms of K ◦a -algebras.
22 PETER SCHOLZE
In order to finish the proof of Theorem 5.2, it suffices to prove the following result.
Theorem 5.10. The functor A 7→ A = A/$ induces an equivalence of categories
K ◦a − Perf ∼
= (K ◦a /$) − Perf.
In other words, we have to prove that a perfectoid K ◦a /$-algebra admits a unique
deformation to K ◦a . For this, we will use the theory of the cotangent complex. Let us
briefly recall it here.
In classical commutative algebra, the definition of the cotangent complex is due to
Quillen, [28], and its theory was globalized on toposes and applied to deformation prob-
lems by Illusie, [22], [23]. To any morphism R → S of rings, one associates a complex
LS/R ∈ D≤0 (S), where D(S) is the derived category of the category of S-modules, and
D≤0 (S) ⊂ D(S) denotes the full subcategory of objects which have trivial cohomology
in positive degrees. The cohomology in degree 0 of LS/R is given by Ω1S/R , and for any
morphisms R → S → T of rings, there is a triangle in D(T ):
T ⊗LS LS/R → LT /R → LT /S → ,
extending the short exact sequence
T ⊗S Ω1S/R → Ω1T /R → Ω1T /S → 0 .
Let us briefly recall the construction. First, one uses the Dold-Kan equivalence to
reinterpret D≤0 (S) as the category of simplicial S-modules modulo weak equivalence.
Now one takes a simplicial resolution S• of the R-algebra S by free R-algebras. Then one
defines LS/R as the object of D≤0 (S) associated to the simplicial S-module Ω1S• /R ⊗S• S.
Just as under certain favorable assumptions, one can describe many deformation
problems in terms of tangent or normal bundles, it turns out that in complete generality,
one can describe them via the cotangent complex. In special cases, this gives back the
classical results, as e.g. if R → S is a smooth morphism, then LS/R is concentrated in
degree 0, and is given by the cotangent bundle.
Specifically, we will need the following results. Fix some ring R with an ideal I ⊂ R
such that I 2 = 0. Moreover, fix a flat R0 = R/I-algebra S0 . We are interested in the
obstruction towards deforming S0 to a flat R-algebra S.
Theorem 5.11 ([22, III.2.1.2.3], [14, Proposition 3.2.9]). There is an obstruction class
in Ext2 (LS0 /R0 , S0 ⊗R0 I) which vanishes precisely when there exists a flat R-algebra
S such that S ⊗R R0 = S0 . If there exists such a deformation, then the set of all
isomorphism classes of such deformations forms a torsor under Ext1 (LS0 /R0 , S0 ⊗R0 I),
and every deformation has automorphism group Hom(LS0 /R0 , S0 ⊗R0 I).
PERFECTOID SPACES 25
(ii) Let R → S be a morphism of Fp -algebras. Let R(Φ) be the ring R with the R-algebra
structure via Φ : R → R, and define S(Φ) similarly. Assume that the relative Frobenius
ΦS/R induces an isomorphism
R(Φ) ⊗LR S → S(Φ)
in D(R). Then LS/R ∼
= 0.
Remark 5.14. Of course, (i) is a special case of (ii), and we will only need part (ii).
However, we feel that (i) is an interesting statement that does not seem to be very
well-known. It allows one to define the ring of Witt vectors W (R) of R simply by saying
that it is the unique deformation of R to a flat p-adically complete Zp -algebra. Also
note that it is clear that Ω1R/Fp = 0 in part (i): Any x ∈ R can be written as y p , and
then dx = dy p = py p−1 dy = 0. This identity is at the heart of this proposition.
Proof. We sketch the proof of part (ii), cf. proof of Lemma 6.5.13 i) in [14]. Let S • be
a simplicial resolution of S by free R-algebras. We have the relative Frobenius map
ΦS • /R : R(Φ) ⊗R S • → S(Φ)
•
.
Note that identifying S k with a polynomial algebra R[X1 , X2 , . . .], the relative Frobenius
map ΦS k /R is given by the R(Φ) -algebra map sending Xi 7→ Xip .
The assumption says that ΦS • /R induces a quasiisomorphism of simplicial R(Φ) -
algebras. This implies that ΦS • /R gives an isomorphism
R(Φ) ⊗LR LS/R ∼
= LS(Φ) /R(Φ) .
On the other hand, the explicit description shows that the map induced by ΦS k /R on
differentials will map dXi to dXip = 0, and hence is the zero map. This shows that
LS(Φ) /R(Φ) ∼
= 0, and we may identify this with LS/R .
In their book [14], Gabber and Ramero generalize the theory of the cotangent complex
to the almost context. Specifically, they show that if R → S is a morphism of K ◦ -
algebras, then LaS/R as an element of D(S a ), the derived category of S a -modules, depends
only the morphism Ra → S a of almost K ◦ -algebras. This allows one to define LaB/A ∈
D≤0 (B) for any morphism A → B of K ◦a -algebras. With this modification, the previous
theorems stay true in the almost world without change.
Remark 5.15. In fact, the cotangent complex LB/A is defined as an object of a derived
category of modules over an actual ring in [14], but for our purposes it is enough to
consider its almost version LaB/A .
Proof. This follows from the almost version of Proposition 5.13, which can be proved
in the same way. Alternatively, argue with B = (A × K ◦a /$)!! , which is a flat K ◦ /$-
algebra such that B/$1/p ∼ = B via Φ. Here, we use the functor C 7→ C!! from [14],
§2.2.25.
LaA/(K ◦ /$)a → LA
a
◦ /$ n )a → LaA ◦ /$ n−1 )a →,
n /(K n−1 /(K
because ∗ commutes with inverse limits and using Lemma 5.3 (iii). Note that the image
of Φ : (A/$)∗ → (A/$)∗ is A∗ /$, because it factors over (A/$1/p )∗ , and the image of
the projection (A/$)∗ → (A/$1/p )∗ is A∗ /$1/p . But
lim A∗ /$ = lim A∗ ,
←− ←−p
Φ x7→x
But we can find x[ , y [ ∈ R[◦ with x − (x[ )] , y − (y [ )] ∈ $R◦ . Then ||x||R = ||x[ ||R[ ,
||y||R = ||y [ ||R[ and ||xy||R = ||x[ y [ ||R[ , and we get the claim.
To see that R is a field, choose x such that x ∈ R◦ , but not in $R◦ , and take x[ as
before. Then by multiplicativity of || · ||R , ||1 − (xx[ )] ||R < 1, and hence (xx[ )] is invertible,
and then also x.
Finally, let us discuss finite étale covers of perfectoid algebras, and finish the proof of
Theorem 3.7.
Proposition 5.22. Let A be a perfectoid K ◦a /$-algebra, and let B be a finite étale
A-algebra. Then B is a perfectoid K ◦a /$-algebra.
Proof. Obviously, B is flat. The statement about Frobenius follows from Theorem 3.5.13
ii) of [14].
In particular, Theorem 4.17 provides us with the following commutative diagram,
where R, A, A, A[ and R[ form a sequence of rings under the tilting procedure.
∼ ∼
Rfét o / Afét o / R[
= =
Afét A[fét fét
∼ ∼ ∼
∼
K − Perf o / (K ◦a /$) − Perf o = / K [ − Perf
= = =
K ◦a − Perf K [◦a − Perf
It follows from this diagram that the functors Afét → Rfét and A[fét → Rfét [ are
fully faithful. A main theorem is that both of them are equivalences: This amounts to
Faltings’s almost purity theorem. At this point, we will prove this only in characteristic
p.
Proposition 5.23. Let K be of characteristic p, let R be a perfectoid K-algebra, and
let S/R be finite étale. Then S is perfectoid and S ◦a is finite étale over R◦a . Moreover,
S ◦a is a uniformly finite projective R◦a -module.
Remark 5.24. We need to define the topology on S here. Recall that if A is any ring with
t ∈ A not a zero-divisor, then any finitely generated A[t−1 ]-module M carries a canonical
topology, which gives any finitely generated A-submodule of M the t-adic topology. Any
morphism of finitely generated A[t−1 ]-modules is continuous for this topology, cf. [14],
Definition 5.4.10 and 5.4.11. If A is complete and M is projective, then M is complete,
as one checks by writing M as a direct summand of a finitely generated free A-module.
In particular, if R is a perfectoid K-algebra and S/R a finite étale cover, then S has a
canonical topology for which it is complete.
Proof. This follows from Theorem 3.5.28 of [14]. Let us recall the argument. Note that
S is a perfect Banach K-algebra. We claim that it is perfectoid. Let S0 ⊂ S be some
finitely generated R◦ -subalgebra with S0 ⊗ K = S. Let S0⊥ ⊂ S be defined as the set of
all x ∈ S such that tS/R (x, S0 ) ⊂ R◦ , using the perfect trace form pairing
tS/R : S ⊗R S → R ;
then S0 and S0⊥ are open and bounded. Let Y be the integral closure of R◦ in S. Then
S0 ⊂ Y ⊂ S0⊥ : Indeed, the elements of S0 are clearly integral over R◦ , and we have
tS/R (Y, Y ) ⊂ R◦ . It follows that Y is open and bounded. As S ◦a = Y a , it follows that
S ◦ is open and bounded, as desired.
Next, we want to check that S ◦a is a uniformly finite projective R◦a -module. For
this, it is enough to prove that there is some n such that for any ∈ m, there are maps
S ◦ → R◦n and R◦n → S ◦ whose composite is multiplication by .
Let e ∈ S ⊗R S be the idempotent showing that S is unramified over N
◦ ◦ N
PnR. Then $ e
is in the image of S ⊗R◦ S in S ⊗R S for some N . Write $ e = i=1 xi ⊗ yi . As
PERFECTOID SPACES 29
N/pm e =
Pn 1/pm 1/pm
Frobenius is bijective, we have $P i=1 xi ⊗ yi for all m. In particular,
for any ∈ m, we can write e = ni=1 ai ⊗ bi for certain ai , bi ∈ S ◦ , depending on .
We get the map S ◦ → R◦n ,
s 7→ (tS/R (s, b1 ), . . . , tS/R (s, bn )) ,
and the map R◦n → S ◦ ,
n
X
(r1 , . . . , rn ) 7→ ai ri .
i=1
One easily checks that their composite is multiplication by , giving the claim.
It remains to see that S ◦a is an unramified R◦a -algebra. But this follows from the
previous arguments, which show that e defines an almost element of S ◦a ⊗R◦a S ◦a with
the desired properties.
∼ ∼ ∼
∼
K − Perf o / (K ◦a /$) − Perf o = / K [ − Perf
= = =
K ◦a − Perf K [◦a − Perf
Moreover, using Theorem 4.17, it follows that all finite étale algebras over A, A or A[
are uniformly almost finitely presented. Let us summarize the discussion.
Theorem 5.25. Let R be a perfectoid K-algebra with tilt R[ . There is a fully faithful
[
functor from Rfét to Rfét inverse to the tilting functor. The essential image of this
functor consists of the finite étale covers S of R, for which S (with its natural topology)
is perfectoid and S ◦a is finite étale over R◦a . In this case, S ◦a is a uniformly finite
projective R◦a -module.
[ ,→ R
In particular, we see that the fully faithful functor Rfét fét preserves degrees. We
will later prove that this is an equivalence in general. For now, we prove that it is an
equivalence for perfectoid fields, i.e. we finish the proof of Theorem 3.7.
Proof. (of Theorem 3.7) Let K be a perfectoid field with tilt K [ . Using the previous
theorem, it is enough to show that the fully faithful functor Kfét[ → K
fét is an equiva-
lence.
Proof using ramification theory. Proposition 6.6.2 (cf. its proof) and Proposition
6.6.6 of [14] show that for any finite extension L of K, the extension L◦a /K ◦a is étale.
Moreover, it is finite projective by Proposition 6.3.6 of [14], giving the desired result.
Proof reducing to the case where K [ is algebraically closed. Let M = K [ be the
c
completion of an algebraic closure of K [ . Clearly, M is complete and perfect, i.e. M
is perfectoid. Let M ] be the untilt of M . Then by Lemma 5.21 and Proposition 3.8,
M ] is an algebraically closed perfectoid field containing K. Any finite extension L ⊂ M
of K [ gives Sthe untilt L] ⊂ M ] , a finite extension of K. It is easy to see that the
union N = L L] ⊂ M ] is a dense subfield. Now Krasner’s lemma implies that N is
algebraically closed. Hence any finite extension F of K is contained in N ; this means
that there is some Galois extension L of K [ such that F is contained in L] . Note that
L] is still Galois, as the functor L 7→ L] preserves degrees and automorphisms. In
particular, F is given by some subgroup H of Gal(L] /K) = Gal(L/K [ ), which gives the
desired finite extension F [ = LH of K [ that untilts to F : The equivalence of categories
shows that (F [ )] ⊂ (L] )H = F , and as they have the same degree, they are equal.
30 PETER SCHOLZE
where K has characteristic 0 by using the result in characteristic p, making use of parts
(i) and (ii) already proved.
Proof. First, we check that the map X → X [ is well-defined and continuous: To check
welldefinedness, we have to see that it maps valuations to valuations. This was already
verified in the proof of Proposition 3.6.
Moreover, the map X → X [ is continuous, because the preimage of the rational subset
f1] ,...,fn]
U ( f1 ,...,f
g
n
) is given by U ( g]
), assuming as in Remark 2.8 that fn is a power of $[
to ensure that f1] , . . . , fn] still generate R.
We have the following description of OX .
Lemma 6.4. Let U = U ( f1 ,...,f
g
n
) ⊂ Spa(R[ , R[+ ) be rational, with preimage U ] ⊂
Spa(R, R+ ). Assume that all fi , g ∈ R[◦ and that fn = $[N for some N ; this is always
possible without changing the rational subspace.
(i) Consider the $-adic completion
! ∞ ! ∞
] 1/p ] 1/p
f fn
R◦ h 1] ,..., i
g g]
of the subring
!1/p∞ !1/p∞
f]
◦ fn] 1
R [ 1] ,..., ] ⊂ R[ ].
g g] g]
1/p∞ 1/p∞
f1] fn]
Then R◦ h g]
,..., g]
ia is a perfectoid K ◦a -algebra.
This shows that we can find elements ri ∈ R+ homogeneous of degree d − di, such that
ri → 0, with
!i
]
X 0 g c
h= $c+1−+(c ) ri ,
1
$c
i∈Z[ p ],0≤i≤1
where we choose some 0 < (c0 ) < (c), (c0 ) ∈ Z[ p1 ]. Choose si ∈ R[+ homogeneous of
degree d − di, si → 0, such that $ divides ri − s]i . Now set
i
X
[ c+1−+(c0 ) gc
gc = gc +
0 ($ ) si .
1
($[ )c
i∈Z[ p ],0≤i≤1
(i) For any f ∈ R and any c ≥ 0, > 0, there exists gc, ∈ R[ such that for all x ∈ X,
we have
]
|f (x) − gc, (x)| ≤ |$|1− max(|f (x)|, |$|c ) .
(ii) For any x ∈ X, the completed residue field k(x)
d is a perfectoid field.
(iii) The morphism X → X [ induces a homeomorphism, identifying rational subsets.
Proof. (i) As any maximal point of Spa(R, R+ ) is contained in Spa(R, R◦ ), and it is
enough to check the inequality at maximal points after increasing slightly, it is enough
to prove this if R+ = R◦ . At the expense of enlarging c, we may assume that f ∈ R◦ ,
and also assume that c is an integer. Further, we can write f = g0] + $g1] + . . . + $c gc] +
$c+1 fc+1 for certain g0 , . . . , gc ∈ R[◦ and fc+1 ∈ R◦ . We can assume fc+1 = 0. Now we
have the map
1/p∞ ∞
KhT0 , . . . , Tc1/p i → R
1/pm 1/pm
sending Ti to (gi )] , and f is the image of T0 + $T1 + . . . + $c Tc , to which we
may apply Lemma 6.5.
(ii), (K of characteristic p) In this case, we know that OX (U )◦a is perfectoid for any
rational subset U . It follows that the $-adic completion of OX,x ◦a is a perfectoid K ◦a -
+
Proof. Corollary 6.7 (iii) and Lemma 6.4 (ii) show that (OX (U ), OX (U )) is a perfectoid
affinoid K-algebra. It can be characterized by the universal property of Proposition 2.14
among all perfectoid affinoid K-algebras, and tilting this universal property shows that
+
its tilt has the analogous universal property characterizing (OX [ (U [ ), OX [
[ (U )) among
is exact. Then ker di is a closed subspace of a K-Banach space, hence itself a K-Banach
space, and di−1 is a surjection onto ker di . By Banach’s open mapping theorem, the
map di−1 is an open map to ker di . This says that the subspace and quotient topologies
on ker di = im di−1 coincide. Now consider the sequence
d◦ Y d◦ Y d◦
0 → OX (X)◦ →0 OX (Ui )◦ →1 OX (Ui ∩ Uj )◦ →2 . . . .
i i,j
By parts (i) and (ii), the quotient topology on im di−1 has $n im d◦i−1 , n ∈ Z, as a basis
of open neighborhoods of 0, and the subspace topology of ker di has $n kerd◦i , n ∈ Z, as
a basis of open neighborhoods of 0. That they agree precisely amounts to saying that
the cohomology group is annihilated by some power of $.
Proposition 6.11. Assume that K is of characteristic p, and that (R, R+ ) is p-finite,
given as the completed perfection of a reduced affinoid K-algebra (S, S + ) of topologically
finite type.
(i) The map X = Spa(R, R+ ) ∼ = Y = Spa(S, S + ) is a homeomorphism identifying ratio-
nal subspaces.
(ii) For any U ⊂ X rational, corresponding to V ⊂ Y , the perfectoid affinoid K-algebra
+
(OX (U ), OX (U )) is equal to the completed perfection of (OY (V ), OY+ (V )).
S
(iii) For any covering X = i Ui by rational subsets, the sequence
Y Y
0 → OX (X)◦a → OX (Ui )◦a → OX (Ui ∩ Uj )◦a → . . .
i i,j
◦a ) = 0 for i > 0.
is exact. In particular, OX is a sheaf, and H i (X, OX
+
Remark 6.12. The last assertion is equivalent to the assertion that H i (X, OX ) is anni-
hilated by m.
Proof. (i) Going to the perfection does not change the associated adic space and rational
subspaces, and going to the completion does not by Proposition 2.11.
(ii) The completed perfection of (OY (V ), OY+ (V )) is a perfectoid affinoid K-algebra. It
+
has the universal property defining (OX (U ), OX (U )) among all perfectoid affinoid K-
algebras.
PERFECTOID SPACES 37
(iii) Note that the corresponding sequence for Y is exact up to some $-power. Hence
after taking the perfection, it is almost exact, and stays so after completion.
Lemma 6.13. Assume K is of characteristic p.
(i) Any perfectoid affinoid K-algebra (R, R+ ) for which R+ is a K ◦ -algebra is the com-
pletion of a filtered direct limit of p-finite perfectoid affinoid K-algebras (Ri , Ri+ ).
(ii) This induces a homeomorphism Spa(R, R+ ) ∼ = lim Spa(Ri , Ri+ ), and each rational
← −
U ⊂ X = Spa(R, R+ ) comes as the preimage of some rational Ui ⊂ Xi = Spa(Ri , Ri+ ).
+
(iii) In this case (OX (U ), OX (U )) is equal to the completion of the filtered direct limit of
+
the (OXj (Uj ), OXj (Uj )), where Uj is the preimage of Ui in Xj for j ≥ i.
(iv) If Ui ⊂ Xi is some quasicompact open subset containing the image of X, then there
is some j such that the image of Xj is contained in Ui .
Proof. (i) For any finite subset I ⊂ R+ , we have the K-subalgebra SI ⊂ R given as the
image of KhTi |i ∈ Ii → R. Then SI is a reduced quotient of KhTi |i ∈ Ii, and we give SI
the quotient topology. Let SI+ ⊂ SI be the set of power-bounded elements; it is also the
set of elements integral over K ◦ hTi |i ∈ Ii by [31], Theorem 5.2. In particular, SI+ ⊂ R+ .
We caution the reader that SI+ is in general not the preimage of R+ in SI .
Now let (RI , RI+ ) be the completed perfection of (SI , SI+ ), i.e. RI+ is the $-adic com-
pletion of limΦ SI+ , and RI = RI+ [$−1 ]. We get an induced map (RI , RI+ ) → (R, R+ ),
−→
and (RI , RI+ ) is a p-finite perfectoid affinoid K-algebra. We claim that R+ /$n =
limI RI+ /$n . Indeed, the map is clearly surjective. It is also injective, since if f1 , f2 ∈ RI+
−→
satisfy f1 − f2 = $n g for some g ∈ R+ , then for some larger J ⊃ I containing g, also
f1 − f2 ∈ $n RJ+ . This shows that R+ is the completed direct limit of the RI+ , i.e.
(R, R+ ) is the completed direct limit of the (RI , RI+ ).
(ii) Let (L, L+ ) be the direct limit of the (RI , RI+ ), equipped with the $-adic topology.
Then one checks by hand that Spa(L, L+ ) ∼ = lim Spa(Ri , Ri+ ), compatible with ratio-
←−
nal subspaces. But then the same thing holds true for the completed direct limit by
Proposition 2.11.
+
(iii) The completion of the direct limit of the (OXj (Uj ), OX j
(Uj )) is a perfectoid affinoid
+
K-algebra, and it satisfies the universal property describing (OX (U ), OX (U )).
(iv) This is an abstract property of spectral spaces and spectral maps. Let Ai be the
closed complement of Ui , and for any j ≥ i, let Aj be the preimage of Ai in Xj . Then
the Aj are constructible subsets of Xj , hence spectral, and the transition maps between
the Aj are spectral. If one gives the Aj the constructible topology, they are compact
topological spaces, and the transition maps are continuous. If their inverse limit is zero,
then one of them has to be zero.
Proposition 6.14. Let K be of any characteristic, and let (R, +
+
S R ) be a perfectoid
affinoid K-algebra, X = Spa(R, R ). For any covering X = i Ui by finitely many
rational subsets, the sequence
Y Y
0 → OX (X)◦a → OX (Ui )◦a → OX (Ui ∩ Uj )◦a → . . .
i i,j
◦a ) = 0 for i > 0.
is exact. In particular, OX is a sheaf, and H i (X, OX
Proof. Assume first that K has characteristic p. We may replace K by a perfectoid
∞
subfield, such as the $-adic completion of Fp (($))($1/p ); this ensures that for any
perfectoid affinoid K-algebra (R, R+ ), the ring R+ is a K ◦ -algebra. Then use Lemma
38 PETER SCHOLZE
6.13 to write X = Spa(R, R+ ) ∼ = lim X = Spa(Ri , Ri+ ) as an inverse limit, with (Ri , Ri+ )
←− i
p-finite. Any rational subspace comes from a finite level, and a cover by finitely many
rational subspaces is the pullback of a cover by finitely many rational subspaces on a
finite level. Hence the almost exactness of the sequence follows by taking the completion
of the direct limit of the corresponding statement for Xi , which is given by Proposition
6.11. The rest follows as before.
In characteristic 0, first use the exactness of the tilted sequence, then reduce modulo
$[ (which is still exact by flatness), and then remark that this is just the original
sequence reduced modulo $. As this is exact, the original sequence is exact, by flatness
and completeness. Again, we also get the other statements.
Definition 6.15. A perfectoid space is an adic space over K that is locally isomorphic
to an affinoid perfectoid space. Morphisms between perfectoid spaces are the morphisms
of adic spaces.
Definition 6.16. We say that a perfectoid space X [ over K [ is the tilt of a perfec-
toid space X over K if there is a functorial isomorphism Hom(Spa(R[ , R[+ ), X [ ) =
Hom(Spa(R, R+ ), X) for all perfectoid affinoid K-algebras (R, R+ ) with tilt (R[ , R[+ ).
Proposition 6.17. Any perfectoid space X over K admits a tilt X [ , unique up to unique
isomorphism. This induces an equivalence between the category of perfectoid spaces over
K and the category of perfectoid spaces over K [ . The underlying topological spaces of X
and X [ are naturally identified. A perfectoid space X is affinoid perfectoid if and only if
its tilt X [ is affinoid perfectoid. Finally, for any affinoid perfectoid subspace U ⊂ X, the
+ +
pair (OX (U ), OX (U )) is a perfectoid affinoid K-algebra with tilt (OX [ (U [ ), OX [
[ (U )).
Proof. This is a formal consequence of Theorem 5.2, Theorem 6.3 and Proposition
2.19. Note that to any open U ⊂ X, one gets an associated perfectoid space with
underlying topological space U by restricting the structure sheaf and valuations to
+
U , and hence its global sections are (OX (U ), OX (U )), so that if U is affinoid, then
+
U = Spa(OX (U ), OX (U )). This gives the last part of the proposition.
Let us finish this section by noting one way in which perfectoid spaces behave better
than adic spaces (cf. Proposition 1.2.2 of [20]).
Proposition 6.18. If X → Y ← Z are perfectoid spaces over K, then the fibre product
X ×Y Z exists in the category of adic spaces over K, and is a perfectoid space.
From the definitions, Proposition 6.17, and Theorem 5.25, we see that f : X → Y is
strongly finite étale, resp. strongly étale, if and only if the tilt f [ : X [ → Y [ is strongly
finite étale, resp. strongly étale. Moreover, in characteristic p, anything (finite) étale is
also strongly (finite) étale.
Lemma 7.3. (i) Let f : X → Y be a strongly finite étale, resp. strongly étale, morphism
of perfectoid spaces and let g : Z → Y be an arbitrary morphism of perfectoid spaces.
Then X ×Y Z → Z is a strongly finite étale, resp. strongly étale, morphism of perfectoid
spaces. Moreover, the map of underlying topological spaces |X ×Z Y | → |X| ×|Z| |Y | is
surjective.
(ii) If in (i), all spaces X = Spa(A, A+ ), Y = Spa(B, B + ) and Z = Spa(C, C + ) are
affinoid, with (A, A+ ) strongly finite étale over (B, B + ), then X ×Y Z = Spa(D, D+ ),
where D = A ⊗B C and D+ is the integral closure of C + in D, and (D, D+ ) is strongly
finite étale over (C, C + ).
(iii) Assume that K is of characteristic p. If f : X → Y is a finite étale, resp. étale,
morphism of adic spaces over k and g : Z → Y is a map from a perfectoid space Z
to Y , then the fibre product X ×Y Z exists in the category of adic spaces over K, is a
perfectoid space, and the projection X ×Y Z → Z is finite étale, resp. étale. Moreover,
the map of underlying topological spaces |X ×Z Y | → |X| ×|Z| |Y | is surjective.
(iv) Assume that in the situation of (iii), all spaces X = Spa(A, A+ ), Y = Spa(B, B + )
and Z = Spa(C, C + ) are affinoid, with (A, A+ ) finite étale over (B, B + ), then X ×Y Z =
Spa(D, D+ ), where D = A ⊗B C, D+ is the integral closure of C + in D and (D, D+ ) is
finite étale over (C, C + ).
Proof. (ii) As A⊗B C is finite projective over C, it is already complete. One easily deduces
the universal property. Also, D◦a is finite étale over C ◦a , as base-change preserves finite
étale morphisms.
(i) Applying the definition of a strongly finite étale map, one reduces the statement about
strongly finite étale maps to the situation handled in part (ii). Now the statement for
strongly étale maps follows, because fibre products obviously preserve open embeddings.
The surjectivity statement follows from the argument of [19], proof of Lemma 3.9 (i).
(iv) Proposition 5.23 shows that D = A ⊗B C is perfectoid. Therefore Spa(D, D+ ) is a
perfectoid space. One easily checks the universal property.
(iii) The finite étale case reduces to the situation considered in part (iv). Again, it is
trivial to handle open embeddings, giving also the étale case. Surjectivity is proved as
before.
Let us recall the following statement about henselian rings.
Proposition 7.4. Let A be a flat K ◦ -algebra such that A is henselian along ($). Then
the categories of finite étale A[$−1 ] and finite étale Â[$−1 ]-algebras are equivalent,
where  is the $-adic completion of A.
Proof. See e.g. [14], Proposition 5.4.53.
We recall that $-adically complete algebras A are henselian along ($), and that if Ai
is a direct system of K ◦ -algebras henselian along ($), then so is the direct limit lim Ai .
−→
In particular, we get the following lemma.
Lemma 7.5. (i) Let Ai be a filtered direct system of complete flat K ◦ -algebras, and let
A be the completion of the direct limit, which is again a complete flat K ◦ -algebra. Then
we have an equivalence of categories
A[$−1 ]fét ∼
= 2 − lim A [$−1 ]fét .
−→ i
PERFECTOID SPACES 41
Using Proposition 5.23, this shows that if K is of characteristic p, then the finite étale
covers of an affinoid perfectoid space X = Spa(R, R+ ) are the same as the finite étale
covers of R.
The same method also proves the following proposition.
Proposition 7.7. Assume that K is of characteristic p. Let f : X → Y be an étale map
of perfectoid spaces. Then for any x ∈ X, there exist affinoid perfectoid neighborhoods
x ∈ U ⊂ X, f (U ) ⊂ V ⊂ Y , and an étale morphism of affinoid noetherian adic spaces
U 0 → V 0 over K, such that U = U 0 ×V 0 V .
Proof. We may assume that X and Y affinoid perfectoid, and that X is a rational
subdomain of a finite étale cover of Y . Then one reduces to the p-finite case by the
same argument as above, and hence to noetherian adic spaces.
42 PETER SCHOLZE
Corollary 7.8. Strongly étale maps of perfectoid spaces are open. If f : X → Y and
g : Y → Z are strongly (finite) étale morphisms of perfectoid spaces, then the composite
g ◦ f is strongly (finite) étale.
Proof. We may assume that K has characteristic p. The first part follows directly from
the previous proposition and the result for locally noetherian adic spaces, cf. [20], Propo-
sition 1.7.8. For the second part, argue as in the previous proposition for both f and
g to reduce to the analogous result for locally noetherian adic spaces, [20], Proposition
1.6.7 (ii).
The following theorem gives a strong form of Faltings’s almost purity theorem.
Theorem 7.9. Let (R, R+ ) be a perfectoid affinoid K-algebra, and let X = Spa(R, R+ )
with tilt X [ .
(i) For any open affinoid perfectoid subspace U ⊂ X, we have a fully faithful functor from
the category of strongly finite étale covers of U to the category of finite étale covers of
OX (U ), given by taking global sections.
(ii) For any U , this functor is an equivalence of categories.
(iii) For any finite étale cover S/R, S is perfectoid and S ◦a is finite étale over R◦a .
Moreover, S ◦a is a uniformly almost finitely generated R◦a -module.
Proof. (i) By Proposition 7.6 and Theorem 5.25, the perfectoid spaces strongly finite étale
over U are the same as the finite étale OX (U )◦a -algebras, which are a full subcategory
of the finite étale OX (U )-algebras.
(ii) We may assume that U = X. Fix a finite étale R-algebra S. First we check that for
any x ∈ X, we can find an affinoid perfectoid neighborhood x ∈ U ⊂ X and a strongly
finite étale cover V → U which gives via (i) the finite étale algebra OX (U ) ⊗R S over
OX (U ).
As a first step, note that we have an equivalence of categories between the direct limit
of the category of finite étale OX (U )-algebras over all affinoid perfectoid neighborhoods
U of x and the category of finite étale covers of the completion k(x)d of the residue field
at x, by Lemma 7.5 (i). The latter is a perfectoid field.
By Theorem 3.7, the categories k(x)
d [
and k(x
fét
[) are equivalent, where k(x[ ) is the
fét
residue field of X [ at the point x[ corresponding to x. Combining, we see that
2 − lim (OX (U ))fét ∼
= 2 − lim (O (U [ ))fét .
−→ −→ X [
x∈U x∈U
In particular, we can find V finite étale over U [ for some U such that the pullbacks of
[
identified over some smaller neighborhood. Shrinking U , we untilt to get the desired
strongly finite étale V → U . S
This shows that there is a cover X = Ui by finitely many rational subsets and
strongly finite étale maps Vi → Ui such that the global sections of Vi are Si = OX (Ui )⊗R
S. Let Si+ be the integral closure of OX +
(Ui ) in Si ; then Vi = Spa(Si , Si+ ).
By Lemma 7.3 (ii), the pullback of Vi to some affinoid perfectoid U 0 ⊂ Ui has the same
description, involving OX (U 0 ) ⊗R S, and hence the Vi glue to some perfectoid space Y
over X, and Y → X is strongly finite étale. By Proposition 7.6, Y is affinoid perfectoid,
i.e. Y = Spa(A, A+ ), with (A, A+ ) an affinoid perfectoid K-algebra. It suffices to show
that A = S. But the sheaf property of OY gives us an exact sequence
Y Y
0→A→ OX (Ui ) ⊗R S → OX (Ui ∩ Uj ) ⊗R S .
i i,j
PERFECTOID SPACES 43
On the other hand, the sheaf property for OX gives an exact sequence
Y Y
0→R→ OX (Ui ) → OX (Ui ∩ Uj ) .
i i,j
Because S is flat over R, tensoring is exact, and the first sequence is identified with the
second sequence after ⊗R S. Therefore A = S, as desired.
(iii) This is a formal consequence of part (ii), Proposition 7.6 and Theorem 5.25.
We see in particular that any (finite) étale morphism of perfectoid spaces is strongly
(finite) étale. Now one can also pullback étale maps between adic spaces in characteristic
0.
Proposition 7.10. Parts (iii) and (iv) of Lemma 7.3 stay true in characteristic 0.
Proof. The same proof as for Lemma 7.3 works, using Theorem 7.9 (iii).
Finally, we can define the étale site of a perfectoid space.
Definition 7.11. Let X be a perfectoid space. Then the étale site of X is the category
Xét of perfectoid spaces which are étale over X, and coverings are given by topological
coverings. The associated topos is denoted Xét∼.
The previous results show that all conditions on a site are satisfied, and that a mor-
phism f : X → Y of perfectoid spaces induces a morphism of sites Xét → Yét . Also, a
morphism f : X → Y from a perfectoid space X to a locally noetherian adic space Y
induces a morphism of sites Xét → Yét .
After these preparations, we get the technical main result.
Theorem 7.12. Let X be a perfectoid space over K with tilt X [ over K [ . Then the
tilting operation induces an isomorphism of sites Xét ∼ [ . This isomorphism is func-
= Xét
torial in X.
Proof. This is immediate.
The almost vanishing of cohomology proved in Proposition 6.14 extends to the étale
topology.
Proposition 7.13. For any perfectoid space X over K, the sheaf U 7→ OU (U ) is a
◦a ) = 0 for i > 0 if X is affinoid perfectoid.
sheaf OX on Xét , and H i (Xét , OX
Proof. It suffices to check exactness of
Y Y
0 → OX (X)◦a → OUi (Ui )◦a → OUi ×X Uj (Ui ×X Uj )◦a → . . .
i i,j
Now we adapt these definitions to the world of usual adic spaces, and to the world
of perfectoid spaces. Assume first that k is a complete nonarchimedean field, and let Σ
be a fan as above. Then we can associate to Σ the adic space XΣad of finite type over k
which is glued out of
Uσad = Spa(khσ ∨ ∩ M i, k ◦ hσ ∨ ∩ M i) .
We note that this is not in general the adic space XΣad over k associated to the variety
XΣ : For example, if XΣ is just affine space, then XΣad will be a closed unit ball. In
general, let XΣ,k◦ be the toric scheme over k ◦ associated to Σ. Let X̂Σ,k◦ be the formal
completion of XΣ,k◦ along its special fibre, which is an admissible formal scheme over
k ◦ . Then XΣad is the generic fibre X̂Σ,k ad
◦ associated to X̂Σ,k ◦ . In particular, if XΣ is
ad
proper, then XΣ = XΣ . ad
H 0 (XΣperf , O(D)) is the free Banach-K-vector space with basis given by {χu }, where u
ranges over u ∈ M [p−1 ] with hu, vi i ≥ −ai .
We have the following comparison statements. Note that any toric variety XΣ comes
with a map ϕ : XΣ → XΣ induced from multiplication by p on M ; the same applies to
XΣad , etc. . For clarity, we use subscripts to denote the field over which we consider the
toric variety.
Theorem 8.5. Let K be a perfectoid field with tilt K [ .
46 PETER SCHOLZE
perf perf
(i) The perfectoid space XΣ,K tilts to XΣ,K [.
perf
(ii) The perfectoid space XΣ,K can be written as
perf ad
XΣ,K ∼ lim XΣ,K .
←−
ϕ
Vét∼ / (X ad )∼
Σ,K [ ét
∼ / (X ad )∼
Uét Σ,K ét
Proof. This is an immediate consequence of our previous results: Part (i) can be checked
on affinoid pieces, where it is an immediate generalization of Proposition 5.20. Part (ii)
can be checked one affinoid pieces again, where it is easy. Then parts (iii) and (iv) follow
from Theorem 7.17, Corollary 7.19 and the preservation of topological spaces and étale
topoi under tilting. Finally, part (v) follows from Proposition 7.16, using the previous
arguments.
Let us denote by π : XΣ,K ad ad
[ → XΣ,K the projection, which exists on topological
spaces and étale topoi. In the following, we restrict to proper smooth toric varieties for
simplicity.
Proposition 8.6. Let XΣ be a proper smooth toric variety. Let ` 6= p be prime. Assume
that K, and hence K [ , is algebraically closed. Then for all i ∈ Z, the projection map π
induces an isomorphism
H i (X ad , Z/`m Z) ∼
Σ,K,ét = H i (X ad [ , Z/`m Z) .
Σ,K ,ét
Proof. This follows from part (iv) of the previous theorem combined with the observation
ad → X ad induces an isomorphism on cohomology with Z/`m Z-coefficients.
that ϕ : XΣ,K Σ,K
Using proper base change, this can be checked on XΣ,κ , where κ is the residue field of K.
But here, ϕ is purely inseparable, and hence induces an equivalence of étale topoi.
We need the following approximation property.
Proposition 8.7. Assume that XΣ,K is proper smooth. Let Y ⊂ XΣ,K be a hypersur-
ad be a small open neighborhood of Y . Then there exists a hypersurface
face. Let Ỹ ⊂ XΣ,K
Z ⊂ XΣ,K [ such that Z ad ⊂ π −1 (Ỹ ). One can assume that Z is defined over a given
dense subfield of K [ .
ai Di be a T -Weil divisor representing Y . Let f ∈ H 0 (XΣ,K , O(D))
P
Proof. Let D =
be the equation with zero locus Y . Consider the graded ring
perf
M M M
H 0 (XΣ,K , O(jD)) = u
u∈M [p−1 ] Kχ ,
d
j∈Z[p−1 ] j∈Z[p−1 ] hu,vi i≥−jai
PERFECTOID SPACES 47
and let R be its completion (with respect to the obvious K ◦ -submodule). Here c
L
denotes the Banach space direct sum. Then as in Proposition 5.20, R is a perfectoid
K-algebra whose tilt is given by the similar construction over K [ . Note that D is given
combinatorially and hence transfers to K [ .
We may assume that Ỹ is given by
ad
Ỹ = {x ∈ XΣ,K | |f (x)| ≤ } ,
for some . In order to make sense of the inequality |f (x)| ≤ , note that XΣ,K and
O(D) have a tautological integral model over K ◦ (by applying the toric constructions
over K ◦ ), which is enough to talk about absolute values: Trivialize the line bundle O(D)
locally on the integral model to interpret f as a function; any two different choices differ
by a unit of K ◦ , and hence give the same absolute value.
Now the analogue of Lemma 6.5 holds true for R, with the same proof. This implies
perf
that we can find g ∈ H 0 (XΣ,K [ , O(D)) such that
perf
π −1 (Ỹ ) = {x ∈ XΣ,K [ | |g(x)| ≤ } .
Replacing g by a large p-power gives a regular function h on XΣ,K [ , such that its zero
locus Z ⊂ XΣ,K [ is contained in π −1 (Ỹ ), as desired.
means that on the invariants V Ik of V = H i (Xk̄ , Q̄` ), all occuring weights are ≤ i. A
similar property holds true for all tensor powers of V , and for all tensor powers of the
dual V ∨ . Then one applies the following lemma from linear algebra, which applies for
all local fields k.
Lemma 9.5. Let V be an `-adic representation of Gk , on which Ik acts unipotently.
Then grNj V is pure of weight i + j for all j ∈ Z if and only if for all j ≥ 0, all weights
on (V ⊗j )Ik are at most ij, and all weights on ((V ∨ )⊗j )Ik are at most −ij.
Our main theorem is the following.
Theorem 9.6. Let k be a local field of characteristic 0. Let Y be a geometrically con-
nected proper smooth variety over k such that Y is a set-theoretic complete intersection
in a projective smooth toric variety XΣ . Then the weight-monodromy conjecture is true
for Y .
∞
Proof. Let $ ∈ k be a uniformizer, and let K be the completion of k($1/p ); then K
is perfectoid. Let K [ be its tilt. Then K [ is the completed perfection of k 0 = Fq ((t)),
where t = $[ . This gives an isomorphism between the absolute Galois groups of K
and k 0 . The notion of weights and the monodromy operator N is compatible with this
isomorphism of Galois groups.
By Theorem 3.6 a) of [21], there is some open neighborhood Ỹ of YKad in XΣ,K ad such
By Corollary 8.8, there is some closed subvariety Z ⊂ XΣ,K [ such that Z ad ⊂ π −1 (Ỹ )
and dim Z = dim Y . Moreover, we can assume that Z is defined over a global field
and geometrically irreducible. Let Z 0 be a projective smooth alteration of Z. We get a
commutative diagram of étale topoi of adic spaces
ad )∼
(XΣ,C
π / (X ad )∼
[ ét Σ,Cp ét
O p O
? ?
(π −1 (Ỹ )C[p )ét
∼ / (ỸC )∼
O O
p ét
?
(ZC0ad ∼
[ )ét (YCad )∼ .
p ét
p
compatible with the cup product. Formally taking the inverse limit and tensoring with
Q̄` , it follows that we get a G-equivariant map
H i (YCp ,ét , Q̄` ) → H i (ZC0 [ ,ét , Q̄` ) .
p
H 2 dim Y (π −1 (Ỹ )C[p ,ét , Z/`m Z) o H 2 dim Y (ỸCp ,ét , Z/`m Z)
∼
=
H 2 dim Y (ZC0 [ ,ét , Z/`m Z) H 2 dim Y (YCp ,ét , Z/`m Z)
p
The isomorphism in the top row is from Proposition 8.6. We can pass to the inverse
limit over m and tensor with Q̄` . If
H 2 dim Y (YCp ,ét , Q̄` ) → H 2 dim Y (ZC0 [ ,ét , Q̄` ) .
p
is not an isomorphism, it is the zero map, and hence the diagram implies that the
restriction map
H 2 dim Y (XΣ,C[p ,ét , Q̄` ) → H 2 dim Y (ZC0 [ ,ét , Q̄` )
p
is the zero map as well. But the dim Y -th power of the first Chern class of an ample
line bundle on XΣ,C[p will have nonzero image in H 2 dim Y (ZC0 [ ,ét , Q̄` ).
p
Now the Poincaré duality pairing implies that H i (YCp ,ét , Q̄` ) is a direct summand of
H i (ZC0 [ ,ét , Q̄` ). By Deligne’s theorem, H i (ZC0 [ ,ét , Q̄` ) satisfies the weight-monodromy
p p
conjecture, and hence so does its direct summand H i (YCp ,ét , Q̄` ).
References
[1] V. G. Berkovich. Spectral theory and analytic geometry over non-Archimedean fields, volume 33 of
Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1990.
[2] S. Bosch, U. Güntzer, and R. Remmert. Non-Archimedean analysis, volume 261 of Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-
Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry.
[3] S. Bosch and W. Lütkebohmert. Formal and rigid geometry. I. Rigid spaces. Math. Ann., 295(2):291–
317, 1993.
[4] P. Boyer. Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura
simples. Invent. Math., 177(2):239–280, 2009.
[5] P. Boyer. Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires. Compos.
Math., 146(2):367–403, 2010.
[6] A. Caraiani. Local-global compatibility and the action of monodromy on nearby cycles.
arXiv:1010.2188.
[7] J.-F. Dat. Théorie de Lubin-Tate non-abélienne et représentations elliptiques. Invent. Math.,
169(1):75–152, 2007.
[8] A. J. de Jong. Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math.,
(83):51–93, 1996.
[9] P. Deligne. Théorie de Hodge. I. In Actes du Congrès International des Mathématiciens (Nice,
1970), Tome 1, pages 425–430. Gauthier-Villars, Paris, 1971.
[10] P. Deligne. La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math., (52):137–252, 1980.
[11] G. Faltings. p-adic Hodge theory. J. Amer. Math. Soc., 1(1):255–299, 1988.
[12] G. Faltings. Almost étale extensions. Astérisque, (279):185–270, 2002. Cohomologies p-adiques et
applications arithmétiques, II.
[13] J.-M. Fontaine and J.-P. Wintenberger. Extensions algébrique et corps des normes des extensions
APF des corps locaux. C. R. Acad. Sci. Paris Sér. A-B, 288(8):A441–A444, 1979.
[14] O. Gabber and L. Ramero. Almost ring theory, volume 1800 of Lecture Notes in Mathematics.
Springer-Verlag, Berlin, 2003.
PERFECTOID SPACES 51
[15] M. Harris and R. Taylor. The geometry and cohomology of some simple Shimura varieties, volume
151 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2001. With an
appendix by Vladimir G. Berkovich.
[16] E. Hellmann. On arithmetic families of filtered ϕ-modules and crystalline representations. 2011.
arXiv:1010.4577.
[17] M. Hochster. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc., 142:43–60,
1969.
[18] R. Huber. Continuous valuations. Math. Z., 212(3):455–477, 1993.
[19] R. Huber. A generalization of formal schemes and rigid analytic varieties. Math. Z., 217(4):513–551,
1994.
[20] R. Huber. Étale cohomology of rigid analytic varieties and adic spaces. Aspects of Mathematics,
E30. Friedr. Vieweg & Sohn, Braunschweig, 1996.
[21] R. Huber. A finiteness result for direct image sheaves on the étale site of rigid analytic varieties. J.
Algebraic Geom., 7(2):359–403, 1998.
[22] L. Illusie. Complexe cotangent et déformations. I. Lecture Notes in Mathematics, Vol. 239. Springer-
Verlag, Berlin, 1971.
[23] L. Illusie. Complexe cotangent et déformations. II. Lecture Notes in Mathematics, Vol. 283. Springer-
Verlag, Berlin, 1972.
[24] L. Illusie. Autour du théorème de monodromie locale. Astérisque, (223):9–57, 1994. Périodes p-
adiques (Bures-sur-Yvette, 1988).
[25] T. Ito. Weight-monodromy conjecture for p-adically uniformized varieties. Invent. Math.,
159(3):607–656, 2005.
[26] T. Ito. Weight-monodromy conjecture over equal characteristic local fields. Amer. J. Math.,
127(3):647–658, 2005.
[27] K. Kedlaya and R. Liu. Relative p-adic Hodge theory, I: Foundations.
http://math.mit.edu/∼kedlaya/papers/relative-padic-Hodge1.pdf.
[28] D. Quillen. On the (co-) homology of commutative rings. In Applications of Categorical Algebra
(Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), pages 65–87. Amer. Math. Soc., Provi-
dence, R.I., 1970.
[29] M. Rapoport and T. Zink. Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltra-
tion und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math., 68(1):21–101, 1982.
[30] S. W. Shin. Galois representations arising from some compact Shimura varieties. Ann. of Math. (2),
173(3):1645–1741, 2011.
[31] J. Tate. Rigid analytic spaces. Invent. Math., 12:257–289, 1971.
[32] R. Taylor and T. Yoshida. Compatibility of local and global Langlands correspondences. J. Amer.
Math. Soc., 20(2):467–493, 2007.
[33] T. Terasoma. Monodromy weight filtration is independent of `. 1998. arXiv:math/9802051.