RIMS Kôkyûroku Bessatsu
Bx (201x), 000–000

Ramification theory and perfectoid spaces — asurveyBy

Shin Hattori∗

Abstract
In 1980s, Deligne proved that, if two complete discrete valuation ﬁelds with perfect residue
ﬁelds are “close” enough, then their absolute Galois groups are isomorphic to each other modulo
certain upper ramiﬁcation subgroups. In this article, we give a brief survey on the author’s
generalization of this result to the case where the residue ﬁelds are imperfect.

§ 1.

Classical ramification theory

This article is a survey of the author’s result on a comparison between ramiﬁcationof complete discrete valuation ﬁelds of mixed and equal characteristics [10], generalizing
Deligne’s theorem for the case of perfect residue ﬁelds [5]. After giving some backgroundin the ﬁrst four sections, the main theorem (Theorem 5.1) and its corollaries are statedin Section 5. We also give a sketch of the proof of the main theorem in Section 6.
Let p be a rational prime and K a complete discrete valuation ﬁeld whose residue
ﬁeld k is of characteristic p. We denote the ring of integers of K by OK . We ﬁx
an algebraic closure K̄ of K, and let K sep be the separable closure of K inside K̄.
We put GK = Gal(K sep /K). Let vK be the additive valuation of K normalized asvK (K × ) = Z, and we extend it naturally to K̄. We denote a uniformizer of K by πK
and the completion of K̄ by C.
Ramiﬁcation theory measures an extent of how far a ﬁnite separable extension L
of K is from unramiﬁed extensions: we can deﬁne a normal subgroup GjK of GK for
any positive rational number j, which is called the j-th (non-log) upper ramiﬁcation
subgroup of GK . For any ﬁnite separable extension L/K, the Galois group GK acts
Received April 20, 201x. Revised September 11, 201x.
2000 Mathematics Subject Classification(s): 11S15
Key Words: ramification, truncated discrete valuation rings, perfectoid spaces∗ Kyushu University, 744 Motooka Nishi-ku, Fukuoka 819-0395 JAPAN.
e-mail: shin-h@math.kyushu-u.ac.jp
c 201x Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
⃝

2

Shin Hattori

continuously on the ﬁnite set FK (L) = HomK-alg. (L, K̄). Then the extension L/K is
unramiﬁed if and only if the group GjK acts trivially on the set FK (L) for any positive
rational number j. The numberc(L/K) = inf{j ∈ Q>0 | GjK acts trivially on FK (L)}can be considered as a measure of ramiﬁcation of the extension L/K, and it is called
the conductor of the extension L/K. We say that the ramiﬁcation of L/K is boundedby j if c(L/K) < j.
For the case where the residue ﬁeld k is perfect, the deﬁnition of the group GjK
is classical (see [17]): Let L/K be a ﬁnite Galois extension with Galois group G =Gal(L/K). For any real number i ≥ −1, we deﬁne the i-th lower ramiﬁcation subgroup
Gi of G by
Gi = {g ∈ G | vL (g(πL ) − πL ) ≥ i + 1}.
This group is not compatible with quotients. Namely, for a Galois extension M/K
inside L with Galois group H = Gal(M/K), the image of Gi by the surjection G → H
is not necessarily equal to Hi . Thus we renumber them to deﬁne GjK by using theHasse-Herbrand function∫ s
dt
φL/K (s) =.
0 [G0 : Gt ]Set ψL/K (j) to be the inverse function of φL/K and put Gj = GψL/K (j−1) (Note thatour deﬁnition is shifted from that of [17] by one). Then it is compatible with quotients.
Deﬁne
GjK = lim Gal(L/K)j ,←−L/K

where the limit is taken over the ﬁltered ordered set of ﬁnite Galois extensions L/K
inside K̄. By the compatibility with quotients, the image of GjK by the natural surjection
GK → Gal(L/K) is equal to Gal(L/K)j . Note that, since the residue ﬁeld k is perfect,the extension L/K can be written as an Eisenstein extension of an unramiﬁed extensionof K. In other words, in the case of perfect residue ﬁelds, we can reduce the study of
ramiﬁcation to extensions L/K such that the OK -algebra OL is generated by a single
element (i.e. monogenic). This property is a key point both to deﬁne Gj and to proveits compatibility with quotients in the classical ramiﬁcation theory.
§ 2.

Deligne’s equivalence

Consider two complete discrete valuation ﬁelds K and F , and suppose that theirresidue ﬁelds are isomorphic to each other. Since the category FE<0+(resp. FE<0+)K
F
of ﬁnite unramiﬁed extensions of K (resp. F ) is equivalent to that of ﬁnite separable

Ramification theory and perfectoid spaces — a survey

3extensions of its residue ﬁeld, we obtain an equivalence of categories FE<0+≃ FE<0+.
K
F<j
Let FEK be the category of ﬁnite separable extensions of K whose ramiﬁcation isbounded by j. In this article, we mainly focus on the following:m
Question 2.1.
Suppose that we have an isomorphism of rings OK /(πK) ≃m
OF /(πF ) for some positive integer m. Then, can we show an equivalence of categories<jFE<jK ≃ FEF for some j?

For the case where the residue ﬁeld of K (and F ) is perfect, this was studied
by Deligne and he proved the following theorem, which gives a striking isomorphism
between the absolute Galois groups of complete discrete valuation ﬁelds of possibly
diﬀerent characteristics modulo ramiﬁcation subgroups:
Let K and F be complete discrete valuation fields with
Theorem 2.2 ([5], 1.3).
perfect residue fields and m a positive integer. Suppose that we have an isomorphismm) ≃ OF /(πFm ). Then there exists an equivalence of categories FE<jof rings OK /(πKK ≃<j
FEF for any j satisfying 0 < j ≤ m. In particular, there exists an isomorphism of
topological groups GK /GjK ≃ GF /GjF .
Suppose moreover that F is of characteristic p. Then for any m, we can ﬁnd amp-adic ﬁeld Km such that there exists an isomorphism of rings OF /(πFm ) ≃ OKm /(πK)
mm
(Indeed, we can take OKm = C(k)[[u]]/(u − p), where C(k) is a Cohen ring of theresidue ﬁeld k of F ). In other words, any complete discrete valuation ﬁeld of equalcharacteristic can be written as a limit of a family of complete discrete valuation ﬁeldsof mixed characteristic. Combining this fact with Theorem 2.2, we see the following:
If the residue ﬁeld is perfect, then we can reduce the study of the absolute Galois
group of a complete discrete valuation ﬁeld of equal characteristic to that of mixed
characteristic! This trick is known as (the Galois side of) the theory of close local
ﬁelds due to Deligne and Kazhdan, and it provides a powerful tool for proving the local
Langlands correspondence for reductive groups over local ﬁelds of equal characteristic
where harmonic analysis is much harder than the mixed characteristic case (see [4] forthe inner forms of SLn , and [6] for GSp4 ).
The key point of the proof of Theorem 2.2 is the fact that ramiﬁcation of a ﬁnite
Galois extension L/K with trivial residue extension can be read oﬀ from the Newton
polygon of a translation of the minimal polynomial f (X) of a uniformizer πL over OK .
Write asf (X + πL ) = a0 X n + a1 X n−1 + · · · + an−1 X.
Consider the subset {(i, vK (ai ))} ⊆ R2 . The Newton polygon NPK (L, πL ) associatedto the extension L/K and the uniformizer πL is by deﬁnition its lower convex hull.
Then the y-intercept of the leftmost slope is equal to c(L/K) [5, Proposition 1.5.1].

4Shin HattoriMore precisely, the Hasse-Herbrand function φL/K can be recovered completely from
P = NPK (L, πL ) by the formula
φL/K (s) = φP (s+1
) − 1,e(L/K)

where e(L/K) is the relative ramiﬁcation index of L/K and φP is the dual polygon ofP:
φP (s) = inf{t ∈ R | y = −sx + t intersects P }(for example, see [7, Section 3]. Note that the corresponding formula there contains anerror). From this we see that if c(L/K) < m, then the Newton polygon NPK (L, πL )m). This allows us to prove that the category FE<jdepends only on OL /(πKK dependsmonly on OK /(πK ), and hence Theorem 2.2 follows.
§ 3.

Residually imperfect case: ramification theory of Abbes-Saito

Next we consider the case where the residue ﬁeld k of K is imperfect. Let L/K
be a ﬁnite Galois extension. The problem is that, in this case, the integer ring OL is
not necessarily monogenic over the integer ring of the maximal unramiﬁed extension
inside L/K, and ramiﬁcation appears not only in the uniformizer πL but also in theinseparable residue extension. Therefore the classical deﬁnition of upper ramiﬁcation
subgroups does not work for this case. A ramiﬁcation theory generalizing the classicalone to such non-monogenic extensions is due to Abbes-Saito [2, 3], for which we brieﬂy
explain their idea.
First we assume again that k is perfect, to illustrate the idea. Let L/K be aﬁnite Galois extension with trivial residue extension and Galois group G. We ﬁx a
uniformizer πL of L and let f (X) be its minimal polynomial over OK , as above. Let
z1 = πL , z2 , . . . , zn be the roots of f (X) in K̄. Put
XCj (OL , πL ) = {x ∈ OC | vK (f (x)) ≥ j}.
Note that it also depends on the base ﬁeld K, and that GK acts naturally on it. We
write it as the disjoint union of discs, as follows. For any x ∈ OC , we havevK (f (x)) =n∑vK (x − zl ).

l=1

Take i such that vK (x − zi ) = maxl=1,...,n vK (x − zl ). Since vK (x − zl ) = vK ((x − zi ) +(zi − zl )), we havevK (x − zi ) ≤ vK (zi − zl ) ⇒ vK (x − zl ) = vK (x − zi ),vK (x − zi ) > vK (zi − zl ) ⇒ vK (x − zl ) = vK (zi − zl )Ramification theory and perfectoid spaces — a survey

and∑vK (f (x)) =∑vK (x − zi ) +l: first case5vK (zi − zl ).

l: second case

Since G acts transitively on the roots of f (X), the set {vK (zi −zl ) | l ̸= i} is independentof i and thus the valuation vK (f (x)) depends only on u = maxl=1,...,n vK (x − zl ). Put
vK (f (x)) = φ̃(u). This is a piecewise linear function passing the origin. On a smooth
point (u, φ̃(u)), the slope of this function is equal to♯{l | u ≤ vK (zi − zl )} = ♯{g ∈ G | vK (g(πL ) − πL ) ≥ u} = ♯Ge(L/K)u−1 .
Hence we obtain the equalityφ̃(u) = φL/K (e(L/K)u − 1) + 1,
and its inverse function ψ̃(j) isψ̃(j) =

1
(ψL/K (j − 1) + 1).
e(L/K)

Put θ = 1/p and write the disc with radius θj centered at z asD(z, θj ) = {x ∈ OC | vK (x − z) ≥ j}.
Then the set XCj (OL , πL ) can be written as∪
XCj (OL , πL ) =D(zl , θψ̃(j) ).

l=1,...,nWe deﬁne an equivalence relation ∼s on OC byz ∼s w ⇔ vK (z − w) ≥ s.
It satisﬁes
z ∼s z ′ ⇔ D(z, θs ) = D(z ′ , θs ),z ≁s z ′ ⇔ D(z, θs ) ∩ D(z ′ , θs ) = ∅.
Note that, for any g, h ∈ G, we haveg(πL ) ∼ψ̃(j) h(πL ) ⇔ g −1 h ∈ Gj .
Thus we obtain the decomposition
XCj (OL , πL ) =⨿g∈G/GjD(g(πL ), θψ̃(j) ).

6Shin HattoriThis decomposition can be interpreted as the decomposition into connected components if we consider both sides as analytic varieties over C. Namely, we can recover
the ramiﬁcation subgroup Gj as the stabilizer in G of any single element of the set
π0 (XCj (OL , πL )). This observation can be generalized to the non-monogenic case, whichis the idea of Abbes-Saito to deﬁne ramiﬁcation subgroups GjK for K with imperfectresidue ﬁeld.
Now we recall their deﬁnition in a slightly diﬀerent, but equivalent, manner fromthe original one [2]. Here we describe their ramiﬁcation theory via adic spaces, in orderto pass to perfectoid spaces (see [10, Section 2]). For this, we brieﬂy explain about thedeﬁnition of adic spaces (see [12, Section 3] and [16, Section 2]).
Let R be a ring. A valuation on R is a multiplicative map | · | : R → Γ ∪ {0}, whereΓ is a totally ordered abelian group with its group structure written multiplicatively,
such that |0| = 0, |1| = 1 and |f + g| ≤ max{|f |, |g|} for any f, g ∈ R. We denote by Γ|·|the subgroup of Γ generated by {|f | | f ̸= 0 ∈ R}. For a topological ring R, a valuation| · | : R → Γ ∪ {0} on R is said to be continuous if the subset {f ∈ R | |f | < γ} is open in
R for any γ ∈ Γ|·| . Two valuations | · | and | · |′ are said to be equivalent if the condition|f | ≤ |g| ⇔ |f |′ ≤ |g|′is satisﬁed for any f, g ∈ R. We denote this equivalence relation by ∼.
Let F be a topological ﬁeld whose topology is given by a non-trivial valuation ofrank one. A topological F -algebra R is called a Tate F -algebra if there exists a subringR0 of R such that {aR0 | a ∈ F × } forms a basis of open neighborhoods of 0. An element
f of a Tate F -algebra R is said to be power-bounded if there exists a ∈ F × such that{f n | n ∈ Z≥0 } ⊆ aR0 . We denote the subring of power-bounded elements in R by R◦ .
Let R+ be an open subring of R which is integrally closed in R and contained in R◦ .
We deﬁneSpa(R, R+ ) = {| · | : R → Γ ∪ {0} | | · | is continuous and |f | ≤ 1 for any f ∈ R+ }/ ∼ .
Hence any point x ∈ Spa(R, R+ ) deﬁnes an equivalence class of valuations on R. We
write a valuation representing x as f 7→ |f (x)|. Note that the group Γ may vary foreach point x ∈ Spa(R, R+ ).
Let f1 , . . . , fr , g be elements of R satisfying (f1 , . . . , fr ) = R. Then the subset{x ∈ Spa(R, R+ ) | |fi (x)| ≤ |g(x)| ̸= 0 for any i}
is called a rational subset of Spa(R, R+ ). We give Spa(R, R+ ) the topology generated
by the rational subsets.
Put π = πK . Let B̃ be a ﬁnite ﬂat OK -algebra. Fix a ﬁnite system of generators
Z = (z1 , . . . , zn ) of the OK -algebra B̃. Consider the surjection OK [X1 , . . . , Xn ] → B̃Ramification theory and perfectoid spaces — a survey

7

deﬁned by Xi 7→ zi and write its kernel as (f1 , . . . , fr ). Let us write as X = (X1 , . . . , Xn )andOC ⟨X⟩ = OC [X1 , . . . , Xn ]∧ , C⟨X⟩ = OC ⟨X⟩[1/π],
where ∧ means the π-adic completion. Put
XCad = Spa(C⟨X⟩, OC ⟨X⟩),
which also depends on n. Deﬁne a rational subset XCj,ad (B̃, Z) of XCad by
XCj,ad (B̃, Z) = {x ∈ XCad | |fi (x)| ≤ |π(x)|j for any i}and putjFK(B̃) = π0 (XCj,ad (B̃, Z)).

Then we can show that this is a ﬁnite GK -set which is independent of the choice of Z
up to an isomorphism and functorial on B̃.
Definition 3.1.

We say that the ramiﬁcation of B̃/OK is bounded by j if
For a ﬁnite extension L/K, we say that the ramiﬁcation of
L/K is bounded by j if the ramiﬁcation of OL /OK is bounded by j.
j(B̃) = rankOK (B̃).
♯FK

This deﬁnition is equivalent to [2, Deﬁnition 6.3] if B̃ is of relative complete intersection over OK and B̃ ⊗OK K is etale over K. Then we have the following compatibility
with base change.
Lemma 3.2 ([1], Proof of Lemme 2.1.5).
Let K ′ /K be an extension of complete
discrete valuation fields which is not necessarily finite. Let B̃ be a finite flat OK -algebra.
Then we have a natural bijection
je(K ′ /K)jFK(B̃) ≃ FK ′(B̃ ⊗OK OK ′ ).

For any ﬁnite Galois extension L/K with Galois group G, we deﬁne the j-th upper
ramiﬁcation subgroup Gj asj
Gj = Ker(G → Aut(FK(OL )))and GjK as their projective limit. In this case, the above deﬁnition of the boundednessof ramiﬁcation is equivalent to what is mentioned earlier. Moreover, for the residuallyperfect case, this subgroup GjK coincides with the one deﬁned before.
As we can see from the above deﬁnition, one of the advantages of Abbes-Saito’s
ramiﬁcation theory is that we can measure ramiﬁcation of ﬁnite ﬂat algebras. Thisadvantage had been used for a study of canonical subgroups of abelian varieties [1, 8,

8Shin Hattori

9, 18, 19]. To pass to ﬁnite ﬂat algebras to measure ramiﬁcation is also a key step inwhat follows.
§ 4.

Ramification of truncated discrete valuation rings

Since we have the higher dimensional ramiﬁcation theory of Abbes-Saito, it isnatural to expect that Theorem 2.2 can be generalized to the residually imperfect case.
mm
The point is to deﬁne an analogue of the category FE<jK over OK /(πK ) ≃ OF /(πF )to bridge between the sides of K and F . In other words, what we need is an intrinsicm) — truncateddeﬁnition of a ramiﬁcation theory of local rings similar to the ring OK /(πKdiscrete valuation rings.
Definition 4.1 ([5], Subsection 1.1, [11], Section 2).
We say that a ring A is atruncated discrete valuation ring if A is a local ring with nilpotent principal maximal
ideal. We refer to a generator of its principal maximal ideal as a uniformizer of A. A
truncated discrete valuation ring A is said to be of length m if the length of A as an
A-module is m.
A typical example of truncated discrete valuation rings of length m is the ringm
OK /(πK). Conversely, for any truncated discrete valuation ring A of length m, we canalways ﬁnd a complete discrete valuation ﬁeld K and a local surjection ι : OK → A
minducing an isomorphism OK /(πK) ≃ A [5, Subsection 1.1]. We refer to such a pair(K, ι) as a lift of A.
A notion of ﬁnite extension of A is deﬁned as follows:
Definition 4.2 ([11], Section 2).
Let A be a truncated discrete valuation ring oflength m. A ﬁnite ﬂat A-algebra B is said to be a ﬁnite extension of truncated discrete
valuation rings over A if B is a truncated discrete valuation ring for m ≥ 2, and B is aﬁeld if m = 1.
For any ﬁnite extension L/K of complete discrete valuation ﬁelds and any intemm
ger m ≥ 2, the OK /(πK)-algebra OL /(πK
) is a ﬁnite extension of truncated discrete
valuation rings in this sense. We have the following converse:
Theorem 4.3 ([11], Proposition 2.2).
Let A be a truncated discrete valuation
ring of length m and B/A a finite extension of truncated discrete valuation rings. Let(K, ι) be a lift of A. Then there exists a finite separable extension L/K with a cocartesian
diagram
/ OL
OK

A


/ B.

Ramification theory and perfectoid spaces — a survey

9Remark. Though the proof of [11, Proposition 2.2] for the claim that we can takea separable L/K has a gap, it can be easily ﬁxed. Note that in [10] we do not use theclaim, since [10, Lemma 4.10 (ii)] is enough for applications.
The idea of Hiranouchi-Taguchi [11] to attack a generalization of Theorem 2.2 isto deﬁne ramiﬁcation of any ﬁnite extension B/A of truncated discrete valuation rings
as ramiﬁcation of L/K as in Theorem 4.3. Namely, for any positive rational numberj ≤ m, they deﬁned that the ramiﬁcation of B/A is bounded by j if the ramiﬁcationof L/K is bounded by j. This notion is independent of the choice of L once we ﬁx a
lift (K, ι) of A, by [7, Lemma 1]. Then they also deﬁned a category FFP<j
A,(K,ι) of ﬁnite
extensions of A whose ramiﬁcation is bounded by j along the lift (K, ι) (in fact, they<jdenote it by FFP≤jA ) and showed that it is naturally equivalent to the category FEK .
If the category FFP<j
A,(K,ι) is independent of the choice of a lift (K, ι), then we canobtain an equivalence of categories generalizing Theorem 2.2 immediately. However,this independence had remained open.
§ 5.

Main theorems

For any complete discrete valuation ﬁeld K of residue characteristic p, we set e(K)to be the absolute ramiﬁcation index of K if K is of characteristic zero and an arbitrary
positive integer if K is of characteristic p. Then the main result of the author’s paper[10] is the following, which settles this problem for the case where pA = 0.
Theorem 5.1 ([10], Theorem 1.1 (i)).
Let L1 /K1 and L2 /K2 be finite extensions of complete discrete valuation fields of residue characteristic p. Let m be a positive
integer satisfying m ≤ mini e(Ki ). Suppose that we have compatible isomorphisms ofmmmm
rings OK1 /(πK) ≃ OK2 /(πK
) and OL1 /(πK) ≃ OL2 /(πK). Then, for any positive
12
12
rational number j ≤ m, the ramification of L1 /K1 is bounded by j if and only if theramification of L2 /K2 is bounded by j.
The assumption m ≤ mini e(Ki ), which means that what we are considering are
only the truncated discrete valuation rings killed by p, is crucial. This is because, tomprove the theorem, we ﬁrst lift A = OKi /(πK) to a complete discrete valuation ﬁeld Fiof equal characteristic using pA = 0, and then compare ramiﬁcation over F and Ki by
passing to perfectoid spaces. The author has no idea of how to drop this assumption,
while he thinks that the p-torsion case is the main case of interest, since it enables us
to switch mixed and equal characteristics.
We also have the following corollaries of Theorem 5.1.
Corollary 5.2.

10

Shin Hattori

1. ([10], Theorem 4.16) If pA = 0, then the category FFP<j
A,(K,ι) is independent of the
choice of a lift (K, ι) of A.
2. ([10], Corollary 4.18) Let K1 and K2 be complete discrete valuation fields, withresidue fields k1 and k2 of characteristic p, respectively. Let j be a positive rational
number satisfying j ≤ mini e(Ki ). Suppose that the fields k1 and k2 are isomorphic
to each other. Then there exists an equivalence of categories<jFE<jK1 ≃ FEK2 .

In particular, there exists an isomorphism of topological groups
GK1 /GjK1 ≃ GK2 /GjK2 .
3. ([10], Theorem 6.2) The functor of higher fields of norms [15] is compatible withramification in the sense of Abbes-Saito.
4. ([10], Theorem 7.2) Suppose char(K) = 0. Let V be a p-adic representation of GK
with finite local monodromy. Define the Artin conductor of V as∑j+j
Art(V ) =j dimQp (V GK /V GK ).
j∈Q>0

If Art(V ) < e(K), then Art(V ) is an integer.
Remark.
1. Corollary 5.2 (3) gives a totally diﬀerent proof of a theorem of Ohkubo [13, Theorem
3.42].
2. In the ramiﬁcation theory of Abbes-Saito, we also have another ramiﬁcation subgroup which generalizes the classical ramiﬁcation subgroup Gj−1
in our notation—Kj
the j-th log ramiﬁcation subgroup GK,log . Theorem 5.1 and Corollary 5.2 can begeneralized to the case of log ramiﬁcation under the slightly stronger assumption ofj ≤ min{e(Ki ) − 2}, except Corollary 5.2 (1) (see [10]).
The reason why we need the stronger assumption is as follows: For the log case, wee(L /K )mhave to compare the elements πLi i i /πKi mod πK
for i = 1, 2. For these two
i
m+1
m+1
elements to be equal, we need to have an isomorphism OK1 /(πK) ≃ OK2 /(πK)
12
and this forces us to have m+1 ≤ e(Ki ). Moreover, the log ramiﬁcation is deﬁned bycounting the number of connected components of an analytic variety whose deﬁning
equations include|(Xne(Li /Ki ) − πKi g)(x)| ≤ |πKi (x)|j+1

Ramification theory and perfectoid spaces — a survey

11with some polynomial g ∈ OKi [X], and this forces j + 1 ≤ m.
On the other hand, the reason why Corollary 5.2 (1) remains open for the log caseis that, if we try to prove a functoriality for any extension L′i /Li of a comparison
result of ramiﬁcation, we encounter an analytic variety whose deﬁning equations
includee(L′i /Ki )|(Xne(L′ /Li ) ′− πKi i′g )(x)| ≤ |πKi (x)|j+e(Li /Li ) .

The power on the right-hand side can be arbitrarily large and we cannot comparemramiﬁcation functorially through modulo πK
with a ﬁxed m.
i
3. Corollary 5.2 (4) is a part of a theorem of Xiao [20, Subsection 1.1, Theorem]. However, the log version of this corollary proves the integrality of the Swan conductorof V for a case which had not been known previously.

§ 6.

Sketch of the proof

In this section, we explain main ideas for the proof of Theorem 5.1. For the residually perfect case, the key point of the proof of Theorem 2.2 is to bridge between
ramiﬁcation theories of complete discrete valuation ﬁelds with possibly diﬀerent characteristics, by extracting the combinatorial objects of Newton polygons which recover
ramiﬁcation and then comparing these objects inside R2 . As we mentioned before, if
the residue ﬁeld k of a complete discrete valuation ﬁeld K is imperfect, then the integer
ring OL of a ﬁnite Galois extension L/K is not necessarily monogenic and we cannot
deﬁne the Newton polygon.
Tropical analytic geometry (see [14]) produces higher dimensional combinatorial
objects generalizing the Newton polygon. Using them, it may be possible to study
lower ramiﬁcation subgroups of Gal(L/K) whose deﬁnition involves the valuations ofdiﬀerences of common zeros of the deﬁning equations of OL over OK as in the classical case. However, in the non-monogenic case, upper ramiﬁcation subgroups do notnecessarily coincide with lower ramiﬁcation subgroups even after renumbering [3, Subsection 2.1, Example]. Thus it is unclear how to deﬁne a combinatorial object which
recovers the Abbes-Saito ramiﬁcation of L/K. The idea to bypass this diﬃculty is ause of perfectoid spaces [16] to bridge between ramiﬁcation theories of mixed and equalcharacteristics.
Let m be a positive integer and j a positive rational number satisfying j ≤ m.
Let A be a truncated discrete valuation ring of length m which is killed by p and B aﬁnite ﬂat A-algebra. Let (K, ι) and (F, ι) be two lifts of A. Suppose that we have a12Shin Hattori

cocartesian diagram
/Bo
O

B̃K
O

B̃O F

(6.1)
OK

ι

/Ao

OF

ι

with B̃K = OL and B̃F = OE for some ﬁnite separable extensions L/K and E/F .
What we have to show is that the ramiﬁcation of L/K is bounded by j if and only if
the ramiﬁcation of E/F is bounded by j. For this, it is enough to show the equality
j♯FK(B̃K ) = ♯FFj (B̃F ).

We identify the residue ﬁeld of A with those of K and F and denote it also by k.
Since pA = 0, we can choose a section k → A of the reduction map A → k. We ﬁx
once and for all such a k-algebra structure of A. Let π̄ be a uniformizer of A. Thenthe map k[[u]] → A sending u to π̄ deﬁnes a lift of A, and by Theorem 4.3, we can ﬁnd
another similar cocartesian diagram over k[[u]]. Thus, by comparing ramiﬁcation overk((u)) with those over K and F , we may assume F = k((u)) and ι(u) = π̄. Let π be a
uniformizer of K satisfying ι(π) = π̄. Let K̄ and C be as before.
For simplicity, we assume that K is of characteristic zero. Fix a Cohen ring C(k)of k. Then we can ﬁnd a local homomorphism C(k) → OK which makes the following
diagram commutative.
/ OK
C(k)

k

ι


/A

By ﬁxing such a local homomorphism, we consider K0 = Frac(C(k)) as a subﬁeld of
K. We also ﬁx a p-basis {b̄λ }λ∈Λ of k and its lift {bλ }λ∈Λ in C(k), and a system of
p-power roots (bλ,l )l≥0 of bλ in K̄ satisfying bλ,0 = bλ and bpλ,l+1 = bλ,l . Let K0′ be thecompletion of the discrete valuation ﬁeld∪

K0 (bλ,l ),λ,lwhich is naturally considered as a subﬁeld of C. Put K ′ = K0′ K, the composite ﬁeld
inside C. Then we have e(K ′ /K) = 1, OK ′ = OK ⊗C(k) OK0′ and the residue ﬁeld k ′ ofK0′ is the perfect closure of k in the residue ﬁeld k̄ of C.
Consider the k-algebra A as a C(k)-algebra by the map C(k) → k → A. Put′
A = A ⊗k k ′ = A ⊗OK OK ′ and similarly for B ′ . Then the lift ι : k[[u]] → A induces

Ramification theory and perfectoid spaces — a survey

13

a lift ι′ : k ′ [[u]] → A′ . Put F ′ = k ′ ((u)). Then we obtain the following cocartesian
diagram/ B′ oB̃K ⊗OK OK ′B̃F ⊗OF OF ′
O
O
O
OK ′

/ A′ oOF ′ .

By Lemma 3.2 and the equalities e(K ′ /K) = e(F ′ /F ) = 1, it is enough to show theequalityjj♯FK′ (B̃K ⊗OK OK ′ ) = ♯FF ′ (B̃F ⊗OF OF ′ ).
Note that the residue ﬁeld k ′ of both K ′ and F ′ is perfect, while the rings B̃K ⊗OK OK ′and B̃F ⊗OF OF ′ are not necessarily normal anymore. Namely, by this base changing
argument, we may assume that the residue ﬁeld k is perfect, at the cost of assumingthat B̃K and B̃F are just finite flat algebras over OK and OF .
Next we choose the completion C♭ of an algebraic closure of F as follows. Deﬁnethe ring OC♭ as the inverse limit ringOC♭ = lim(OC /(π m ) ← OC /(π m ) ← · · · ),←−where all the transition maps are given by x 7→ xp . Here we consider the leftmost entry
in the limit as the zeroth entry, and write any element x of OC♭ as x = (x0 , x1 , . . .). We
can see that the reduction modulo π m induces a bijectionlim(OC ← OC ← · · · ) → lim(OC /(π m ) ← OC /(π m ) ← · · · ),←−←−where all the transition maps in the former limit are also given by x 7→ xp . Composing
this bijection with the zeroth projection, we obtain a natural multiplicative map (·)♯ :OC♭ → OC , which can be written explicitly as follows: for x = (x0 , x1 , . . .) ∈ OC♭ , takel
any lift x̂l of xl in OC and put x♯ = liml→∞ x̂pl . This map extends to a multiplicative
map C♭ = Frac(OC♭ ) → C. The ﬁeld C♭ is an algebraically closed complete valuationﬁeld whose integer ring is OC♭ and additive valuation is vC♭ (x) = vK (x♯ ).
p
We ﬁx a system (πl )l≥0 of p-power roots of π satisfying π0 = π and πl+1
= πl . This♯
gives the element π = (π0 , π1 , . . .) ∈ OC♭ with π = π. Since we are assuming that k
is perfect, the residue ﬁeld of the maximal unramiﬁed extension K nr of K in K̄ is k̄,and we have a unique multiplicative section [·] : k̄ → OK nr of the reduction map. Using
this, we consider OC♭ as a k̄-algebra by the map
2

x 7→ ([x], [x1/p ], [x1/p ], . . .).
We also consider OC♭ as an OF -algebra by this map and u 7→ π. Then we see that theﬁeld C♭ is the completion of an algebraic closure of F . Moreover, we have a commutative

14

Shin Hattori

diagram of k-algebras
/ OC♭

OF
ι


A

(6.2)

pr0

ι−1 ≀


/ OK̄ /(π m ).

OK /(π m )Let us return to the diagram (6.1) with ﬁnite ﬂat algebras B̃K and B̃F , as weare assuming. Let Z̄ = (z̄1 , . . . , z̄n ) be a ﬁnite system of generators of the A-algebra
B. Consider the surjection A[X] → B deﬁned by Xi 7→ z̄i , and write its kernel asI¯ = (f¯1 , . . . , f¯r ). Let fi be a lift of f¯i to the ring OK [X] by the map ι : OK → A, andlet fi be its lift to the ring OF [X] by the map ι : OF → A. Let zi be a lift of z̄i to B̃K .
Then the set Z = (z1 , . . . , zn ) is a ﬁnite system of generators of the OK -algebra B̃K .
Let I be the kernel of the surjection OK [X] → B̃K deﬁned by Z. Then we haveI + π m OK [X] = (f1 , . . . , fr ) + π m OK [X].
Since j ≤ m, the rational subset XCj,ad (B̃K , Z) is equal to
XCj,ad (B, Z̄) = {x ∈ XCad | |fi (x)| ≤ |π(x)|j for any i}.
Similarly, the rational subset we have to consider on the side of F is equal toadj
XCj,adfor any i}.
♭ (B, Z̄) = {x ∈ XC♭ | |fi (x)| ≤ |π(x)|To compare the sets of connected components of these two adic spaces over C and
C , we pass to perfectoid spaces [16]. Put
♭

1/pl

l

OC [X 1/p ] = OC [X1l, . . . , Xn1/p ],

∞

1/pl

OC [X 1/p ] = lim OC [X1−→l, . . . , Xn1/p ]

l∈Z≥0

∞

l

and denote their π-adic completions by OC ⟨X 1/p ⟩ and OC ⟨X 1/p ⟩. We also putllC⟨X 1/p ⟩ = OC ⟨X 1/p ⟩[1/π],

∞

∞

l

∞

C⟨X 1/p ⟩ = OC ⟨X 1/p ⟩[1/π].
∞l

We deﬁne the rings OC♭ ⟨X 1/p ⟩, OC♭ ⟨X 1/p ⟩, C♭ ⟨X 1/p ⟩ and C♭ ⟨X 1/p ⟩ on the side of∞
F similarly, using π instead of π. Then the ring C⟨X 1/p ⟩ is a perfectoid C-algebra with∞tilt C♭ ⟨X 1/p ⟩ [16, Deﬁnition 5.1, Proposition 5.20]. In particular, we have a canonicalring isomorphism∞∞∞OC♭ ⟨X 1/p ⟩ ≃ lim(OC ⟨X 1/p ⟩/(π m ) ← OC ⟨X 1/p ⟩/(π m ) ← · · · ),←−

15

Ramification theory and perfectoid spaces — a survey

where all the transition maps are x 7→ xp , such that the composite with the zeroth
projection is equal to∞∞∞OC♭ ⟨X 1/p ⟩ → OC♭ ⟨X 1/p ⟩/(π m ) ≃ OC ⟨X 1/p ⟩/(π m ),1/pl

where the right isomorphism is deﬁned by pr0 : OC♭ /(π m ) ≃ OC /(π m ) and Xi

7→

1/pl
Xi
[10, Lemma 3.2]. Moreover, the reduction modulo π m induces a bijection∞∞lim(OC ⟨X 1/p ⟩ ← OC ⟨X 1/p ⟩ ← · · · )←−∞∞→ lim(OC ⟨X 1/p ⟩/(π m ) ← OC ⟨X 1/p ⟩/(π m ) ← · · · ),←−as in the case of OC and OC♭ . Composing these bijections with the zeroth projection,
we obtain a continuous multiplicative map∞∞(·)♯ : C♭ ⟨X 1/p ⟩ → C⟨X 1/p ⟩[16, Proposition 5.17]. From the commutative diagram (6.2), we also obtain the congruencefi♯ ≡ fi mod π m .

(6.3)
Put
l

l

ad
XC,l
= Spa(C⟨X 1/p ⟩, OC ⟨X 1/p ⟩),

∞

∞

ad
XC,∞
= Spa(C⟨X 1/p ⟩, OC ⟨X 1/p ⟩)and similarly for XCad♭ ,l , XCad♭ ,∞ . By [16, Theorem 6.3], we have a homeomorphismadτ : XC,∞→ XCad♭ ,∞∞addeﬁned by |f (τ (x))| = |f ♯ (x)| for any x ∈ XC,∞
and any f ∈ C♭ ⟨X 1/p ⟩.
j,ad
Now we denote by XC,∞ (B, Z̄) the inverse image of the rational subset XCj,ad (B, Z̄) ⊆ad→ XCad . This is the rational subset
XCad by the natural projection p∞,0 : XC,∞ad
{x ∈ XC,∞| |fi (x)| ≤ |π(x)|j for any i}ad. Similarly, we deﬁne XCj,adof the adic space XC,∞
♭ ,∞ (B, Z̄) to be the inverse image ofad
ad♭
XCj,ad♭ (B, Z̄) by the projection p∞,0 : XC♭ ,∞ → XC♭ . This is equal to{x ∈ XCad♭ ,∞ | |fi (x)| ≤ |π(x)|j for any i}.
Thus we have a commutative diagram
j,ad/ X ad(B, Z̄) 
XC,∞
C,∞



XCj,ad (B, Z̄) τ∼/ X ad♭

C ,∞

p∞,0


/ X ad
C



o

p♭∞,0

XCad♭ o? _ X j,ad
(B, Z̄)♭C ,∞
? _ X j,ad
♭ (B, Z̄)C

whose two squares are cartesian. Then Theorem 5.1 follows from the proposition below:

16

Shin Hattori

Proposition 6.1.
j,ad
1. ([10], Lemma 3.5) τ −1 (XCj,ad♭ ,∞ (B, Z̄)) = XC,∞ (B, Z̄).

2. ([10], Lemma 3.6) The projections p∞,0 and p♭∞,0 induce bijectionsj,ad
π0 (XCj,ad♭ ,∞ (B, Z̄)) → π0 (XC♭ (B, Z̄)).

j,ad
π0 (XC,∞(B, Z̄)) → π0 (XCj,ad (B, Z̄)),

Outline of proof. The ﬁrst part of the proposition follows from the congruence(6.3) and the assumption j ≤ m, since|fi (x)| ≤ |π(x)|j ⇔ |fi♯ (x)| ≤ |π ♯ (x)|j = |π(x)|j .
For the second part, the assertion on p♭∞,0 follows from the fact that this projection isa homeomorphism. Consider the assertion on p∞,0 . By a limit argument, it is enoughadto show a similar assertion for the projection pl,0 : XC,l
→ XCad for any l ≥ 0.
j,ad
j,ad
Put XC,l(B, Z̄) = p−1
l,0 (XC (B, Z̄)). Note that the map pl,0 is surjective, andalso open since it is ﬂat and ﬁnitely presented. It is enough to show that for any
connected component C of XCj,ad (B, Z̄), its inverse image p−1l,0 (C) is connected. Sincej,adthe adic spaces XCj,ad (B, Z̄) and XC,l
(B, Z̄) are locally of ﬁnite type, the numbersof their connected components are ﬁnite and the components are open. The inversej,adimage p−1l,0 (C) is the disjoint union of some connected components of XC,l (B, Z̄). If it
had more than one connected components, then the intersection of their images in C
would be a nonempty open subset and thus contain a classical point (namely, a pointof XCad deﬁned by the valuation f 7→ p−vK (f (x)) for some closed point x ∈ Spec(C⟨X⟩)).
Hence it suﬃces to show that, for any classical point x in XCj,ad (B, Z̄) deﬁned by the
map Xi 7→ xi with xi ∈ OC , any two points y, y ′ ∈ p−1
l,0 (x) are contained in the samej,adconnected component of XC,l
(B, Z̄).
Consider the polydisc

U = {z ∈ XCad | |(Xi − xi )(z)| ≤ |π(z)|m for any i}.
Since j ≤ m, this polydisc is an open neighborhood of x contained in XCj,ad (B, Z̄). Its
inverse image p−1l,0 (U ) is equal toad{z ∈ XC,l| |(Xi − xi )(z)| ≤ |π(z)|m for any i}.

Since m ≤ e(K), we have
1/pl|(Xi − xi )(z)| ≤ |π(z)|m ⇔ |(Xi

1/pl

− xi

l

)(z)| ≤ |π(z)|m/p .

Hence, the inverse image p−1l,0 (U ) is also a polydisc, thus connected, which is containedj,adin XC,l(B, Z̄) and contains both y and y ′ . This concludes the proof.

Ramification theory and perfectoid spaces — a survey

17

References[1] A. Abbes and A. Mokrane: Sous-groupes canoniques et cycles évanescents p-adiques pourles variétés abéliennes, Publ. Math. Inst. Hautes Etudes Sci. 99 (2004), 117–162.
[2] A. Abbes and T. Saito: Ramiﬁcation of local ﬁelds with imperfect residue ﬁelds I, Amer.
J. Math. 124 (2002), 879–920.
[3] A. Abbes and T. Saito: Ramiﬁcation of local ﬁelds with imperfect residue ﬁelds II, Kazuya
Kato’s ﬁftieth birthday. Doc. Math. 2003, Extra Vol., 5–72 (electronic).
[4] A.-M. Aubert, P. Baum, R. Plymen and M. Solleveld: The local Langlands correspondence
for inner forms of SLn , preprint, arXiv:1305.2638.
[5] P. Deligne: Les corps locaux de caractéristique p, limites de corps locaux de caractéristique0, Representations of reductive groups over a local ﬁeld, 119–157, Travaux en Cours,
Hermann, Paris, 1984.
[6] R. Ganapathy: The local Langlands correspondence for GSp4 over local function ﬁelds,
preprint, arXiv:1305.6088.
[7] S. Hattori: Ramiﬁcation of a ﬁnite ﬂat group scheme over a local ﬁeld, J. Number Theory
118 (2006), no. 2, 145–154.
[8] S. Hattori: Canonical subgroups via Breuil-Kisin modules, Math. Z. 274 (2013), no. 3–4,
933–953.
[9] S. Hattori: Canonical subgroups via Breuil-Kisin modules for p = 2, J. Number Theory
137 (2014) 142–159.
[10] S. Hattori: Ramiﬁcation theory and perfectoid spaces, Compos. Math. 150 (2014), no. 5,
798–834.
[11] T. Hiranouchi and Y. Taguchi: Extensions of truncated discrete valuation rings, Pure
Appl. Math. Q. 4 (2008), no. 4, part 1, 1205–1214.
[12] R. Huber: Continuous valuations, Math. Z. 212 (1993), no. 3, 455–477.
[13] S. Ohkubo: On diﬀerential modules associated to de Rham representations in the imperfectresidue ﬁeld case, preprint, arXiv:1307.8110v2.
[14] J. Rabinoﬀ: Tropical analytic geometry, Newton polygons, and tropical intersections, Adv.
Math. 229 (2012), no. 6, 3192–3255.
[15] A. J. Scholl: Higher ﬁelds of norms and (ϕ, Γ)-modules, Doc. Math. (2006) Extra Vol.,
685–709 (electronic).
[16] P. Scholze: Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), no. 1,
245–313.
[17] J.-P. Serre: Corps Locaux, Publications de l’Université de Nancago, No. VIII. Hermann,
Paris, 1968.
[18] Y. Tian: Canonical subgroups of Barsotti-Tate groups, Ann. of Math. (2) 172 (2010), no.
2, 955–988.
[19] Y. Tian: An upper bound on the Abbes-Saito ﬁltration for ﬁnite ﬂat group schemes andapplications, Algebra Number Theory 6 (2012), no. 2, 231–242.
[20] L. Xiao: On ramiﬁcation ﬁltrations and p-adic diﬀerential equations II: mixed characteristic case, Compos. Math. 148 (2012), no. 2, 415–463.