INTEGRAL p-ADIC HODGE THEORY AND
RAMIFICATION OF CRYSTALLINE REPRESENTATIONSSHIN HATTORI
Abstract. This is a survey on integral p-adic Hodge theory, especiallyon the Fontaine-Laffaille theory, and a ramification bound for crystalline
representations due to Abrashkin and Fontaine, based on the author’s
talks at the spring school “Classical and p-adic Hodge theories”.

Contents
1. Introduction
2. Fontaine-Laffaille modules
2.1. Definition of Fontaine-Laffaille modules
2.2. Associated Galois representations
2.3. Relation to crystalline cohomology
2.4. Ramification of torsion crystalline representations
2.5. Classification of finite flat group schemes
3. Further developments
3.1. Applications and generalizations of the Fontaine-Laffailletheory3.2. Wach modules
3.3. Breuil modules
3.4. Breuil-Kisin classification of finite flat group schemes
3.5. Kisin modules and (ϕ, Ĝ)-modules
3.6. Ramification theory
4. Sketch of proofs
4.1. Cutting off higher divided powers
4.2. Classification of simple objects∗
4.3. The fullness of Tcrys
4.4. Ramification bound
References

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Date: May 9, 2019.
The former title of the article was “Ramification of crystalline representations” which
was also the title of the author’s talks at the spring school.
1

2

SHIN HATTORI

1. Introduction
Let p be a rational prime, k a perfect field of characteristic p and W =W (k) the Witt ring for k. Put Wn = W/pn W for any n ∈ Z>0 . We write
σ for the p-th power Frobenius maps of various p-torsion rings, and also
for the natural lift (x0 , x1 , . . .) 7→ (xp0 , xp1 , . . .) to W of the Frobenius map
on k. Put K0 = Frac(W ). Let K be a finite totally ramified extension of
K0 of degree e with integer ring OK , maximal ideal mK and uniformizer
πK . We fix an algebraic closure K̄ of K and let k̄ be the residue fieldof K̄, which is an algebraic closure of k. Let vp and vK be the p-adic
valuations on K̄ normalized as vp (p) = 1 and vK (πK ) = 1, respectively.
Put GK = Gal(K̄/K). Let C be the completion of K̄ and OC its integer
ring. Let K ur be the maximal unramified extension of K inside K̄ and
IK = Gal(K̄/K ur ) the inertia subgroup of GK . For any extension of rings
A ⊆ B and any free B-module M of rank d, we say an A-submodule N of
M is an A-lattice of M if it is a free A-module of rank d which generates
the B-module M .
A p-adic GK -representation is by definition a finite dimensional Qp -vector
space with a continuous Qp -linear GK -action for the p-adic topology. On theother hand, we also have the category of Zp -modules with a continuous Zp linear GK -action for the p-adic topology, which is denoted by RepZ (GK ).
p
It is this “integral” variant of the category of p-adic GK -representations that
we are concerned with in this article.
Recall that p-adic Hodge theory has the following two aspects:(i) to study, or classify if possible, nice p-adic GK -representations (suchas Hodge-Tate, de Rham, semi-stable and crystalline representations), using semi-linear algebraic data (such as filtered (ϕ, N )modules).
(ii) to study a relation between p-adic etale cohomology of algebraic
varieties over K and other cohomology theories on the differentialside, such as de Rham, log-crystalline and crystalline cohomology.
Integral p-adic Hodge theory is its variant of Zp - or p-power torsion coefficients. Namely, its aim can be summarized as:(i) to study, or classify if possible, Zp -lattices and subquotients of nicep-adic GK -representations, using semi-linear algebraic data which
are in a sense lattices and subquotients of the data attached top-adic GK -representations (such as filtered (ϕ, N )-modules).
(ii) to study a relation between etale cohomology of Zp - or p-power torsion coefficients of algebraic varieties over K and other cohomology
theories on the differential side, such as de Rham, log-crystalline
and crystalline cohomology.
Example 1.1.
(i) Let A be an Abelian variety over K of dimension
g. Let Tp (A) = limn A[pn ](K̄) be the Tate module of A and put←−
Vp (A) = Tp (A) ⊗Zp Qp . The module Tp (A) is a free Zp -module ofINTEGRAL p-ADIC HODGE THEORY

3

rank 2g on which GK acts continuously (for the p-adic topology).
Similarly, the module Vp (A) is a p-adic GK -representation of dimension 2g, and moreover it is de Rham with Hodge-Tate weights in[0, 1]. The module Tp (A) is a GK -stable Zp -lattice in Vp (A), and the
GK -module A[pn ](K̄) is isomorphic to the quotient Tp (A)/pn Tp (A).
These subquotients of Vp (A) are typical examples of what integral
p-adic Hodge theory studies on the aspect (i).
More generally, let V be a de Rham GK -representation. Thenwe want to study any GK -stable Zp -lattice T in V and the quotient
T /T 0 for any such lattices T ⊇ T 0 in V .
(ii) Let X be a proper smooth scheme over K. Then the etale cohomoli (X , Z ) and H i (X , Z/pn Z) are what we want toogy groups Het
p
et
K̄
K̄
study on the aspect (ii).
The integral theory captures more subtle information than the p-adic
counterpart, such as ramification of Galois representations. We can passfrom the theory of Zp -coefficients both to that of Qp -coefficients by invertingp, and to that of Fp -coefficients by reducing modulo p. For example, let V
be a semi-stable GK -representation with associated filtered (ϕ, N )-module
D and T a GK -stable Zp -lattice in V . By integral p-adic Hodge theory,
we can associate with T a semilinear algebraic data M . The data M is a
finitely generated module over a coefficient ring which is a W -algebra, andM is endowed with additional structures (such as a Frobenius map). We canrecover the filtered (ϕ, N )-module D via M ⊗W K0 , and the data M/pM
has information of the GK -representation T /pT . In some cases M is defined
as a W -lattice in D as we will see in §2, while it is not the case in general.
Downsides are the following: In the integral case, the coefficient ring ofsemi-linear algebraic data is not a field, unlike the coefficient field K0 of filtered (ϕ, N )-modules. Actually, it is not even a complete discrete valuation
ring but a larger W -algebra, unless the base field K is absolutely unramified(namely, K = K0 ), and the theory becomes more complicated compared tothe p-adic theory. Moreover, it is often very hard to describe explicitly the
semilinear algebraic data associated with a given integral GK -representation.
In this article, we will focus on the classical integral theory of FontaineLaffaille [FL82], which treats the subquotients of crystalline GK -representations
with Hodge-Tate weights in [0, p − 1] for the case where K is absolutely unramified. We introduce basic definitions and state main theorems of thetheory in §2. We also give a sketch of proofs for some of the main theorems(Theorem 2.9 and Theorem 2.19) in §4. In fact, now we have an integral theory without these three restrictions (crystalline representations, small rangeof Hodge-Tate weights, absolutely unramified base field) thanks to works ofmany people including Breuil, Kisin and Liu. We will discuss such recent
developments of the integral theory in §3.
The author would like to thank Victor Abrashkin, Yoshiyasu Ozeki, Takeshi
Saito and Yuichiro Taguchi for helpful comments on an earlier draft of the4

SHIN HATTORI

article. He also would like to thank the anonymous referee for comments
and suggestions, which helped improving the article. The interested reader isrecommended also to consult other survey articles [Bre02, BMs02, BC14+],
to which this article owes very much.
2. Fontaine-Laffaille modules
Let the notation be as in §1. Let V be a p-adic GK -representation and∗ (V ) = Hom
D = Dcrys
Qp [GK ] (V, Bcrys ) the associated filtered ϕ-module. The
module D is a K0 -vector space satisfying dimK0 (D) ≤ dimQp (V ). We say
V is crystalline if the equality holds. Put DK = D ⊗K0 K. The Hodge-Tate• 6= 0. Note that here we adopt theweights are the integers i such that gri DK
convention such that the cyclotomic character Qp (1) has the Hodge-Tateweight one.
Remark 2.1. This dual convention to the usual Dcrys (V ) = (V ⊗Qp Bcrys )GK
is preferred in integral p-adic Hodge theory. A reason for this dual preference∗ (V ) is a Hom group, and so isseems to be the following: The module Dcrys∗ (D) = Homits quasi-inverse Vcrys
K0 ,ϕ,Fil (D, Bcrys ). In the integral theory, we
use a similar Hom group to construct Galois representations from semilinear
algebraic data, as we will see later. One of the advantages of Hom and Ext
compared to ⊗ and Tor is that we can study them in a more explicit way, byusing the Yoneda extension. This makes the analysis of the associated Galois
representations easier especially in the integral theory, where the coefficientring of semilinear algebraic data is not a field and we cannot always resort
to dimension calculation or flatness. However, in many cases we can develop
covariant integral theories and they are also useful (for example, see §2.3).
Now we concentrate on the case of K = K0 . Let V be a crystalline GK0 ∗ (V ).
representation with Hodge-Tate weights in [0, p − 1] and put D = Dcrys
The module D is a K0 -vector space of dimension dimQp (V ) endowed witha decreasing filtration {Di }i∈Z and a σ-semilinear bijection ϕ : D → D
satisfying the admissibility condition.
The idea of the integral p-adic Hodge theory of Fontaine-Laffaille is to classify GK0 -stable Zp -lattices in V and their quotients by ϕ-stable W -lattices
M in D and their quotients. We have to impose on M an integral version ofadmissibility;
namely, the conditions that ϕ(M ∩ Di ) ⊆ pi M for any i andP−ithat i∈Z p ϕ(M ∩ Di ) = M . In fact, these conditions originated from astructure of integral crystalline cohomology [Fon83, 1.3, Proposition]. Thisidea leads to the following definitions.
2.1. Definition of Fontaine-Laffaille modules.
Definition 2.2.
(i) A Fontaine-Laffaille module over W is a triplet(M, {M i }i∈Z , {ϕiM }i∈Z ), which we denote abusively by M , consisting of• a W -module M ,INTEGRAL p-ADIC HODGE THEORY

5• a decreasingSfiltration {MTi }i∈Z of M by W -submodules satisfying M = i∈Z M i and i∈Z M i = 0,
• a family of σ-semilinear maps {ϕiM : M i → M }i∈Z which makes
the diagram
M i+1
_

ϕi+1
M



Mi

/M


ϕiM

×p

/M

commutative for any i. We often drop the subscript M in ϕiMif no confusion occurs.
(ii) A morphism of Fontaine-Laffaille modules f : M → N over W isa W -linear map f : M → N between the underlying W -modules
satisfying f (M i ) ⊆ N i and f ◦ ϕiM = ϕiN ◦ f for any i.
(iii) A sequence of Fontaine-Laffaille modules 0 → L → M → N → 0 isexact if the sequence on the underlying W -modules 0 → L → M →N → 0 is exact, and also the sequence induced on the i-th filtration
0 → Li → M i → N i → 0 is exact for any i.
We denote the category of Fontaine-Laffaille modules over W by MFW .
We also denote by MFk its full subcategory consisting of those killed by
p. With the above notion of exact sequence, the categories MFW and MFk
are exact categories in the sense of Quillen [Qui73]. Thus we can define
Extn groups via the Yoneda extension and we have long exact sequences of
Hom and Ext for these categories. Though Yoneda Extn groups in an exact
category may be a proper class in general, we can see that Ext1MFW (M, N )and Ext1MFk (M, N ) are sets, since any equivalence class of such an extension
is represented by a Fontaine-Laffaille module whose underlying set is N ×M .
For any W -module M , put Mσ = M ⊗W,σ−1 W . We often identify theunderlying module of Mσ with that of M and think that the action of W on
Mσ is twisted by σ from that on M . We also identify the data {ϕiM }i∈Z in
the above definition with a family of W -linear maps {ϕiM : M i → Mσ }i∈Z
making the diagram there commutative. Note that if M is of finite length,
then so is Mσ and lgW (M ) = lgW (Mσ ).
We define a category of torsion Fontaine-Laffaille modules satisfying atorsion version of the admissibility condition, as follows.
[a,b]Definition 2.3.
(i) For any integers a ≤ b, we denote by MFW thefull subcategory of MFW consisting of M such that M a = M andM b+1 = 0.
(ii) We denote by MFfW,tor the full subcategory of MFW consisting of
M such that• Mof finite length,
P is a iW -modulei) = M .
•
ϕ
(M
σ
i∈Z

6

SHIN HATTORI[a,b]

The full subcategory of MFW
f,[a,b]denoted by MFW,tor .

defined by the same conditions isf,[0,p−1]0f,[0,p−1](iii) We denote by MFW,tor
the full subcategory of MFW,tor consisting of M such thatf,[0,p−1]• There exists no non-zero quotient M → N in MFW,tor satisfying N = N p−1 .
0This category is denoted in [FL82, Théorème 6.1] by MFf,ptor . Notef,[0,p−2]
f,[0,p−1]0
that MFW,tor ⊆ MFW,tor .
[a,b](iv) Similar full subcategories of MFk are denoted by MFk
f,[a,b]f,[0,p−1]0
MFk
and MFk.

, MFfk ,Finally, we define the category of free objects over W , which classifies
GK0 -stable Zp -lattices in crystalline GK0 -representations.
fd,[a,b]Definition 2.4. For any integers a ≤ b, we denote by MFW
the full[a,b]subcategory of MFW consisting of M such that
• M is a free W -module of finite rank,• P
M i is a direct summand of the W -module M for any i,ii•i∈Z ϕ (M ) = Mσ .
The sign fd stands for the French word fortement divisible, or strongly divisible (Definition 2.10).
f,[a,b]What is striking for the category MFW,tor is the following.
Proposition 2.5 ([FL82], 1.10 (b)). Any morphism f : M → N of thef,[a,b]category MFW,tor is strict with filtrations. Namely, we havef (M i ) = f (M ) ∩ N i
for any i.
[a,b]Proof. First let M be any object of MFW and define a W -module M̄ by
the exact sequence
Lb
Lbl θM /
l
/
/ M̄
/ 0,
0
l=a+1 M
l=a M
where θM is given by(xa+1 , . . . , xb ) 7→ (xa+1 , −pxa+1 + xa+2 , . . . , −pxb−1 + xb , −pxb ).
Then the mapbM
l=a

M l → Mσ ; (ya , . . . , yb ) 7→bXϕl (yl )l=ainduces a W -linear map ϕM : M̄ → Mσ . These constructions are functorial,
and the functor M 7→ M̄ is exact by the snake lemma. If the underlyingINTEGRAL p-ADIC HODGE THEORY

7

W -module M is of finite length, then we have the equalitylgW (M ) = lgW (M̄ ).
f,[a,b]Moreover, such M is contained in MFW,tor if and only if ϕM is surjective,
and from the above length equality, it is equivalent to saying that ϕM is
injective or bijective.
To prove the proposition, first we assume that f is injective, and identify
M with a submodule of N . By the commutative diagram
/ N̄

M̄
o



Mσ





o

/ Nσ

the induced map M̄ → N̄ is injective (note that here we cannot use theexactness of M 7→ M̄ , since the cokernel object N/M is well-defined only if
M l = M ∩ N l for any l). On the other hand, suppose that there exists anelement x ∈ (N i ∩ M ) \ M i . Take j < i such that x ∈ M j \ M j+1 . Since M
is p-power torsion, we have ps x ∈/ M j+1 and ps+1 x ∈ M j+1 for some integerss ≥ 0. Replacing x by p x, we may assume px ∈ M j+1 . Put
y = (0, . . . , 0, x, −px, 0, . . . , 0) ∈

b
M

M l,

l=a

where x is at the j-th entry. Then y ∈/ Im(θM ) since x ∈/ M j+1 , whileij+1y ∈ Im(θN ) since x ∈ N ⊆ N. Thus the image of y in M̄ is non-zero,
while the image in N̄ is zero. This contradicts the injectivity of M̄ → N̄ .
For a general f , put L = f (M ) and Li = f (M i ). Since Ker(M i →iL ) = Ker(f ) ∩ M i and f ◦ ϕiM = ϕiN ◦ f , the map ϕiM induces a map
ϕiL : Li → L for any i. Then the triplet L = (L, {Li }i∈Z , {ϕiL }i∈Z ) is an[a,b]object of MFW whose underlying W -module is of finite length, and thenatural maps M → L → N are morphisms of this category. Now we havethe commutative diagram
/ L̄
M̄
o



Mσ


/ / Lσ ,which yields the surjectivity of the map L̄ → Lσ . This implies that L ∈f,[a,b]
MFW,tor . Applying the former half of the proof to the injection L → N , weobtain f (M i ) = f (M ) ∩ N i .
f,[a,b]Corollary 2.6.
(i) ([FL82], Proposition 1.8) The category MFW,tor is
Abelian.
f,[a,b](ii) ([Win84], Proposition 1.4.1 (ii)) For any object M ∈ MFW,tor , theW -submodule M i ⊆ M is a direct summand for any i.

8SHIN HATTORI
f,[a,b]

Proof. For the first assertion, let f : M → N be a morphism of MFW,tor andput Ker(f )i = Ker(f ) ∩ M i . Then, by using the exact sequence/ Ker(f )

0

/M

/L

/0

with L as in the proof of Proposition 2.5, we can show that the triplet(Ker(f ), {Ker(f )i }i∈Z , {ϕiM |Ker(f )i }i∈Z )f,[a,b]is an object of MFW,tor , just as in the proof of the latter part of Proposition
2.5. Moreover, since the map ϕiN induces a map ϕ̄iN : N i /f (M i ) → N/f (M ),Proposition 2.5 implies that the Fontaine-Laffaille module(Coker(f ), {N i /f (M i )}i∈Z , {ϕ̄iN }i∈Z )f,[a,b]is well-defined, and it is an object of MFW,tor similarly. These construcf,[a,b]tions give the kernel and cokernel of f in MFW,tor . To show this category
is Abelian amounts to showing that the filtrations {f (M i )}i∈Z (coimage filtration) and {f (M ) ∩ N i }i∈Z (image filtration) on f (M ) coincide, which is
Proposition 2.5.
On the other hand, applying Proposition 2.5 to the map pn : M → M
yields pn M i = M i ∩ pn M for any n, which implies the second assertion by
an Ext calculation.

2.2. Associated Galois representations. In this subsection, we will atf,[0,p−1]∗ (M ) in a similar way
tach to any object M ∈ MFW,tor a GK0 -module Tcrys∗
to Vcrys (D), using Acrys instead of Bcrys . Recall that we have the projective
limit ringR = lim(OC /pOC ← OC /pOC ← · · · ),←−where every transition map is the p-th power map. For any element x =l(x0 , x1 , . . .) ∈ R, take any lift x̂l ∈ OC of xl and put x(m) = liml→∞ x̂pm+l ∈OC , which is independent of the choice of lifts. Then the ring R is a complete valuation ring of characteristic p with algebraically closed fraction field
whose valuation is given by vR (x) = vp (x(0) ) for any element x ∈ R. Thisring admits a natural action of GK0 induced by the action on each entry.
Moreover, using the natural inclusion W (k̄) → OC , we consider R as a k2
algebra by the map k → R given by a 7→ ([a], [a1/p ], [a1/p ], . . .). Then the
Witt ring W (R) has a natural W -algebra structure.
We also have a natural W -linear, GK0 -equivariant surjection θ : W (R) →Pl (l)OC sending (r0 , r1 , . . .) ∈ W (R) to ∞l=0 p rl . Choose a system of p-powerl+1
lroots of −p in OC satisfying ((−p)1/p )p = (−p)1/p for any l. Put
2

β = (−p, (−p)1/p , (−p)1/p , · · · ) ∈ R
and ξ = p + [β] ∈ W (R). Then we have Ker(θ) = (ξ). The ring Acrys is bydefinitionξnAcrys = W (R)[ | n ∈ Z≥0 ]∧ ,n!

INTEGRAL p-ADIC HODGE THEORY

9where ∧ means the p-adic completion.
The natural GK0 -action on R induces a GK0 -action on Acrys . Other additional structures on Acrys are defined as follows: The p-th power map R → R
induces a natural map ϕ : W (R) → W (R) by (r0 , r1 , . . .) 7→ (r0p , r1p , . . .) andit extends to a ring endomorphism ϕ : Acrys → Acrys . Note thatp−1  X
1 p l
ξpϕ(ξ) = p + [β p ] = p + (ξ − p)p = p(1 + (−1)p pp−1 +ξ (−p)p−l + )
p l
p
l=1

ξp
+ p(· · · )).
= p(1 +p
Let Fili Acrys be the closure of the ideal of Acrys generated by ξ n /n! for any
n ≥ i. Then we have ϕ(Fili Acrys ) ⊆ pi Acrys for i = 0, 1, . . . , p − 1. Put
ϕi = p−i ϕ|Fili Acrys for such i. The above equality impliesξp
mod pAcrys .
p
All these structures are compatible with the GK0 -action. By putting −i(i < 0) Acrys p ϕ0 (i < 0)ϕi(0 ≤ i ≤ p − 1)
Aicrys =Fili Acrys (0 ≤ i ≤ p − 1) and ϕiAcrys =0(i ≥ p)0(i ≥ p)(1)

ϕ1 (ξ) = 1 +we consider the ring Acrys as a Fontaine-Laffaille module over W with anatural GK0 -action. On the other hand, since Acrys /Fili Acrys is p-torsion free
for any i, the natural map Fili Acrys /pn Fili Acrys → Acrys /pn Acrys is injective.
Put Fili (Acrys /pn Acrys ) = Fili Acrys /pn Fili Acrys . We also have the induced
map ϕi : Fili (Acrys /pn Acrys ) → Acrys /pn Acrys from ϕi : Fili Acrys → Acrys .
Using these induced structures, we consider Acrys /pn Acrys andAcrys,∞ = Acrys ⊗W (K0 /W ) ' lim Acrys /pn Acrys−→nas Fontaine-Laffaille modules over W with a natural GK0 -action similarly,
by setting i-th filtration to be zero for any i ≥ p.
f,[0,p−1]
Let M be an object of MFW,tor . We define∗
Tcrys(M ) = HomMFW (M, Acrys,∞ ).

This is a W -module with a GK0 -action induced by the natural action on
Acrys,∞ . This construction is functorial on M , and we obtain a functor
f,[0,p−1]∗
Tcrys: MFW,tor→ RepZ (GK0 ).
pRemark 2.7. In [FL82, Théorème 3.3], an associated GK0 -module with
M is defined as U (M ) = Ext1MFW (M, SFL ) by using the subring SFL =p

W (R)[ ξp ] of Acrys which they denote by S [FL82, Lemme 5.4] and it isidentified by [FL82, Lemme 3.8] with HomMFW (M, SFL,∞ ), where SFL,∞ =SFL ⊗W (K0 /W ). In fact, we can show that the natural map SFL,∞ →10SHIN HATTORI∗ (M ) (Remark
Acrys,∞ induces an isomorphism of GK0 -modules U (M ) ' Tcrys
4.13).

Note that M 7→ M ⊗W W (k̄) defines a functor
f,[0,p−1]

MFW,torf,[0,p−1]→ MFW (k̄),tor .

The effect of this base extension on the associated Galois representation is
the following. Let IK0 be the inertia subgroup of GK0 , which we identify
with the absolute Galois group of K̂0ur = Frac(W (k̄)). By using the natural∗ (M 0 )embedding W (k̄) → OC , we identify the associated IK0 -module Tcrys
f,[0,p−1]

of an object M 0 ∈ MFW (k̄),tor with
HomMFW (k̄) (M 0 , Acrys,∞ ).
From this we can see:f,[0,p−1]

Lemma 2.8 ([FL82], 3.11). For any object M ∈ MFW,tormap∗∗(M ⊗W W (k̄))
Tcrys(M )|IK0 → Tcrys
is an isomorphism of IK0 -modules.

, the natural

Now the first main theorem of this article is the following:∗
Theorem 2.9.
(i) ([FL82], Théorème 3.3 (ii)) The functor Tcrys
isexact and faithful.
f,[0,p−1](ii) ([FL82], Théorème 3.3 (i)) For any M ∈ MFW,tor , the Zp -module∗ (M ) is of finite length. Moreover, the invariant factors of the
Tcrys∗ (M ) and the W -module M are the same.
Zp -module Tcrys∗(iii) ([FL82], Théorème 6.1 (ii)) The restriction of Tcrys
to the full subf,[0,p−1]0

category MFW,tor

is full.
f,[0,p−1]0∗: MFW,tor(iv) The essential image of the restricted functor Tcrys
RepZ (GK0 ) is stable under subquotients.

→p

Next we consider the case where M is free over W . Let r be an integer
fd,[0,r]. Then the associatedsatisfying 0 ≤ r ≤ p − 1 and M an object of MFW
GK0 -representation with M is defined by∗T̂crys(M ) = HomMFW (M, Acrys ).
f,[0,r]

Note that M/pn M is an object of MFW,tor . From Theorem 2.9 (ii) and∗ (M ) is a free Z -module of rank
M ' limn M/pn M , we can show that T̂crysp←−equal to rankW (M ).
Moreover, put D = M ⊗W K0 , Di = M i ⊗W K0 and ϕ = ϕ0 ⊗ 1. These
define on D a structure of a filtered ϕ-module over K0 , which is admissible
by a theorem of Laffaille [Laf80, Théorème 3.2]. From the definition, we canshow that the natural map∗∗+
T̂crys(M ) ⊗Zp Qp → Vcrys(D) = HomK0 ,ϕ,Fil (D, Bcrys)

INTEGRAL p-ADIC HODGE THEORY

11

is an isomorphism of GK0 -representations.
Conversely, let us start from a crystalline GK0 -representation V with∗ (V ), which is a filtered ϕ-module
Hodge-Tate weights in [0, r]. Put D = Dcrys
over K0 .
Definition 2.10. A strongly divisible lattice in D is a W -submodule M of
D such that
• M is free of finite rank and M ⊗W K0 = D,
i
i
• ϕ(M
P ∩1 D ) ⊆ p Mi for any i,•i∈Z pi ϕ(M ∩ D ) = Mσ .
For such M , put M i = M ∩ Di and ϕi = p1i ϕ|M i . With these structures
fd,[0,r]we consider M as an object of MFW, and the structure of a filtered ϕimodule on D can be recovered by D = M i ⊗W K0 and ϕ = ϕ0 ⊗ 1. Then
we can show the following.
Theorem 2.11.
(i) ([FL82], Théorème 8.4 (i)) For any r ∈ {0, . . . , p−fd,[0,r], the p-adic GK0 -representation V =1} and any M ∈ MFW∗T̂crys (M ) ⊗Zp Qp is crystalline with Hodge-Tate weights in [0, r] sat∗ (V ) ' M ⊗isfying Dcrys
W K0 as filtered ϕ-modules over K0 .
(ii) ([Bre99], Proposition 3) Suppose r < p − 1. Let V be a crystalline GK0 -representation with Hodge-Tate weights in [0, r] and put∗ (V ). Then the functor T̂ ∗D = Dcrys
crys induces an anti-equivalence
between the category of strongly divisible lattices in D and that of
GK0 -stable Zp -lattices in V .
In fact, the first assertion is now a part of the “weakly admissible implies
admissible” theorem of Colmez-Fontaine [CF00, Théorème A]. The second
assertion follows from Theorem 2.9, except the essential surjectivity. For
this, by the theorem of Laffaille [Laf80, Théorème 3.2], D contains a strongly
divisible lattice M . Using Theorem 2.9 (iv), we can modify M to produce thestrongly divisible lattice corresponding with a given GK0 -stable Zp -latticein V .
Definition 2.12. A GK0 -representation T̄ ∈ RepZ (GK0 ) is said to be torp
sion crystalline with Hodge-Tate weights in [0, r] if there exist a crystalline
GK0 -representation V with Hodge-Tate weights in [0, r] and GK0 -stable Zp lattices T ⊇ T 0 in V such that T̄ is isomorphic as a GK0 -module to T /T 0 .
Corollary 2.13 ([BMs02], Theorem 3.1.3.3). For any r ∈ {0, . . . , p−2}, thef,[0,r]∗functor Tcrys
induces an anti-equivalence between the category MFW,tor andthe full subcategory of RepZ (GK0 ) consisting of torsion crystalline GK0 p
representations with Hodge-Tate weights in [0, r].

12SHIN HATTORIFor this, first we can show by using Corollary 2.6 (ii) that any object
f,[a,b]M ∈ MFW,tor has a resolution in MFW
0

/ L0

/L

/M

/0

fd,[a,b]with objects L, L0 ∈ MFW[Win84, Proposition 1.6.3]. This and Theorem
2.11 show that the functor in Corollary 2.13 is well-defined, since we have
∗ (M ) ' T̂ ∗ (L0 )/T̂ ∗ (L). Theorem 2.9 and Theorem 2.11 imply that
Tcrys
crys
crys
it is an anti-equivalence.
As we will see in §4, the proof in [FL82] of Theorem 2.9 is based on af,[0,p−1]
and an explicit
classification of the simple objects of the category MFk
description of their associated Galois representations, for the case where k is
algebraically closed (Propositions 4.7 and 4.9). They also have the followinginteresting corollary. Let h be a positive integer. We define a characterbyθh : IK0 → F×phhg(p1/(p −1) )θh (g) =mod mK̄ ∈ µph −1 (k̄) = F×
,
h −1)
ph
1/(p
p
1

which is independent of the choice of a (ph − 1)-st root p ph −1 of p. It
factors through the quotient IK0 /PK0 by the wild inertia subgroup PK0 .
The character θh is called the fundamental character of level h.
Theorem 2.14 ([FL82], Proposition 4.4 and Théorème 5.3). Let r be an
f,[0,r]integer satisfying 0 ≤ r ≤ p−1 and M an object of MFW,tor . Then the semi∗ (M )|simplification of the IK0 -module Tcrys
IK0 is a direct sum of charactersof the form
i +pi1 +···+ph−1 ih−1

θh0with integers h ≥ 1 and ij ∈ [0, r].
In fact, each character appearing in the semi-simplification comes from a
Jordan-Hölder factor N of the Fontaine-Laffaille module M ⊗W W (k̄) overW (k̄), and the ij ’s in the theorem are exactly the integers i satisfying N i 6=
N i+1 . Thus we have a complete understanding of the semi-simplification∗ (M )|of Tcrys
IK0 in terms of M . As for the extension structure of these∗ (M ) (Theorem
characters, we will prove an estimate of ramification of Tcrys
2.19).
2.3. Relation to crystalline cohomology. Let X be a proper smooth
scheme over W and put Xn = X ×W Spec(Wn ). For any r ∈ {0, 1, . . . , p−1},
r (X ) = H r (X , Ω• ) a
we can give the r-th de Rham cohomology group HdR
n
n
Xn
[0,r]

structure as an object of MFW , as follows. The i-th filtration for 0 ≤ i ≤ r
is the usual Hodge filtration:r
HdR(Xn )i = H r (Xn , σ≥i Ω•Xn ),INTEGRAL p-ADIC HODGE THEORY

13

where σ≥i Ω•Xn is the truncation (0 → · · · → 0 → ΩiXn → Ωi+1
Xn → · · · ) of thede Rham complex of Xn over Wn . Note that we have the isomorphism[i]

H r (Xn , σ≥i Ω•Xn ) ' H r ((Xn /Wn )crys , JXn /Wn ),where the right-hand side is the r-th crystalline cohomology group of the ith divided power of the natural ideal sheaf JXn /Wn on the crystalline site of
Xn /Wn [Bert74, Chapitre V, Corollaire 2.3.7]. Hence we obtain a crystalliner (X ) → H r (X ), and by [FMe87, II, 2.3] we can
Frobenius map ϕ0 : HdRnndRalso define a σ-semilinear map
rϕi : H r (Xn , σ≥i Ω•Xn ) → HdR(Xn ).

Theorem 2.15 ([FMe87], II, Corollary 2.7 (ii)). With these structures,r (X ) is an object of MFf,[0,r] .
HdR
n
W,tor
f,[0,r]For any M ∈ MFW,tor , the map Fill Acrys ⊗W M i−l → Acrys ⊗W M is
injective, since Acrys /Fill Acrys is p-torsion free and the W -submodule M i−l of
M is a direct summand by Corollary 2.6 (ii). Using this, we give Acrys ⊗W M
the tensor product structure as a Fontaine-Laffaille module. Namely, we put
X(Acrys ⊗W M )i =Alcrys ⊗W M i−ll∈Z

and ϕiAcrys ⊗W M =i−lll∈Z ϕAcrys ⊗ ϕM . We definePr

Tcrys (M ) = ((Acrys ⊗W M )r )ϕ =1 (−r),
where (−r) means the Tate twist. Note that we can show Tcrys (M ) '∗ (M )∨ [Bre98b, Proposition 3.2.1.7]. Then we can state a torsion version
Tcrysof the comparison isomorphism between etale and de Rham cohomology.
Theorem 2.16 ([FMe87], III, Remark 6.4). For any r ∈ {0, . . . , p − 2},there exists a natural isomorphism of GK0 -modulesrr(Xn )) ' Het(X ×W Spec(K̄), Z/pn Z).
Tcrys (HdR

From this and Theorem 2.14, we obtain the following corollary, which
settles a conjecture of Serre [Ser72, 1.13] on tame characters appearing in
torsion etale cohomology of proper smooth schemes over OK , for the casewhere K is absolutely unramified.
Corollary 2.17 ([FMe87], I, 3.2). Let X be as above. Then the semir (X × Spec(K̄), Z/pn Z)|
simplification of the IK0 -module Het
W
IK0 is a direct
sum of characters of the form−(i0 +pi1 +···+ph−1 ih−1 )θhwith integers h ≥ 1 and ij ∈ [0, r].

14

SHIN HATTORI

2.4. Ramification of torsion crystalline representations. For any finite Galois extension L/K in K̄ and any j ∈ R≥−1 , we have the j-th upper numbering ramification subgroups Gal(L/K)j ⊆ Gal(L/K) and GjK =(j)limL/K Gal(L/K)j ⊆ GK [Ser68, Chapitre IV]. Put GK = Gj−1
K . We also
←−
S
0
(j+)
j
define Gj+
j 0 >j GK , and similarly for GK .
K to be the closure of the union(0+)

Then GK is equal to the inertia subgroup IK , and G0+
K is equal to the wildinertia subgroup PK . For any finite Galois extension L/K in K̄, the value(j)uL/K = inf{j ∈ R>0 | GK ⊆ GL }measures the extent of ramification of L/K. For example, uL/K = 0 if andonly if L/K is unramified and uL/K = 1 if and only if it is tamely ramified and not unramified. It is also related to the different DL/K [Fon85,
Proposition 1.3], and used as a local term to calculate the different andthe discriminant of a finite extension of number fields. In algebraic number theory, imposing local constraints in terms of ramification everywhere
often yields the finiteness, and even the non-existence, of number fields. For
example, the Hermite-Minkowski theorem states that for any integer d andany finite set S of primes, there exist only finitely many number fields ofdegree d which are unramified outside S, and that there exists no nontrivial
unramified extension over Q.
More generally, a GK -representation V is said to be unramified (resp.
tamely ramified) if IK (resp. PK ) acts trivially on V . We want to study
ramification (resp. wild ramification) of V , namely how far V is from unramified (resp. tamely ramified) ones. For any p-adic GK -representation Vof finite local monodromy (in other words, when IK acts on V through afinite quotient), the Artin and Swan conductors
X
X
(j+)
(j)
j+
j
Art(V ) =j dimQp V GK /V GK , Sw(V ) =j dimQp V GK /V GK ,
j>0

j>0

which are integers by the Hasse-Arf theorem, measure the extent of ramification and wild ramification of V , respectively. They are also used, for
any number field F with absolute Galois group GF and any p-adic GF representation V which factors through a finite quotient, to compute theglobal Artin conductor of V
Y
Art(V |GF )p ,Art(V ) =pp∈Spec(OF )which appears in the functional equation of the Artin L-function of V .
If V is not of finite local monodromy (say V is Hodge-Tate with a non-zero
Hodge-Tate weight), these definitions of conductors do not work well. When
V is de Rham, one way to study ramification is to pass to the Weil-Deligne
representation Dpst (V ), by which any crystalline representation is consideredunramified. For any number field F and any p-adic GF -representation V
which is unramified outside a finite set of places and de Rham at any place

INTEGRAL p-ADIC HODGE THEORY

15

dividing p, by using Dpst (V ) at places over p and Grothendieck’s monodromy
theorem at places over primes l 6= p, we can attach to V a GFv -representationof finite local monodromy for any finite place v of F and can define the global
Artin conductor Art(V ) for V .
For the case where F = Q and V is of dimension two over a coefficient field
which is a finite extension of Qp , Fontaine-Mazur [FMa95] conjectured thatsuch V is modular, namely it is isomorphic to the p-adic GQ -representation
attached to an elliptic modular form of level equal to the global Artin conductor Art(V ) and weight determined by the Hodge-Tate weights of V |GQp .
Such modularity theorems often give finiteness or non-existence results for
global objects, since the set of normalized Hecke eigenforms of fixed level
and weight is finite, and even empty for some cases. One of the examples ofsuch results is the proof of Fermat’s last theorem [TW95, Wil95] via proving modularity of elliptic curves over Q (the Shimura-Taniyama conjecture).
The Fontaine-Mazur conjecture was proved in many cases by works of Kisin[Kis10] and Emerton [Eme11+].
Another way to study ramification of a p-adic GK -representation V , which
we will consider in this article, is to take a GK -stable Zp -lattice T in V
and consider its quotients T /pn T . Define the finite extension Ln /K by
GLn = Ker(GK → Aut(T /pn T )). Then by a theorem of Sen [Sen72], there
exists a constant c such that for any n we havee(n − c) ≤ uLn /K ≤ e(n + c).
Though uLn /K itself depends on the choice of a lattice T , we may consider
that such an estimate of ramification reflects how good V is. In fact, for thecase where V is crystalline with Hodge-Tate weights in [0, 1], the GK -module
T /pn T is isomorphic to the one attached to a finite flat (commutative) groupscheme over OK , and we have the following upper bound of ramification:
Theorem 2.18. ([Fon85], Théorème A and Corollaire) Let G be a finite flat(j)group scheme over OK which is killed by pn . Then the subgroup GK acts
trivially on the GK -module G(K̄) for any j > e(n + 1/(p − 1)). Namely, if
we define a finite Galois extension L/K by GL = Ker(GK → Aut(G(K̄))),then we have1
uL/K ≤ e(n +).
p−11Moreover, we have vK (DL/K ) < e(n + p−1), where DL/K is the different of
L/K.
Fontaine’s proof makes an essential use of the fact that G is locally ofcomplete intersection over OK , while the special case of n = 1 and e = 1
was obtained independently by Abrashkin [Abr87] via the theory of Honda
systems (§2.5). Combining this theorem with the Odlyzko bound of the root
discriminant of a number field, Abrashkin and Fontaine deduced that there
exists no Abelian variety with everywhere good reduction over Q, and evenover a couple of other small number fields [Abr87, Fon85]. Note that Faltings

16

SHIN HATTORI[Fal83] proved that for any number field F , any finite set S of places of F
and any positive integer g, there exist finitely many isomorphism classes ofAbelian varieties over F of dimension g which has good reduction outside
S. These generalizations of the Hermite-Minkowski theorem to Abelianvarieties had originally been conjectured by Shafarevich.
On the other hand, Fontaine also conjectured that, for any proper smooth(j)scheme X over OK , the subgroup GK acts trivially on the GK -modulei
Het(X ×OK Spec(K̄), Z/pn Z)

for any j > e(n + i/(p − 1)) [Fon85, Remarques 2.2]. Though this conjectureis still wide open, Abrashkin and Fontaine independently gave a proof forthe case where K = K0 , i < p − 1 and n = 1. Theorem 2.16 reduces it to
the following theorem on ramification of Galois representations associatedwith Fontaine-Laffaille modules, which is the second main theorem of thisarticle.
Theorem 2.19 ([Abr89b], §2, Assertion 8.1 and [Fon93], Théorème 2). Letf,[0,r]r ∈ {0, . . . , p − 2} be an integer and M an object of the category MFk.
(j)∗ (M ) for any
Then the subgroup GK0 acts trivially on the GK0 -module Tcrys
j > 1 + r/(p − 1). Namely, if we define a finite extension L/K0 by GL =∗ (M ))), then we haveKer(GK0 → Aut(Tcrysrr, vK0 (DL/K0 ) < 1 +.
uL/K0 ≤ 1 +p−1
p−1
f,[0,p−1]0Moreover, this also holds for r = p − 1 and M ∈ MFk.

For 0 ≤ r ≤ p − 2, Corollary 2.13 implies that the same assertions as
Theorem 2.19 hold for any torsion crystalline GK0 -representation T̄ with
Hodge-Tate weights in [0, r] such that pT̄ = 0. Note that uL/K0 = 0 for∗ (M ) is unramified
r = 0, since Corollary 4.6 shows that the GK0 -module Tcrys
f,[0,0]for any M ∈ MFk.

2.5. Classification of finite flat group schemes. Another applicationof Fontaine-Laffaille modules is a classification theory of finite flat groupschemes killed by some p-power over W . First recall the classification offinite group schemes over k via Dieudonné modules. Let Ḡ be a finite groupscheme over k which is killed by some p-power. Let Ḡ (p) be the pull-backof Ḡ by the Frobenius map Spec(σ) : Spec(k) → Spec(k). Let FḠ : Ḡ →Ḡ (p) and VḠ : Ḡ (p) → Ḡ be the (relative) Frobenius and the Verschiebung
homomorphisms of Ḡ. On the other hand, a Dieudonné module over W isa W -module M endowed with a σ-semilinear endomorphism F and a σ −1 semilinear endomorphism V satisfying F V = V F = p. Then there exists ananti-equivalenceḠ 7→ M ∗k (Ḡ)
from the category of finite group schemes over k which are killed by somep-power to that of Dieudonné modules over W whose underlying W -module

INTEGRAL p-ADIC HODGE THEORY

17

is of finite length, in a way that F and V correspond with FḠ and VḠ , respectively [Fon77, III, Théoremè 1]. Moreover, M ∗k induces an anti-equivalence
between the category of p-divisible groups over W and that of Dieudonnémodules over W whose underlying W -module is free of finite rank [Fon77,
Proposition 6.1].
The crystalline Dieudonné theory of Grothendieck-Messing [Gro74, Mes72]
suggested that finite flat group schemes and p-divisible groups G over W
should be classified by a pair of the Dieudonné module associated with thespecial fiber Ḡ = G ×W Spec(k) and its submodule (the Hodge filtration).
Fontaine constructed such a classification via Honda systems, which werenamed after a work of Honda [Hon70] (see [Fon77, IV, Théorème 1] for pdivisible groups, and [Con99, Fon75] for finite flat group schemes, where weneed the additional assumption that group schemes are unipotent or p 6= 2).
Using this, Fontaine-Laffaille also obtained the following:
Theorem 2.20 ([FL82], 9.11 and Proposition 9.12). Suppose p 6= 2. Then
there exists an anti-equivalence ILM from the category of finite flat groupf,[0,1]schemes over W killed by some p-power to the category MFW,tor , with anatural isomorphism of GK0 -modules∗
G(K̄) ' Tcrys(ILM(G)).

The same assertion holds for any p if we restrict to the category of finite
f,[0,1]0
flat unipotent group schemes and MFW,tor .
The Fontaine-Laffaille module M = ILM(G) for a finite flat group scheme
G over W is defined as follows: the underlying W -module M is M =M ∗k (Ḡ) ⊗W,σ W for the special fiber Ḡ. The first filtration M 1 is the image by V : M ∗k (Ḡ) → M ∗k (Ḡ) ⊗W,σ W of the Hodge filtration L ⊆ M ∗k (Ḡ).
The restricted map V |L is injective, and the Frobenius structure on M isdefined by ϕ0 = F and ϕ1 = V −1 .
3. Further developments
3.1. Applications and generalizations of the Fontaine-Laffaille theory. Typical applications of the Fontaine-Laffaille theory are as follows:(i) Studying crystalline deformation rings, especially to prove modularity of Galois representations for number fields unramified over p[BeK13, CHT08, DFG04, FMa95, Fuj06+, Ram93, Tay03, Tay06,
Tay08, TW95, Wes04, Wil95].
(ii) Studying the reduction modulo p of GK0 -stable Zp -lattices in crystalline GK0 -representations, and of Galois representations attached
to modular forms of level prime to p [BMz02, Dim05], including ageneralization of Serre’s conjecture [Ser87] on modularity of mod p
GQ -representations to totally real number fields unramified over p[Gee11].

18

SHIN HATTORI(iii) Proving a rareness of global objects satisfying strong local constraints everywhere [Jos99], including a vanishing of Hodge numbers for algebraic varieties over Q with everywhere good reduction[Abr89b, Fon93].
Integral p-adic Hodge theory itself has also been highly developed andused for applications where original theories stated above are not available—
under ramification of the base field K or with larger range of Hodge-Tateweights.
First we remark that the theory of Honda systems was generalized to thecase where the base field is ramified by Fontaine himself [Fon77] to obtain aclassification of p-divisible groups over OK for the case of e < p − 1, and also
their classification up to isogeny for any e. For finite flat group schemes over
OK , a classification via Honda systems was obtained by Conrad [Con99] forthe case of e < p − 1 and applied to the proof of a case of the ShimuraTaniyama conjecture which was not treated by Taylor-Wiles [CDT99].
Note that Theorem 2.9 (iii) does not hold for r = p − 1, a counterexample
with p = 2 being given by the isomorphism of GK0 -modules µ2 → Z/2Z.
By considering a category of filtered GK0 -modules, Abrashkin constructed∗a modified version of the functor Tcrys
which is fully faithful [Abr89a] anddeduced the above vanishing of Hodge numbers for weight p − 1 [Abr89b].
On the other hand, Faltings [Fal89] established a version of the FontaineLaffaille theory in families, in which the coefficient ring of Fontaine-Laffaillemodules is not W but the structure sheaf of a proper smooth scheme overW , and generalized Theorem 2.16 and Theorem 2.20 to a relative situation.
The assumption r ≤ p−1 is crucial for the Fontaine-Laffaille theory, since
the map p−i ϕ on Fili Acrys is well-defined only for i ≤ p − 1. This was one ofthe difficulties for studying subquotients of crystalline representations withlarger range of Hodge-Tate weights. Integral theories without any restriction
on Hodge-Tate weights have been developed in two ways: theories of Wachmodules and Kisin modules.
S
3.2. Wach modules. Consider the infinite extension K(ζp∞ ) = n K(ζpn )of K obtained by adjoining all the p-power roots of unity. It is a Galois extension over K which is arithmetically profinite with Galois group ΓK , and thus
the theory of fields of norms of Fontaine-Wintenberger [Win83] can be applied. This theory shows that the Galois group GK(ζp∞ ) = Gal(K̄/K(ζp∞ ))is isomorphic to the absolute Galois group of the field of Laurent power series l((π)) over the residue field l of K(ζp∞ ). Thus the category of GK(ζp∞ ) modules is equivalent to that of Gl((π)) -modules.
Fontaine [Fon90] showed that the latter is also equivalent to a category of
ϕ-modules over the p-adic completion OE of the ring of Laurent power series
W (l)[[π]][1/π]. Here, for any ring B endowed with a ring endomorphism
ϕ : B → B, a ϕ-module over B is a B-module M with a ϕ-semilinear
map ϕM : M → M which we often denote abusively by ϕ. We can define anatural Frobenius map ϕ on the ring OE , and we consider ϕ-modules over OE

INTEGRAL p-ADIC HODGE THEORY

19

with respect to this map. The ring OE has a natural ΓK -action, and he also
showed that the category of GK -modules which are finite over Zp (resp. Qp )is equivalent to a category of such ϕ-modules over OE (resp. E = Frac(OE ))with a compatible ΓK -action, which are called (ϕ, ΓK )-modules.
For the case of K = K0 , we can study crystalline GK0 -representations
and their subquotients with this construction: The theory of Wach modules [Berg04, Col99, Wac96, Wac97] tells us that for any crystalline GK0 representation V with non-negative Hodge-Tate weights, there exists a nice(ϕ, ΓK0 )-stable W [[π]][1/p]-lattice N(V ) in its associated (ϕ, ΓK0 )-module
D(V ) over E such that V 7→ N(V ) gives an equivalence of categories. Moreover, there also exists an equivalence between the category of GK0 -stable
Zp -lattices in V and a category of (ϕ, ΓK0 )-stable W [[π]]-lattices in N(V ).
Wach modules are typically used for a study of the reduction modulo pof crystalline GK0 -representations with larger range of Hodge-Tate weights[Berg12, BLZ04, CD11, Dou10]. Another interesting feature of Wach modules is a relation to Iwasawa theory [Ben00, BB08, Berg03, LLZ10, LLZ11,
LeZ12, LoZ13].
The theory of Wach modules generalizes to a case where K is absolutely
ramified [KR09] by using the Lubin-Tate extension for K instead of the cyclotomic extension K0 (ζp∞ ), to give a classification of GK -stable OF -lattices
in a class of crystalline GK -representations with coefficients in a finite extension F of Qp which is called F -analytic. Such a Lubin-Tate version of(ϕ, ΓK0 )-modules has also been investigated in an attempt to generalize thep-adic Langlands correspondence for GL2 (Qp ) to p-adic local fields otherthan Qp [Berg13, FX13].
3.3. Breuil modules. On the other hand, for any integer r satisfying
0 ≤ r ≤ p − 1, let us think of generalizing the Fontaine-Laffaille theory
to semi-stable representations, by naı̈vely adding a monodromy operator
fd,[0,r]N to Fontaine-Laffaille modules M ∈ MFW. Then dividing the usual
relation N ϕ = pϕN by pi+1 yields N ϕi+1 = ϕi N for any i, which forces
us to impose the Griffiths transversality N (M i+1 ) ⊆ M i on M , and thus
N (Di+1 ) ⊆ Di on the filtered (ϕ, N )-module D = M ⊗W K0 . However,admissible filtered (ϕ, N )-modules over K do not satisfy it in general. Thisdifficulty was resolved by Breuil [Bre97], who proved that the category offiltered (ϕ, N )-modules over K is equivalent to a category of filtered (ϕ, N )modules over a ring SK0 satisfying the Griffiths transversality. Here the ringSK0 is defined as follows: Let πK be a uniformizer of K and E(u) ∈ W [u]the minimal polynomial of πK over W . Then we put
E(u)n| n ∈ Z≥0 ]∧ , SK0 = S ⊗W K0 ,S = W [u][n!
where ∧ means the p-adic completion. The ring S has a natural divided
power filtration Fili S for i ≥ 0, a σ-semilinear Frobenius endomorphism ϕsending u to up , a derivation N satisfying N (u) = −u, and a ϕ-semilinear
map ϕi : Fili S → S for 0 ≤ i ≤ p − 1, which define a similar structure

20

SHIN HATTORIon SK0 . For any filtered (ϕ, N )-module D over K, the equivalence is given
by D 7→ D = D ⊗K0 SK0 with the induced ϕ, N and a filtration definedusing the isomorphism D/E(u)D ' DK . Note that the ring S/pn S had also
appeared in a work of Kato [Kat94].
Using this construction, Breuil generalized the Fontaine-Laffaille theory to
the case of semi-stable representations with Hodge-Tate weights in [0, r]. Let
r+1
0 = D
D be a filtered (ϕ, N )-module over K satisfying DK= 0.
K and DK
His idea is to consider S-lattices in D = D ⊗K0 SK0 and their subquotients,
instead of W -lattices in D itself. In other words, we consider a category offiltered S-modules M with a Frobenius structure and a derivation, which
are called Breuil modules. It turned out that we have to keep only the last
filtration Filr M in the definition of filtered S-modules, to make a variant of
Proposition 2.5 hold. Then Theorem 2.9 and Theorem 2.11 were generalized
to this semi-stable variant by Breuil [Bre98a, Bre99] (when e = 1, r < p − 1)
and Caruso [Car06] (when er < p − 1).
Theorem 2.14 and Theorem 2.16 were also generalized under these assumptions [Bre98a, Bre98b, Car06, Car08], which yield the aforementioned
conjecture of Serre on tame characters appearing in torsion etale cohomology, even for the case of semi-stable reduction. Though such a comparison
isomorphism for integral or torsion coefficients as in Theorem 2.16 does notseem to exist for general e and r, Bhatt-Morrow-Scholze [BMS18] proved
certain comparison results between etale and crystalline cohomology of integral coefficients without any assumption on e and r.
Note that the control on tame characters mentioned above was applied toshow the non-existence or a rareness of global objects having good properties
everywhere [Oze11, Oze13b, OT14]. Moreover, a classification of finite flatgroup schemes and p-divisible groups over OK via Breuil modules was also
obtained by Breuil himself [Bre00] for p ≥ 3.
Breuil modules have also been applied to modularity theorems [Sav04,
Sav05], including the complete proof of the Shimura-Taniyama conjecture[BCDT01], to a study of the reduction modulo p of semi-stable representations [CS09, CS10] and to that of generalized Serre conjectures [EGH13,
Gee11, GLS12].
3.4. Breuil-Kisin classification of finite flat group schemes. Anotherbreakthrough on integral p-adic Hodge theory was also brought by Breuil.
In search of a classification of finite flat group schemes over OK which isvalid for p = 2, he proposed the idea of a classification via ϕ-modules overW [[u]] and, using the classification via Breuil modules [Bre00], proved sucha classification for the case where group schemes are killed by p ≥ 3 [Bre98+,
Bre02].
One of the advantages of this theory is that we can control the GK module G(K̄) associated with a finite flat group scheme G over OK very
easily. In fact, choose a system of p-powerroots (πn )n≥0 of πK satisfyingSpπ0 = πK , πn+1
= πn and put K∞ = n K(πn ). This is an arithmeticallyINTEGRAL p-ADIC HODGE THEORY

21

profinite non-Galois extension over K and thus, via the theory of fields ofnorms [Win83], the category of GK∞ -modules is equivalent to a category of
ϕ-modules over OE = W [[u]][1/u]∧ . Let G be a finite flat group scheme over
OK which is killed by p ≥ 3, and let M be the ϕ-module over OE /pOE =k((u)) associated with the GK∞ -module G(K̄)|GK∞ . Then Breuil provedthat G is classified by a ϕ-stable k[[u]]-lattice M in M such that the cokernelof the map 1 ⊗ ϕ : k[[u]] ⊗ϕ,k[[u]] M → M is killed by E(u), and thus we canrecover G(K̄)|GK∞ from M. We do not lose information by the restriction to
GK∞ , since he also proved that the restriction functor G(K̄) 7→ G(K̄)|GK∞from the category of GK -modules attached to finite flat group schemes overOK killed by some p-power to that of GK∞ -modules is fully faithful.
Kisin established such a classification valid for any finite flat group schemes
killed by some p-power and any p-divisible groups over OK for p ≥ 3[Kis06, Kis09a], and combining it with his modified Taylor-Wiles patchingargument, he generalized modularity theorems to the case where the basenumber field is highly ramified over p [Kis09a]. The ϕ-module M is often
referred to as the Breuil-Kisin module associated with G.
Breuil-Kisin’s classification has been generalized in various ways. Firstly,
the classification theory of finite flat group schemes and p-divisible groups
over OK was completed, after a work of Kisin [Kis09b] treating the casewhere p = 2 and group schemes are unipotent, independently by Kim[Kim12], Lau [Lau14] and Liu [Liu13]. We also have a similar classification of finite flat group schemes and p-divisible groups over regular rings[Kim15, Lau10, VZ10], whereas the theory of displays [Zin02] provides aclassification theory of formal p-divisible groups applicable for more generalbase rings, via much more complicated semilinear algebraic data.
Let us mention that Scholze-Weinstein [SW13] gave a striking classification of p-divisible groups over OC by certain linear algebraic data, notsemi-linear algebraic data like Breuil and Breuil-Kisin modules. For any
p-divisible group G over OC , we have the pair of the Tate module Tp (G)and an injection of C-vector spaces Lie(G) ⊗OC C(1) → Tp (G) ⊗Zp C, where
Lie(G) denotes the Lie algebra of G and C(1) the Tate twist of C. They
proved that the category of such pairs is equivalent to that of p-divisiblegroups over OC .
The Breuil-Kisin classification of finite flat group schemes turned out
to be useful also for studying the Hecke operators at places dividing p on
the space of p-adic modular forms, via an analysis of canonical subgroups[Hat13, Hat14a, Kas13, Tia14].

3.5. Kisin modules and (ϕ, Ĝ)-modules. Secondly, for any r ≥ 0, Kisin
studied ϕ-modules M over W [[u]] such that the cokernel of the map 1⊗ϕM :W [[u]]⊗ϕ,W [[u]] M → M is killed by E(u)r , which are often called Kisin modules, and also those endowed with an endomorphism N on (M/uM) ⊗W K0 ,which are called (ϕ, N )-modules over W [[u]]. He proved that the category

22

SHIN HATTORI

0 = D
of admissible filtered (ϕ, N )-modules D satisfying DK
K can be considered as a full subcategory of the isogeny category of (ϕ, N )-modules overW [[u]]. This new classification of semi-stable GK -representations was used
to prove the existence of potentially semi-stable deformation rings [Kis08],
which led to proving many cases of the Breuil-Mézard conjecture and the
Fontaine-Mazur conjecture [Kis09c] and to a construction of the p-adic local
Langlands correspondence for GL2 (Qp ) [Kis10].
Kisin’s theory also enables us to have a control on GK∞ -stable Zp -lattices
in semi-stable GK -representations. Moreover, for r ≤ p − 1, by tensoring thering S over W [[u]] along the map ϕ with Kisin modules, we obtain Breuil
modules without derivation, and to recover not only the GK∞ -action but
the GK -action amounts to giving it a derivation. Using these features, Liu
generalized Theorem 2.11 to semi-stable representations with Hodge-Tateweights in [0, p − 2] which is valid for any K, in terms of Breuil modules[Liu08] (the unipotent case with Hodge-Tate weights in [0, p − 1] was shownby [Gao17a]). Kisin’s theory was generalized to the case where the residue
field k is imperfect [BT08], using a framework of p-adic Hodge theory forimperfect residue fields [Bri06].
On the other hand, since K∞ /K is not Galois, it is not so easy to study
the action of the whole Galois group GK on semi-stable GK -representations
via Kisin’s theory, unless the range of Hodge-Tate weights is small. Liuovercame this difficulty, by considering the Galois closure K∞ (ζp∞ )/K of
K∞ , its Galois group Ĝ = Gal(K∞ (ζp∞ )/K) and a category of ϕ-modules
M over W [[u]] with a Ĝ-action on a scalar extension M⊗W [[u]],ϕ R̂, which arecalled (ϕ, Ĝ)-modules [Liu07, Liu10]. He proved an anti-equivalence between
a category of (ϕ, Ĝ)-modules and the category of GK -stable Zp -lattices insemi-stable GK -representations generalizing Theorem 2.11, without any restriction on Hodge-Tate weights nor on the base field K. It was also applied
to a study of generalized Serre conjectures [GLS14]. For the case of K = K0 ,the GK0 -stable Zp -lattices in crystalline GK0 -representations are classified
both by Wach modules and (ϕ, Ĝ)-modules, and the attached Wach module
to such a lattice can be recovered from the attached (ϕ, Ĝ)-module [Liu18].
More generally, we can develop a variant of the theory of (ϕ, ΓK0 )-modules
using Ĝ instead of ΓK0 [Car13, Oze13a, Tav11].
Using these developments on integral p-adic Hodge theory, Theorem 2.19
was also generalized for subquotients of semi-stable representations, whichhave a worse ramification bound [BMs02, Hat09, Car13, CL11]. Abrashkinproved a similar ramification bound and applied it to a vanishing of Hodge
numbers of algebraic varieties over Q with everywhere good reduction except
semi-stable reduction at 3 [Abr13]. He also found a new method to prove
such a ramification bound of GK -representations, by using a non-canonicalisomorphism between two complete discrete valuation fields of equal characteristic, and generalized Theorem 2.19 to the case where Fontaine-Laffaillemodules are not killed by p [Abr14, Abr15].

INTEGRAL p-ADIC HODGE THEORY

23

3.6. Ramification theory. The classical ramification theory as reviewed
in §2.4 has been typically applied to:(i) Definitions and calculations of invariants of global objects via local terms concerning ramification, such as the global Artin conductor and the Grothendieck-Ogg-Shafarevich formula to compute
Euler-Poincaré characteristics of l-adic sheaves over algebraic curves[Gro77].
(ii) Proving the non-existence or a rareness of global objects satisfying strong local constraints everywhere, such as Abelian varieties [Abr87, BrK01, BrK04, BrK14, Fon85, Schf03, Schf05, Schf12,
Ver13], algebraic varieties [Abr89b, Abr13, Fon93] and Galois representations [Bru99, Bru05, Jos99, Moo00, Moo03, MT03, Tat94],
generalizing the Hermite-Minkowski theorem.
(iii) The theory of fields of norms of Fontaine-Wintenberger, to give an
isomorphism between the absolute Galois group of an arithmeticallyprofinite extension of a complete discrete valuation field of mixed
characteristic with perfect residue field and that of a complete discrete valuation field of equal characteristic [Win83].
(iv) The theory of close local fields, including a comparison of absolute Galois groups of complete discrete valuation fields of mixed
and equal characteristics with the same perfect residue field modulo ramification subgroups [Del84], especially to deduce the local
Langlands correspondence of reductive groups for the equal characteristic case from the mixed characteristic case [ABPS16, Gan15]
by combining with [Kaz86].
(v) A local version of the Grothendieck conjecture, which states thatany local field can be recovered from its absolute Galois group andramification subgroups [Abr00, Abr10, Moc97].
In the classical ramification theory as in [Ser68], it is crucial that theresidue field k is perfect and thus for any finite extension L/K, the OK algebra OL is generated by a single element. In general, for any finite Galois
extension L/K, we can define the i-th lower numbering ramification subgroup
Gal(L/K)i = Ker(Gal(L/K) → Aut(OL /mi+1
L )).
For the case where k is perfect, we can prove, by using the monogenicityof OL , that the j-th upper numbering ramification subgroup Gal(L/K)j =Gal(L/K)ψL/K (j) is compatible with quotients, where ψL/K is the Herbrandfunction. This enables us to take the projective limit to define the ramification subgroup GjK of GK . However, when k is imperfect, the lower numbering ramification subgroups are not necessarily compatible with quotientseven after renumbering.
After various attempts to define appropriate ramification subgroups of
GK when k is imperfect, including especially a work of Kato for Abelian
extensions [Kat89], different definitions were proposed independently by24

SHIN HATTORI

Abbes-Saito [AS02, AS03] (geometric definition via connected components
of analytic varieties), Borger [Bor04] (reducing to the case of perfect k via
the universal residual perfection), Kedlaya [Ked07a] (differential definition
via (ϕ, ∇)-modules over the Robba ring) and Zhukov [Abr02, Zhu03] (forhigher dimensional local fields, via induction from lower dimensional ones).
Each definition has its own advantages and applications. Abbes-Saito’s
definition was used to define characteristic cycles of etale sheaves on higher
dimensional algebraic varieties to compute their Euler-Poincaré characteristics [Sai09, Sai15]. One of the advantages of their definition is that it
also gives a ramification theory of finite flat algebras over OK . With thisframework, Theorem 2.18 can be shown easily from the fact that multiplicative groups are minimal in the category of finite flat group schemes overOK [Hat06]. This advantage was also used to prove the overconvergenceof canonical subgroups of Abelian varieties [AM04, Hat13, Hat14a, Tia10,
Tia12] and a generalization of the theory of close local fields to the case ofimperfect k [Hat14b].
Kedlaya’s definition is more flexible about base extensions than AbbesSaito’s. One of its consequences is that we can show the integrality of the
Swan conductor generalizing the classical Hasse-Arf theorem much moreeasily [Ked07a], via a reduction to the case of perfect k by adding p-basis
to k and “rotate” in the enlarged space to make ramification concentrateon the uniformizer direction. This differential version of ramification theory
was used to prove a conjecture of Shiho [Shi02, Conjecture 3.1.8] on semistable reduction of overconvergent F -isocrystals on smooth varieties over aperfect field of characteristic p [Ked07b, Ked08, Ked09, Ked11]. In Zhukov’s
definition, ramification subgroups are multi-indexed, and it is suitable fora detailed study of the structure of GK . It was used for a proof of a twodimensional analogue of the Grothendieck conjecture [Abr03] and a generalization of the theory of fields of norms to higher dimensional local fields[Abr07]. Note that the theory of higher fields of norms was also obtained indifferent ways for more general settings [And06, Schl06] and generalized tothe theory of perfectoid spaces [Schz12].
Among these different definitions, the comparison between Abbes-Saito’sand Kedlaya’s has been well studied. Note that a differential interpretationof the classical ramification theory had been investigated, for example, by[Cre00, Ked05, Mat02, Tsu98]. For the case of imperfect k, after works of
Matsuda [Mat04] and Chiarellotto-Pulita [CP09], Xiao proved that thesetwo definitions (and also Borger’s for the equal characteristic case) yield thesame Artin and Swan conductors for almost all the cases [Xia10, Xia12],from which the integrality of the conductors for Abbes-Saito’s definition
follows.

INTEGRAL p-ADIC HODGE THEORY

25

4. Sketch of proofs
In this section, we sketch the proofs of Theorem 2.9 and Theorem 2.19,
following [FL82] and [Fon93]. The key ideas for the proofs are twofold:First, by a dévissage argument, we reduce a large part of the proofs to af,[0,p−1]
study of simple objects in the category MFk̄, which we can classify
completely. Second, passing from the period ring Acrys /pAcrys to OK̄ /pOK̄
by using the first projection pr1 : R → OK̄ /pOK̄ , we reduce a study of the∗ (M ) to a study of equations over W and their solutions in
GK0 -module Tcrys
OK̄ . The latter enables us especially to prove the aforementioned estimateof ramification. Let us mention that another proof of the Fontaine-Laffailletheory was given by Gao [Gao17b].
Note that, as we remarked before (Remark 2.7), the original arguments
in [FL82] used the subring SFL instead of Acrys , while almost verbatim arguments are valid also for our period ring Acrys which is more standard thesedays. To mediate these two period rings and also to make the notation
simpler, in this article we will mainly work on the ringA = Im(SFL /pSFL → Acrys /pAcrys ).
4.1. Cutting off higher divided powers. Put RDP = Acrys /pAcrys . We
[0,p−1]as before. Note that a divided
consider this ring as an object of MFk
power structure on this ring is induced from that of Acrys : we put the l-th
divided power γl (x) of x ∈ Fil1 RDP = Fil1 Acrys /pFil1 Acrys by
γl (x) = γl (x̂) mod pAcrys =x̂lmod pAcrysl!

with any lift x̂ of x in Fil1 Acrys . The ring RDP is also the divided powerenvelope of the surjection pr0 : R → OK̄ /pOK̄ by [BO78, Remarks 3.20 (8)].
Recall that we have defined the element β = (−p, (−p)1/p , . . .) of the ring
R before. The surjection pr1 : R → OK̄ /pOK̄ induces a σ −1 -semilinear
isomorphism of rings R/β p R → OK̄ /pOK̄ sending β to (−p)1/p . Since β p =p!γp (β) = 0 in the ring RDP , we can consider the ring RDP as an (OK̄ /pOK̄ )algebra by the map OK̄ /pOK̄ ' R/β p R → RDP . For any divided power
ideal I of a ring and x ∈ I, we put δ(x) = (p − 1)!γp (x) and write δ l (x) =(δ ◦ · · · ◦ δ)(x) for the l-times iteration of δ. Then we can describe thestructure of the ring RDP as follows:
Lemma 4.1. The ring homomorphism(OK̄ /pOK̄ )[Y1 , Y2 , . . .]/(Y1p , Y2p , . . .) → RDP
sending Yl to δ l (β) is a σ-semilinear bijection, by which we identify bothsides.

26

SHIN HATTORI

Proof. Put ξ = p + [β] ∈ W (R) as before. It is enough to show a similar
W (R)-linear map
W (R)[Y1 , Y2 , . . .]/(ξ p −pY1 , Y1p −pY2 , . . .) → W (R)DP = W (R)[ξn| n ∈ Z≥0 ]n!

defined by Yl 7→ δ l (ξ) is an isomorphism. From the definition, we see thatthe map is surjective, and isomorphic after inverting p. Thus it is enoughto show that the ring on the left-hand side is p-torsion free. Putp− pYl ).
Bl = W (R)[Y1 , . . . , Yl ]/(ξ p − pY1 , . . . , Yl−1

For any ring B without p-torsion and any element x ∈ B, if the image of x
is a regular element of B/pB, then B/xB is p-torsion free. Since the imageof ξ p is a regular element of R, we see inductively that Bl [Yl+1 ]/(Ylp − pYl+1 )is p-torsion free. Taking the inductive limit yields the claim.
Using this identification, we can explicitly describe additional structuresof RDP such as GK0 -action, filtration and Frobenius maps (see [FL82, 5.9],p
where they consider the subring SFL = W (R)[ ξp ] of Acrys with similar explicit description modulo p). The action of g ∈ GK0 is written as
XXg(al1 ,...,ln )Y1l1 · · · Ynln ,g(al1 ,...,ln Y1l1 · · · Ynln ) =l1 ,...,ln

l1 ,...,ln

where al1 ,...,ln ∈ OK̄ /pOK̄ . Indeed, from the definition of the divided power
structure on RDP , we have δ(x) = δ(x̂) mod p for any x ∈ Fil1 RDP , where x̂
is any lift of x in Fil1 Acrys . Putting g(β) = βεa with ε = (1, ζp , ζp2 , . . .) anda ∈ Zp , we haveg(δ(β)) = δ(g(ξ)) mod p = δ(ξ + [β]([ε]a − 1)) mod p
 
p−1
X
−1 p
ξ l ([β]([ε]a − 1))p−l + [β]p δ([ε]a − 1) mod p= δ(ξ) +
p
l
l=1
 
p−1
X
p
−1 p
= δ(β) + βp(εa − 1)p−l + β p δ(εa − 1) = δ(β),
ll=1since β p = 0 in RDP . Thus we obtain g(Y1 ) = Y1 and g(Yl ) = Yl for any l.
Moreover, the ideal Fili RDP of RDP is written as
Fili RDP = ((−p)i/p , Yl | l ≥ 1) (i ≤ p − 1),

Filp RDP = (Yl | l ≥ 1).
[0,p−1]by putting (RDP )i =Recall that we consider RDP as an object of MFk
i DP
DP
p
Fil R for 0 ≤ i ≤ p − 1 and (R ) = 0. For any i ∈ {0, . . . , p − 1}, write
any element x in the i-th filtration asXx = x0 +xl Yl + (higher terms)l≥1

INTEGRAL p-ADIC HODGE THEORY

27

with x0 ∈ (−p)i/p (OK̄ /pOK̄ ) and xl ∈ OK̄ /pOK̄ . Let x̂0 be any lift of x0 in
OK̄ . From the equality (1) in §2.2, we obtain
 x̂p
i
0

(i ≤ p − 2),i (1 + Y1 ) (−p)px̂0
ϕi (x) =(1 + Y1 )i + xp1(i = p − 1, p ≥ 3),
(−p)i

p

 x̂0 (1 + Y )i + P xp (i = p − 1, p = 2).
1
l≥1 l
(−p)i
Remark 4.2. In particular, we have ϕ1 ((−p)1/p ) = 1 + Y1 in the ring RDP .
This implies that, contrary to what is stated in [Wac97, 2.1.3.4], there exists
no non-trivial ideal I of RDP stable under ϕp−1 such that ϕ1 ((−p)1/p ) ≡1 mod I. Indeed, this forces Y1 ∈ I and thus I contains ϕp−1 (Y1 ) = 1.
Now we put
A = (OK̄ /pOK̄ )[Y1 ]/(Y1p ) ⊆ RDP[0,p−1]and give it the induced structure as an object of MFk
Namely, we put the i-th filtration Ai of A as
Ai = ((−p)i/p , Y1 ) (i ≤ p − 1),from RDP .

Ap = 0and the i-th Frobenius map of A aspp−1
 x̂0 (1 + Y1 )iX(i ≤ p − 2),
(−p)i
i
l
p
ϕ (x0 + x1 Y1 +xl Y1 ) =x̂
p0
(1 + Y1 )i + x1 (i = p − 1).
l=2
(−p)i
Then we have ϕi (Fili RDP ) ⊆ A for any i ≤ p − 1. Consider the quotient[0,p−1]
RDP /A as an object of MFk
with the induced structure from RDP . It
satisfies (RDP /A)p−1 = RDP /A and ϕp−1 = 0 on RDP /A. Then the following∗ (M )
lemma enables us to replace RDP by the simpler ring A to study Tcrys
for the case of pM = 0.
f,[0,p−1]

Lemma 4.3. For any object M ∈ MFk
HomMFk (M, RDP, we have/A) = Ext1MFk (M, RDP /A) = 0.

In particular, we have∗
Tcrys
(M ) = HomMFk (M, A),

Ext1MFk (M, RDP ) = Ext1MFk (M, A).

This follows from the lemma below:
Lemma 4.4. Let r ∈ {0, . . . , p − 1} be an integer and M an object off,[0,r]MFk. Let N be an object of MFk such that N r = N and that for any
x ∈ N , the iteration (ϕrN ◦ · · · ◦ ϕrN )(x) of any sufficiently large times is zero.
Then we have HomMFk (M, N ) = Ext1MFk (M, N ) = 0.
Proof. Fix a k-basis e1 , . . . , eh of M with en ∈ M in \ M in +1 and write as
ϕiMn (en ) =

h
X
m=1

cm,n em28

SHIN HATTORI

with P = (cm,n ) ∈ GLh (k). Let f : M → N be a morphism of MFk and put
f (en ) = xn ∈ N in = N r = N . Then the equality ϕiN ◦ f = f ◦ ϕiM implies(x1 , . . . , xh ) = Φ(x1 , . . . , xh )P −1
with Φ = (ϕiN1 , . . . , ϕiNh ). By assumption, the l-th iteration Φl (x1 , . . . , xh ) iszero for any sufficiently large l and thus we obtain f = 0 by recursion.
Next consider an extension in MFk
/E

/N

0

/M

/ 0.

Choose a lift ên of en to E in . We have
ϕiEn (ên ) =hXcm,n êm + bn

m=1

with some bn ∈ N . Then the extension splits if and only if there exists
an ∈ N in satisfying
ϕiEn (ên + an ) =hXcm,n (êm + am ),m=1

which is equivalent to
bn + ϕiNn (an ) =hXcm,n am .

m=1

Thus we obtain(a1 , . . . , ah ) = (b1 , . . . , bh )P −1 + Φ(a1 , . . . , ah )P −1 .
By recursion, we see that the equation has the unique solution
∞
X
l
l−1
(a1 , . . . , ah ) =Φl (b1 , . . . , bh )(P −1 )σ (P −1 )σ · · · P −1 ,l=0since the right-hand side is a finite sum.

Let F be any algebraic extension of K0 in K̄. We put
AF = (OF /pOF )[Y1 ]/(Y1p ) ⊆ A = AK̄
[0,p−1]and give this subring the induced structure as an object of MFk
f,[0,p−1]
A. For M ∈ MFk, we define

from∗Tcrys,F(M ) = HomMFk (M, AF ).
∗ (M ) on which the absolute Galois group G acts
This is a subset of Tcrys
Ftrivially.
f,[0,r]∗, the module Tcrys,F(M ) can also
For r ∈ {0, . . . , p − 2} and M ∈ MFk
be obtained by using a far simpler ring than AF . PutrbF = {x ∈ OF | vp (x) >}.
p−1

INTEGRAL p-ADIC HODGE THEORY

29

[0,r]

We consider the quotient OF /bF as an object of MFki(OF /bF )i = {x ∈ OF | vp (x) ≥ }/bF ,p

ϕi (x) =by puttingx̂p
mod bF(−p)i

for any 0 ≤ i ≤ r and (OF /bF )r+1 = 0. Here x̂ denotes any lift of x in OF .
f,[0,r], the surjecLemma 4.5. For any r ∈ {0, . . . , p − 2} and any M ∈ MFktion AF → OF /bF defined by Y1 7→ 0 induces an isomorphism∗
Tcrys,F(M ) → HomMFk (M, OF /bF ).

Proof. Since r < p − 1, the map ϕrAF kills Y1 . Moreover, for any x̂ ∈ OF
satisfying vp (x̂) = r/(p − 1) + ε with some ε > 0, we have vp (x̂p /(−p)r ) ≥r/(p − 1) + pε. Thus the ideal I = bF AF + (Y1 ) of AF , which is contained in
the r-th filtration ArF , is stable under ϕrAF and, for any x ∈ I, the iteration(ϕrAF ◦ · · · ◦ ϕrAF )(x) of any sufficiently large times is zero. Thus, by considering I as an object of MFk with the induced structure from AF , Lemma 4.4
yields HomMFk (M, I) = Ext1MFk (M, I) = 0. Since I = Ker(AF → OF /bF ),the long exact sequence of Hom and Ext implies the lemma.
For r = 0, the ideal bK̄ is equal to the maximal ideal mK̄ and we obtainthe following corollary.
f,[0,0]Corollary 4.6. For any M ∈ MFk
ified.

∗ (M ) is unram, the GK0 -module Tcrys

4.2. Classification of simple objects. Let h be a positive integer and i a
map Z/hZ → Z. We write in = i(n) for any n ∈ Z/hZ. We suppose that i isof period h. This means that h is the minimum among the positive integers
h0 satisfying in+h0 = in for any n.
For such h and i, we define an object M (h; i) of the category MFfk̄ byM
M (h; i) =k̄enn∈Z/hZ

with a basis {en }n∈Z/hZ and
M (h; i)l =Mk̄en ,ϕin (en ) = en−1 .

in ≥lIn particular, en ∈ M (h; i)in \ M (h; i)in +1 . When h = 1 and i(0) = j ∈ Z,
we write M (h; i) as M (1; j).
We define an Fph -action on this object by[a] ∈ EndMFk̄ (M (h; i)),−n[a](en ) = ap

enfor any a ∈ Fph . Then we can show the following proposition.
Proposition 4.7.
(i) ([FL82], Proposition 4.4) An object M ∈ MFfk̄ issimple if and only if M is isomorphic to M (h; i) for some h ∈ Z>0
and i : Z/hZ → Z of period h.

30

SHIN HATTORI(ii) ([FL82], Lemme 4.9) Let (h, i) be a pair of a positive integer h anda map i : Z/hZ → Z of period h. Let (h0 , i0 ) be a similar pair.
Then HomMFk̄ (M (h; i), M (h0 ; i0 )) = 0 unless h = h0 and i = i0 up
to a shift (namely, there exists an integer m satisfying in = i0n+m
for any n). In the latter case the two objects are isomorphic, andEndMFk̄ (M (h; i)) ' Fph via a 7→ [a].
(iii) ([FL82], Lemme 6.3, 6.4) Ext1MFk̄ (M, M (1; p − 1)) = 0 for any M ∈f,[0,p−1]

MFk̄.

The assertions (i) and (ii) follow from a semilinear algebraic argument
whose key point is that any σ h -semilinear automorphism of a finite dimensional k̄-vector space has a non-zero fixed vector. The last assertion is shownby dévissage, after explicitly constructing a split of any extension of M (h; i)by M (1; p − 1) for i : Z/hZ → {0, . . . , p − 1} as in the proof of Lemma 4.4.
f,[0,p−1]
is in the full
Proposition 4.7 (iii) implies that an object M of MFk̄f,[0,p−1]0

subcategory MFk̄
if and only if every Jordan-Hölder factor of M is
not isomorphic to M (1; p − 1). This leads to the following corollary.
Corollary 4.8 ([FL82], Théorème 6.1 (i)). Let0/ M0f,[0,p−1]

be an exact sequence of MFW,tor
and only if so are M 0 and M 00 .

/M

/ M 00/0f,[0,p−1]0. Then M is an object of MFW,tor
f,[0,p−1]0

Proof. For this, it is enough to show that if M ∈ MFW,tor
f,[0,p−1]0
W (k̄) ∈ MFW (k̄),tor .

ifthen M ⊗W

Suppose that there exists a non-zero quotient M ⊗W

W (k̄) → N satisfying N = N p−1 . Then we can find a finite extension
k 0 /k in k̄ such that this quotient is a base extension of a quotient M ⊗W
f,[0,p−1]
W (k 0 ) → N 0 in MFW (k0 ),tor satisfying N 0 = (N 0 )p−1 . Since k 0 /k is finite, thef,[0,p−1]

latter quotient can be viewed as a quotient in MFW,tor . In this category
M ⊗W W (k 0 ) is the direct sum of finite copies of M . The image N0 of acopy of M in N 0 is non-zero, and satisfies N0 = N0p−1 by Proposition 2.5,
which is the contradiction.

Next we study the Galois representations associated with the simple obf,[0,p−1]
jects of MFk̄, which is one of the key ingredients of [FL82].
Proposition 4.9 ([FL82], Théorème 5.1). Let h be a positive integer andi : Z/hZ → {0, 1, . . . , p − 1} a map of period h.
(i) Ext1MFk̄ (M (h; i), RDP ) = 0.
∗ (M (h; i)) is an F (ii) Via the Fph -action on M (h; i), the module Tcrys
phvector space of dimension one.

INTEGRAL p-ADIC HODGE THEORY

31∗ (M (h; i)) is given by the character(iii) The IK0 -action on Tcrys
i +pi1 +···ph−1 ih−1

θh0,
: IK0 → F×
phwhere θh is the fundamental character of level h.
Proof. By Lemma 4.3, we may replace RDP with A to compute Hom andExt1 groups. First consider any extension
/A

0

/E

/ M (h; i)/0in MFk̄ . Let {en }n∈Z/hZ be the basis of M (h; i) as before, and take a lift ênof en in E in . We have ϕin (ên ) = ên−1 − dn−1 with some dn−1 ∈ A. Thenthe extension splits if and only if there exists un ∈ Ain such thatϕin (ên + un ) = (ên−1 + un−1 )for any n ∈ Z/hZ, which is equivalent toϕin (un ) − un−1 = dn−1 .
Since this equation is linear, to prove that it has a solution, we may assume
that dn = 0 except a single index n = n0 . By permuting the indices, we may
assume n0 = 0. On the other hand, a k̄-linear map M (h; i) → A defined by
en 7→ un ∈ Ain gives a morphism of MFk̄ if and only if
ϕin (un ) − un−1 = 0
for any n ∈ Z/hZ. Hence we reduced ourselves to showing that the equation un ∈ Ain ,ϕin (un ) − un−1 = 0 (n 6= 1),(2)
 in
ϕ (u1 ) − u0 = d
has a solution for any d ∈ A in the case of the assertion (i), and has exactly
ph solutions for d = 0 in the case of the assertion (ii).
Putun = an + bn Y1 +

p−1
X
l=2

in

cn,l Y1l ,d = d0 + d1 Y1 +p−1
Xdl Y1ll=2

with some an ∈ (−p) p (OK̄ /pOK̄ ) and bn , cn,l , dl ∈ OK̄ /pOK̄ . We also put
ε(n) = 0 for n 6= p − 1 and ε(p − 1) = 1. Then the equation (2) is equivalenttoin an ∈ (−p) p (OK̄ /pOK̄ ), bn , cn,l ∈ OK̄ /pOK̄ ,p ânin (1 + Y1 )in + ε(in )bpn − un−1 = 0 (n 6= 1)(−p)
d (n = 1).

32

SHIN HATTORI

From this we see that cn,l ’s are uniquely determined by an ’s, and the solutions of the equation (2) correspond bijectively with the solutions ofinp (O /pO ), b ∈ O /pO ,

a
∈
(−p)
n
n

K̄
K̄
K̄
K̄



 âpn0 (n 6= 1)p+ ε(in )bn − an−1 =(−p)in(3)d0 (n = 1),



p

0
(n
=
6
1)
â

 in nin − bn−1 =(−p)
d1 (n = 1).
First let us assume M (h; i) 6= M (1; p − 1), and suppose d = 0. Then we
can show ε(in )bpn = 0 for any n, by a valuation calculation [FL82, 5.11 (iii)].
Thus bn ’s are also uniquely determined by an ’s in this case. For an ’s, we canshow that any solution (an ) of the equation (3) uniquely lifts to a solution
in OK̄ of the equation of degree p
Xnp− Xn−1 = 0 for any n(−p)inover W [FL82, Lemme 5.12].
Let K0tr be the maximal tamely ramified extension of K0 in K̄. For anyρ ∈ Z(p) , we write ρ = a/b with a ∈ Z and b ∈ Z>0 satisfying p - b.
Using [Bou56, §7, n◦ 4, Proposition 5], we fix once and for all (−p)ρ ∈ K0tr
00such that ((−p)ρ )b = (−p)a and (−p)ρ (−p)ρ = (−p)ρ+ρ for any such ρ, ρ0 .
∗ (M (h; i)) =Then, from the above equation, we see that the module Tcrys
HomMFk̄ (M (h; i), RDP ) is equal to{(en 7→ λp−n(−p)ρn (1 + Y1 )in+1 ) | λ ∈ Fph }h−1+···+pinwith ρn = in+1 +pin+2, which settles the assertions (ii) and (iii) forph −1
this case. The assertion (i) can be shown by constructing a solution of theequation (3) similarly from a solution in OK̄ of an equation over W [FL82,
5.11 (i), (ii)].
The case of M (1; p − 1) needs an extra care [FL82, 5.14]. We also havean equation over W to lift solutions, while its degree is p2 and lifts are notunique. Nevertheless, we can show for the case of d = 0 that the solutions of
the lifted equation have exactly p distinct reductions modulo p, which yields
the proposition for this case.


Now the assertions of Theorem 2.9 except the fullness follow formally.
∗ (M )
Indeed, Lemma 2.8 reduces assertions on the module structure of Tcrys
to the case of k = k̄. Moreover, for any object N ∈ MFk and any extension0/ Acrys,∞/E/N/ Acrys,∞×p/0

in MFW , the exact sequence0/ Acrys /pAcrys/ Acrys,∞/0INTEGRAL p-ADIC HODGE THEORY

33

and the snake lemma yield the extension0/ Acrys /pAcrys/ E[p]/N/0

in MFk with E[p] = Ker(p : E → E), and the induced map
Ext1MFW (N, Acrys,∞ ) → Ext1MFk (N, Acrys /pAcrys )∗is an injection. Using this, the exactness of Tcrys
in Theorem 2.9 (i) folf,[0,p−1]

lows by showing Ext1MFW (k̄) (M, Acrys,∞ ) = 0 for any M ∈ MFW (k̄),tor by∗dévissage from Proposition 4.9 (i). By combining this exactness of Tcrys
with Proposition 4.9 (ii), we obtain∗lgZp (Tcrys(M )) = lgW (M )f,[0,p−1]

for any M ∈ MFW,tor. Applying this to M [pl ] = Ker(pl : M → M ) for
f,[0,p−1]

any l shows Theorem 2.9 (ii). For any morphism f : M → N of MFW,tor∗ (f ) = 0, applying the above length equality to Im(f ) showssatisfying Tcrys∗
in Theorem 2.9 (i). Thus Theorem
f = 0 and settles the faithfulness of Tcrys∗
2.9 (iv) follows from Corollary 4.8 modulo the fullness of Tcrys(Theorem 2.9(iii)).
∗ . Now we prove Theorem 2.9 (iii). First we may
4.3. The fullness of Tcrys∗
assume k = k̄ by a Galois descent argument using the faithfulness of Tcrys[FL82, 6.2]. By Proposition 4.7 and Proposition 4.9, the natural map∗∗
HomMFW (k̄) (M, N ) → HomZp [IK0 ] (Tcrys(N ), Tcrys(M ))is an isomorphism if M and N are simple. In order for a dévissage argument
to imply the theorem, we need to show that the natural map∗∗
Ext1MFW (k̄) (M, N ) → Ext1Zp [IK ] (Tcrys(N ), Tcrys(M ))0f,[0,p−1]0

is an injection for any objects M, N ∈ MFW (k̄),tor . To prove this injectivity∗
and the fullness of Tcrys
simultaneously, by dévissage and an induction on
the length we may assume that M and N are simple. By Theorem 2.9(ii), we only have to consider the extensions killed by p. For this case,∗ (M ) and T ∗ (N ) are tamely ramified by Proposition 4.9 (iii),
since Tcrys
crys∗ (N ) by T ∗ (M ) splits
Maschke’s theorem implies that an extension of Tcrys
crys
if and only if the extension is also tamely ramified. Thus we are reduced toshowing the following lemma.
Lemma 4.10 ([FL82], 6.11). Suppose k = k̄. Let M be an object off,[0,p−1]0∗ (M ) is tamely rami. Then M is semi-simple if and only if Tcrys
MFk̄
fied.
∗ (M )GF for
To prove the lemma, we need to describe the fixed part Tcrys
any algebraic extension F/K0 , which is also one of the key steps for the
proof of Theorem 2.19. This is essentially done again by uniquely lifting

34

SHIN HATTORI∗ (M ) to a solution in O
each element of Tcrys
K̄ of an equation over W andG
Fusing (OK̄ ) = OF , as follows.

Proposition 4.11 ([FL82], Proposition 6.7). Suppose k = k̄. Let M be an
f,[0,p−1]0
object of MFk̄and F/K0 an algebraic extension in K̄. Then GF acts
∗ (M ) if and only if T ∗ (M ) = T ∗trivially on Tcrys
crys
crys,F (M ).
Proof. We proceed by induction on the length of the Fontaine-Laffaille module M . When M is simple, this follows from the explicit description of∗ (M ) given in the proof of Proposition 4.9. Suppose that we have an
Tcrys
f,[0,p−1]0

exact sequence in MFk̄
0

/N

/M

/ M (h; i)/0∗ (M ). Then G
and that GF acts trivially on Tcrys
F also acts trivially on∗∗∗
Tcrys (N ) and thus Tcrys (N ) = Tcrys,F (N ) by the induction hypothesis. Take
the basis (en )n∈Z/hZ of M (h; i) as before and a lift ên ∈ M in of en . Thenϕin (ên ) = ên−1 + dn−1
∗ (M ). Sincewith some dn−1 ∈ N . Let f : M → A be an element of Tcrys∗ (N ) = T ∗f |N ∈ Tcrys
crys,F (N ), we can write asf (dn ) = αn + βn Y1 +p−1Xγn,l Y1ll=2with some αn , βn , γn,l ∈ OF /pOF . Putf (ên ) = an + bn Y1 +p−1
Xcn,l Y1l ∈ Ain

l=2
in
p

with an ∈ (−p) (OK̄ /pOK̄ ) and bn , cn,l ∈ OK̄ /pOK̄ . From ϕin (f (ên )) =f (ϕin (ên )), we obtain
 
in
âpn
cn−1,l =− γn−1,ll (−p)in
and thus cn,l ’s are uniquely determined by an ’s in a way that if an ∈ OF /pOF
for any n, then cn,l ’s are also in OF /pOF . Moreover, we see that to give∗ (M ) with f (d ) being as above is the same as to give a solutionf ∈ Tcrysn(an , bn )n∈Z/hZ of the equation
in
 an ∈ (−p) p (OK̄ /pOK̄ ), bn ∈ OK̄ /pOK̄ ,âpn+ ε(in )bpn − an−1 = αn−1 ,(−p)in in âpni − bn−1 = βn−1 .
(−p) nLet α̂n and β̂n be any lifts of αn and βn in OF , respectively. Then we
can show [FL82, Lemme 6.8] that any solution (an , bn )n∈Z/hZ of the above

INTEGRAL p-ADIC HODGE THEORY

35

equation uniquely lifts to a solution (xn , yn )n∈Z/hZ in OK̄ of the equation
over OF( Xpn+ ε(in )Ynp − Xn−1 = α̂n−1 ,(−p)inpXnin (−p)in − Yn−1 = β̂n−1 .

Since (g(an ), g(bn )) = (an , bn ) for any g ∈ GF by assumption, the uniquenessof the lifting implies (g(xn ), g(yn )) = (xn , yn ) for any g ∈ GF . Thus xn , yn ∈(OK̄ )GF = OF and an , bn ∈ OF /pOF for any n. Hence we obtain f (ên ) ∈ AF∗
and f ∈ Tcrys,F(M ).

For any ρ ∈ Z(p) ∩ [0, 1[, there exists a unique pair (h, i) of h ∈ Z>0 andi : Z/hZ → {0, . . . , p − 1} of period h satisfying
ρ=i1 + pi2 + · · · + ph−2 ih−1 + ph−1 i0.
ph − 1This gives a bijection between Z(p) ∩ [0, 1[ and the set of such (h, i)’s except(1, p − 1). Putω̄ρ = (−p)ρ (1 + Y1 )i1 ∈ AiK0 tr ,0

where (−p)ρ is chosen as before.

i

Then we have ϕi0 (ω̄ρ ) = ω̄pρ−i0 ∈ AKh−1
tr
0

with
pρ − i0 =

i0 + pi1 + · · · + ph−2 ih−2 + ph−1 ih−1
∈ Z(p) ∩ [0, 1[.
ph − 1

Repeating this, we see that the k̄-subspace generated byω̄ρ , ϕi0 (ω̄ρ ), ϕih−1 (ϕi0 (ω̄ρ )), . . .
f,[0,p−1]

which is isomorphic to M (h; i).
defines an object of MFk̄Put
Ass = Spank̄ {ω̄ρ | ρ ∈ Z(p) ∩ [0, 1[} ⊆ AK0tr
and we give this subspace the induced structure from AK0tr as an object ofMFk̄ . Then we see that Ass is isomorphic to the direct sum of representatives
f,[0,p−1]0
of all isomorphism classes of simple objects of MFk̄. Moreover, we canf,[0,p−1]0

show by induction on the length that M ∈ MFk̄
only if the natural injection

is semi-simple if and∗
HomMFk̄ (M, Ass ) → Tcrys(M )is bijective.
Thus we are reduced to comparing AK0tr and Ass . Consider the quotient
AK0tr /Ass '

p−1
M
l=1

(OK0tr /pOK0tr )Y1l .

36

SHIN HATTORI

Then we have AK0tr /Ass = (AK0tr /Ass )p−1 . Moreover, a simple calculation as
in [FL82, 6.10] shows that ϕp−1 = 0 on this quotient. By Lemma 4.4, we
obtain HomMFk̄ (M, AK0tr /Ass ) = 0 and thus
HomMFk̄ (M, Ass ) = HomMFk̄ (M, AK0tr ).
The equality shows that M is semi-simple if and only if HomMFk̄ (M, AK0tr ) =∗ (M ), which is the same as saying that T ∗ (M ) is tamely ramified by
Tcrys
crys
Proposition 4.11. This concludes the proofs of Lemma 4.10 and Theorem
2.9 (iii).
4.4. Ramification bound. To prove Theorem 2.19, we need a lemma of
Fontaine (which was improved later by Yoshida) relating ramification to aramified variant of formal smoothness, as follows. For any algebraic extension E/K in K̄ and any non-negative real number m, putam
E/K = {x ∈ OE | vK (x) ≥ m}.
For any finite Galois extension L/K, consider the property For any algebraic extension E/K, if there exists
an OK -algebra homomorphism OL → OE /am(Pm )L/K
E/K ,
then there exists a K-algebra injection L → E.
Note that we do not require the map L → E to be a lift of the given map
OL → OE /am
E/K . The property (Pm )L/K is related to uL/K by the following
lemma.
Lemma 4.12 ([Fon85], Proposition 1.5, [Yos10], Proposition 3.3).
uL/K = inf{m ∈ R≥0 | (Pm )L/K holds}.
For this, Krasner’s lemma implies that (Pm )L/K holds for any m > uL/K .
Conversely, by explicitly constructing counterexamples for (Pm )L/K , we canshow that if (Pm )L/K holds, then m > uL/K − e−1
L/K , where eL/K is the relative ramification index of the extension L/K [Fon85]. Furthermore, we cankill the error term by using arbitrarily large tamely ramified base extensions[Yos10].
f,[0,r]and L/K0 be as in Theorem 2.19. Since
Now let r < p − 1, M ∈ MFk∗ (M ), by Lemma 2.8
the theorem concerns only about the IK0 -action on Tcrys
we may assume k = k̄. Since r < p − 1, Lemma 4.5 gives the identification
∗
∗
Tcrys
(M ) = HomMFk̄ (M, OK̄ /bK̄ ), Tcrys,F(M ) = HomMFk̄ (M, OF /bF )for any algebraic extension F/K0 in K̄. By Lemma 4.12, it is enough tocheck the property (Pm )L/K0 for any m > 1 + r/(p − 1).
Let E/K0 be an algebraic extension in K̄, m a real number satisfyingm > 1 + r/(p − 1) and η : OL → OE /am
E/K0 a W -algebra homomorphism.

INTEGRAL p-ADIC HODGE THEORY

37

Let P (X) be the minimal polynomial over W of a uniformizer πL of L. We
write aseL/K0 −1
X
eL/K0
P (X) = X+
pcs X ss=0

∈ W ×.

with some cs ∈ W such that c0
Let x̂ be a lift of η(πL ) in OE .
Since P (η(πL )) = 0 in the ring OE /am,
E/K0 we have P (x̂) + δ = 0 with someδ ∈ OE satisfying vp (δ) ≥ m > 1 + r/(p − 1). From the Newton polygon of
the polynomial P (X) + δ, we obtain vp (x̂) = e−1
L/K0 = vp (πL ). Hence we seethat η induces an injectionη̄ : OL /bL → OE /bE
and that η̄ respects filtrations. We claim that η̄ also respects ϕi ’s. Indeed,
note that the i-th filtration (OL /bL )i is spanned over k̄ by
eL/K0{πLj | j ∈ Z, j ≥i}.
p
For such j, put pj = eL/K0 i + l. Then−1eϕ (πLj ) =
i

L/K0
X
πLpj
l
mod
b
=
π
(
L
L
(−p)i

cs πLs )i mod bL ,

X
x̂pj
l
mod
b
=
x̂
(
E
(−p)i

δ
cs x̂s + )i mod bE .
p

s=0
eL/K0 −1ϕi (η̄(πLj )) =s=0

Since pδ ∈ bE , we obtain η̄(ϕi (πLj )) = ϕi (η̄(πLj )), which proves the claim.
Thus the map η̄ defines an injection of the category MFk̄ and induces aninjection∗∗
Tcrys,L(M ) → Tcrys,E
(M ).
∗
∗
Now Proposition 4.11 shows Tcrys (M ) = Tcrys,L (M ). Thus we also have∗
∗ (M ) = T ∗Tcrys
crys,E (M ) and GE acts trivially on Tcrys (M ). From the definition of L, we obtain GE ⊆ GL and L ⊆ E, namely the property (Pm )L/K0
holds for any m > 1 + r/(p − 1). This concludes the proof of Theorem 2.19
for r < p − 1.
f,[0,p−1]0
For the case of r = p − 1 and M ∈ MFk, it is enough to check theproperty (Pm )L/K0 for any m > 2 = 1 + r/(p − 1). Consider a W -algebra
homomorphism η : OL → OE /am
E/K0 . We see in the same way as abovethat the induced injection η̄ : OL /pOL → OE /pOE and Y1 7→ Y1 define
an injection AL → AE which is compatible with filtrations and Frobenius∗∗
structures. Thus we have an injection Tcrys,L(M ) → Tcrys,E(M ) also in thiscase. Hence Proposition 4.11 yields the property (Pm )L/K0 and the desired
ramification bound.
Remark 4.13. As we mentioned before, [FL82] uses the subring SFL =p
W (R)[ ξp ] instead of Acrys to construct a GK0 -representation associated with

38

SHIN HATTORI
f,[0,p−1]

M ∈ MFW,tor. We can show that these two constructions are naturally
f,[0,p−1]

isomorphic: we claim that for any M ∈ MFW,tor, the natural map∗
HomMFW (M, SFL,∞ ) → HomMFW (M, Acrys,∞ ) = Tcrys(M )is bijective andExt1MFW (M, SFL,∞ ) → Ext1MFW (M, Acrys,∞ )is injective. Indeed, by dévissage we may assume pM = 0. By [FL82,
Lemme 5.7], we have a natural identification (OK̄ /pOK̄ )[Y1 ] ' SFL /pSFL
defined by Y1 7→ δ(ξ) mod p. Since ϕp−1 (Y1p ) = 0 in this ring [FL82, 5.9],
Lemma 4.4 implies that the natural map SFL /pSFL → A induces a bijection∗(M )HomMFk (M, SFL /pSFL ) ' HomMFk (M, A) = Tcrys

and an injection
Ext1MFk (M, SFL /pSFL ) → Ext1MFk (M, A) = Ext1MFk (M, Acrys /pAcrys ).
The former one settles the assertion on Hom. Moreover, as in the proof of
Theorem 2.9 (i), we have a natural injection
Ext1MFW (M, SFL,∞ ) → Ext1MFk (M, SFL /pSFL )and the assertion on Ext1 follows from the commutative diagram
Ext1MFW (M, SFL,∞ ) Ext1MFW (M, Acrys,∞ ) / Ext1

/pSFL )MFk (M, S
 FL/ Ext1_

MFk (M, Acrys /pAcrys ).

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[Xia10]L. Xiao: On ramification filtrations and p-adic differential modules, I: the equalcharacteristic case, Algebra Number Theory 4 (2010), no. 8, 969–1027.
[Xia12]
L. Xiao: On ramification filtrations and p-adic differential equations, II: mixed
characteristic case, Compos. Math. 148 (2012), no. 2, 415–463.
[Yos10] M. Yoshida: Ramification of local fields and Fontaine’s property (Pm ), J. Math.
Sci. Univ. Tokyo 17 (2010), no. 3, 247–265.
[Zhu03] I. Zhukov: On ramification theory in the case of an imperfect residue field, Mat.
Sb. 194 (2003), no. 12, 3–30; translation in Sb. Math. 194 (2003), no. 11-12,
1747–1774.
[Zin02]
T. Zink: The display of a formal p-divisible group, Cohomologies p-adiques etapplications arithmétiques I, Astérisque 278 (2002), 127–248.
Email address: hattoris@tcu.ac.jp
Department of Natural Sciences, Tokyo City University