Instructions for use A uthor(s ) Hattori,S hin
C itation Hokkaido University Preprint S eries in Mathematics, 917: 1-30
Is s ue D ate 2008-07-16
D O I 10.14943/84066
D oc UR L http://hdl.handle.net/2115/69724
T ype bulletin (article)
F ile Information pre917.pdf
ON A RAMIFICATION BOUND OF SEMI-STABLE TORSION REPRESENTATIONS OVER A LOCAL FIELD
SHIN HATTORI
Abstract. Let p be a rational prime, k be a perfect field of char-acteristic p, W = W(k) be the ring of Witt vectors, K be a finite totally ramified extension of Frac(W) of degree e and r be a non-negative integer satisfyingr < p−1. LetV be a semi-stablep-adicGK -representation with Hodge-Tate weights in{0, . . . , r}. In this paper, we prove the upper numbering ramification group G(Kj) forj > u(K, r, n) acts trivially on the mod pn representations associated to V, where u(K,0, n) = 0, u(K,1, n) = 1 +e(n+ 1/(p−1)) and u(K, r, n) = 1−p−ne(K(ζ
p)/K)−1+e(n+r/(p−1)) forr >1.
1. Introduction
Let p be a rational prime, k be a perfect field of characteristic p, W =
W(k) be the ring of Witt vectors andKbe a finite totally ramified extension ofK0 = Frac(W) of degreee=e(K). Let the maximal ideal ofKbe denoted
bymK, an algebraic closure of K by ¯K and the absolute Galois group ofK
byGK = Gal( ¯K/K). We normalize the valuationvK ofKasvK(p) =eand
extend this to ¯K. Let G(Kj) denote the j-th upper numbering ramification group in the sense of [7]. Namely, we put G(Kj) =GjK−1, where the latter is the upper numbering ramification group defined in [15].
Let XK be a proper smooth scheme over K and put XK¯ = XK ×K
¯
K. Consider the r-th ´etale cohomology group H´etr(XK¯,Qp) and its GK
-stableZp-lattices L ⊇ L′. In [7], Fontaine conjectured the upper numbering
ramification groupG(Kj) acts trivially on theGK-moduleL/L′ forj > e(n+ r/(p−1)) ifXKhas good reduction and this module is killed bypn. Fore= 1
and r < p−1, this conjecture was proved independently by himself ([8], for
n= 1) and Abrashkin ([2], for anyn), using the theory of Fontaine-Laffaille ([10]) and the comparison theorem of Fontaine-Messing ([11]) between thep -adic ´etale cohomology groups ofXK and the crystalline cohomology groups
of the reduction ofXK. From this result, Fontaine also showed some rareness
of a proper smooth scheme over Q with everywhere good reduction ([8, Th´eor`eme 1]). In fact, they proved this ramification bound for the torsion
Date: June 16, 2008.
Supported by 21st Century COE program “Mathematics of Nonlinear Structure via Singularity” at Department of Mathematics, Hokkaido university.
representations of the crystallinep-adic representations ofGK with
Hodge-Tate weights in{0, . . . , r}in the case where K is absolutely unramified. On the other hand, for a semi-stablep-adic representationV with Hodge-Tate weights in the same range, a similar ramification bound for e = 1 and n= 1 is obtained by Breuil (see [4, Proposition 9.2.2.2]). He showed, assuming the Griffiths transversality which in general does not hold, that if
e= 1 and r < p−1, then the ramification group G(Kj) acts trivially on the mod p representations ofV forj >2 + 1/(p−1).
In this paper, we prove a version of the result of Breuil for the case where
K is absolutely ramified, under the conditionr < p−1. Our main theorem is the following.
Theorem 1.1. Let r be a non-negative integer such that r < p−1. Let
V be a semi-stable p-adic GK-representation with Hodge-Tate weights in
{0, . . . , r} and L ⊇ L′ be GK-stable Zp-lattices in V. Suppose that the
quotient L/L′ is killed by pn. Then the j-th upper numbering ramification
group G(Kj) acts trivially on the GK-module L/L′ for j > u(K, r, n), where
u(K, r, n) =
0 (r= 0),
1 +e(n+p−11) (r= 1),
1−pne(K(1ζ
p)/K)+e(n+
r
p−1) (r >1)
and e(K(ζp)/K) denotes the relative ramification index of the extension K(ζp)/K.
We can check that this bound is sharp for r ≤ 1 (Remark 5.13). From this theorem and [7, Proposition 1.3], we have the following corollary.
Corollary 1.2. Let the notation be as in the theorem andLbe the finite ex-tension ofK cut out by theGK-moduleL/L′. LetDL/K denote the different
of the extension L/K. Then we have the inequality
vK(DL/K)< u(K, r, n)
for r >0 and vK(DL/K) = 0 for r = 0.
For the proof of Theorem 1.1, we essentially follow a beautiful argument of Abrashkin ([2]). We may assumep≥3 andr≥1. Thanks to Liu’s theorem ([14]) on the GK-stable Zp-lattices in semi-stable p-adic representations, it
is enough to bound the ramification of theGK-module Tst∗,π(Mn) = HomS,Filr,φ
r,N(Mn,Aˆst,∞),
whereMn is apn-torsion object of a category Modr,φ,N/S∞ of filtered (φr, N
)-modules overSdefined by Breuil ([3]) and ˆAst,∞is ap-adic period ring. We
may also assume ζp ∈K and consider the finite Galois extension Fn=K(π1/p
n
, ζpn+1)
Then we bound the ramification of Ln over K. For this, we show that to
study this GFn-module we can use a variant over a smaller coefficient ring
Σ of filtered (φr, N)-modules overS. In precise, let E(u) be the Eisenstein
polynomial of a uniformizer π of K overW and we set Σ =W[[u, E(u)p/p]].
This ring Σ is small enough for the method of Abrashkin, in which he uses filtered modules of Fontaine-Laffaille ([10]) whose coefficient ring is W, to work also in the case whereK is absolutely ramified.
Acknowledgments. The author would like to pay his gratitude to Iku Nakamura and Takeshi Saito for their warm encouragements. He wants to thank Manabu Yoshida for kindly allowing him to include the proof of Proposition 5.6. He also wants to thank Ahmed Abbes, Takeshi Tsuji, Xavier Caruso and especially Yuichiro Taguchi, for useful discussions and comments.
2. Filtered (φr, N)-modules of Breuil
In this section, we recall the theory of filtered (φr, N)-modules over S
of Breuil, which is developed by himself and most recently by Caruso and Liu (see for example [3], [5], [14], [6]). In what follows, we always take the divided power envelope of aW-algebra with the compatibility condition with the natural divided power structure onpW.
Let p≥3 be a rational prime and σ be the Frobenius endomorphism of
W. We fix once and for all a uniformizerπ of K and a system{πn}n∈Z≥0 of
p-power roots of π such thatπ0=π and πn=πpn+1 for anyn. Let E(u) be
the Eisenstein polynomial of π overW and set S = (W[u]PD)∧, where PD means the divided power envelope and this is taken with respect to the ideal (E(u)), and ∧ means the p-adic completion. The ring S is endowed with the σ-semilinear endomorphism φ : u 7→ up and a natural filtration FiltS
induced by the divided power structure such that φ(FiltS) ⊆ ptS for any non-negative integer t. We setφt=p−tφ|FiltS and c=φ1(E(u))∈S×. Let
N denote the W-linear derivation onS defined by the formula N(u) =−u. We also define a filtration, φ,φt,N on Sn=S/pnS similarly.
Let r ∈ {0, . . . , p−2} be an integer. Set ′Modr,φ,N/S to be the category consisting of the following data:
• anS-moduleMand its S-submodule FilrMcontaining FilrS.M, • aφ-semilinear mapφr: FilrM → Msatisfying
φr(srm) =φr(sr)φ(m)
for anysr∈FilrSandm∈ M, where we setφ(m) =c−rφr(E(u)rm),
• aW-linear mapN :M → Msuch that
– N(sm) =N(s)m+sN(m) for any s∈S and m∈ M,
– the following diagram is commutative:
FilrM φr
−−−−→ M
E(u)N y
ycN
FilrM −−−−→
φr
M,
and the morphisms of′Modr,φ,N
/S are defined to be theS-linear maps
preserv-ing Filr and commuting with φr and N. The category defined in the same
way but dropping the dataN is denoted by′Modr,φ
/S. These categories have
obvious notions of exact sequences. Let Modr,φ,N/S
1 denote the full
subcate-gory of′Modr,φ,N
/S consisting ofMsuch thatMis free of finite rank overS1
and generated as an S1-module by the image of φr. We write Modr,φ,N/S∞ for
the smallest full subcategory which contains Modr,φ,N/S
1 and is stable under
extensions. We let Modr,φ,N/S denote the full subcategory consisting of M such that
• theS-moduleMis free of finite rank and generated by the image of
φr,
• the quotient M/FilrMisp-torsion free.
We define full subcategories Modr,φ/S
1, Mod r,φ
/S∞ and Mod
r,φ
/S of ′Mod r,φ /S in a
similar way. For ˆM ∈ Modr,φ,N/S (resp. Modr,φ/S), the quotient ˆM/pnMˆ has a natural structure as an object of Modr,φ,N/S
∞ (resp. Mod
r,φ /S∞).
Forp-torsion objects, we also have the following categories. Consider the
k-algebrak[u]/(uep)=∼S1/FilpS1and let this algebra be denoted by ˜S1. The
algebra ˜S1 is equipped with the natural filtration,φandN induced by those
of S. Namely, FiltS˜1 = uetS˜1, φ(u) = up and N(u) = −u. Let ′Modr,φ,N/S˜ 1
denote the category consisting of the following data:
• an ˜S1-module ˜Mand its ˜S1-submodule FilrM˜ containinguerM,˜
• a φ-semilinear mapφr: FilrM →˜ M,˜
• a k-linear mapN : ˜M →M˜ such that
– N(sm) =N(s)m+sN(m) for any s∈S˜1 and m∈M,˜ – ueN(FilrM)˜ ⊆FilrM,˜
– the following diagram is commutative:
FilrM˜ φr
−−−−→ M˜
ueN y
ycN
FilrM −−−−→˜
φr
and whose morphisms are defined as before. Its full subcategory Modr,φ,N
/S˜1
is defined by the following condition:
• As an ˜S1-module, ˜Mis free of finite rank and generated by the image
of φr.
We define categories′Modr,φ
/S˜1 and Mod r,φ
/S˜1 similarly.
Let D be a weakly admissible filtered (φ, N)-module over K satisfying Fil0DK =DK and Filr+1DK = 0. SetSK0 =S⊗WK0 andD=D⊗K0SK0.
Then theSK0-moduleDis equipped with the naturalφ-semilinear mapφ⊗σ
and K0-linear derivation N ⊗1 + 1⊗N, which are denoted by φ and N,
respectively. We define a filtration onD inductively by Fil0D=D and
Fili+1D={x∈ D |N(x)∈FiliDand fπ(x)∈Fili+1DK},
where fπ :D → DK is induced by the map S → OK sending u to π. An S-submodule ˆM of D is said to be a strongly divisible lattice of D if the following conditions are satisfied:
• theS-module ˆMis free of finite rank, • M ⊗ˆ W K0=D,
• Mˆ is stable underφand N,
• φ(FilrM)ˆ ⊆prM, where we set Filˆ rMˆ = ˆM ∩FilrD.
We putφr=p−rφ|FilrMˆ. Then theS-module ˆMis generated byφr(FilrM)ˆ
([3, Proposition 2.1.3]) and we can consider ˆMas an object of Modr,φ,N/S . LetAcrys and ˆAstbep-adic period rings. These are constructed as follows.
SetR to be the ring
R= lim
←−(OK¯/pOK¯ ← OK¯/pOK¯ ← · · ·),
where every arrow is the p-power map. For an element x = (xi)i∈Z≥0 ∈R
and an integer n≥0, we set
x(n)= lim
m→∞xˆ pm
n+m ∈ OC,
where ˆxi is a lift of xi inOK¯ and OC is the p-adic completion of OK¯. Let
vp denote the valuation ofOC normalized as vp(p) = 1. Then the ring R is
a complete valuation ring whose valuation of an element x ∈R is given by
vR(x) =vp(x(0)). We define a natural ring homomorphism θby θ:W(R)→ OC
(x0, x1, . . .)7→ X
n≥0
pnx(nn).
Then Acrys is thep-adic completion of the divided power envelope ofW(R)
with respect to the principal ideal Ker(θ) and ˆAst is thep-adic completion
of the divided power polynomial ringAcryshXioverAcrys. We setAcrys,∞= Acrys⊗WK0/W and ˆAst,∞= ˆAst⊗WK0/W. Putπ = (πn)n∈Z≥0 ∈R, where
are considered asS-algebras by the ring homomorphismsS→Aˆstand ˆAst →
Acrys which are defined by u 7→[π]/(1 +X) and X 7→ 0, respectively. The
ringAcrys is endowed with a natural filtration induced by the divided power
structure, a natural Frobenius endomorphism φ and the φ-semilinear map
φt=p−tφ|FiltA
crys. With these structures, Acrys and Acrys,∞ are considered
as objects of′Modr,φ
/S. Moreover, the absolute Galois groupGKacts naturally
on these two rings. As for ˆAst, its filtration is defined by
FiltAˆst=
X
i≥0
ai Xi
i!
¯ ¯ ¯ ¯
ai ∈Filt−iAcrys, lim i→∞ai= 0
and the Frobenius structure ofAcrys extends to ˆAst by
φ(X) = (1 +X)p−1, φt=p−tφ|FiltAˆst.
We write N also for the Acrys-linear derivation on ˆAst defined by N(X) =
1 +X. The rings ˆAst and ˆAst,∞are objects of′Modr,φ,N/S . TheGK-action on Acrys naturally extends to an action on ˆAst. Indeed, the action ofg ∈ GK
on ˆAst is defined by the formula
g(X) = [ε(g)](1 +X)−1,
where g(πn) = εn(g)πn and ε(g) = (εn(g))n∈Z≥0 ∈ R with the abusive
notation as above.
These rings have other descriptions, as follows. For an integern≥1, put
Wn=W/pnW and letWn(OK¯/pOK¯) be the ring of Witt vectors of lengthn
associated toOK¯/pOK¯. We define aWn-algebra structure onWn(OK¯/pOK¯)
by twisting the naturalWn-algebra structure byσ−n. Then the natural ring
homomorphism
θn:Wn(OK¯/pOK¯)→ OK¯/pnOK¯
(a0, . . . , an−1)7→ n−1 X
i=0
piˆapin−i,
where ˆai is a lift ofai inOK¯, isWn-linear. Let us denote
WnPD(OK¯/pOK¯)
the divided power envelope ofWn(OK¯/pOK¯) with respect to the ideal Ker(θn).
This ring is considered as an S-algebra by u 7→ [πn]. This ring also has a
natural filtration defined by the divided power structure, and a naturalGK
induces on this ring a φ-semilinear Frobenius endomorphism, which is de-noted also byφ. Then, by theS-linear transition maps
WnPD+1(OK¯/pOK¯)→WnPD(OK¯/pOK¯)
(a0, . . . , an)7→(ap0, . . . , apn−1),
these S-algebras form a projective system compatible with all structures. Using this transition map, aφ-semilinear map
φr: FilrWnPD(OK¯/pOK¯)→WnPD(OK¯/pOK¯)
is defined by setting φr(x) to be the image of p−rφ(ˆx), where ˆx is a lift of x in FilrWPD
n+r(OK¯/pOK¯). By definition, the maps φr are also compatible
with the transition maps. The S-algebra WPD
n (OK¯/pOK¯) is considered as
an object of ′Modr,φ
/S. Then we have a natural isomorphism in
′Modr,φ /S
Acrys/pnAcrys→WnPD(OK¯/pOK¯)
(x0, . . . , xn−1)7→(x0,n, . . . , xn−1,n),
where we set xi = (xi,k)k∈Z≥0.
Similarly, the divided power polynomial ring
WnPD(OK¯/pOK¯)hXi
over WPD
n (OK¯/pOK¯) is considered as an S-algebra by u 7→ [πn]/(1 +X).
This ring has a natural filtration coming from the divided power structure. We define aGK-action on this ring by
g(X) = [εn(g)](1 +X)−1.
We also define aφ-semilinear Frobenius endomorphism, which we also write as φ, by φ(X) = (1 +X)p−1 and a WnPD(OK¯/pOK¯)-linear derivation N
by N(X) = 1 +X. These rings form a projective system of S-algebras compatible with all structures by the transition maps defined by the maps above andX 7→X. We define φ-semilinear maps
φr : FilrWnPD(OK¯/pOK¯)hXi →WnPD(OK¯/pOK¯)hXi
compatible with the transition maps as before. TheS-algebraWPD
n (OK¯/pOK¯)hXi
is considered as an object of ′Modr,φ,N
/S and there exists a natural
isomor-phism in′Modr,φ,N /S
ˆ
Ast/pnAˆst →WnPD(OK¯/pOK¯)hXi
(x0, . . . , xn−1)7→(x0,n, . . . , xn−1,n) X 7→X
which isGK-linear.
Put Kn = K(πn) and K∞ = ∪nKn. For M ∈ Modr,φ,N/S∞ , we define a GK-moduleTst∗,π(M) to be
Tst∗,π(M) = HomS,Filr,φ
When Mis killed bypn, we have a natural identification ofGK-modules Tst∗,π(M) = HomS,Filr,φ
r,N(M, W PD
n (OK¯/pOK¯)hXi).
Note that theGK-module on the right-hand side is independent of the choice
of πk fork > n. Since the natural map
WnPD(OK¯/pOK¯)hXi →WnPD(OK¯/pOK¯)
X 7→0
is by definition GKn-linear, we also have a GKn-linear isomorphism ([3,
Lemme 2.3.1.1])
Tst∗,π(M)|GKn →HomS,Filr,φr(M, W PD
n (OK¯/pOK¯)).
A variant of filtered (φr, N)-modules over S is also introduced by Breuil
and Kisin, and developed also by Caruso and Liu (see for example [12], [13], [14], [6]). Put S = W[[u]] and let φ : S → S be the σ-semilinear
Frobenius endomorphism defined by φ(u) = up. Let ′Modr,φ/S denote the category consisting of the following data:
• an S-moduleM,
• a φ-semilinear map M→M, which is denoted also byφ, such that
the cokernel of the map 1⊗φ:φ∗M→Mis killed byE(u)r,
and whose morphisms are defined as before. The full subcategory of′Modr,φ /S
consisting ofM such thatMis free of finite rank overS/pS(resp. overS)
is denoted by Modr,φ/S
1 (resp. Mod r,φ
/S). We let Mod r,φ
/S∞ denote the smallest full subcategory which contains Modr,φ/S
1 and is stable under extensions, as
before. Then we have an exact functor ([6, Proposition 2.1.2], see also [12, Proposition 1.1.11])
MS∞ : Mod
r,φ
S∞ →Mod
r,φ S∞. For M∈Modr,φS
∞, the filtered φr-module M=MS∞(M) over S is defined as follows:
• M=S⊗φ,SM,
• FilrM= Ker(M1→⊗φS⊗SM→(S/FilrS)⊗SM),
• φr : FilrM 1⊗φ
→ FilrS⊗SM φr⊗1
→ S⊗φ,SM=M.
We writeMSfor the functor Modr,φ/S→Modr,φ/S defined similarly.
Finally, letDandD be as above and ˆMbe a strongly divisible lattice in D. The S-module Mn = ˆM/pnMˆ has a natural structure as an object of
Modr,φ,N/S
∞ . We set aGK-module ˆT
∗
st,π( ˆM) to be
ˆ
Tst∗,π( ˆM) = HomS,Filr,φ
r,N( ˆM,Aˆst).
Then we have an exact sequence of GK-modules
0−→Tˆst∗,π( ˆM) p
n
The GK-module ˆTst∗,π( ˆM) is naturally considered as a GK-stable Zp-lattice
in V∗
st(D). By Liu’s theorem ([14, Theorem 2.3.5]), the functor ˆTst∗,π gives
an anti-equivalence of categories between the category of strongly divisible lattices inDand the category ofGK-stableZp-lattices inVst∗(D). Moreover,
for such a latticeL, its corresponding strongly divisible lattice ˆMinDis in the essential image of the functorMS([14, Subsection 3.5]).
3. Filtered φr-modules over Σ
In this section, we define another variant of filtered φr-modules over S
and prove its properties.
Letp≥3 be a rational prime andr be an integer such that 0≤r < p−1. Consider the W-algebra Σ = W[[u, Y]]/(E(u)p −pY) as in [3, Subsection
3.2]. We regard Σ as a subring of S by the map sending Y to E(u)p/p. Then the elementc=φ1(E(u))∈S×is contained in Σ×. We define on Σ a
σ-semilinear Frobenius endomorphism φby φ(u) = up and φ(Y) = pp−1cp. Put FiltΣ = (E(u)t, Y) for 0≤ t≤p−1 and FilpΣ = (Y). Then we have φ(FiltΣ) ⊆ ptΣ for 0 ≤ t ≤ p−1. We put φt = p−tφ|FiltΣ. We also set
Σn = Σ/pnΣ and put on this ring the natural structures induced by those
of Σ.
We define a category ′Modr,φ
/Σ of filtered φr-modules over Σ to be the
category consisting of the following data:
• a Σ-moduleM and its Σ-submodule FilrM containing FilrΣ.M, • aφ-semilinear map FilrM →Msatisfyingφr(srm) =φr(sr)φ(m) for
anysr∈FilrΣ andm∈M, where we set φ(m) =c−rφr(E(u)rm).
and the morphisms are defined in the same manner as′Modr,φ/S. This category has a natural notion of exact sequences. We define its full subcategory Modr,φ/Σ1 to be the category consisting ofM which is free of finite rank and generated by the image of φr as a Σ1-module. We also let Modr,φ/Σ∞ denote
the smallest full subcategory of′Modr,φ
/Σ which contains Mod r,φ
/Σ1 and is stable
under extensions. Moreover, we define a full subcategory Modr,φ/Σ of′Modr,φ /Σ
to be the category consisting of M such that
• the Σ-moduleM is free of finite rank and generated by the image of
φr,
• the quotientM/FilrM isp-torsion free.
Then we see that for ˆM ∈ Modr,φ/Σ, the quotient ˆM /pnMˆ is naturally con-sidered as an object of Modr,φ/Σ
∞.
The natural ring isomorphism Σ1/FilpΣ1 ∼= ˜S1 defines a functor T0 :
Modr,φ/Σ1 → Modr,φ
/S˜1 by M 7→ M/Fil pΣ
1.M. Then [3, Proposition 2.2.1.3]
Lemma 3.1. Let M be an object of Modr,φ/Σ1 of rank dover Σ1. Then there exists a basis{e1, . . . , ed}of M such thatFilrM = Σ1ur1e1⊕ · · · ⊕Σ1urded+
FilpΣ1.M for some integers r1, . . . , rd with 0≤ri ≤er for any i.
Then we can show the following lemma just as in the proof of [3, Lemme 2.3.1.3].
Lemma 3.2. The functor
M 7→HomΣ,Filr,φ
r(M, Acrys,∞) from Modr,φ/Σ
∞ to the category of GK∞-modules is exact.
ForM ∈Modr,φ/Σ1, we can show as in the case of the category Modr,φ/S
1 that
there is an isomorphism ofGK1-modules
HomΣ,Filr,φ
r(M,(OK¯/pOK¯)
PD)→Hom ˜
S1,Filr,φr(T0(M),OK¯/pOK¯),
whereOK¯/pOK¯ is considered as an object of′Modr,φ˜
S1 by the natural
isomor-phism
(OK¯/pOK¯)PD/Filp(OK¯/pOK¯)PD→ OK¯/pOK¯.
Thus [3, Lemme 2.3.1.2] implies that, for such a Σ1-moduleM, we have
#HomΣ,Filr,φ
r(M,(OK¯/pOK¯)
PD) =pd,
whered= dimΣ1M.
For the category Modr,φ/Σ, we can show the following lemma just as in the proof of [14, Proposition 4.1.2].
Lemma 3.3. Let Mˆ be in Modr,φ/Σ. Then there exists α1, . . . , αd ∈Mˆ such
thatFilrMˆ = Σα1⊕ · · · ⊕Σαd+ FilpΣ.Mˆ, E(u)rMˆ ⊆Σα1⊕ · · · ⊕Σαd and
the elements e1 =φr(α1), . . . , ed=φr(αd) form a basis of Mˆ.
Corollary 3.4. Let Mˆ be in Modr,φ/Σ and A be a Σ-algebra which has a structure as an object of ′Modr,φ/Σ. Let C∈Md(Σ)be the matrix such that
(α1, . . . , αd) = (e1, . . . , ed)C
with the notation of the previous lemma. Then a Σ-linear homomorphism
f : ˆM →A preserving Filr also commutes with φr if and only if
φr(f(e1, . . . , ed)C) = (f(e1), . . . , f(ed)).
Proof. Suppose that the latter condition holds. By assumption, we have
for someC′ ∈M
d(Σ). We claim that f commutes withφ. Indeed, we have φ(f(e1, . . . , ed)) =c−rφr(E(u)r(f(e1), . . . , f(ed)))
=c−rφr((f(α1), . . . , f(αd))C′)
=c−rf(e1, . . . , ed)φ(C′)
=c−rf(φr(α1, . . . , αd))φ(C′)
=c−rf(φr(E(u)r(e1, . . . , ed))) =f(φ(e1, . . . , ed)).
This implies φr◦f =f◦φr also on FilpΣ.Mˆ.
¤
Corollary 3.5. Let Mˆ and A be as above and J ⊆FilrA be an ideal of A
such that φr(J) ⊆ J. We can consider the Σ-algebra A/J naturally as an
object of′Modr,φ/Σ. Suppose that for any x∈J, there existst∈Z≥0 such that
φtr(x) = 0. Then we have an isomorphism
HomΣ,Filr,φ
r( ˆM , A)→HomΣ,Filr,φr( ˆM , A/J).
Proof. Letf : ˆM →A/J be an element of the abelian group on the right-hand side and ˆxi be an lift of f(ei) in A. By the previous corollary, it is
enough to show that for any (ˆc1, . . . ,cˆd) ∈ Jd, there is a unique solution
(ˆy1, . . . ,yˆd)∈Jd of the equation
(ˆc1, . . . ,cˆd) + (φr(ˆy1), . . . , φr(ˆyd))φ(C) = (ˆy1, . . . ,yˆd).
By assumption, thed-tuple
t X
i=0
(φir(ˆc1), . . . , φir(ˆcd))φi(C)φi−1(C)· · ·φ(C)
is stable for sufficiently large t and we see that this limit gives a unique
solution of the equation. ¤
For an S-module M in Modr,φ
/S∞ (resp. Mod r,φ
/S), we associate to it a
Σ-moduleM ∈′Modr,φ
/Σ as follows:
• M = Σ⊗φ,SM,
• FilrM = Ker(M 1→⊗φΣ⊗SM→(Σ/FilrΣ)⊗SM),
• φr : FilrM 1 ⊗φ
→ FilrΣ⊗SM φr⊗1
→ Σ⊗φ,SM=M.
We can check that this defines an exact functor Modr,φ/S
∞ →Mod
r,φ
/Σ∞ (resp. Modr,φ/S → Modr,φ/Σ) as in the proof of [12, Proposition 1.1.11]. We let this functor be denoted by MS∞ (resp. MS).
Proposition 3.6. Let Mbe an object ofModr,φ
/S∞ which is killed by p
n. Set
M =MS∞(M) and M= MS∞(M). Then there exists a natural
isomor-phism of GKn-modules
HomΣ,Filr,φ
r(M, W PD
n (OK¯/pOK¯))→HomS,Filr,φ
r(M, W PD
Proof. By definition,M=S⊗ΣM and we have a natural isomorphism
HomΣ(M, WnPD(OK¯/pOK¯))→HomS(M, WnPD(OK¯/pOK¯)).
Letfbe an element of HomΣ,Filr,φ
r(M, W PD
n (OK¯/pOK¯)) andf′be the image
of f in the right-hand side of the above isomorphism. Let us check that f′
preserves Filr and commutes withφr. Sincef′ isS-linear, it maps FilpS.M
into FilpWPD
n (OK¯/pOK¯). For x∈FilrM ∩Im(M → M), the commutative
diagram whose right vertical arrow is an isomorphism
M = Σ⊗φ,SM 1⊗φ
−−−−→ Σ⊗SM −−−−→ Σ/FilrΣ⊗SM
y
y
y
M=S⊗φ,SM 1⊗φ
−−−−→ S⊗SM −−−−→ S/FilrS⊗SM
implies x∈Im(FilrM →FilrM) and thusf′(x)∈FilrWPD
n (OK¯/pOK¯).
As for the compatibility withφr, again by theS-linearity off′ it suffices
to showf′(φ
r(x)) =φr(f′(x)) forx∈FilrM ∩Im(M → M) = Im(FilrM →
FilrM). This follows from the commutative diagram
FilrM −−−−→φr M −−−−→f WnPD(OK¯/pOK¯)
y
y
° ° °
FilrM −−−−→ Mφr f ′
−−−−→ WnPD(OK¯/pOK¯).
Hence the map in the proposition is well-defined and injective. To prove the bijectivity, by devissage we may assume that pM= 0. Then both sides
of this injection have the same cardinality by the above remark. Thus the
proposition follows. ¤
4. A method of Abrashkin
In this section, we study theGKn-module HomΣ,Filr,φr(M, W PD
n (OK¯/pOK¯))
following Abrashkin ([2]).
Let p ≥ 3 and 0 ≤ r < p−1 be as before. Consider the Lubin-Tate logarithm
l(X) =X+X
p
p +· · ·+ Xpn
pn +· · ·
and put ψ(X) = l−1(log(1 +X)). Then ψ gives a strict isomorphism of
formal groups between the formal group associated to the logarithm l(X) and the multiplicative group ˆGm overZp. We fix a system of p-power roots of unity {ζpn}n∈Z
≥0 such that ζp 6= 1 and ζpn =ζ p
pn+1 for anyn, and set an
element ε of R to be (ζpn)n∈Z
≥0. Then the elements [ε]−1 and [ε
1/p]−1
are topologically nilpotent in W(R) and the element of W(R)
is a generator of the principal ideal Ker(θ). The elementZ =ψ([ε]−1)p−1/p
of Acrys is topologically nilpotent and φ(t) is contained in the subset
p(1 +ZW(R)[[Z]]) of Acrys ([2, Subsection 1.8]). We set
ˆ
A=W(R)[[Z]]⊆Acrys.
Lemma 4.1. The element tp/p of Acrys is contained in the subring Aˆ and topologically nilpotent in this subring.
Proof. Putt′ = ([ε]−1)/([ε1/p]−1). This is another generator of Ker(θ).
We have
([ε]−1)p−1
p =
(t′)p−1
p ·([ε
1/p]−1)p−1
and θ([ε1/p]−1) =ζ
p −1. Take an element a ∈W(R)× such that θ(a) =
(ζp−1)p−1/p. Then we have
([ε]−1)p−1
p =a(t
′)p−1+b(t′)p/p
for some b ∈ W(R)×. Indeed, to show b ∈ W(R)×, it suffices to check
that the element ([ε1/p]−1)p−1−paof Ker(θ) also generates this ideal. This
follows from the fact that the 0-th entry (ε1/p−1)p−1 of this element satisfies
vR((ε1/p−1)p−1) = 1. Then we see that (t′)p/p is topologically nilpotent
because so ist′ inW(R). ¤
In the following, we set the elementa in the proof of the lemma to be
a=
p−2 X
k=1
p−1((−1)p−1−kp−1Ck−1)[ε1/p]k,
wherep−1Ck= (p−1)!/(k!(p−1−k)!) is the binomial coefficient. Note that
the coefficient of [ε1/p]k in each term is an integer.
From this lemma, we can consider the ring ˆA as a Σ-algebra byu7→[π]. Put FiliAˆ = (ti, Z) for 0 ≤ i ≤ p−1. The Frobenius endomorphism φ
of Acrys preserves ˆA and satisfies φ(FiliAˆ) ⊆ piAˆ for 0 ≤ i ≤ p−1. Set
φr =p−rφ|FilrAˆ. Then we can consider the ring ˆA also as an object of the
category ′Modr,φ
/Σ, and similarly for ˆAn = ˆA/pnAˆ and ˆA∞ = ˆA⊗W K0/W.
The absolute Galois group GK∞ acts naturally on these Σ-algebras. The following lemma is used implicitly in [2].
Lemma 4.2. We have a natural decomposition
ˆ
A1=R/(tp)⊕(Z).
[x]∈ W(R) is contained in pAˆ, then its image in Acrys/pAcrys is zero. We
have an isomorphism of R-algebras
R[Y1, Y2, . . .]/(tp, Y1p, Y2p, . . .)→Acrys/pAcrys
which sends Yi to the image of tp
i
/(pi)!. Thus the inequality v
R(x) ≥ p
holds. Conversely, ifvR(x)≥p, then we have
[x] =w(ψ([ε]−1)p−1) +pw′
for somew, w′∈W(R) and this implies [x]∈pAˆ.
Let us consider the commutative diagram ofR-algebras
R/(tp) //
$
$
J J J J J J J J J
ˆ
A1
²
²
ˆ
A1/(Z).
By definition, the left downward arrow is surjective. We claim that this arrow is an isomorphism. Indeed, let x be in the kernel of this surjection. From the proof of Lemma 4.1, we see that the image ofZ in the ring on the left-hand side of the above isomorphism can be written as a′tp−1+b′Y
1 for
somea′, b′ ∈R×. By assumption, in this ring, we have
x=c1(a′tp−1+b′Y1) +c2(a′tp−1+b′Y1)2+· · ·+cp−1(a′tp−1+b′Y1)p−1
for some elements c1, . . . , cp−1 of R. Then we see that ci = 0 for anyi and
vR(x)≥p. This concludes the proof. ¤
Since r < p−1, from this lemma we can show the following lemma as in the proof of [3, Lemme 2.3.1.3].
Lemma 4.3. The functor
M 7→HomΣ,Filr,φ
r(M,A∞ˆ ) from Modr,φ/Σ
∞ to the category of GK∞-modules is exact.
Corollary 4.4. For any M ∈Modr,φ/Σ
∞, the natural map
HomΣ,Filr,φ
r(M,A∞ˆ )→HomΣ,Filr,φr(M, Acrys,∞) is an isomorphism of GK∞-modules.
Proof. By Lemma 3.2 and Lemma 4.3, we may assume pM = 0. Consider the commutative diagram of rings
ˆ
A1 //
%
%
J J J J J J J J J J J
Acrys/pAcrys
²
²
whose downward arrows are defined by modulo Filp−1 of the rings ˆA1 and
Acrys/pAcrys, respectively. Since r < p−1, we have φr(Filp−1Aˆ1) = 0 and
similarly for the ring Acrys/pAcrys. Thus these two surjections induce on
the ringR/(tp−1) the same structure of a filteredφ
r-module over Σ. Hence,
as in the proof of Corollary 3.5, we see from Lemma 3.1 that we have a commutative diagram
HomΣ,Filr,φ
r(M,Aˆ1)
/
/
*
*
V V V V V V V V V V V V V V V V V
HomΣ,Filr,φ
r(M, Acrys/pAcrys)
²
²
HomΣ,Filr,φ
r(M, R/(t
p−1))
whose downward arrows are isomorphisms. This concludes the proof. ¤
We sketch the proof of the following lemma stated in [2, Subsection 3.2].
Lemma 4.5. The natural inclusionW(R)→Aˆ induces an isomorphism of
W(R)-algebras Wn(R)/(ψ([ε]−1)p−1)→Aˆn/(Z).
Proof. For a subringB ofAcrys, put
I[s]B ={x∈B |φi(x)∈FilsAcrys for anyi}
as in [9, Subsection 5.3]. Then we have I[s]W(R) = ([ε]−1)sW(R) and the natural ring homomorphism
W(R)/I[s]W(R)→Acrys/I[s]Acrys
is an injection ([9, Proposition 5.1.3, Proposition 5.3.5]). Since the element
Z is contained in the ideal I[p−1]A
crys, this injection factors as
W(R)/I[p−1]W(R)→A/ˆ (Z)→Acrys/I[p−1]Acrys.
Hence the former arrow is an isomorphism and the lemma follows. ¤
Since the ideal (Z) of ˆAn satisfies the condition of Corollary 3.5, the
Σ-algebra ˆAn/(Z) is naturally considered as an object of′Modr,φ/Σ. We also give
the ringWn(R)/(ψ([ε]−1)p−1) the structures of a Σ-algebra and a filtered φr-module over Σ induced by those of ˆAn/(Z). The map
Σ→Wn(R)/(ψ([ε]−1)p−1)
sends the element u ∈ Σ to the image of [π] in the ring on the right-hand side. Put v =t′/E([π])∈ W(R)× with the notation of Lemma 4.1. As for
the element Y ∈Σ, the equality
Y =−ab−1v−1E([π])p−1+wb−1v−pZ
holds in ˆA, where a and b are the elements in W(R)× as in the proof of
Consider the surjective ring homomorphism
R→ OK¯/pOK¯
x= (x0, x1, . . .)7→xn
and the induced surjectionWn(R)→Wn(OK¯/pOK¯). Let
J ={(x0, . . . , xn−1)∈Wn(R) |vR(xi)≥pn for anyi }
be the kernel of the latter surjection.
Lemma 4.6. The ideal J is contained in the ideal (ψ([ε]−1)p−1) of the ring Wn(R).
Proof. Write ([ε]−1)p−1 also asx= (x0, . . . , xn−1)∈Wn(R) withvR(x0) =
p. Take an element z = (z0, . . . , zn−1) in the ideal J. We construct y ∈
Wn(R) such that xy = z. By induction, it is enough to show that if z0 =
· · · = zi−1 = 0 for some 0 ≤ i ≤ n−1 and (x0, . . . , xi)(0, . . . ,0, yi) =
(0, . . . ,0, zi) inWi+1(R), then x(0, . . . ,0, yi,0, . . . ,0)∈J. Let us write this
element as (0, . . . ,0, wi, . . . , wn−1) withwi =zi. We havevR(yi)≥pn−pi+1.
In the ring of Witt vectors Wn(Fp[X0, . . . , Xn−1, Y0, . . . , Yn−1]), the k-th
entry of the vector
(X0, . . . , Xn−1)(0, . . . ,0, Yi,0, . . . ,0)
isXkp−iiYipk−i for anyk≥i. Thus we havevR(wk)≥pn. ¤
Note that the elements [ζpn]−1 and [ζpn+1]−1 is nilpotent inWn(OK¯/pOK¯).
By the above lemma, we have an isomorphism of rings
Wn(R)/(ψ([ε]−1)p−1)→Wn(OK¯/pOK¯)/(ψ([ζpn]−1)p−1).
We give the ring on the right-hand side the structure of a filteredφr-module
over Σ induced by this isomorphism.
PutFn=Kn(ζpn+1). For an algebraic extension F of Fn, let us consider
the ideals
¯
mn,F ={(x0, . . . , xn−1)∈Wn(OF/pOF) |xi ∈mF/pOF for anyi } mn,F ={(x0, . . . , xn−1)∈Wn(OF) |xi ∈mF for anyi }
of Wn(OF/pOF) and Wn(OF), respectively. The elements [ζpn]−1 and
[ζpn+1]−1 are topologically nilpotent inWn(OF) and we define an element
ˆ
t∈Wn(OF) to be
ˆ
t=ψ([ζpn]−1)/ψ([ζpn+1]−1).
Note that these elements are non-zero divisors of Wn(OF). Let the ring Wn(OF/pOF)/ψ([ζpn]−1)rm¯n,F
be denoted by ¯An,F,r+. We also put ¯mn= ¯mn,K¯, mn=mn,K¯ and ¯An,r+=
¯
An,K,r¯ +.
For an algebraic extension F ofK, we put
Note that the ring OF/bF is killed byp. WhenF contains Fn, we also put
¯
A′n,F,r+ =Wn(OF/bF)/ψ([ζpn]−1)rm¯n,F.
Then, for 0≤r < p−1, we have natural isomorphisms of rings
Wn(OF)/ψ([ζpn]−1)rmn,F →A¯n,F,r+ →A¯′n,F,r+.
Indeed, as in the proof of Lemma 4.6, we can show that both of the kernels of the maps Wn(OF) → Wn(OF/pOF) and Wn(OF) → Wn(OF/bF) are
contained in the ideal ψ([ζpn]−1)rmn,F of the ring Wn(OF). We often
identify these rings. We also put ¯A′
n,r+= ¯A′n,K,r¯ +.
Write Zn for the image of the element Z of Acrys in WnPD(OK¯/pOK¯).
Then we have a commutative diagram of Σ-algebras
ˆ An ² ² ² ² / /A
crys/pnAcrys
≀
²
²
WnPD(OK¯/pOK¯)
²
²
²
²
Wn(R)/(ψ([ζpn]−1)p−1) ∼ //
≀
²
²
ˆ
An/(Z)
( ( R R R R R R R R R R R R R
Wn(OK¯/pOK¯)/(ψ([ζpn]−1)p−1) //
²
²
²
²
WnPD(OK¯/pOK¯)/(Zn),
¯
An,r+
≀ ² ² ¯ A′ n,r+
where all vertical arrows are surjections satisfying the condition of Corollary 3.5. Thus we see that this is also a commutative diagram in ′Modr,φ
/Σ. Note
that these rings and homomorphisms are independent of the choice of a system{ζpn}n∈Z
≥0. Moreover, let ˆM be in Mod r,φ
/Σ and put Mn= ˆM /pnMˆ.
Then, by Corollary 3.5 and Corollary 4.4, we have a natural isomorphism of abelian groups
HomΣ,Filr,φ
r(Mn, W PD
n (OK¯/pOK¯))≃HomΣ,Filr,φ
r(Mn,A¯n,r+).
To study GFn-actions on both sides of this isomorphism, we need the
following proposition.
Proof. We have the equality
([ε1/p]−1)p−1=pa−E([π])bv
inW(R). By definition, we see that the images of the elementsa, [ε1/p] and
E([π]) inWn(OK¯/pOK¯) are contained in the subringWn(OFn/pOFn). Write
β for the productbv ∈W(R). Letan, βn and αn denote the images of the
elementsa,β andpa−([ε1/p]−1)p−1in the ringW
n(OK¯/pOK¯), respectively.
Then the element αn is also contained in the subringWn(OFn/pOFn). Now
we have the equality
E([πn])βn=αn.
Note that any element β′
n ∈ Wn(OK¯/pOK¯) satisfying the same equality
is invertible and thus the elements (β′
n)−1E([πn]) are equal to each other.
Since Y =−anβ−n1E([πn])p−1 in ¯An,r+, it suffices to construct an element
β′
n in the ringWn(OFn/pOFn) such that the equalityE([πn])β
′
n=αnholds.
Take a lift ˆαn of αn in Wn(OFn). Since E([πn]) ∈ Wn(OK¯) divides every
element in the kernel of the surjectionWn(OK¯)→Wn(OK¯/pOK¯), we have
E([πn]) ˆβn′ = ˆαn for some ˆβn′ ∈Wn(OK¯). Then the element ˆβ′n is contained
in the subring Wn(OFn) and we set β
′
n to be the image of ˆβn′.
By a similar argument, we can also check that the ring ¯An,F,r+is a subring
of ¯An,r+ and coincides with the image of Wn(OF) in ¯An,r+. This concludes
the proof. ¤
Lettnand ¯tnbe the images of t∈W(R) inWn(OK¯/pOK¯) and ¯An,r+ (or
¯
A′
n,r+), respectively. Then ˆtis a lift oftn and ¯tn toWn(OK¯) by the natural
surjections
Wn(OK¯)→Wn(OK¯/pOK¯)→A¯n,r+.
Note that we defined the filtration of ¯An,r+ as FilrA¯n,r+ = ¯trnA¯n,r+.
Lemma 4.8. Let x¯ be in FilrA¯n,r+. Then we have
φr(¯x) =φ(y) mod ψ([ζpn]−1)rm¯n,
where y is any element of Wn(OK¯/pOK¯) such that the element trny is a lift of x¯. In particular, the right-hand side of the above equality is independent of the choice of a system {ζpn}n∈Z
≥0. Similar assertions also hold for the ring
Wn(OK¯/pOK¯)/(ψ([ζpn]−1)p−1).
Proof. Since the filteredφr-module structure of ¯An,r+ is induced from that
of ˆA and φr(tr) ≡1 mod Z in ˆA, we see that there exists an element y as
in the lemma.
To prove the independence of the choice of a lift, let z = (z0, . . . , zn−1)
p−[pn], where pk=p1/p
k
. This implies
pnz0 ∈pOK¯
zp0+pn−1z1 ∈pOK¯
.. .
znp−2+p1zn−1 ∈pOK¯
and vp(zk) ≥1−1/pn−k for 0≤k≤n−1. Repeating this, we see that if z is killed bytrn, thenvp(zk)≥1−r/pn−k. For such an elementz, we have φ(z) = 0 in the ring Wn(OK¯/pOK¯).
Lety1 and y2 be elements as in the lemma. Then we have
y1−y2=ψ([ζpn+1]−1)rw+z,
wherew∈m¯nand z is an element as above. The Frobenius endomorphism φsends the element on the right-hand side to an element which is contained in the idealψ([ζpn]−1)rm¯n. Thus the assertions for the ring ¯An,r+ follows.
We can show the assertion for the ring Wn(OK¯/pOK¯)/(ψ([ζpn]−1)p−1)
similarly. ¤
From this lemma and Proposition 4.7, we see that the naturalGFn-actions
on the ringsWn(OK¯/pOK¯)/(ψ([ζpn]−1)p−1), ¯An,r+ and ¯A′
n,r+ are
compat-ible with the filtered φr-module structures over Σ. In the commutative
diagram above, the lowest horizontal arrow and lower right vertical arrow areGFn-linear by definition. Hence we have shown the following proposition.
Proposition 4.9. Let Mˆ be in Modr,φ/Σ and put Mn = ˆM /pnMˆ. Then we
have an isomorphism of GFn-modules
HomΣ,Filr,φ
r(Mn,A¯n,r+)≃HomΣ,Filr,φr(Mn, W PD
n (OK¯/pOK¯)).
Let e1, . . . , ed be a basis of ˆM as in Lemma 3.3 and C = (ci,j) ∈Md(Σ)
be the associated matrix representing φr as in Corollary 3.4. Then the
underlyingGFn-set of the GFn-module
HomΣ,Filr,φ
r(Mn,A¯n,r+)
is identified with the set ofd-tuples (¯x1, . . . ,x¯d) in ¯An,r+ such thatc1,ix¯1+
· · ·+cd,ix¯d∈FilrA¯n,r+ for anyiand the following equality holds:
(1)
φr(c1,1x¯1+· · ·+cd,1x¯d) = ¯x1
.. .
φr(c1,dx¯1+· · ·+cd,dx¯d) = ¯xd.
We choose a lift (ˆci,j)∈ Md(Wn(OFn)) of the image of C in Md( ¯An,r+) by
the natural ring homomorphism
Wn(OK¯)→Wn(OK¯/pOK¯)→A¯n,r+.
Fix a polynomial Φi ∈Z[X0, . . . , Xn−1] such that Φi ≡Xip mod p. This
particular, set B to be the polynomial ring Z[X0, . . . , Xn−1, Y0, . . . , Yn−1].
PutX = (X0, . . . , Xn−1) and Y = (Y0, . . . , Yn−1) in the ringWn(B). Then
we see that there exists elementsU0, . . . , Un−1andU0′, . . . , Un′−1of the
poly-nomial ringB such that
Φ(X+Y) = Φ(X) + Φ(Y) + (pU0, . . . , pUn−1),
Φ(XY) = Φ(X)Φ(Y) + (pU0′, . . . , pUn′−1)
in the ringWn(B).
Proposition 4.10. Every solution(¯x1, . . . ,x¯d) inA¯n,r+ of the equation(1) such that c1,ix¯1+· · ·+cd,ix¯d∈FilrA¯n,r+ for any iuniquely lifts to ad-tuple
(ˆx1, . . . ,xˆd) in Wn(OK¯) such that ˆc1,ixˆ1+· · ·+ ˆcd,ixˆd ∈tˆrWn(OK¯) for any
iand the following equality holds:
(2)
Φ((ˆc1,1xˆ1+· · ·+ ˆcd,1xˆd)/tˆr) = ˆx1 ..
.
Φ((ˆc1,dxˆ1+· · ·+ ˆcd,dxˆd)/tˆr) = ˆxd.
Proof. Fix a lift ˆxi of ¯xi toWn(OK¯). Then we have
Φ((ˆc1,1xˆ1+· · ·+ ˆcd,1xˆd)/ˆtr) = ˆx1+ ([ζpn]−1)rδˆ1
.. .
Φ((ˆc1,dxˆ1+· · ·+ ˆcd,dxˆd)/ˆtr) = ˆxd+ ([ζpn]−1)rδˆd
for some ˆδ1, . . . ,δˆd ∈ mn. It suffices to show that there exists a unique d-tuple (ˆy1, . . . ,yˆd) in mn such that
Φ((ˆc1,i(ˆx1+ ([ζpn]−1)ryˆ1) +· · ·+ ˆcd,i(ˆxd+ ([ζpn]−1)ryˆd))/ˆtr)
= ˆxi+ ([ζpn]−1)ryˆi
for anyi. For this, we need the following lemma.
Lemma 4.11. Let N be a complete discrete valuation field and mN be the
maximal ideal ofN. Letǫ1, . . . , ǫdbe inmN. LetP1, . . . , PdandP1′. . . , Pd′ be
elements ofON[[Y1, . . . , Yd]]such that Pi ∈(Y1, . . . , Yd)2. Then the equation
Y1−P1(Y1, . . . , Yd)−ǫ1P1′(Y1, . . . , Yd)) = 0
.. .
Yd−Pd(Y1, . . . , Yd)−ǫdPd′(Y1, . . . , Yd)) = 0
has a unique solution in mN.
Proof. By assumption, we see that for any integer l ≥ 1, a d-tuple y = (y1, . . . , yd) in mN/mlN satisfying the above equation lifts uniquely to a d
Let us write as ˆyi = (ˆyi,0, . . . ,yˆi,n−1). Since the image of Φ(([ζpn+1]−1)r)
in ¯An,r+ is divisible by ([ζpn]−1)r, we can find ˆb∈Wn(OK¯) such that
Φ(([ζpn]−1)r/ˆtr) = ([ζpn]−1)rˆb.
Then there exists polynomials Ui,m over OK¯ of the indeterminates Y =
(Yi,m)1≤i≤d,0≤m≤n−1 such that the equation we have to solve is
ˆ
xi+ ([ζpn]−1)ryˆi = ˆxi+ ([ζpn]−1)rˆδi
+ ([ζpn]−1)rˆb(Φ(ˆci,1)Φ(ˆy1) +· · ·+ Φ(ˆci,d)Φ(ˆyd))
+ (pUi,0(ˆy), . . . , pUi,n−1(ˆy))
for anyi, where we put ˆy= (ˆyi,m)1≤i≤d,0≤m≤n−1. Note that, for any elements
P0, . . . , Pn−1 of the polynomial ring OK¯[Y], we can uniquely find elements
Q0, . . . , Qn−1 of this ring such that the coefficients of these polynomials are
in the maximal idealmK¯ and the equality
(pP0, . . . , pPn−1) = ([ζpn]−1)r(Q0, . . . , Qn−1)
holds in the ring of Witt vectors Wn(OK¯[Y]). Therefore, this equation is
equivalent to the equation
ˆ
yi= ˆδi+ ˆb(Φ(ˆci,1)Φ(ˆy1) +· · ·+ Φ(ˆci,d)Φ(ˆyd))
+ (Vi,0(ˆy), . . . , Vi,n−1(ˆy)),
whereVi,m is a polynomial ofY overOK¯ whose coefficients are in the
max-imal ideal mK¯. From the definition of Φ, we see that the elements ˆyi,m is a
solution of a system of equations
Yi,m−Pi,m(Y)−ǫi,mPi,m′ (Y) = 0
satisfying the condition of Lemma 4.11 for a sufficiently large finite extension
N ofK. Then, by this lemma, we can solve the equation uniquely inmK¯. ¤
Let F be an algebraic extension of Fn = Kn(ζpn+1) and consider the
ring ¯An,F,r+. By Lemma 4.7, we can consider this ring as a Σ-algebra and
also as an object of ′Modr,φ
/Σ by putting FilrA¯n,F,r+ = ¯trnA¯n,F,r+ and for
¯
x∈FilrA¯n,F,r+,
φr(¯x) =φ(y) mod ψ([ζpn]−1)rm¯n,F,
where y is any element of Wn(OF/pOF) such that the elementtrny is a lift
of ¯x. ForMn= ˆM /pnMˆ ∈Modr,φ/Σ∞ as before, let us set
Tcrys∗ ,πn,F(Mn) = HomΣ,Filr,φ
r(Mn,A¯n,F,r+).
We see that
¯
An,r+= ¯An,K,r¯ += [
F/Fn
¯
in′Modr,φ
/Σ and thus we have a natural identification of abelian groups
Tcrys∗ ,π
n,K¯(Mn) = [
F/Fn
Tcrys∗ ,πn,F(Mn).
The absolute Galois group GFn acts on the abelian group on the left-hand
side.
Lemma 4.12. For an algebraic extensionF ofFn, the fixed partTcrys∗ ,π
n,K¯(Mn)
GF
is equal to T∗
crys,πn,F(Mn).
Proof. From Proposition 4.10, we see that the elements of T∗
crys,πn,K¯(Mn)
correspond bijectively to the solutions of the equation (2) in Wn(OK¯)
sat-isfying the condition on Filr. The uniqueness assertion of this proposition shows that g ∈ GF fixes a solution in Wn(OK¯) if and only if g fixes its
image in ¯An,r+. Hence a solution is fixed byGF if and only if this solution
is contained in the image of Wn(OF). Thus the lemma follows. ¤
Corollary 4.13. LetLnbe the finite Galois extension ofFncorresponding to
the kernel of the mapGFn →Aut(T
∗
crys,πn,K¯(Mn)). Then an algebraic exten-sionF of Fn containsLnif and only if#Tcrys∗ ,πn,F(Mn) = #T
∗
crys,πn,K¯(Mn). Proof. An algebraic extensionF ofFn containsLn if and only if the action
of GF on Tcrys∗ ,πn,K¯(Mn) is trivial. By Lemma 4.12, this is equivalent to T∗
crys,πn,F(Mn) =T
∗
crys,πn,K¯(Mn). ¤
5. Ramification bound
In this section, we prove Theorem 1.1. TakeGK-stableZp-latticesL ⊇ L′
inV such thatL′ ⊇pnL. Since theG
K-moduleL/L′is a quotient ofL/pnL,
we may assume L′ =pnL. If r = 0, then the G
K-module V is unramified
and the theorem is trivial. Thus we may assumer≥1 andp≥3. LetLbe the finite Galois extensions of K corresponding to the kernel of the map
GK →Aut(L/pnL).
It is enough to show that, for the greatest upper ramification breakuL(ζp)/K
of the Galois extensionL(ζp)/K, the inequality uL(ζp)/K ≤u(K, r, n)
holds. Since the Herbrand function is transitive and the finite Galois ex-tension K(ζp) is tamely ramified over K, we may assume ζp ∈ K. We fix
a uniformizer π of K and a system {πn}n∈Z≥0 as before. Then, by Liu’s
theorem ([14, Theorem 2.3.5]), it suffices to show the following.
Theorem 5.1. Let r be an integer such that 1 ≤ r < p−1 and Mˆ be the strongly divisible lattice corresponding to L. Put Mn = ˆM/pnM ∈ˆ
Modr,φ,N/S
∞ . Then G
(j)
LetLnbe the finite Galois extension ofFn=Kn(ζpn+1) corresponding to
the kernel of the map
GFn →Aut(T
∗
st,π(Mn)).
Since Fn is Galois over K, the extension Ln is also a Galois extension of K. Let ˆM be the object of the category Modr,φ
/S such that MS( ˆM) ≃
ˆ
M. From Proposition 3.6 and Proposition 4.9, we see that Ln is also the
finite extension of Fn cut out by theGFn-moduleT
∗
crys,πn,K¯(Mn) for Mn=
MS( ˆM)/pnMS( ˆM). It is enough to prove the inequality
uLn/K ≤u(K, r, n) =
(
1 +e(n+p−11) (r= 1), pn−1
pn +e(n+p−r1) (r >1).
Before proving this, we state some general lemmas to calculate the ramifica-tion bound. LetN be a complete discrete valuation field of positive residue characteristic,vN be its valuation normalized as vN(N×) =Z and Nsep be
its separable closure.
Lemma 5.2. Let f(T) ∈ ON[T] be a separable monic polynomial and z1, . . . , zd be the zeros off in ONsep. Suppose that the set {vN(zk−zi)|k=
1, . . . , d, k6=i} is independent ofi. Put
s= X
k=1,...,d k6=i
vN(zk−zi) and α= sup k=1,...,d
k6=i
vN(zk−zi),
which are independent of i by assumption. If j > s+α, then we have the decomposition
{x∈ ONsep |vN(f(x))≥j}= a
i=1,...,d
{x∈ ONsep |vN(x−zi)≥j−s}.
Otherwise, the set on the left-hand side contains
{x∈ ONsep | vN(x−zi)≥α},
which contains at least two zeros of f.
Proof. A verbatim argument in the proof of [1, Lemma 6.6] shows the claim.
¤
Corollary 5.3. Let f(T) be as above and put B=ON[T]/(f(T)). Suppose
that the algebra B is finite flat and of relative complete intersection over
ON. Let us write theN-algebra N′ =B⊗ONN as the productN1× · · · ×Nt of finite separable extensions N1, . . . , Nt of N. If j > s+α, then the j-th
upper numbering ramification group ([1]), which we let be denoted by G(Nj), is contained in GNi for any i. Moreover, if N
′ is a field and B coincides
with ON′, then j > s+α if and only if G(j)
Proof. From the previous lemma, the conductorc(B) of the ON-algebra B
([1, Proposition 6.4]) is equal tos+α. Thus we have the inequality
c(ON1 × · · · × ONt)≤c(B) =s+α
by the definition of the conductor and a functoriality of the functor Fj
defined in [1]. This implies the corollary. ¤
Corollary 5.4. Consider the finite Galois extension Fn =Kn(ζpn+1) of K and let uFn/K denote the greatest upper ramification break of Fn/K. Then we have the equality
uFn/K = 1 +e(n+
1
p−1).
Proof. Note that we are assuming thatζp is contained inK. Applying the
previous corollary to the Eisenstein polynomialf(T) =Tpn
−π shows that
j > 1 +e(n+ 1/(p−1)) if and only if G(Kj) ⊆ GKn. Similarly, putting
f(T) =Tpn−ζp, we see that ifj > e(n+ 1/(p−1)), then G(Kj) ⊆GK(ζpn+1).
Since GFn =GKn ∩GK(ζpn+1), we conclude that j >1 +e(n+ 1/(p−1)) if
and only if G(Kj) ⊆GFn. ¤
Remark 5.5. Note that this argument also shows the equality
uKn(ζpn)/K = 1 +e(n+ 1
p−1) without assumingζp ∈K.
Next we assume that the residue field of N is perfect. For an algebraic extensionF ofN, we put
aj
F/N ={x∈ OF |vN(x)≥j}.
For a finite Galois extension Qof N, we write uQ/N for the greatest upper ramification break ([7]) ofQ/N. Let us consider the property
(Pj)
for any algebraic extensionF of N, if there exists anON-algebra homomorphismOQ→ OF/ajF/N,
then there exists anN-algebra injectionQ→F
for j ∈R≥0, as in [7, Proposition 1.5]. Then we have the following
propo-sition, which is due to Yoshida. Here we reproduce his proof for the conve-nience of the reader.
Proposition 5.6 ([16]).
uQ/N = inf{j ∈R≥0 |the property (Pj) holds }.
Proof. By [7, Proposition 1.5 (i)], it is enough to show that the property (Pj) does not hold for j = uQ/N −(e′)−1 with an arbitrarily large e′ > 0.
extension ofN such thatQ∩N′ =N and putQ′ =QN′. Note that we have uQ′/N =uQ/N by assumption. From this proposition in [7], we see that for some algebraic extensionF ofN, there exists anON-algebra homomorphism
OQ′ → OF/aj
F/N forj=uQ/N−e(Q′/N)−1but noN-algebra injectionQ′ → F. SinceQ/N is wildly ramified, we see that e(Q/N)uQ/N −1 > e(Q/N).
Hence we haveuQ/N−e(Q′/N)−1>1 =u
N′/N and there exists anN-algebra injection N′ → F also by this proposition. Thus there exists no N-algebra injection Q → F and the property (Pj) for Q/N does not hold. Since we
can choose an arbitrarily largeN′ as above, the proposition follows. ¤
We see from Proposition 5.6 that to bound the greatest upper ramification breakuLn/K, it is enough to show the following proposition.
Proposition 5.7. Let F be an algebraic extension of K. If j > u(K, r, n)
and there exists an OK-algebra homomorphism
η:OLn → OF/a
j F/K,
then there exists a K-algebra injection Ln→F.
Proof. By assumption, we have j > er/(p−1) and bF ⊇ aj
F/K. Thus η
induces anOK-algebra homomorphism
OLn → OF/bF.
Sinceηalso induces anOK-algebra homomorphismOFn → OF/a
j
F/K and r≥1, from Corollary 5.4 and [7, Proposition 1.5] we get aK-linear injection
Fn → F. Thus we see that F contains πn and ζpn+1. More precisely, we
have the following lemma.
Lemma 5.8. For some integers i and i′ such that i′ ≡ 1 mod p, we have η(πn) ≡πnζpin mod bF and η(ζpn+1) ≡ζi
′
pn+1 mod bF. Moreover, there
ex-ists g∈GK such that g(πn) =πnζpin and g(ζpn+1) =ζi
′
pn+1.
Proof. Since the map η is OK-linear, the equality η(πn)p
n
= π holds in OF/ajF/K. Set ˆx to be a lift ofη(πn) in OF. Then we have
vK(ˆxp
n
−π) =
pn−1 X
i=0
vK(ˆx−πnζpin)≥j.
Let us apply Lemma 5.2 tof(T) =Tpn−π. Then, with the notation of the lemma, we have
s=ne+p
n−1
pn and α=
1
pn + e p−1. Since j−s > er/(p−1) by assumption, we have
ˆ
x≡πnζpin mod bF
Let h(T) be the minimal polynomial of ζpn+1 over OK. Since h divides
Tpn −ζp, the OK-algebra B′ = OK[T]/(h(T)) is also finite flat of relative
complete intersection and theK-algebraB′⊗OKKis ´etale. The Galois group
Gal(K(ζpn+1)/K) acts transitively on the set of zeros of h. Hence h also
satisfies the conditions of Lemma 5.2. Let s′ andα′ be as in this lemma for h. Then we haves′ ≤neandα′ ≤e/(p−1). This impliesj−s′ > er/(p−1).
By this lemma, there exists an elementg′∈Gal(K(ζ
pn+1)/K) such that the
elementg′(ζ
pn+1) =ζi
′
pn+1 satisfies
η(ζpn+1)≡g′(ζpn+1) modbF.
Since Kn∩K(ζpn+1) =K (see for example [14, Lemma 5.1.2]), we can find
an elementg ∈GK such thatg(πn) =πnζpin and g(ζpn+1) =g′(ζpn+1). This
concludes the proof. ¤
Lemma 5.9. TheOK-algebra homomorphism η induces an OK-algebra
in-jection
ηb:OLn/bLn→ OF/bF.
Proof. We write the Eisenstein polynomial of a uniformizerπLn of Ln over
OK as
P(T) =Te′+c1Te
′−1
+· · ·+ce′−1T+ce′, where e′ = e(L
n/K). Then z = η(πLn) satisfies P(z) = 0 in OF/a
j F/K.
Let ˆz be a lift of z in OF. Since j > 1, we have vF(ˆz) =e(F/K)/e′. The
condition i > e(Ln)r/(p−1) is equivalent to the condition
vF(ˆzi)>
e(Ln)r p−1 ·
e(F/K)
e′ = e(F)r
p−1.
Thus the claim follows. ¤
Since Ln containsFn, we can consider the ring
¯
A′n,Ln,r+=Wn(OLn/bLn)/ψ([ζpn]−1)
rm¯ n,Ln
and similarly ¯A′
n,F,r+ forF. We give these rings structures of Σ-algebras as
follows. The ring ¯A′
n,Ln,r+ is considered as a Σ-algebra by using the system
{πn}n∈Z≥0 we chose ofp-power roots ofπ, as in the previous section. On the
other hand, usingiandi′ in Lemma 5.8, put ˜π
n=πnζpin and ˜ζpn+1 =ζi
′
pn+1.
Then we consider the ring ¯A′
n,F,r+ as a Σ-algebra by using a system of p
-power roots ofπ containing ˜πn. We define Filr and φr of these rings in the
same way as before.
Lemma 5.10. The induced ring homomorphism
¯
η: ¯A′n,Ln,r+→A¯′n,F,r+
Proof. Firstly, we check that ¯η is Σ-linear. By definition, this homomor-phism commutes with the action of the element u ∈Σ. To show the com-patibility with the elementY ∈Σ, let us consider the commutative diagram
Wn(OLn/pOLn) −−−−→ Wn(OF/pOF)
y
y
Wn(OLn/bLn)
ηb
−−−−→ Wn(OF/bF)
y
y
¯
A′n,Ln,r+ −−−−→η¯ A¯′n,F,r+,
where the horizontal arrows are induced byη. Note that we haveηb(πn) = ˜πn
and ηb(ζpn+1) = ˜ζpn+1. Put β ∈W(R)× as in the proof of Proposition 4.7.
Namely, the element β is the solution in W(R) of the equation
E([π])β =pa−([ε1/p]−1)p−1,
where the element a∈ W(R) is as in the remark after Lemma 4.1. Let an
andβndenote the images ofaandβ inWn(OLn/pOLn), respectively. Then
the element βn is a solution of the equation
E([πn])βn=pan−([ζpn+1]−1)p−1.
Similarly, we define elements ˜anand ˜βnofWn(OF/pOF) using ˜πnand ˜ζpn+1.
By definition, the element ˜βn is a solution of the equation E([˜πn]) ˜βn=p˜an−([˜ζpn+1]−1)p−1.
Now what we have to show is the equality
¯
η(anβ−n1E([πn])p−1) = ˜anβ˜n−1E([˜πn])p−1
in the ring ¯A′
n,F,r+. Since the element an of Wn(OLn/pOLn) is a
lin-ear combination of the elements 1,[ζpn+1], . . . ,[ζpn+1]p−1 over Z, we have
¯
η(an) = ˜an in ¯An,F,r′ +. Thus the elements ˜βn and ¯η(βn) satisfy the same
equation in ¯A′
n,F,r+. Since these two elements are invertible, we see that
¯
η(βn)−1E([˜πn]) = ˜βn−1E([˜πn]) and the equality holds. Since the diagram
above is compatible with the Frobenius endomorphisms, we see from the definition that ¯ηalso preserves Filrand commutes withφrof both sides. ¤
Thus the homomorphism ¯η induces a homomorphism of abelian groups
Tcrys∗ ,Ln,πn(Mn)→Tcrys∗ ,F,˜πn(Mn).
Then the following lemma, whose proof is omitted in [2, Subsection 3.13], implies that this homomorphism is an injection. We insert here a proof of this lemma for the convenience of the reader.
Lemma 5.11. The ring homomorphismη¯: ¯A′
n,Ln,r+→A¯
′
Proof. For an algebraic extension N of Fn, let us write ¯A′N for the ring
¯
A′n,N,r+. Note that the ring ¯A′N/pA¯′N is isomorphic to the ring
ON/{x∈ ON |vp(x)> r
pn−1(p−1)}.
As in the proof of Lemma 5.9, we see that the homomorphism ¯η induces an injection
¯
A′Ln/pA¯′Ln →A¯′F/pA¯′F.
Thus it is enough to show the exactness of the sequence
0→A¯′N/pmA¯′N ×→pA¯′N/pm+1A¯′N →A¯′N/pA¯′N →0.
Let ¯x and ¯y be in ¯A′
N such that px¯ = pm+1y¯. Let ˆx = (ˆx0, . . . ,xˆn−1) and
ˆ
y = (ˆy0, . . . ,yˆn−1) be lifts of ¯x and ¯y in the ring Wn(ON), respectively. In
this ring, we have
(0,xˆp0,xˆ1p, . . . ,) = (0, . . . ,0,yˆ0pm+1,yˆ1pm+1, . . .) + ([ζpn]−1)rz,ˆ
where ˆz is in the ideal mn,N. From this equality we see that vp(ˆx0) >
r/(pn−1(p−1)) and ˆ
x= ([ζpn]−1)rwˆ+ (0,xˆ′1,xˆ′2, . . .)
for some ˆw∈mn,N and ˆx′i ∈ ON. The image of the first term on the
right-hand side in ¯A′
N is zero. Hence we may assume ˆx0 = 0. Repeating this, we
can see that ¯x∈pmA¯′
N and the above sequence is exact. ¤
Now Corollary 4.13 shows that the abelian group Tcrys∗ ,Ln,πn(Mn) is of
order pnd, where d = dimQpV. This implies that the the abelian group
T∗
crys,F,π˜n(Mn) is also of orderp
nd. Letg ∈G
K be as in Lemma 5.8. Then
we have the following lemma.
Lemma 5.12. The GFn-action on T
∗
crys,K,¯ π˜n(Mn) is the conjugate of the action on T∗
crys,K,π¯ n(Mn) by the element g.
Proof. Let an,a˜n and βn,β˜n be the elements of Wn(OK¯/pOK¯) as in the
proof of Lemma 5.10. Let us consider the composite
Σ→A¯′n,r+→g A¯′n,r+
of the ring homomorphism defined byu7→[πn] andY 7→ −anβn−1E([πn])p−1,
and the map induced byg. We claim that this is the natural ring homomor-phism defined by ˜πn. For this, we only have to check that this composite
sends the elementY ∈Σ to−˜anβ˜n−1E([˜πn]). Since the equality E([πn])βn=pan−([ζpn+1]−1)p−1
holds in the ring ¯A′
n,r+ on the source of the above map g, we have
