RAMIFICATION CORRESPONDENCE OF FINITE FLATGROUP SCHEMES OVER EQUAL AND MIXEDCHARACTERISTIC LOCAL FIELDSSHIN HATTORI
Abstract. Let p > 2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fraction
field of the Witt ring of k. Let G and H be finite flat commutative groupschemes killed by p over OK and k[[u]], respectively. In this paper, weshow the ramification subgroups of G and H in the sense of Abbes-Saito
are naturally isomorphic to each other when they are associated to thesame Kisin module.

1. Introduction
Let p be a rational prime, k be a perfect field of characteristic p, W =W (k) be the Witt ring of k and K be a finite totally ramified extension of
Frac(W ) of degree e. Let φ denote the Frobenius endomorphism of W . We
fix once and for all an algebraic closure K̄ of K, a uniformizer π of K and apsystem of its p-power roots {πn }n∈Z≥0 in K̄ with π0 = π and πn = πn+1
. Put
Kn = K(πn ), K∞ = ∪n Kn , GK = Gal(K̄/K) and GK∞ = Gal(K̄/K∞ ). By
the theory of norm fields ([25]), there exist a complete discrete valuation
field X ' k((u)) of characteristic p with residue field k and an isomorphismof groups
GK∞ ' GX = Gal(X sep /X ),where X sep is a separable closure of X . A striking feature of this isomorphism
is its compatibility with the upper ramification subgroups of both sides up
to a shift by the Herbrand function of K∞ /K ([25, Corollaire 3.3.6]).
On the other hand, Breuil ([6]) introduced linear algebraic data over aring S, which are now called as Breuil modules, and proved a classificationof finite flat (commutative) group schemes over OK via these data for p >2. Based on his works, Kisin ([18], [19]) simplified this classification byreplacing Breuil’s data by φ-modules over W [[u]], which are referred as Kisin
modules. Let us consider the case where finite flat group schemes are killedby p. Put S1 = k[[u]], which is isomorphic to the k-algebra OX . We let
φ also denote the absolute Frobenius endomorphism of the ring S1 . For a
non-negative integer r, let Modr,φ/S1 be the category of free S1 -modules M
of finite rank endowed with a φ-semilinear map φM : M → M such that theDate: January 2, 2011.
Supported by Grant-in-Aid for Young Scientists B-21740023.
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SHIN HATTORI

cokernel of the map 1 ⊗ φM : S1 ⊗φ,S1 M → M is killed by uer . Then there
exists an anti-equivalence of categories G(−) from Mod1,φ/S1 to the categoryof finite flat group schemes over OK killed by p. It is well-known that we
also have an anti-equivalence H(−) from Modr,φ/S1 to a category of finite flat
generically etale group schemes over OX killed by their Verschiebung ([15]).
Hence a correspondence between finite flat group schemes over OK and OX
is obtained, and if finite flat group schemes G over OK and H over OX arein correspondence with each other, then their generic fiber Galois modules
G(OK̄ ) and H(OX sep ) are also in correspondence via the theory of norm
fields. Namely, for an object M of Mod1,φ/S1 , we have an isomorphism of
GK∞ -modules
G(M)(OK̄ )|GK∞ → H(M)(OX sep )([19, Proposition 1.1.13]). From this, we can show that the Galois modules G(M)(OK̄ ) and H(M)(OX sep ) have exactly the same greatest upper
ramification jump in the classical sense (see also [1]).
Besides the classical ramification theory of their generic fibers, finite flatgroup schemes over a complete discrete valuation ring have their own ramification theory, which was discovered by Abbes-Saito ([3], [4]) and AbbesMokrane ([2]). Such a finite flat group scheme G has filtrations of upper
ramification subgroups {G j }j∈Q>0 ([2]) and lower ramification subgroups{Gi }i∈Q≥0 ([12], [16]) as in the classical ramification theory of local fields.
For simplicity, let K be a complete discrete valuation field as above andconsider a finite flat group scheme G over OK . Then, using the upper ramification filtration of G, we can bound the classical greatest upper ramification jump of the generic fiber GK -module G(OK̄ ) ([16]) and also describe
completely the semi-simplification of the restriction to the inertia subgroupof this GK -module ([17]). Moreover, the canonical subgroup of a possibly
higher dimensional abelian scheme A over OK is found in the upper ramification filtration of A[pn ] ([2], [23], [24]), while the canonical subgroup isalso found in the Harder-Narasimhan filtration of A[pn ] defined by Fargues([11], [12]).
In this paper, we establish the following correspondence of the ramification filtrations between finite flat group schemes over OK and OX which
is similar to that of the classical ramification jumps of their generic fiber
Galois modules stated above.
Theorem 1.1. Let p > 2 be a rational prime and K and X be as before.
Let M be an object of the category Mod1,φ/S1 . Then the natural isomorphismof GK∞ -modules G(M)(OK̄ )|GK∞ → H(M)(OX sep ) induces isomorphisms of
the upper and the lower ramification subgroups
G(M)j (OK̄ )|GK∞ → H(M)j (OX sep ),G(M)i (OK̄ )|GK∞ → H(M)i (OX sep )for any j ∈ Q>0 and i ∈ Q≥0 .

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

3This theorem enables us to reduce the study of ramification of finite flatgroup schemes over OK killed by p to the case where the base is a complete
discrete valuation ring of equal characteristic. This makes calculations oframification of finite flat group schemes over OK , for example as in [17, Section 5], much easier. We remark that, for the Harder-Narasimhan filtration,such a correspondence of filtrations of G(M) and H(M) follows easily fromthe definition.
A key idea to prove the theorem is to switch from the upper ramification
filtration to the lower ramification filtration via Cartier duality, which theauthor learned from works of Tian ([23]) and Fargues ([12]). Let G be a finite
flat group scheme over OK killed by p and G ∨ be its Cartier dual. Thenthey showed that the upper ramification subgroup G j (OK̄ ) is the orthogonal
subgroup of the lower ramification subgroup (G ∨ )i (OK̄ ) for some i via the
Cartier pairing
G(OK̄ ) × G ∨ (OK̄ ) → Z/pZ(1).
We prove a version of this theorem for the group scheme H(M) over OX .
Since we are in characteristic p, usual Cartier dual does not preserve thegeneric etaleness of finite flat group schemes. Instead, we use duality theories
of Caruso ([9]) and Liu ([20]) for Breuil modules and Kisin modules. Thisrequires us to check compatibilities of various duality theories, though it
is straightforward to carry out. Thus we reduce ourselves to proving thecorrespondence of the lower ramification subgroups of G(M) and H(M).
This is a consequence of the fact that, up to the base changes from K and
X to the extensions generated by p-th roots of their uniformizers π and u,
the schemes G(M) modulo π e and H(M) modulo ue become isomorphic toeach other, not as group schemes but as pointed schemes. We prove thisfact by using Breuil’s explicit computation of the aﬃne algebra of a finite
flat group scheme over OK killed by p in terms of its corresponding Breuil
module ([6, Section 3]), after showing that his classification of finite flatgroup schemes is compatible with the base change from K to Kn .
Acknowledgments. The author would like to thank Victor Abrashkin,Toshiro Hiranouchi, Wansu Kim and Yuichiro Taguchi for stimulating discussions, and Xavier Caruso for kindly answering his questions on the paper[9].
2. Review of Cartier duality theory of Caruso and Liu
Let p > 2 be a rational prime and K be a complete discrete valuation
field of mixed characteristic (0, p) with perfect residue field k, as in Section
1. It is well-known that finite flat group schemes over OK killed by somep-power are classified by linear algebraic data, Breuil modules ([6]) or Kisin
modules ([18], [19]). For these data, corresponding notions of duality to
Cartier duality for finite flat group schemes are introduced by Caruso and
Liu ([8], [9], [10], [20]), which play key roles in the integral p-adic Hodge

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SHIN HATTORI

theory. In this section, we recall the definitions of these data and their
theories of Cartier duality.
2.1. Breuil and Kisin modules. Let E(u) ∈ W [u] be the Eisenstein polynomial of the uniformizer π over W . Put F (u) = p−1 (ue − E(u)). Thisdefines units in the rings S = W [[u]] and S1 = k[[u]]. The φ-semilinear
continuous ring endomorphisms of these rings defined by u 7→ up are also
denoted by φ. Let r be a non-negative integer. Then a Kisin module over
S of E-height ≤ r is an S-module M endowed with a φ-semilinear map
φM : M → M such that the cokernel of the map 1 ⊗ φM : S ⊗φ,S M → M
is killed by E(u)r . We write φM also as φ if there is no risk of confusion.
A morphism of Kisin modules over S is an S-linear map which is compatible with φ’s of the source and the target. The Kisin modules over S of
E-height ≤ r form a category with an obvious notion of exact sequences.
r,φWe let Modr,φ/S1 (resp. Mod/S ) denote its full subcategory consisting of M
which is free of finite rank over S1 (resp. S). We also let Modr,φ/S∞ denoteits full subcategory consisting of M such that M is a finite S-module which
is p-power torsion and u-torsion free.
r,φr,φFor r < p − 1, we have categories Modr,φ/S1 , Mod/S∞ and Mod/S of Breuil
modules defined as follows. Let S be the p-adic completion of the divided
power envelope W [u]DP of W [u] with respect to the ideal (E(u)) and thecompatibility condition with the canonical divided power structure on pW .
The ring S has a natural filtration Fili S defined as the closure in S of theideal generated by E(u)j /j! for integers j ≥ i. The φ-semilinear continuous
ring endomorphism of S defined by u 7→ up is also denoted by φ. For 0 ≤ i ≤p − 1, we have φ(Fili S) ⊆ pi S and put φi = p−i φ|Fili S . These filtration andφi ’s induce a similar structure on the ring Sn = S/pn S. Put c = φ1 (E(u)) ∈S × . Then we let 0 Modr,φ/S denote the category of S-modules M endowedwith an S-submodule Filr M containing (Filr S)M and a φ-semilinear map
φr,M : Filr M → M satisfying φr,M (sr m) = c−r φr (sr )φr,M (E(u)r m) for
any sr ∈ Filr S and m ∈ M. A morphism of this category is defined to be ahomomorphism of S-modules compatible with Filr ’s and φr ’s. We drop thesubscript M of φr,M if no confusion may occur. The category 0 Modr,φ/S hasan obvious notion of exact sequences. Its full subcategory consisting of M
such that M is a free S1 -module of finite rank and the image φr,M (Filr M)r,φgenerates the S-module M is denoted by Modr,φ/S1 . We let Mod/S∞ denoter,φthe smallest full subcategory of 0 Modr,φ/S containing Mod/S1 and stable under

extensions, and Modr,φ/S denote the full subcategory consisting of M such that
M is a free S-module of finite rank, the S-module M/Filr M is p-torsion
free and the image φr,M (Filr M) generates the S-module M.
r,φThe categories Modr,φ/S∞ and Mod/S∞ for r < p − 1 are in fact equiva-r,φlent. We define an exact functor MS : Modr,φ/S∞ → Mod/S∞ by putting

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

5

MS (M) = S ⊗φ,S M with
1⊗φFilr MS (M) = Ker(S ⊗φ,S M → (S/Filr S) ⊗S M),
1⊗φ

φr ⊗1

φr : Filr MS (M) → Filr S ⊗S M → S ⊗φ,S M = MS (M).
Then the functor MS is an equivalence of categories ([10, Theorem 2.3.1]).
r,φSimilarly, we have an equivalence of categories Modr,φ/S → Mod/S ([10, Theorem 2.2.1]), which is denoted also by MS .
The categories of Breuil and Kisin modules have natural duality theories([10, Subsection 2.4]). For objects M and M of the categories Modr,φ/S∞∨∨
and Modr,φ/S∞ , we let M and M denote their dual objects as in loc. cit.,
respectively. Put∞∏
λ=φi (E(u)/E(0)) ∈ S × .

i=1
Then the multiplication by λ−r defines a natural isomorphism

τM : MS (M∨ ) → MS (M)∨of the category Modr,φ/S∞ . We can also extend these duality theories to ther,φcategories Modr,φ/S and Mod/S with the same compatibility with the functor
MS .

2.2. The associated Galois representations and duality. Next we recall constructions of the associated Galois representations to Breuil andKisin modules and their duality theories. Let vK be the valuation on K
which is normalized as vK (π) = 1 and we extend it naturally to K̄. Set
ÕK̄ = OK̄ /pOK̄ and C to be the completion of K̄. Consider the ringR = lim(ÕK̄ ← ÕK̄ ← · · · ),←−where the transition maps are defined by x 7→ xp . For an element x =n(x0 , x1 , . . .) ∈ R with xi ∈ ÕK̄ , we put x(m) = limn→∞ x̂pn+m ∈ OC , wherex̂i is a lift of xi in OK̄ . This is independent of the choice of lifts. Thenthe ring R is a complete valuation ring of characteristic p with valuationvR (x) = vK (x(0) ). We put m≥i
R = {x ∈ R | vR (x) ≥ i} and similarly for>imR . Define an element π of R by π = (π, π1 , π2 , . . .), where we abusively
write πn also for its image in ÕK̄ . The W -algebras R and W (R) have natural
S-algebra structures defined by the continuous map S → W (R) which sends
u to the Teichmüller lift [π] of the element π. Note that the identification
GK∞ ' GX stated in Section 1 is given by the action of GK∞ on the ring R
and the inclusion k[[u]] → R defined by u 7→ π ([7, Subsection 3.3]).
We set OE to be the p-adic completion of S[1/u] and put E = Frac(OE ).
We extend the inclusion S → W (R) to an inclusion OE → W (Frac(R)). The
maximal unramified extension of E in the field W (Frac(R))[1/p] is denoted
ur . We put Sur = O
by E ur and its closure in the same field by Ec
ur ∩ W (R)dE

6SHIN HATTORI

inside the ring W (Frac(R)). The Galois group GK∞ acts naturally on the
S-algebra Sur . For an object M ∈ Modr,φ/S∞ , its associated GK∞ -moduleTS∗ (M) is by definitionTS∗ (M) = HomS,φ (M, Qp /Zp ⊗ Sur ).
If the S-module M is killed by pn , then we have a natural identificationTS∗ (M) ' HomS,φ (M, Wn (R)) ([14, Proposition 1.8.3]). Similarly, for an
object M of the category Modr,φ/S , its associated GK∞ -module is defined asTS∗ (M) = HomS,φ (M, Sur ).
Consider a natural W -algebra surjection θ : W (R) → OC defined by∞∑(i)θ((z0 , z1 , . . .)) =pi z i ,i=0where zi is an element of R. The p-adic completion of the divided powerenvelope of W (R) with respect to the ideal Ker(θ) is denoted by Acrys . The
ring Acrys has a Frobenius endomorphism φ and a GK -action induced bythose of R, and also a filtration induced by the divided power structure.
The W -algebra homomorphism W [u] → W (R) defined by u 7→ [π] induces
a map S → Acrys , by which we consider the ring Acrys as an S-algebra. For
0 ≤ r ≤ p − 2, the ring Acrys has a natural structure as an object of 0 Modr,φ/S
by putting φr = p−r φ|Filr Acrys .
Let M be an object of Modr,φ/S∞ . Then we also have the associated GK∞ module∗
Tcrys(M) = HomS,Filr ,φr (M, Qp /Zp ⊗ Acrys ).
If M is killed by p, then we have a natural identification∗
Tcrys(M) ' HomS,Filr ,φr (M, RDP ),where RDP is the divided power envelope of R with respect to the ideal m≥eR
and we identify this ring with Acrys /pAcrys . Similarly, for an object M of
the category Modr,φ/S , we put∗
Tcrys(M) = HomS,Filr ,φr (M, Acrys ).
r,φ∗The functors TS∗ and Tcrys
from Modr,φ/S∞ and Mod/S∞ to the category of

GK∞ -modules are exact. For an object M of Modr,φ/S1 , we have the equality∗∗ .
dimFp (TS (M)) = rankS1 (M) and a similar assertion also holds for Tcrysr,φr,φThen, for an object M of the category Mod/S∞ or Mod/S , we have an
isomorphism of GK∞ -modules∗TS∗ (M) → Tcrys(MS (M))defined by f 7→ (s ⊗ m 7→ sφ(f (m))) ([21, Lemma 3.3.4]).
∗ (M), let us also fix a
To describe duality theories for TS∗ (M) and Tcrys
system {ζpn }n∈Z≥0 of p-power roots of unity in K̄ such that ζ1 = 1, ζp 6= 1

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

7

and ζpn = ζppn+1 , and set an element ε ∈ R to be ε = (1, ζp , ζp2 , . . .) with anabusive notation as before. Define an element t of Acrys byt = log([ε]) =∞∑(−1)i−1i=1([ε] − 1)i.
i

Put c0 = p−1 E(0) ∈ W × . Let S(r) = Se be the object of the categoryrMod1,φ/S of rank one with a basis e satisfying φ(e) = (E(u)/c0 ) e. Similarly,
we define an object S(r) of the category Mod1,φ/S of rank one by S(r) = Se =r
Fil S(r) and φr (e) = e. Then the Breuil module MS (S(r)) is isomorphic
to S(r) via the multiplication by λ−r . Thus we have isomorphisms of GK∞ modules∗∗TS∗ (S(r)) → Tcrys(MS (S(r))) → Tcrys(S(r)) = Zp (e 7→ tr ).

Their composite is given by f 7→ (e 7→ λr φ(f (e))). Set an element t ∈ Acrys
to beE(u) −1 −1) λ t.
t=(c0
Then the element t is contained in the subring Sur ([20, Subsection 3.2]).
∨
Let M be an object of the category Modr,φ/S and M be its dual object.
Then as in [10, Subsection 2.4], we have a natural perfect pairing of GK∞ modules
h , iM : TS∗ (M) × TS∗ (M∨ ) → TS∗ (S(r)) ' Zp tr ⊆ Sur .
Similarly, for an object M of the category Modr,φ/S , we also have a naturalperfect pairing of GK∞ -modules∗∗∗
h , iM : Tcrys(M) × Tcrys(M∨ ) → Tcrys(S(r)) ' Zp tr ⊆ Acrys .

Moreover, we can see that the following diagram is commutative for M =MS (M):TS∗ (M) × TS∗ (M∨ )ooh , iM/ Zp tr∗ (M (M)) × T ∗ (M (M∨ ))
Tcrys
S
S
crys

o λr φO∗ (τ )TcrysoM∗ (M (M)) × T ∗ (M (M)∨ )
Tcrys
S
S
crys


/ Zp tr .

h , iMWe also have similar perfect pairings h , iM and h , iM for the categoriesr,φModr,φ/S∞ and Mod/S∞ , which are compatible with each other as above ([10]).

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SHIN HATTORI

3. Cartier duality for upper and lower ramification subgroups
Let G be a finite flat generically etale (commutative) group scheme over
the ring of integers of a complete discrete valuation field. Abbes-Mokrane([2]) initiated a study of ramification of G using a ramification theory ofAbbes-Saito ([3], [4]). As in the classical ramification theory of local fields,
G has upper and lower ramification subgroups ([12], [16]). When the basefield is of mixed characteristic, Tian proved that the upper and the lower
ramification subgroups correspond to each other via usual Cartier duality if
G is killed by p ([23]), and Fargues gave a much simpler proof of this theorem([12]). In this section, after briefly recalling the ramification theory of finite
flat group schemes, we show a variant of Tian’s theorem for a complete
discrete valuation field of equal characteristic p with perfect residue field,
using the duality techniques presented in the previous section instead of
Cartier duality of finite flat group schemes.
3.1. Ramification theory of finite flat group schemes. In this subsection, we let K denote a complete discrete valuation field, π a uniformizerof K, OK the ring of integers and K sep a separable closure of K. Let vK
be the valuation of K normalized as vK (π) = 1 and extend it naturally to>isep | vK (x) ≥ i} and similarly for m sep .
K sep . We put m≥i
K sep = {x ∈ OKK
sep
We also put GK = Gal(K /K).
Let G = Spec(B) be a finite flat generically etale group scheme over OK .
Then G is locally of complete intersection over OK ([6, Proposition 2.2.2])
and we have a natural surjection of GK -modules G(OK sep ) → F j (B) forj ∈ Q>0 , where F j (B) is the set of geometric connected components of thej-th tubular neighborhood of B with a group structure induced by that of
G ([2, Subsection 2.3]). We define the j-th upper ramification subgroup G jof G for j ∈ Q>0 to be the scheme-theoretic closure in G of the kernel of thissurjection. On the other hand, for i ∈ Q≥0 , the scheme-theoretic closure in
G of the subgroup Ker(G(OK sep ) → G(OK sep /m≥i
K sep )) is denoted by Gi andcalled the i-th lower ramification subgroup of G. We also put∪∪ 0
Gi0 (OK sep )G j+ (OK sep ) =G j (OK sep ), Gi+ (OK sep ) =j 0 >ji0 >iand set G j+ and Gi+ to be their scheme-theoretic closures in G, respectively.
As in the classical case, the upper (resp. lower) ramification subgroups
are compatible with quotients (resp. subgroups). Namely, for a faithfully
flat homomorphism G → G 00 of finite flat group schemes over OK , the imageof G j (OK sep ) in G 00 (OK sep ) coincides with (G 00 )j (OK sep ) ([2, Lemme 2.3.2]).
From the definition, we also see that for a closed immersion G 0 → G offinite flat group schemes over OK , the subgroup G 0 (OK sep ) ∩ Gi (OK sep ) coincides with (G 0 )i (OK sep ). In addition, for a finite extension L/K of relative
ramification index e0 , we have natural isomorphisms
0(OL ×OK G)je → OL ×OK G j ,(OL ×OK G)ie0 → OL ×OK Gi

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

9

of finite flat group schemes over OL .
Suppose that K is of mixed characteristic (0, p) and G is killed by pn .
Then we have G j = 0 for j > e(n + 1/(p − 1)), where e is the absolute
ramification index of K ([16, Theorem 7]). Let G ∨ be the Cartier dual of G
and consider the Cartier pairing
h , iG : G(OK̄ ) × G ∨ (OK̄ ) → Z/pn Z(1).
When G is killed by p, we have the following duality theorem for the upper
and the lower ramification subgroups of G ([23, Theorem 1.6], [12, Proposition 6]).
Theorem 3.1. Let K be a complete discrete valuation field of mixed characteristic (0, p) with absolute ramification index e and G be a finite flat groupscheme over OK killed by p. For j ≤ pe/(p − 1), we have an equality
G j (OK̄ )⊥ = (G ∨ )l(j)+ (OK̄ )of subgroups of G ∨ (OK̄ ), where ⊥ means the orthogonal subgroup with respect
to the pairing h , iG and l(j) = e/(p − 1) − j/p.
3.2. Kisin modules and finite flat group schemes of equal characteristic. Consider the complete discrete valuation field X = k((u)) withuniformizer u and perfect residue field k. We embed OX = S1 = k[[u]] into
R by u 7→ π as before. Then R is the completion of the ring of integers ofan algebraic closure of X . We let X sep denote the separable closure of X in
Frac(R).
Let Y be a finite extension of X in Frac(R) of relative ramification index
e0 and let φ also denote the absolute Frobenius endomorphism of Y. We
identify OY with l[[v]] for a finite extension l of k. Define a category Modr,φ/OY
to be the category of free OY -modules N of finite rank endowed with a φsemilinear map φN : N → N such that the cokernel of the map 1 ⊗ φN :OY ⊗φ,OY N → N is killed by uer .
r,φFor an object M of Modr,φ/OX = Mod/S1 , we consider the OY -module
OY ⊗OX M with the φ-semilinear map φ ⊗ φM as an object of Modr,φ/OY . Thisr,φdefines a base change functor OY ⊗OX − : Modr,φ/S1 → Mod/OY . We also

have a dual object N∨ for an object N of the category Modr,φ/OY , which isdefined similarly to the duality theory of Modr,φ/S1 . Moreover, for an object
M of Modr,φ/S1 , we have a natural isomorphism
OY ⊗OX M∨ → (OY ⊗OX M)∨
of Modr,φ/OY , by which we identify both sides.
For a finite flat group scheme H over a base scheme of characteristicp, we let FH and VH denote the Frobenius and the Verschiebung of H,
respectively. We say that a finite flat group scheme H over OY is v-height10

SHIN HATTORI≤ s if its Verschiebung VH is zero and the cokernel of the natural map
VH∨ : OY ⊗φ,OY Lie(H∨ ) → Lie(H∨ )is killed by v s . The category of finite flat group schemes over OY of v-height≤s≤ s is denoted by CO. Then we have an anti-equivalence of categories
Y
0

≤ee r
HY (−) : Modr,φ/OY → COY([15, Théorème 7.4]). The group scheme HY (N) is defined as a functor over
OY by
A 7→ HomOY ,φ (N, A),
where we consider an OY -algebra A as a φ-module over OY with the absolute
Frobenius endomorphism of A. If we choose a basis e1 , . . . , ed of N and takea matrix A = (ai,j ) ∈ Md (OY ) satisfying
φ(e1 , . . . , ed ) = (e1 , . . . , ed )A,
then HY (N) is isomorphic to the additive group scheme over OY defined by
the system of equations
Xip −

d
∑

aj,i Xj = 0 (i = 1, . . . , d).

j=1

For an object M of Modr,φ/S1 , we also have a natural isomorphism
OY ×OX HX (M) → HY (OY ⊗OX M)
of finite flat group schemes over OY . We drop the subscript Y of HY if there
is no risk of confusion.
The following lemma is a variant of the scheme-theoretic closure for finite
flat group schemes.
Lemma 3.2. Let M be an object of Modr,φ/S1 and L be a GX -stable subgroupof TS∗ (M). Then there exists a surjection M → M00 of Modr,φ/S1 such that∗00∗
the image of the corresponding injection TS (M ) → TS (M) coincides with
L. A surjection M → M00 satisfying this property is unique up to a unique
isomorphism.
Proof. This follows from [20, Lemma 2.3.6]. Indeed, let Modφ/X denote thecategory of etale φ-modules over X ([14, Section A1]). We have an equivalence of categories T∗ from Modφ/X to the category of finite GX -modules
over Fp defined by T∗ (M ) = (X sep ⊗X M )φ=1 . For an object M of Modφ/X ,we also put T ∗ (M ) = HomX ,φ (M, X sep ). Then the natural map
T∗ (M ) → HomFp (T ∗ (M ), Fp )

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

11

is an isomorphism of GX -modules. Set M = X ⊗S1 M. We have a natural
isomorphism of GX -modules TS∗ (M) → T ∗ (M ) and let M 00 be the quotientof M corresponding to the surjectionT∗ (M ) → HomFp (T ∗ (M ), Fp ) → HomFp (L, Fp ).
Then the Kisin module M00 = Im(M → M → M 00 ) satisfies the desired
property.

Since the finite flat group scheme H(M) is generically etale, the group
H(M)(OX sep ) can be identified with the group H(M)(R) and we have the jth upper ramification subgroup H(M)j (OX sep ) = H(M)j (R) of H(M). We
also have the i-th lower ramification subgroup
H(M)i (OX sep ) = Ker(H(M)(OX sep ) → H(M)(OX sep /m≥i
X sep )),
which we identify with
H(M)i (R) = Ker(H(M)(R) → H(M)(R/m≥i
R ))
≥i
∗
by using the injection OX sep /m≥i
X sep → R/mR . Since H(M)(R) = TS (M),the pairing h , iM of Subsection 2.2 induces a perfect pairing

h , iM : H(M)(R) × H(M∨ )(R) → R.
Then the main theorem of this section is the following.
j
Theorem 3.3. Let M be an object of Modr,φ/S1 . Then we have H(M) = 0
for j > per/(p − 1). Moreover, for j ≤ per/(p − 1), we have the equality

H(M)j (R)⊥ = H(M∨ )lr (j)+ (R)of subgroups of H(M∨ )(R), where ⊥ means the orthogonal subgroup withrespect to the pairing h , iM and lr (j) = er/(p − 1) − j/p.
Proof. We proceed as in the proof of [12, Proposition 6]. Let Y be a finite
separable extension of X in X sep of relative ramification index e0 and put
MY = OY ⊗S1 M. Then we have a commutative diagram
H(M)(R) × H(M∨ )(R)
o



o



h , iM

HY (MY )(R) × HY (M∨Y )(R)/R/ R,

h , iMY

where h , iMY is a perfect pairing defined similarly to the pairing h , iM
and the vertical arrows are isomorphisms. Since H(M∨ ) is generically etale,
after making a finite separable base change and replacing e by ee0 , we may
assume that the GX -action on H(M∨ )(R) is trivial.
Let x∨ be an element of H(M∨ )(R) and consider the surjection M∨ → N∨∨
of the category Modr,φ/S1 corresponding to the subspace Fp x ⊆ H(M )(R)

12

SHIN HATTORIby Lemma 3.2. Then we have the commutative diagram
H(M)(R) × H(M∨ )(R)h , iMO

H(N∨ )(R) × H(N)(R)h , iN/R/ R.

Thus, by the compatibility with quotients (resp. subgroups) of the upper(resp. lower) ramification subgroups, the theorem follows from Lemma 3.4
below.

Lemma 3.4. Let N be an object of Modr,φ/S1 which is free of rank one over∨j
S1 . Then we have H(N ) (R) = 0 if j > per/(p − 1) and H(N)i (R) = 0 if
i > er/(p − 1). For j ≤ per/(p − 1), the subgroup H(N∨ )j (R) is zero if andonly if H(N)lr (j)+ (R) = H(N)(R).
Proof. Let n be a basis of N and n∨ be its dual basis of N∨ . Put φN (n) =∨er−s a0 n∨
us an with 0 ≤ s ≤ er and a ∈ S×
1 . Then we have φN∨ (n ) = u
×0
with some a ∈ S1 . Hence the defining equations of H(N) and H(N∨ ) are
X p − us aX = 0 and X p − uer−s a0 X = 0, respectively. By a calculation asin [16, Section 3], we see that{
H(N∨ )(R) (j ≤ p(er − s)/(p − 1))∨ j
H(N ) (R) =0(j > p(er − s)/(p − 1)),{
H(N)(R) (i ≤ s/(p − 1))
H(N)i (R) =0(i > s/(p − 1))and the first assertion follows. Moreover, for j ≤ per/(p − 1), we havelr (j) ≥ 0 andH(N∨ )j (R) = 0 ⇔ j > p(er − s)/(p − 1)
⇔ lr (j) < s/(p − 1) ⇔ H(N)lr (j)+ (R) = H(N)(R).

By the previous lemma, we also have the following corollary.
Corollary 3.5. Let M be an object of Modr,φ/S1 . Then the i-th lower ramification subgroup H(M)i vanishes for i > er/(p − 1).
Proof. As in the proof of Theorem 3.3, we may assume that the GX -module
H(M)(R) is trivial. For i > er/(p − 1), take an element x ∈ H(M)i (R)and consider the quotient M → N corresponding to the subspace Fp x ⊆H(M)(R). By Lemma 3.4, we have H(N)i (R) = 0 and thus x = 0.
Remark 3.6. Let K be a complete discrete valuation field of mixed characteristic (0, p) and G be a finite flat group scheme over OK killed by p.
Then, by using the usual scheme-theoretic closure of finite group schemes,
we can easily see that the i-th lower ramification subgroup Gi vanishes fori > e/(p − 1), just as in the proof of Corollary 3.5.

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

13

4. Comparison of ramification
Let p > 2 be a rational prime and K be a complete discrete valuation
field of mixed characteristic (0, p) with perfect residue field k, as in Section
2. Then it is known that there exists an anti-equivalence G(−) from the category Mod1,φ/S1 to the category of finite flat group schemes over OK killed by p.

≤e
On the other hand, we also have the anti-equivalence H(−) : Mod1,φ/S1 → COX
defined in Section 3 and an isomorphism of GK∞ -modules

εM : G(M)(OK̄ )|GK∞ → H(M)(R).
In this section, we prove that this isomorphism is compatible with the upper
and the lower ramification subgroups of both sides. For the proof, after recalling the definitions of the anti-equivalence G(−) and the isomorphism εM([6], [18], [19]), we show that these are compatible with the base changes inside K∞ /K and dualities on both sides. Then, by the duality theorems presented in Section 3, we reduce ourselves to comparing the lower ramification
subgroups of both sides, which is achieved by constructing an isomorphism
as pointed schemes between reductions of G(M) and H(M).
4.1. Breuil-Kisin classification. In this subsection, we briefly recall theclassification of finite flat group schemes and p-divisible groups over OK due
to Breuil and Kisin ([6], [18], [19]) and its properties. Their classification
theorem is as follows.
Theorem 4.1.
(1) There exists an anti-equivalence of categories G(−)
from Mod1,φ/S to the category of p-divisible groups over OK with anatural isomorphism of GK∞ -modules
εM : Tp G(M)|GK∞ → TS∗ (M),where Tp G(M) = limn G(M)[pn ](OK̄ ) is the p-adic Tate module of←−the p-divisible group G(M).
(2) There exists an anti-equivalence of categories G(−) from Mod1,φ/S∞
to the category of finite flat group schemes over OK killed by somep-power with a natural isomorphism of GK∞ -modules
εM : G(M)(OK̄ )|GK∞ → TS∗ (M).
(3) Let M be an object of the category Mod1,φ/S∞ and take a resolution of
Kisin modulesf

0 → M1 → M2 → M → 0,
where Mi is an object of the category Mod1,φ/S . Put G = G(M) andΓi = G(Mi ). Then we have an exact sequence of fppf sheaves
G(f )0 → G → Γ2 → Γ1 → 0

14SHIN HATTORI

which induces the commutative diagram with exact rows00

/ Tp Γ2/ Tp Γ1/ G(O )
K̄

εM2

εM1

εM



/ T ∗ (M2 )S/ T ∗ (M1 )S/ T ∗ (M)
S

/0

/ 0.

We have two definitions of the functor G(−) and the isomorphism εM .
One is putting G(−) = GrB (MS (−)) with the anti-equivalence GrB (−) of[6], as in [19]. The isomorphism εM is constructed in the proof of [19, Proposition 1.1.13]. Then we can prove Theorem 4.1 (3) by using [10, Proposition
2.1.3 and Theorem 2.3.4]. The other is given in [18], as follows. For a pdivisible group Γ over OK , set ModK (Γ) to be the section D∗ (Γ)(S→OK ) of
the contravariant Dieudonné crystal D∗ (Γ) on the divided power thickening
S → OK defined by u 7→ π. The S-module ModK (Γ) is endowed with thenatural Frobenius and the Hodge filtration. Then it is shown that this defines an anti-equivalence from the category of p-divisible groups over OK toK
the category Mod1,φ/S with a quasi-inverse Gr ([18, Proposition A.6]). Put
G(−) = GrK (MS (−)) and define G(−) for the category Mod1,φ/S∞ by taking

a resolution as in Theorem 4.1 (3). For an object M of the category Mod1,φ/S ,put M = MS (M) and Γ = G(M). Consider the divided power thickening
Acrys → OC induced by the map θ : W (R) → OC and we let FAcrys denote the section of a crystalline sheaf F on this thickening. Then we define
the isomorphism εM in this case to be the composite of the isomorphism∗ (M) and the natural isomorphism ε∗TS∗ (M) → Tcrys
M : Tp Γ → Tcrys (M)induced by the isomorphism∗
Tp Γ → Tcrys(D∗ (Γ)(S→OK ) ) = HomS,Fil1 ,φ1 (D∗ (Γ)Acrys , D∗ (Qp /Zp )Acrys )sending a homomorphism g : Qp /Zp → Γ of p-divisible groups over OC toD∗ (g) ([13, Theorem 7]). We can see that these two definitions are naturally
isomorphic to each other.
We need an explicit construction of the isomorphism εM for an object M of
Mod1,φ/S1 ([19, Proposition 1.1.13]). Let H1 be the divided power polynomial
ring RDP hu−πi over RDP . Consider the n-th projection prn : R → ÕK̄ . The
map pr0 induces a surjection H1 = RDP hu − πi → ÕK̄ defined by u 7→ π,
which is a divided power thickening of ÕK̄ . The map H1 → RDP defined by
u 7→ π induces an isomorphism of GK∞ -modules
G(M)(OK̄ ) = HomS,Fil1 ,φ1 (M, H1 ) → HomS,Fil1 ,φ1 (M, RDP )(see the proof of [6, Lemme 5.3.1]) and the isomorphism εM is the composite∗ (M).
of this map and the natural isomorphism TS∗ (M) → Tcrys
Let M be an object of Mod1,φ/S1 and put M = MS (M). In [6, Section3], Breuil gave an explicit description of the aﬃne algebra RM of the finite

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

15

flat group scheme G(M) = GrB (M) in terms of M. Let e1 , . . . , ed be an
adapted basis of M ([6, Définition 2.1.2.6]) such that
Fil1 M = ur1 S1 e1 ⊕ · · · ⊕ urd S1 ed + (Filp S)M,
φ1 (ur1 e1 , . . . , urd ed ) = (e1 , . . . , ed )Gwith a matrix G ∈ GLd (S1 ) (note that we adopt the transpose of the notation in [6]). Put S̃1 = k[u]/(uep ) and identify this k-algebra with S1 /Filp S1 .
Assume that the image G̃ of G in GLd (S̃1 ) is contained in the subgroup
GLd (k[up ]/(uep )). Consider the ring homomorphisms
k[u]/(uep ) → OK1 /pOK1 ← OK1 ,where the first map is the φ−1 -semilinear isomorphism defined by u 7→ π1 .
We choose a lift Gπ = (ai,j ) of G̃ by this map to GLd (OK ) using the assumption on G and put
OK [X1 , . . . , Xd ]).
RM = (pp
π e−r1 ∑d
π e−rd ∑dX1 + F (π) ( j=1 aj,1 Xj ), . . . , Xd + F (π) ( j=1 aj,d Xj )Then we have an isomorphism of p-adic formal schemes Spf(RM ) ' G(M).
The induced map
Ψ : HomOK -alg. (RM , OK̄ ) → G(M)(OK̄ ) = HomS,Fil1 ,φ1 (M, H1 )is defined as follows. Let f : RM → OK̄ be an element of the left-handside. Set xi = f (Xi ) and x̄i to be its image in ÕK̄ . Let (ÕK̄ )DP be the≥e/pdivided power envelope of the ring ÕK̄ with respect to the ideal mK̄ .
Then the map pr1 induces an isomorphism pr1 : RDP → (ÕK̄ )DP . We let
yi denote the inverse image of x̄i by this isomorphism. Then Ψ(f ) is theunique element of the right-hand side which is congruent to the S-linear
map (ei 7→ yi ) modulo Filp H1 (see the proof of [6, Proposition 3.1.5]). From
this description, we see that the zero section of the group scheme Spec(RM )is defined by X1 = · · · = Xd = 0.
4.2. Compatibility with a base change and Cartier duality. The
functor G(−) is compatible with the base changes inside K∞ /K. Moreover,Caruso proved its compatibility with Cartier duality ([9]). In this subsection,
we briefly present a proof of these facts, along with similar compatibilitiesof the isomorphism εM . First we recall the following lemma.
Lemma 4.2. For a p-divisible group Γ over OK , we put Vp Γ = Qp ⊗ Tp Γ.
Then the functor Γ 7→ Vp Γ|GK∞ from the category of p-divisible groups over
OK to the category of p-adic GK∞ -representations is fully faithful.
Proof. This follows from Tate’s theorem ([22, Subsection 4.2, Corollary 1])
and [18, Corollary 2.1.14].
Let us show the compatibility with the base change. Put S0 = W [[v]] andlet φ : S0 → S0 denote the natural φ-semilinear map defined by v 7→ v p .

16SHIN HATTORI

Note that the polynomial E 0 (v) = E(v p ) is the Eisenstein polynomial of
the uniformizer πn ∈ Kn . Consider the categories Mod1,φ, Mod1,φ/S0 and
/S0
n

∞

1

0
0
Mod1,φ/S0 of Kisin modules over S of E -height ≤ 1. Using the W -algebra

homomorphism S0 → W (R) defined by v 7→ [π 1/p ], we define a similarfunctor TS∗ 0 to TS∗ . The homomorphism of W -algebras S → S0 defined byn1,φ0
u 7→ v p induces natural functors (−)0 : Mod1,φ/S∞ → Mod/S0 and (−) :
n

∞

1,φ
Mod1,φ/S → Mod/S0 by

M 7→ M0 = S0 ⊗S M,

φM0 = φ ⊗ φM .

Then we have a natural isomorphism of GK∞ -modules TS∗ (M) → TS∗ 0 (M0 ).
On the other hand, these categories classify finite flat group schemes killed
by some p-power and p-divisible groups over OKn by Theorem 4.1, and we
let G 0 (−) denote the anti-equivalences of the theorem over OKn .
Proposition 4.3.

(1) Let M be an object of the category Mod1,φ/S and

M0 be the associated object of the category Mod1,φ/S0 . Then there exists
a natural isomorphism
G 0 (M0 ) → OKn ×OK G(M)of p-divisible groups over OKn which makes the following diagram
commutative:
Tp G(M)|GK∞
o

∼

K∞

o



TS∗ (M)/ Tp G 0 (M0 )|G∼/ T ∗ 0 (M0 ).
S

0(2) Let M be an object of the category Mod1,φ/S∞ and M be the asso-

ciated object of the category Mod1,φ/S0∞ . Then there exists a natural
isomorphism
G 0 (M0 ) → OKn ×OK G(M)of finite flat group schemes over OKn which makes the following
diagram commutative:
G(M)(OK̄ )|GK∞
o

∼

o



TS∗ (M)/ G 0 (M0 )(O )|G
K̄
K∞

∼


/ T ∗ 0 (M0 ).
S

Proof. The assertion (1) follows from Lemma 4.2, and (2) follows from (1) bytaking a resolution of M as in Theorem 4.1 (3). Note that the isomorphism
G 0 (M0 ) → OKn ×OK G(M) is independent of the choice of a resolution ([18,Lemma 2.3.4]).


RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

17

Next we show the compatibility with Cartier duality. For the element
ur n urt ∈ Sur , we let tn denote its image in the ring Sur
n = S /p S .
Proposition 4.4.
(1) Let M be an object of the category Mod1,φ/S and∨M be its dual object. Then there exists a natural isomorphism
G(M)∨ → G(M∨ ) of p-divisible groups over OK such that the induced mapε∨MδM : Tp (G(M)∨ ) → Tp G(M∨ ) →TS∗ (M∨ )fits into the commutative diagram of GK∞ -modules/ Zp (1)

Tp G(M) × Tp (G(M)∨ )
εM o



δM o



TS∗ (M) × TS∗ (M∨ )o/ Zp t,h , iM

where the top arrow is the Cartier pairing of G(M) and the rightvertical arrow is the isomorphism defined by (ζpn )n∈Z≥0 7→ t.
n(2) Let M be an object of the category Mod1,φ/S∞ which is killed by p
and M∨ be its dual object. Then there exists a natural isomorphism
G(M)∨ → G(M∨ ) of finite flat group schemes over OK such that theinduced mapε∨MδM : G(M)∨ (OK̄ ) → G(M∨ )(OK̄ ) →TS∗ (M∨ )fits into the commutative diagram of GK∞ -modules/ Z/pn Z(1)G(M)(OK̄ ) × G(M)∨ (OK̄ )
εM o

δM o





TS∗ (M) × TS∗ (M∨ )

h , iMo/ (Z/pn Z)tn ,where the top arrow is the Cartier pairing of G(M) and the rightvertical arrow is the isomorphism defined by ζpn 7→ tn .
Proof. Let us prove the first assertion. Put M = MS (M). By Lemma
4.2, it suﬃces to construct a natural isomorphism of GK∞ -modules δM :∗ (M∨ ) which fits into the commutative diagram
Tp (G(M)∨ ) → Tcrys
Tp G(M) × Tp (G(M)∨ )εM o/ Zp (1)δM o

∗ (M) × T ∗ (M∨ )
Tcrys
crys



h , iMo/ Zp t,where the right vertical arrow is the isomorphism defined by (ζpn )n∈Z≥0 7→ t.
∗
Put Γ = G(M) and D∗ (Γ)∨
Acrys = HomAcrys (D (Γ)Acrys , Acrys ). Consider the18

SHIN HATTORI

natural isomorphismΦΓ : D∗ (Γ)∨ → D∗ (Γ∨ )of [5, Section 5.3]. From [5, Proposition 5.3.6], we see that this induces anisomorphism M∨ → D∗ (Γ∨ )(S→OK ) of the category Mod1,φ/S . Thus the dual∨
map ΦΓ induces an isomorphism of GK∞ -modules
Φ∨

∼

∗
∗
δM : Tp (Γ∨ ) → Tcrys(D∗ (Γ∨ )(S→OK ) ) →Γ Tcrys(M∨ ).

Then the functoriality of the map ΦΓ reduces the proof of the commutativity of the diagram above to showing the following claim. For any homomorphism x : Qp /Zp → µp∞ of p-divisible groups over OC , we havea system of p-power roots of unity {xn (1)}n∈Z≥0 in K̄ and the elementx(1) = (x0 (1), x1 (1), . . .) of the ring R. Then we claim that the map∗Φµp∞ : D∗ (µp∞ )∨Acrys → D (Qp /Zp )Acrys = Acrys

sends the element D∗ (x) of the left-hand side to the element log([x(1)]) of
the right-hand side for any x : Qp /Zp → µp∞ . This claim follows easily fromthe definition of the map ΦΓ .
Next we prove the second assertion. Take a resolution 0 → M1 → M2 →M → 0 of M as in Theorem 4.1 (3). Then this induces a resolution 0 →∨∨M∨
2 → M1 → M → 0 by the snake lemma. Put G = G(M) and Γi =G(Mi ). By the snake lemma, we have an epimorphism w : Γ1 [pn ] → G
whose dual map w∨ fits into the commutative diagram with exact rows0

/ G(M∨ )


/ G∨

0

/ G(M∨ )/ G(M∨ )

o

o

1

w∨

2




/ Γ∨

/ Γ∨
1

2

/0

/ 0.

Thus we get an isomorphism G(M∨ ) → G ∨ , which is independent of the
choice of a resolution ([18, Lemma 2.3.4]). On the other hand, we can checkthe commutativity of the diagram of the Cartier pairings
Tp Γ1/ Hom(Tp (Γ∨ ), Zp (1))

/ Hom(G ∨ (O ), Z/pn Z(1))
K̄

G(OK̄ )

1and of a similar diagram for TS∗ (M). Hence we can prove the compatibility
with the duality pairings by the functoriality of the connecting homomorphism of the snake lemma.

4.3. Proof of the main theorem. Now we prove Theorem 1.1. By Theorem 3.1, Theorem 3.3 and Proposition 4.4 (2), the assertion for the upper
ramification subgroups is reduced to showing that the isomorphism
δM : G(M)∨ (OK̄ ) → TS∗ (M∨ ) = H(M∨ )(R)RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

19

induces an isomorphism of the lower ramification subgroups(G(M)∨ )i (OK̄ ) → H(M∨ )i (R)for any i ∈ Q≥0 . By the definition of the map δM , it is enough to show theassertion of Theorem 1.1 for the lower ramification subgroups. Namely, foran object M of the category Mod1,φ/S1 , we reduced ourselves to showing thenatural map
εM : G(M)(OK̄ ) → TS∗ (M) = H(M)(R)
induces an isomorphism of the i-th lower ramification subgroups for any i.
For this, by Proposition 4.3 (2) and replacing K1 by K, we may assume thate is divisible by p and the entries of a representing matrix of φM is containedin the subring k[[up ]] of S1 . Note that, by Corollary 3.5 and Remark 3.6,
the i-th lower ramification subgroups of both sides vanish for i > e/(p − 1).
Thus we are reduced to showing the theorem below.
Theorem 4.5. Let M be an object of Mod1,φ/S1 . Suppose that e is divisible by
p and the entries of a representing matrix of φM is contained in the subringk[[up ]] of S1 . Consider the isomorphism of k-algebras k[u]/(ue ) → OK /pOK
defined by u 7→ π, by which we identify both sides. Then there exists an
isomorphism of schemesηM : (OK /pOK ) ×OK G(M) → (k[u]/(ue )) ×k[[u]] H(M)which preserves the zero section and makes the following diagram commutative for any non-negative rational number i ≤ e:
G(M)(OK̄ )εM



G(M)(OK̄ /m≥i
)
K̄

ηM

/ H(M)(R)/ H(M)(R/m≥i ).
R

Here the bottom arrow is induced by the isomorphism pr0 : R/m≥i
R →e ) → O /pO .
OK̄ /m≥i
lying
over
the
isomorphismk[u]/(uKKK̄
Proof. Let m1 , . . . , md be a basis of M such that we can write as
φM (m1 , . . . , md ) = (m1 , . . . , md )A
for some A ∈ Md (k[[up ]]). We can take matrices P, Q ∈ GLd (k[[up ]]) suchthat
P AQ = diag(ue−r1 , . . . , ue−rd )for some non-negative integers ri divisible by p with ri ≤ e. Here diag(a1 , . . . , ad )denotes the diagonal matrix whose (i, i)-th entry is ai . Set a basis n1 , . . . , ndof M to be (n1 , . . . , nd ) = (m1 , . . . , md )φ−1 (Q). Then we haveφM (n1 , . . . , nd ) = (n1 , . . . , nd )φ−1 (Q)−1 P −1 diag(ue−r1 , . . . , ue−rd ).

20

SHIN HATTORI

Thus the object M = MS (M) is described as
Fil1 M = SpanS (ur1 ⊗ n1 , . . . , urd ⊗ nd ) + (Filp S)M,
φ1 (ur1 ⊗ n1 , . . . , urd ⊗ nd ) = (1 ⊗ n1 , . . . , 1 ⊗ nd )G,
where G = cQ−1 φ(P )−1 ∈ GLd (S1 ) with c = φ1 (E(u)) as before. Take
lifts P̂ and Q̂ of the matrices P and Q in GLd (W [[up ]]), respectively. Put
Ĝ = φ(−F (u))Q̂−1 φ(P̂ )−1 ∈ GLd (W [[up ]]). Since we have the equalityc = φ(−F (u)) in the ring S̃1 , the images of the matrices G and Ĝ in GLd (S̃1 )coincide with each other. We write this image as G̃, which is contained in
the subgroup GLd (k[up ]/(uep )).
Note that we have a commutative diagram of W -algebras
u7→πW [[u]]
/ k[u]/(ue )k[[u]]/ OK∼

/ OK /pOK .

Consider the composite map∼

W [[u]] → OK /pOK → OK1 /pOK1 → k[u]/(uep ) = S̃1 ,where the last arrow is the φ-semilinear isomorphism defined by π1 7→ u.
Then the image of the matrix −F (u)φ−1 (Q̂)−1 P̂ −1 by this composite map
coincides with G̃. Let ai,j (u) ∈ W [[u]] be the (i, j)-th entry of this matrix.
From the explicit description of the aﬃne algebra RM of G(M) = GrB (M)recalled in Subsection 4.1, we see that RM is defined by the system of
equations over OK
π e−ri ∑(aj,i (π)Xj ) (i = 1, . . . , d),
F (π)
d

Xip +

j=1

where ai,j (π) denotes the image of ai,j (u) by the map W [[u]] → OK defined
as in the above diagram. On the other hand, the defining equations of H(M)over k[[u]] are
u
Xip +

e−ri

F (u)

(

d
∑

āj,i (u)Xj ) (i = 1, . . . , d),j=1

where āi,j (u) denotes the image of ai,j (u) by the natural map W [[u]] →k[[u]], and the zero section of H(M) is by definition X1 = · · · = Xd = 0.
This implies that there exists an isomorphism
ηM : (OK /pOK ) ×OK G(M) → (k[u]/(ue )) ×k[[u]] H(M)of schemes over the isomorphism k[u]/(ue ) ' OK /pOK defined by Xi 7→ Xi .
Thus, for i ≤ e, we get a bijection) → H(M)(R/m≥iηM : G(M)(OK̄ /m≥iR )K̄satisfying ηM (0) = 0.

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

21

To prove the compatibility of εM and ηM , let us consider the diagram/ (Õ )DP o
K̄

OK̄


ÕK̄

o

pr1
∼

φ

RDP o

R


pr0
∼

R/m≥e
R .

Let x = (x1 , . . . , xd ) be an element of Spec(RM )(OK̄ ) and z = (ni 7→ zi ) be
the corresponding element of TS∗ (M) = HomS,φ (M, R) via the compositeεMTS∗ (M).
Spec(RM )(OK̄ ) ' G(M)(OK̄ ) →Let yi ∈ R be the element such that pr1 (yi ) coincides with the image x̄iof xi in ÕK̄ . Then, in the ring RDP , we have φ(zi ) − yi ∈ Filp RDP . Put
yi = (yi,0 , yi,1 , . . .) and zi = (zi,0 , zi,1 , . . .) with yi,j , zi,j ∈ ÕK̄ . Since thenatural map R → RDP induces an isomorphismDPR/m≥ep/Filp RDP
R →R

and the kernel of the map pr1 : R → ÕK̄ coincides with the ideal m≥ep
R , wephave yi,1 = zi,1
= zi,0 . This implies x̄i = zi,0 and the compatibility of εMand ηM as in the theorem follows. Hence we conclude the proof of Theorem
1.1.
Note that we have also shown the following corollary.
Corollary 4.6. Let M be an object of the category Mod1,φ/S1 . Consider thepk-algebra k[[v]] as a k[[u]]-algebra by the map u 7→ v . By the k-algebra
isomorphism k[[v]]/(v ep ) → OK1 /pOK1 defined by v 7→ π1 , we identify bothsides. Then we have an isomorphism(OK1 /pOK1 ) ×OK G(M) → (k[[v]]/(v ep )) ×k[[u]] H(M)of schemes over k[[v]]/(v ep ) ' OK1 /pOK1 preserving the zero section.
Remark 4.7. The way we have proved Theorem 1.1 is based on switchingfrom the upper to the lower ramification subgroups via duality. The author
wonders if we can prove the theorem in an “upper” way, namely by constructing a natural isomorphism between the sets of geometric connectedcomponents of tubular neighborhoods of G(M) and H(M) using the similarity of their aﬃne algebras, even though they are in diﬀerent characteristics.
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E-mail address: shin-h@math.kyushu-u.ac.jp
Faculty of Mathematics, Kyushu University