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 ([30]), there exist a complete discrete valuationﬁeld 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 ([30, Corollaire 3.3.6]).
On the other hand, Breuil ([7]) 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.
He ([6], [8]) also simpliﬁed this classiﬁcation by replacing these data by ϕmodules over W [[u]], which are referred as Kisin modules since the latterclassiﬁcation was reproved and investigated further by Kisin ([20], [21]).
Let us consider the case where ﬁnite ﬂat group schemes are killed by p. Put
S1 = k[[u]], which is isomorphic to the k-algebra OX . We let ϕ also denotethe 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
Date: February 14, 2012.
Supported by Grant-in-Aid for Young Scientists B-21740023.
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endowed with a ϕ-semilinear map ϕM : M → M such that the cokernel ofthe map 1 ⊗ ϕM : S1 ⊗ϕ,S1 M → M is killed by uer . Then there exists ananti-equivalence of categories G(−) from Mod1,ϕ/S1 to the category of ﬁnite
ﬂat group schemes over OK killed by p. It is well-known that, for any r,
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([16]). 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 are in correspondence with each other, then their generic ﬁber Galois
modules G(OK̄ ) and H(OX sep ) are also in correspondence via the theory ofnorm ﬁelds. Namely, for an object M of Mod1,ϕ/S1 , we have an isomorphismof GK∞ -modules
G(M)(OK̄ )|GK∞ → H(M)(OX sep )([21, 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 ([14], [17]) 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 ) ([17]) and also describe
completely the semi-simpliﬁcation of the restriction to the inertia subgroupof this GK -module ([18]). 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], [28], [29]), while the canonical subgroup isalso found in the Harder-Narasimhan ﬁltration of A[pn ] deﬁned by Fargues([13], [14]).
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 )RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

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for any j ∈ Q>0 and i ∈ Q≥0 .
This 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 of
ramiﬁcation of ﬁnite ﬂat group schemes over OK , for example as in [18, 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 ([28]) and Fargues ([14]). 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 a duality theory of Liu ([24]) for Kisin modules. This requires us to check compatibilitiesof these two duality theories, though it is straightforward to carry out. Thus
we reduce ourselves to proving the correspondence 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 to each other, not as group schemes but as pointed
schemes. We prove this fact by using Breuil’s explicit computation of theaﬃne algebra of a ﬁnite ﬂat group scheme over OK killed by p in termsof its corresponding Breuil module ([7, Section 3]), after showing that hisclassiﬁcation of ﬁnite ﬂat group schemes is compatible with the base change
from K to Kn .
It should be mentioned that a classiﬁcation of ﬁnite ﬂat group schemes
via Kisin modules is also proved for p = 2 by Kisin ([22]) for the case of
unipotent ﬁnite ﬂat group schemes and independently by Kim ([19]), Lau([23]) and Liu ([26]) for the general case. However, the author does not knowwhether a similar correspondence of ramiﬁcation between characteristic zeroand two holds for ﬁnite ﬂat group schemes with non-trivial multiplicative
parts.
2. Review of Cartier duality theory for Kisin modules
Let p > 2 be a rational prime and K be a complete discrete valuationﬁeld 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 ([7]) or Kisin

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modules ([6], [8], [20], [21]). For these data, corresponding notions of duality
to Cartier duality for ﬁnite ﬂat group schemes are introduced by Caruso and
Liu ([9], [10], [11], [24]), which play key roles in the integral p-adic Hodge
theory. In this section, we recall the deﬁnitions of these data and the theoryof Cartier duality for Kisin modules.
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.
The categories of Kisin modules have a natural duality theory ([11, Sub∨section 2.4]). For an object M of the category Modr,ϕ/S∞ , we let M denoteits dual object as in loc. cit.. We give here an explicit description of the
duality theory of the category Modr,ϕ/S1 for the convenience of the reader.
Let M be an object of this category. By deﬁnition, the S1 -module M∨ is
HomS (M, S1 ). Choose a basis e1 , . . . , ed of the free S1 -module M and let∨e∨
1 , . . . , ed denote its dual basis. Deﬁne a matrix A ∈ Md (S1 ) byϕM (e1 , . . . , ed ) = (e1 , . . . , ed )A.
= p−1 E(0) ∈ W × .

Put c0given byThen the ϕ-semilinear map ϕM∨ : M∨ → M∨ is∨∨∨
r t −1ϕM∨ (e∨1 , . . . , ed ) = (e1 , . . . , ed )(E(u)/c0 ) ( A) .

r,ϕr,ϕFor r < p − 1, we also have categories Modr,ϕ/S1 , Mod/S∞ and Mod/S ofBreuil modules deﬁned as follows. Let S be the p-adic completion of thedivided power envelope W [u]DP of W [u] with respect to the ideal (E(u))
and the compatibility 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 the ideal generated by E(u)j /j! for integers j ≥ i. The ϕsemilinear 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 and ϕi ’s induce a similar structure on
the ring Sn = S/pn S. Put c = ϕ1 (E(u)) ∈ S × . Then we let ′ Modr,ϕ/S denote the category of S-modules M endowed with an S-submodule Filr M

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

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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 a homomorphism of S-modules
compatible with Filr ’s and ϕr ’s. We drop the subscript M of ϕr,M if noconfusion may occur. The category ′ Modr,ϕ/S has an 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) generates the S-moduler,ϕM is denoted by Modr,ϕ/S1 . We let Mod/S∞ denote the smallest full subr,ϕcategory of ′ Modr,ϕ/S containing Mod/S1 and stable under extensions, andModr,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 theimage 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 equivalent.

r,ϕWe deﬁne an exact functor MS : Modr,ϕ/S∞ → Mod/S∞ by putting 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 ([11, Theorem 2.3.1]).
r,ϕSimilarly, we have an equivalence of categories Modr,ϕ/S → Mod/S ([11, Theorem 2.2.1]), which is denoted also by 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 ([8, 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

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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
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)) ([15, 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 the 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 ′ 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∗from Modr,ϕThe functors TS∗ and Tcrys/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,ϕThen, for an object M of the category Modr,ϕ/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))) ([25, Lemma 3.3.4]).

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

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To describe the duality theory for TS∗ (M), let us also ﬁx a system {ζpn }n∈Z≥0of p-power roots of unity in K̄ such that ζ1 = 1, ζp ̸= 1 and ζpn = ζppn+1 ,and set an element ε ∈ R to be ε = (1, ζp , ζp2 , . . .) with an abusive notation
as before. Deﬁne an element t of Acrys byt = log([ε]) =∞∑(−1)i−1i=1

Put c0([ε] − 1)i.
i= p−1 E(0) ∈ W × andλ=∞∏ϕi (E(u)/E(0)) ∈ S × .

i=1

Let S(r) = Se be the object of the category Mod1,ϕ/S of rank one with a basisre satisfying ϕ(e) = (E(u)/c0 ) e. Similarly, we deﬁne an object S(r) of thercategory Mod1,ϕ/S of rank one by S(r) = Se = Fil S(r) and ϕr (e) = e. Thenthe Breuil module MS (S(r)) is isomorphic to S(r) via the multiplicationby 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 −1t=() λ t.
c0
Then the element t is contained in the subring Sur ([24, Subsection 3.2]).
∨
Let M be an object of the category Modr,ϕ/S and M be its dual object.
Then as in [11, Subsection 2.4], the evaluation map M×M∨ → S(r) induces
a natural perfect pairing of GK∞ -modules⟨ , ⟩M : TS∗ (M) × TS∗ (M∨ ) → TS∗ (S(r)) ≃ Zp tr ⊆ Sur ,which gives an isomorphism of GK∞ -modulesTS∗ (M∨ ) → HomZp (TS∗ (M), Zp tr ).
We also have a similar perfect pairing ⟨ , ⟩M for the category Modr,ϕ/S∞ ([11]).
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 ([14], [17]). 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 ([28]), and Fargues gave a much simpler proof of this theorem([14]). In this section, after brieﬂy recalling the ramiﬁcation theory of ﬁnite

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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
We also put GK = Gal(K sep /K).
Let G = Spec(B) be a ﬁnite ﬂat generically etale group scheme over OK .
Then G is locally of complete intersection over OK ([7, 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 j of
G for j ∈ Q>0 to be the scheme-theoretic closure in G of (the ﬁnite subgroupscheme of G × Spec(K) associated to) the kernel of this surjection. On theother hand, for i ∈ Q≥0 , the scheme-theoretic closure in G of the kernel ofthe natural homomorphism G(OK sep ) → G(OK sep /m≥i
K sep ) is denoted by Gi
and called the i-th lower ramiﬁcation subgroup of G. In particular, we havethe equality
Gi (OK sep ) = Ker(G(OK sep ) → G(OK sep /m≥i
K sep )).
We also put
G j+ (OK sep ) =∪′G j (OK sep ),j ′ >jGi+ (OK sep ) =∪Gi′ (OK sep )i′ >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 ′′ of ﬁnite ﬂat group schemes over OK , the imageof G j (OK sep ) in G ′′ (OK sep ) coincides with (G ′′ )j (OK sep ) ([2, Lemme 2.3.2]).
From the deﬁnition, we also see that for a closed immersion G ′ → G ofﬁnite ﬂat group schemes over OK , the subgroup G ′ (OK sep ) ∩ Gi (OK sep ) coincides with (G ′ )i (OK sep ). In addition, for a ﬁnite extension L/K of relative
ramiﬁcation index e′ , we have natural isomorphisms′(OL ×OK G)je → OL ×OK G j ,(OL ×OK G)ie′ → OL ×OK Gi

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 ([17, Theorem 7]). Let G ∨ be the Cartier dual of G

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

9and consider the Cartier pairing⟨ , ⟩G : 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 ([28, Theorem 1.6], [14, 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 ⟨ , ⟩G 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
e′ 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≤ s if its Verschiebung VH is zero and the cokernel of the natural map
VH∨ : OY ⊗ϕ,OY Lie(H∨ ) → Lie(H∨ )10SHIN HATTORI

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
′

≤ee r
HY (−) : Modr,ϕ/OY → COY([16, 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 → M′′ of Modr,ϕ/S1 such that∗′′∗the image of the corresponding injection TS (M ) → TS (M) coincides with
L. A surjection M → M′′ satisfying this property is unique up to a unique
isomorphism.
Proof. This follows from [24, Lemma 2.3.6]. Indeed, let Modϕ/X denote thecategory of etale ϕ-modules over X ([15, 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 )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 ′′ be the quotientof M corresponding to the surjectionT∗ (M ) → HomFp (T ∗ (M ), Fp ) → HomFp (L, Fp ).

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

11

Then the Kisin module M′′ = Im(M → M → M ′′ ) 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 ⟨ , ⟩M of Subsection 2.2 induces a perfect pairing⟨ , ⟩M : 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 ⟨ , ⟩M and lr (j) = er/(p − 1) − j/p.
Proof. We proceed as in the proof of [14, Proposition 6]. Let Y be a ﬁnite
separable extension of X in X sep of relative ramiﬁcation index e′ and put
MY = OY ⊗S1 M. Then we have a commutative diagram
H(M)(R) × H(M∨ )(R)≀≀

⟨ , ⟩MHY (MY )(R) × HY (M∨Y )(R)/R/ R,⟨ , ⟩MY

where ⟨ , ⟩MY is a perfect pairing deﬁned similarly to the pairing ⟨ , ⟩M
and the vertical arrows are isomorphisms. Since H(M∨ ) is generically etale,
after making a ﬁnite separable base change and replacing e by ee′ , 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)by Lemma 3.2. Then we have the commutative diagram
H(M)(R) × H(M∨ )(R)⟨ , ⟩MO

H(N∨ )(R) × H(N)(R)⟨ , ⟩N

/R

/ R.

12

SHIN HATTORI

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 a′ n∨
us an with 0 ≤ s ≤ er and a ∈ S×
1 . Then we have ϕN∨ (n ) = u
×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 a′ X = 0, respectively. By a calculation asin [17, 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.
4. Comparison of ramification
Let p > 2 be a rational prime and K be a complete discrete valuationﬁeld 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.

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

13≤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([7], [20], [21]), 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], [7], [8], [20], [21]) and its properties. Their
classiﬁcation theorem is as follows.
Theorem 4.1.
(1) There exists an anti-equivalence of categories G(−)
1,ϕfrom Mod/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
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.

14

SHIN HATTORI

We have two deﬁnitions of the functor G(−) and the isomorphism εM . One
is putting G(−) = GrB (MS (−)) with the anti-equivalence GrB (−) of [7], asin [21]. The isomorphism εM is constructed in the proof of [21, Proposition
1.1.13]. Then we can prove Theorem 4.1 (3) by using [11, Proposition 2.1.3
and Theorem 2.3.4]. The other is given in [20], as follows. For a p-divisiblegroup Γ over OK , set ModK (Γ) to be the section D∗ (Γ)(S→OK ) of the contravariant Dieudonné crystal D∗ (Γ) ([5]) 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 ([20, 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) ([12, 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 ([21, Proposition 1.1.13]). Let H1 be the divided power polynomial
ring RDP ⟨u−π⟩ over RDP . Consider the n-th projection prn : R → ÕK̄ . The
map pr0 induces a surjection H1 = RDP ⟨u − π⟩ → 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 [7, Lemme 5.3.1]) and the isomorphism εM is the composite∗ (M).
of this map and the natural isomorphism TS∗ (M) → Tcrys
1,ϕLet M be an object of Mod/S1 and put M = MS (M). In [7, Section3], Breuil gave an explicit description of the aﬃne algebra RM of the ﬁnite
ﬂat group scheme G(M) = GrB (M) in terms of M. Let e1 , . . . , ed be an
adapted basis of M ([7, 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 [7]). Put S̃1 = k[u]/(uep ) and identify this k-algebra with S1 /Filp S1 .

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

15

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 = (∑∑e−r
e−r
X1p + πF (π)1 ( dj=1 aj,1 Xj ), . . . , Xdp + πF (π)d ( dj=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 [7, 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 ([10]). In thissubsection, we brieﬂy present a proof of these facts, along with similar compatibilities of the isomorphism εM . First we recall the following lemma.
Lemma 4.2. The functor Γ 7→ Tp Γ|GK∞ from the category of p-divisiblegroups over OK to the category of p-adically continuous GK∞ -representations
is fully faithful.
Proof. This follows from Tate’s theorem ([27, Subsection 4.2, Corollary 1])
and [8, Theorem 3.4.3].
□
Let us show the compatibility with the base change. Put S′ = W [[v]] andlet ϕ : S′ → S′ denote the natural ϕ-semilinear map deﬁned by v 7→ v p .
nNote that the polynomial E ′ (v) = E(v p ) is the Eisenstein polynomial of
the uniformizer πn ∈ Kn . Consider the categories Mod1,ϕ, Mod1,ϕ/S′ and
/S′
1

∞

′
′
Mod1,ϕ/S′ of Kisin modules over S of E -height ≤ 1. Using the W -algebranhomomorphism S′ → W (R) deﬁned by v 7→ [π 1/p ], we deﬁne a similarfunctor TS∗ ′ to TS∗ . The homomorphism of W -algebras S → S′ deﬁned by

16

SHIN HATTORI

1,ϕ′u 7→ v p induces natural functors (−)′ : Mod1,ϕ/S∞ → Mod/S′ and (−) :
n

∞

1,ϕ
Mod1,ϕ/S → Mod/S′ by

M 7→ M′ = S′ ⊗S M,

ϕM ′ = ϕ ⊗ ϕ M .

Then we have a natural isomorphism of GK∞ -modules TS∗ (M) → TS∗ ′ (M′ ).
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 ′ (−) denote the anti-equivalences of the theorem over OKn .
1,ϕProposition 4.3. Let M be an object of the category Mod1,ϕ/S (resp. Mod/S∞ )1,ϕand M′ be the associated object of the category Mod1,ϕ/S′ (resp. Mod/S′∞ ).
Then there exists a natural isomorphism

G ′ (M′ ) → OKn ×OK G(M)of p-divisible groups (resp. finite flat group schemes) over OKn which respectively makes the following diagrams commutative:
Tp G(M)|GK∞≀G(M)(OK̄ )|GK∞

∼

∼

K∞


/ T ∗ ′ (M′ ),S/ G ′ (M′ )(O )|G
K̄
K∞
≀



TS∗ (M)/ Tp G ′ (M′ )|G≀TS∗ (M)≀∼∼/ T ∗ ′ (M′ ).
S

Proof. The assertion for the category Mod1,ϕ/S follows from Lemma 4.2, andthis implies the assertion for Mod1,ϕ/S∞ by taking a resolution of M as in
Theorem 4.1 (3). Note that the isomorphism G ′ (M′ ) → OKn ×OK G(M) is
independent of the choice of a resolution ([20, Lemma 2.3.4]).
□
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 .
1,ϕProposition 4.4. Let M be an object of the category Mod1,ϕ/S (resp. Mod/S∞ )and M∨ be its dual object. Then there exists a natural isomorphism G(M)∨ →G(M∨ ) of p-divisible groups (resp. finite flat group schemes) over OK suchthat the induced mapε∨MTS∗ (M∨ )δM : Tp (G(M)∨ ) → Tp G(M∨ ) →ε∨MTS∗ (M∨ ))(resp. δM : G(M)∨ (OK̄ ) → G(M∨ )(OK̄ ) →RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

17

respectively makes the following diagrams of GK∞ -modules commutative:/ Zp (1)

Tp G(M) × Tp (G(M)∨ )
εM ≀

δM ≀







TS∗ (M) × TS∗ (M∨ )⟨ , ⟩M/ Zp t,/ Z/pn Z(1)G(M)(OK̄ ) × G(M)∨ (OK̄ )
εM ≀

≀

δM ≀







TS∗ (M) × TS∗ (M∨ )≀/ (Z/pn Z)tn .

⟨ , ⟩M

Here the top arrows are the Cartier pairings and the right vertical arrows
are the isomorphisms induced by (ζpn )n∈Z≥0 7→ t.
∨
Proof. For an object M of the category Mod1,ϕ/S , deﬁne δM : Tp (G(M) ) →∗∨TS (M ) to be the unique isomorphism which ﬁts into the commutative
diagram of GK∞ -modules

Tp (G(M)∨ )/ HomZ (Tp (G(M)), Zp (1))

/ HomZ (T ∗ (M), Zp t),

pTS∗ (M∨ )p

S

where the top (resp. the bottom) arrow is induced by the Cartier pairing(resp. the pairing ⟨ , ⟩M ) and the right vertical arrow is induced by εM
and the natural isomorphism Zp (1) → Zp t as in the proposition. Then the∨∨isomorphism ε−1M∨ ◦ δM deﬁnes the desired isomorphism G(M) → G(M )uniquely by Lemma 4.2.
Next let M be an object of the category Mod1,ϕ/S∞ . 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 withexact rows0

/ G(M∨ )


0

/ G∨

/ G(M∨ )/ G(M∨ )

≀

≀

1

w∨



/ Γ∨
1

2


/ Γ∨
2

/0

/ 0.

Thus we get an isomorphism G(M∨ ) → G ∨ , which is independent of the
choice of a resolution ([20, Lemma 2.3.4]). On the other hand, we can check18

SHIN HATTORI

the commutativity of the diagram of the Cartier pairings
T p 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, the assertion for the upper
ramiﬁcation subgroups is reduced to showing that the isomorphism
δM : G(M)∨ (OK̄ ) → TS∗ (M∨ ) = H(M∨ )(R)
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 and replacing K1 by K, we may assume that e
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
K̄ηM

/ H(M)(R)/ H(M)(R/m≥i ).
R

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

19

Here the bottom arrow is induced by the isomorphism pr0 : R/m≥i
R →≥ieOK̄ /mK̄ lying over the isomorphism k[u]/(u ) → OK /pOK .
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 ).
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]]/ k[u]/(ue )

/ 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

20

SHIN HATTORI

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]] aredue−ri ∑pXi +(āj,i (u)Xj )
F (u)
j=1

(i = 1, . . . , d),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
ηM : G(M)(OK̄ /m≥i
) → H(M)(R/m≥i
R )K̄
satisfying ηM (0) = 0.
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 compositeεM
Spec(RM )(OK̄ ) ≃ G(M)(OK̄ ) →TS∗ (M).

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)

RAMIFICATION CORRESPONDENCE OF FINITE FLAT GROUP SCHEMES

21

of schemes over k[[v]]/(v ep ) ≃ OK1 /pOK1 preserving the zero section.
Remark 4.7. Let M be an object of the category Modr,ϕ/S1 . For j ∈ Q>0j(resp. i ∈ Q≥0 ), let M (resp. Mi ) be the object which corresponds via theanti-equivalence H(−) to the closed subgroup scheme H(M)j (resp. H(M)i )of H(M). These objects deﬁne coﬁltrations {Mj }j∈Q>0 and {Mi }i∈Q≥0 of
M in the category Modr,ϕ/S1 . Note that, for a ﬁnite ﬂat group scheme over a
discrete valuation ring, its ﬁnite ﬂat closed subgroup scheme is determinedby the generic ﬁber. Therefore, for r = 1, Theorem 1.1 and [8, Theorem
3.4.3] imply that the quotient M → Mj (resp. M → Mi ) also corresponds
via the anti-equivalence G(−) to the closed subgroup scheme G(M)j (resp.
G(M)i ) of G(M). They can be considered as “upper and lower ramiﬁcation
coﬁltrations” of the Kisin module M.
Remark 4.8. 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.
Acknowledgments. The author would like to thank Victor Abrashkin,Toshiro Hiranouchi, Wansu Kim and Yuichiro Taguchi for stimulating discussions, Xavier Caruso for kindly answering his questions on the paper [10]and the anonymous referee for helpful suggestions to improve this paper.
This work was supported by Grant-in-Aid for Young Scientists B-21740023.
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E-mail address: shin-h@math.kyushu-u.ac.jp
Faculty of Mathematics, Kyushu University