On a ramification bound of torsion semi-stable representations over a local field
aa r X i v : . [ m a t h . N T ] J un ON A RAMIFICATION BOUND OF TORSIONSEMI-STABLE REPRESENTATIONS OVER A LOCALFIELD
SHIN HATTORI
Abstract.
Let p be a rational prime, k be a perfect field of charac-teristic p , W = W ( k ) be the ring of Witt vectors, K be a finite to-tally ramified extension of Frac( W ) of degree e and r be a non-negativeinteger satisfying r < p −
1. In this paper, we prove the upper num-bering ramification group G ( j ) K for j > u ( K, r, n ) acts trivially on the p n -torsion semi-stable G K -representations with Hodge-Tate weights in { , . . . , r } , where u ( K, , n ) = 0, u ( K, , n ) = 1 + e ( n + 1 / ( p − u ( K, r, n ) = 1 − p − n + e ( n + r/ ( p − < r < p − Introduction
Let p be a rational prime, k be a perfect field of characteristic p , W = W ( k ) be the ring of Witt vectors and K be a finite totally ramified extensionof K = Frac( W ) of degree e = e ( K ). We normalize the valuation v K of K as v K ( p ) = e and extend this to any algebraic closure of K . Let themaximal ideal of K be denoted by m K , an algebraic closure of K by ¯ K and the absolute Galois group of K by G K = Gal( ¯ K/K ). Let G ( j ) K denotethe j -th upper numbering ramification group in the sense of [10]. Namely,we put G ( j ) K = G j − K , where the latter is the upper numbering ramificationgroup defined in [18].Consider a proper smooth scheme X K over K and put X ¯ K = X K × K ¯ K .Let L ⊇ L ′ be G K -stable Z p -lattices in the r -th etale cohomology group H r ´et ( X ¯ K , Q p ) such that the quotient L / L ′ is killed by p n . In [10], Fontaineconjectured the upper numbering ramification group G ( j ) K acts trivially onthe G K -modules L / L ′ and H r ´et ( X ¯ K , Z /p n Z ) for j > e ( n + r/ ( p − X K has good reduction. For e = 1 and r < p −
1, this conjecture was provedindependently by himself ([11], for n = 1) and Abrashkin ([3], for any n ),using the theory of Fontaine-Laffaille ([13]) and the comparison theorem ofFontaine-Messing ([14], see also [5] and [7]) between the etale cohomologygroups of X K and the crystalline cohomology groups of the reduction of X K . From these results, they also showed some rareness of a proper smooth Date : November 19, 2018.Supported by 21st Century COE program “Mathematics of Nonlinear Structure viaSingularity” at Department of Mathematics, Hokkaido university, and by the JSPS Inter-national Training Program (ITP). scheme over Q with everywhere good reduction ([11, Th´eor`eme 1], [2, Section7]). In fact, they proved this ramification bound for the torsion crystallinerepresentations of G K with Hodge-Tate weights in { , . . . , r } in the casewhere K is absolutely unramified.On the other hand, for a torsion semi-stable representation with Hodge-Tate weights in the same range, a similar ramification bound for e = 1and n = 1 is obtained by Breuil (see [7, Proposition 9.2.2.2]). He showed,assuming the Griffiths transversality which in general does not hold, that if e = 1 and r < p −
1, then the ramification group G ( j ) K acts trivially on themod p semi-stable representations for j > / ( p − G K with Hodge-Tate weights in { , . . . , r } with no as-sumption on e but under the assumption r < p −
1. Let π be a uniformizerof K , E ( u ) ∈ W [ u ] be the Eisenstein polynomial of π over W and S be the p -adic completion of the divided power envelope of W [ u ] with respect to theideal ( E ( u )). Consider a category Mod r,φ,N/S ∞ of filtered ( φ r , N )-modules overthe ring S and a G K -module T ∗ st ,π ( M ) = Hom S, Fil r ,φ r ,N ( M , ˆ A st , ∞ )for M ∈
Mod r,φ,N/S ∞ , where ˆ A st , ∞ is a p -adic period ring ([6]). Then our maintheorem is the following. Theorem 1.1.
Let r be a non-negative integer such that r < p − . Let M be an object of the category Mod r,φ,N/S ∞ which is killed by p n . Then the j -thupper numbering ramification group G ( j ) K acts trivially on the G K -module T ∗ st ,π ( M ) for j > u ( K, r, n ) , where u ( K, r, n ) = r = 0) , e ( n + p − ) ( r = 1) , − p n + e ( n + rp − ) (1 < r < p − . We can check that this bound is sharp for r ≤ Corollary 1.2.
Let the notation be as in the theorem and L be the finiteextension of K cut out by the G K -module T ∗ st ,π ( M ) . Namely, the finiteextension L is defined by G L = Ker( G K → Aut( T ∗ st ,π ( M ))) . Let D L/K denote the different of the extension
L/K . Then we have theinequality v K ( D L/K ) < u ( K, r, n ) for r > and v K ( D L/K ) = 0 for r = 0 . AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 3
Combining these results with a theorem of Liu ([17, Theorem 2.3.5]) or atheorem of Caruso ([8, Th´eor`eme 1.1]), we will show the corollary below.
Corollary 1.3.
Let r be a non-negative integer such that r < p − . Thenthe same bounds as in Theorem 1.1 and Corollary 1.2 are also valid for thetorsion G K -modules of the following two cases:(1) the G K -module L / L ′ , where L ⊇ L ′ are G K -stable Z p -lattices ina semi-stable p -adic representation V with Hodge-Tate weights in { , . . . , r } such that L / L ′ is killed by p n .(2) the G K -module H r ´ et ( X ¯ K , Z /p n Z ) , where X K is a proper smooth alge-braic variety over K which has a proper semi-stable model over O K and r satisfies er < p − for n = 1 and e ( r + 1) < p − for n > . For the proof of Theorem 1.1, we basically follow a beautiful argumentof Abrashkin ([3]). We may assume p ≥ r ≥
1. Consider the finiteGalois extension F n = K ( π /p n , ζ p n +1 )of K whose upper ramification is bounded by u ( K, r, n ). Let L n be thefinite Galois extension of F n cut out by T ∗ st ,π ( M ) | G Fn . Then we bound theramification of L n over K . For this, we show that to study this G F n -modulewe can use a variant over a smaller coefficient ring Σ of filtered ( φ r , N )-modules over S . In precise, we setΣ = W [[ u, E ( u ) p /p ]] . This ring Σ is small enough for the method of Abrashkin, in which he usesfiltered modules of Fontaine-Laffaille ([13]) whose coefficient ring is W , towork also in the case where K is absolutely ramified. Acknowledgments.
The author would like to pay his gratitude to IkuNakamura and Takeshi Saito for their warm encouragements. He wantsto thank Manabu Yoshida for kindly allowing him to include the proof ofProposition 5.6. He also wants to thank Ahmed Abbes, Xavier Caruso,Takeshi Tsuji, and especially Victor Abrashkin and Yuichiro Taguchi, foruseful discussions and comments. Finally, he is very grateful to the refereefor his or her valuable comments to improve this paper.2.
Filtered ( φ r , N ) -modules of Breuil In this section, we recall the theory of filtered ( φ r , N )-modules over S of Breuil, which is developed by himself and most recently by Caruso andLiu (see for example [6], [8], [17], [9]). In what follows, we always takethe divided power envelope of a W -algebra with the compatibility conditionwith the natural divided power structure on pW .Let p be a rational prime and σ be the Frobenius endomorphism of W .We fix once and for all a uniformizer π of K and a system { π n } n ∈ Z ≥ of p -power roots of π in ¯ K such that π = π and π n = π pn +1 for any n . Let E ( u ) SHIN HATTORI be the Eisenstein polynomial of π over W and set S = ( W [ u ] PD ) ∧ , wherePD means the divided power envelope and this is taken with respect to theideal ( E ( u )), and ∧ means the p -adic completion. The ring S is endowedwith the σ -semilinear endomorphism φ : u u p and a natural filtrationFil t S induced by the divided power structure such that φ (Fil t S ) ⊆ p t S for0 ≤ t ≤ p −
1. We set φ t = p − t φ | Fil t S and c = φ ( E ( u )) ∈ S × . Let N denotethe W -linear derivation on S defined by the formula N ( u ) = − u . We alsodefine a filtration, φ , φ t and N on S n = S/p n S similarly.Let r ∈ { , . . . , p − } be an integer. Set ′ Mod r,φ,N/S to be the categoryconsisting of the following data: • an S -module M and its S -submodule Fil r M containing Fil r S · M , • a φ -semilinear map φ r : Fil r M → M satisfying φ r ( s r m ) = φ r ( s r ) φ ( m )for any s r ∈ Fil r S and m ∈ M , where we set φ ( m ) = c − r φ r ( E ( u ) r m ), • a W -linear map N : M → M such that – N ( sm ) = N ( s ) m + sN ( m ) for any s ∈ S and m ∈ M , – E ( u ) N (Fil r M ) ⊆ Fil r M , – the following diagram is commutative:Fil r M φ r −−−−→ M E ( u ) N y y cN Fil r M −−−−→ φ r M , and the morphisms of ′ Mod r,φ,N/S are defined to be the S -linear maps preserv-ing Fil r and commuting with φ r and N . The category defined in the sameway but dropping the data N is denoted by ′ Mod r,φ/S . These categories haveobvious notions of exact sequences. Let Mod r,φ,N/S denote the full subcate-gory of ′ Mod r,φ,N/S consisting of M such that M is free of finite rank over S and generated as an S -module by the image of φ r . We write Mod r,φ,N/S ∞ forthe smallest full subcategory which contains Mod r,φ,N/S and is stable underextensions. We let Mod r,φ,N/S denote the full subcategory consisting of M such that • the S -module M is free of finite rank and generated by the image of φ r , • the quotient M / Fil r M is p -torsion free.We define full subcategories Mod r,φ/S , Mod r,φ/S ∞ and Mod r,φ/S of ′ Mod r,φ/S in asimilar way. For ˆ
M ∈
Mod r,φ,N/S ( resp. Mod r,φ/S ), the quotient ˆ M /p n ˆ M hasa natural structure as an object of Mod r,φ,N/S ∞ ( resp. Mod r,φ/S ∞ ). AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 5
For p -torsion objects, we also have the following categories. Consider the k -algebra k [ u ] / ( u ep ) ∼ = S / Fil p S and let this algebra be denoted by ˜ S . Thealgebra ˜ S is equipped with the natural filtration, φ and N induced by thoseof S . Namely, Fil t ˜ S = u et ˜ S , φ ( u ) = u p and N ( u ) = − u . Let ′ Mod r,φ,N/ ˜ S denote the category consisting of the following data: • an ˜ S -module ˜ M and its ˜ S -submodule Fil r ˜ M containing u er ˜ M , • a φ -semilinear map φ r : Fil r ˜ M → ˜ M , • a k -linear map N : ˜ M → ˜ M such that – N ( sm ) = N ( s ) m + sN ( m ) for any s ∈ ˜ S and m ∈ ˜ M , – u e N (Fil r ˜ M ) ⊆ Fil r ˜ M , – the following diagram is commutative:Fil r ˜ M φ r −−−−→ ˜ M u e N y y cN Fil r ˜ M −−−−→ φ r ˜ M , and whose morphisms are defined as before. Its full subcategory Mod r,φ,N/ ˜ S is defined by the following condition: • As an ˜ S -module, ˜ M is free of finite rank and generated by the imageof φ r .We define categories ′ Mod r,φ/ ˜ S and Mod r,φ/ ˜ S similarly. Then we can showas in the proof of [4, Proposition 2.2.2.1] that the natural functor M 7→M / Fil p S · M induces equivalences of categories T : Mod r,φ,N/S → Mod r,φ,N/ ˜ S and T : Mod r,φ/S → Mod r,φ/ ˜ S .For r = 0, let Mod φ/W ∞ be the category consisting of the following data: • a finite torsion W -module ˜ M , • a σ -semilinear automorphism φ : ˜ M → ˜ M .Let κ be the kernel of the natural surjection S → W defined by u S ( M , S/κS ) = 0 for any M ∈
Mod ,φ/S ∞ , the proofs of [8, Lemme2.2.7, Proposition 2.2.8] work also for the category Mod ,φ,N/S ∞ and we have acommutative diagram of categoriesMod ,φ,N/S ∞ % % KKKKKKKKKK / / Mod ,φ/S ∞ (cid:15) (cid:15) Mod φ/W ∞ , where the downward arrows and horizontal arrow are defined by M 7→M /κ M and forgetting N respectively and these three arrows are equiva-lences of categories. SHIN HATTORI
Let A crys and ˆ A st be p -adic period rings. These are constructed as follows.Put ˜ O ¯ K = O ¯ K /p O ¯ K . Set R to be the ring R = lim ←− ( ˜ O ¯ K ← ˜ O ¯ K ← · · · ) , where every arrow is the p -power map. For an element x = ( x i ) i ∈ Z ≥ ∈ R and an integer n ≥
0, we set x ( n ) = lim m →∞ ˆ x p m n + m ∈ O C , where ˆ x i is a lift of x i in O ¯ K and O C is the p -adic completion of O ¯ K . Let v p denote the valuation of O C normalized as v p ( p ) = 1. Then the ring R isa complete valuation ring whose valuation of an element x ∈ R is given by v R ( x ) = v p ( x (0) ). We define a natural ring homomorphism θ by θ : W ( R ) → O C ( x , x , . . . ) X n ≥ p n x ( n ) n . Then A crys is the p -adic completion of the divided power envelope of W ( R )with respect to the principal ideal Ker( θ ) and ˆ A st is the p -adic completionof the divided power polynomial ring A crys h X i over A crys . We set A crys , ∞ = A crys ⊗ W K /W and ˆ A st , ∞ = ˆ A st ⊗ W K /W . Put π = ( π n ) n ∈ Z ≥ ∈ R , wherewe abusively let π n denote the image of π n ∈ O ¯ K in ˜ O ¯ K . These rings areconsidered as S -algebras by the ring homomorphisms S → ˆ A st and ˆ A st → A crys which are defined by u [ π ] / (1 + X ) and X
0, respectively. Thering A crys is endowed with a natural filtration induced by the divided powerstructure, a natural Frobenius endomorphism φ and the φ -semilinear map φ t = p − t φ | Fil t A crys . With these structures, A crys and A crys , ∞ are consideredas objects of ′ Mod r,φ/S . Moreover, the absolute Galois group G K acts naturallyon these two rings. As for ˆ A st , its filtration is defined byFil t ˆ A st = X i ≥ a i X i i ! (cid:12)(cid:12)(cid:12)(cid:12) a i ∈ Fil t − i A crys , lim i →∞ a i = 0 and the Frobenius structure of A crys extends to ˆ A st by φ ( X ) = (1 + X ) p − ,φ t = p − t φ | Fil t ˆ A st . We write N also for the A crys -linear derivation on ˆ A st defined by N ( X ) =1 + X . The rings ˆ A st and ˆ A st , ∞ are objects of ′ Mod r,φ,N/S . The G K -action on A crys naturally extends to an action on ˆ A st . Indeed, the action of g ∈ G K on ˆ A st is defined by the formula g ( X ) = [ ε ( g )](1 + X ) − , AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 7 where g ( π n ) = ε n ( g ) π n and ε ( g ) = ( ε n ( g )) n ∈ Z ≥ ∈ R with the abusivenotation as above.These rings have other descriptions, as follows. For an integer n ≥ W n = W/p n W and let W n ( ˜ O ¯ K ) be the ring of Witt vectors of length n associated to ˜ O ¯ K . We define a W n -algebra structure on W n ( ˜ O ¯ K ) bytwisting the natural W n -algebra structure by σ − n . Then the natural ringhomomorphism θ n : W n ( ˜ O ¯ K ) → O ¯ K /p n O ¯ K ( a , . . . , a n − ) n − X i =0 p i ˆ a p n − i i , where ˆ a i is a lift of a i in O ¯ K , is W n -linear. Let us denote W PD n ( ˜ O ¯ K ) thedivided power envelope of W n ( ˜ O ¯ K ) with respect to the ideal Ker( θ n ). Thisring is considered as an S -algebra by u [ π n ]. This ring also has a naturalfiltration defined by the divided power structure, and a natural G K -modulestructure. The Frobenius endomorphism of the ring of Witt vectors induceson this ring a φ -semilinear Frobenius endomorphism, which is denoted alsoby φ . Then, by the S -linear transition maps W PD n +1 ( ˜ O ¯ K ) → W PD n ( ˜ O ¯ K )( a , . . . , a n ) ( a p , . . . , a pn − ) , these S -algebras form a projective system compatible with all the structures.Using this transition map, a φ -semilinear map φ r : Fil r W PD n ( ˜ O ¯ K ) → W PD n ( ˜ O ¯ K )is defined by setting φ r ( x ) to be the image of p − r φ (ˆ x ), where ˆ x is a lift of x in Fil r W PD n + r ( ˜ O ¯ K ). By definition, the maps φ r are also compatible withthe transition maps. The S -algebra W PD n ( ˜ O ¯ K ) is considered as an object of ′ Mod r,φ/S . Then we have a natural isomorphism in ′ Mod r,φ/S A crys /p n A crys → W PD n ( ˜ O ¯ K )( x , . . . , x n − ) ( x ,n , . . . , x n − ,n ) , where we set x i = ( x i,k ) k ∈ Z ≥ for x i ∈ R .Similarly, the divided power polynomial ring W PD n ( ˜ O ¯ K ) h X i over W PD n ( ˜ O ¯ K )is considered as an S -algebra by u [ π n ] / (1 + X ). This ring has a naturalfiltration coming from the divided power structure. We define a G K -actionon this ring by g ( X ) = [ ε n ( g )](1 + X ) − . We also define a φ -semilinear Frobenius endomorphism, which we also writeas φ , by φ ( X ) = (1+ X ) p − W PD n ( ˜ O ¯ K )-linear derivation N by N ( X ) =1 + X . These rings form a projective system of S -algebras compatible with SHIN HATTORI all the structures by the transition maps defined by the maps above and X X . We define φ -semilinear maps φ r : Fil r W PD n ( ˜ O ¯ K ) h X i → W PD n ( ˜ O ¯ K ) h X i compatible with the transition maps as before. The S -algebra W PD n ( ˜ O ¯ K ) h X i is considered as an object of ′ Mod r,φ,N/S and there exists a natural isomor-phism in ′ Mod r,φ,N/S ˆ A st /p n ˆ A st → W PD n ( ˜ O ¯ K ) h X i ( x , . . . , x n − ) ( x ,n , . . . , x n − ,n ) X X which is G K -linear.Put K n = K ( π n ) and K ∞ = ∪ n K n . For M ∈
Mod r,φ,N/S ∞ , we define a G K -module T ∗ st ,π ( M ) to be T ∗ st ,π ( M ) = Hom S, Fil r ,φ r ,N ( M , ˆ A st , ∞ ) . When M is killed by p n , we have a natural identification of G K -modules T ∗ st ,π ( M ) = Hom S, Fil r ,φ r ,N ( M , W PD n ( ˜ O ¯ K ) h X i ) . Note that the G K -module on the right-hand side is independent of the choiceof π k for k > n . Since the natural map W PD n ( ˜ O ¯ K ) h X i → W PD n ( ˜ O ¯ K ) X G K n -linear, we also have a G K n -linear isomorphism ([6, Lemme 2.3.1.1]) T ∗ st ,π ( M ) | G Kn → Hom S, Fil r ,φ r ( M , W PD n ( ˜ O ¯ K )) . On the other hand, for r = 0, the proof of [8, Proposition 2.3.13] shows thatthe G K -module T ∗ st ,π ( M ) is unramified for any M ∈
Mod ,φ,N/S ∞ .A variant of filtered ( φ r , N )-modules over S is also introduced by Breuiland Kisin, and developed also by Caruso and Liu (see for example [15],[16], [17], [9]). Put S = W [[ u ]] and let φ : S → S be the σ -semilinearFrobenius endomorphism defined by φ ( u ) = u p . Let ′ Mod r,φ/ S denote thecategory consisting of the following data: • an S -module M , • a φ -semilinear map M → M , which is denoted also by φ , such thatthe cokernel of the map 1 ⊗ φ : φ ∗ M → M , where we set φ ∗ M = S ⊗ φ, S M , is killed by E ( u ) r ,and whose morphisms are defined as before. The full subcategory of ′ Mod r,φ/ S consisting of M such that M is free of finite rank over S /p S ( resp. over S )is denoted by Mod r,φ/ S ( resp. Mod r,φ/ S ). We let Mod r,φ/ S ∞ denote the smallestfull subcategory which contains Mod r,φ/ S and is stable under extensions, as AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 9 before. Then we have an exact functor ([9, Proposition 2.1.2], see also [15,Proposition 1.1.11]) M S ∞ : Mod r,φ/ S ∞ → Mod r,φ/S ∞ . For M ∈ Mod r,φ/ S ∞ , the filtered φ r -module M = M S ∞ ( M ) over S is definedas follows: • M = S ⊗ φ, S M , • Fil r M = Ker( M ⊗ φ → S ⊗ S M → ( S/ Fil r S ) ⊗ S M ), • φ r : Fil r M ⊗ φ → Fil r S ⊗ S M φ r ⊗ → S ⊗ φ, S M = M .We write M S for the functor Mod r,φ/ S → Mod r,φ/S defined similarly.3.
Filtered φ r -modules over ΣIn this section, we define another variant Mod r,φ/ Σ ∞ of the category Mod r,φ/S ∞ over a subring Σ of the ring S , and prove that they are categorically equiv-alent.Let p be a rational prime and r be an integer such that 0 ≤ r < p − W -algebra Σ = W [[ u, Y ]] / ( E ( u ) p − pY ) as in [6, Subsection3.2]. We regard Σ as a subring of S by the map sending Y to E ( u ) p /p .Then the element c = φ ( E ( u )) ∈ S × is contained in Σ × . We define on Σ a σ -semilinear Frobenius endomorphism φ by φ ( u ) = u p and φ ( Y ) = p p − c p .Put Fil t Σ = ( E ( u ) t , Y ) for 0 ≤ t ≤ p − p Σ = ( Y ). Then we have φ (Fil t Σ) ⊆ p t Σ for 0 ≤ t ≤ p −
1. We put φ t = p − t φ | Fil t Σ . We also setΣ n = Σ /p n Σ and put on this ring the natural structures induced by thoseof Σ.We define a category ′ Mod r,φ/ Σ of filtered φ r -modules over Σ to be thecategory consisting of the following data: • a Σ-module M and its Σ-submodule Fil r M containing Fil r Σ · M , • a φ -semilinear map φ r : Fil r M → M satisfying φ r ( s r m ) = φ r ( s r ) φ ( m )for any s r ∈ Fil r Σ and m ∈ M , where we set φ ( m ) = c − r φ r ( E ( u ) r m ),and whose morphisms are defined in the same manner as ′ Mod r,φ/S . This cat-egory has a natural notion of exact sequences. We define its full subcategoryMod r,φ/ Σ to be the category consisting of M which is free of finite rank andgenerated by the image of φ r as a Σ -module. We also let Mod r,φ/ Σ ∞ denotethe smallest full subcategory of ′ Mod r,φ/ Σ which contains Mod r,φ/ Σ and is stableunder extensions. Moreover, we define a full subcategory Mod r,φ/ Σ of ′ Mod r,φ/ Σ to be the category consisting of M such that • the Σ-module M is free of finite rank and generated by the image of φ r , • the quotient M/ Fil r M is p -torsion free. Then we see that for ˆ M ∈ Mod r,φ/ Σ , the quotient ˆ M /p n ˆ M is naturally con-sidered as an object of Mod r,φ/ Σ ∞ .The natural ring isomorphism Σ / Fil p Σ ∼ = ˜ S defines a functor T , Σ :Mod r,φ/ Σ → Mod r,φ/ ˜ S by M M/ Fil p Σ · M . Then just as in the case of thefunctor T : Mod r,φ/S → Mod r,φ/ ˜ S ([4, Proposition 2.2.2.1]), we can show thefollowing lemma. Lemma 3.1.
The functor T , Σ : Mod r,φ/ Σ → Mod r,φ/ ˜ S is an equivalence ofcategories. On the other hand, [6, Proposition 2.2.1.3] and Nakayama’s lemma showthe following.
Lemma 3.2.
Let M be an object of Mod r,φ/ Σ of rank d over Σ . Then thereexists a basis { e , . . . , e d } of M such that Fil r M = Σ u r e ⊕ · · · ⊕ Σ u r d e d +Fil p Σ · M for some integers r , . . . , r d with ≤ r i ≤ er for any i . Then we can show the following lemma just as in the proof of [6, Lemme2.3.1.3].
Lemma 3.3.
The functor M Hom Σ , Fil r ,φ r ( M, A crys , ∞ ) from Mod r,φ/ Σ ∞ to the category of G K ∞ -modules is exact. For M ∈ Mod r,φ/ Σ , we can show as in the case of the category Mod r,φ/S thatthere is an isomorphism of G K -modulesHom Σ , Fil r ,φ r ( M, ( ˜ O ¯ K ) PD ) → Hom ˜ S , Fil r ,φ r ( T , Σ ( M ) , ˜ O ¯ K ) , where ˜ O ¯ K is considered as an object of ′ Mod r,φ/ ˜ S by the natural isomorphism( ˜ O ¯ K ) PD / Fil p ( ˜ O ¯ K ) PD → ˜ O ¯ K . Thus [6, Lemme 2.3.1.2] implies the following.
Lemma 3.4.
For M ∈ Mod r,φ/ Σ , we have Σ , Fil r ,φ r ( M, ( ˜ O ¯ K ) PD ) = p d , where d = dim Σ M . For the category Mod r,φ/ Σ ∞ , we have the following lemma. Lemma 3.5.
Let M be in Mod r,φ/ Σ ∞ . Then there exists α , . . . , α d ∈ Fil r M such that Fil r M = Σ α + · · · + Σ α d + Fil p Σ · M and the elements e = φ r ( α ) , . . . , e d = φ r ( α d ) form a system of generators of M . AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 11
Proof.
By induction and Lemma 3.2, we may assume that there exists anexact sequence of the category Mod r,φ/ Σ ∞ → M ′ → M → M ′′ → M ′ and M ′′ . Let α ′ , . . . , α ′ l ′ ( resp. α ′′ , . . . , α ′′ l ′′ )be elements of Fil r M ′ ( resp. Fil r M ′′ ) as in the lemma. Let α l ∈ Fil r M bea lift of α ′′ l . Then the elements α ′ , . . . , α ′ l ′ , α , . . . , α l ′′ satisfy the conditionin the lemma for M . (cid:3) Corollary 3.6.
Let M be an object of Mod r,φ/ Σ ∞ and C ∈ M d (Σ) be a matrixsatisfying ( α , . . . , α d ) = ( e , . . . , e d ) C with the notation of the previous lemma. Let A be an object of ′ Mod r,φ/ Σ .Then a Σ -linear homomorphism f : M → A preserving Fil r also commuteswith φ r if and only if φ r ( f ( e , . . . , e d ) C ) = ( f ( e ) , . . . , f ( e d )) . Proof.
Suppose that the latter condition holds. Then we have φ r ( f ( α i )) = f ( φ r ( α i )) for any i . We only have to check the equality φ r ◦ f = f ◦ φ r onFil p Σ · M . Suppose that this equality holds on the submodule p l +1 Fil p Σ · M .For m ∈ M , we can take m ′ ∈ Fil p Σ · M such that E ( u ) r m = P i s i α i + m ′ .Let s be in Fil p Σ. Then we have f ( φ r ( p l sm )) = p l φ r ( s ) c − r X i φ ( s i ) f ( φ r ( α i )) + p l φ r ( s ) c − r f ( φ r ( m ′ )) . Since φ r (Fil p Σ) ⊆ p Σ, this equals to φ r ( f ( p l sm )) by assumption. Thus thelemma follows by induction. (cid:3) Corollary 3.7.
Let M and A be as above and J ⊆ Fil r A be a Σ -submoduleof A such that φ r ( J ) ⊆ J . We can consider the Σ -module A/J naturally asan object of ′ Mod r,φ/ Σ . Suppose that for any x ∈ J , there exists t ∈ Z ≥ suchthat φ tr ( x ) = 0 . Then the natural homomorphism of abelian groups Hom Σ , Fil r ,φ r ( M, A ) → Hom Σ , Fil r ,φ r ( M, A/J ) is an isomorphism.Proof. The proof is similar to [3, Subsection 2.2]. We consider the Σ-submodule J as an object of the category ′ Mod r,φ/ Σ by putting Fil r J = J .By devissage, it is enough to show that, for any M ∈ Mod r,φ/ Σ , we haveExt ′ Mod r,φ/ Σ ( M, J ) = 0 and the map in the corollary is an isomorphism. Forthe first assertion, let0 / / J / / E / / M / / be an extension in the category ′ Mod r,φ/ Σ . Let e i , α i and C be as in Corollary3.6 such that e , . . . , e d form a basis of M . Let ˆ e i ∈ E be a lift of e i ∈ M .Then we have (ˆ e , . . . , ˆ e d ) C ∈ (Fil r E ) ⊕ d and φ r ((ˆ e , . . . , ˆ e d ) C ) = (ˆ e + δ , . . . , ˆ e d + δ d )for some δ , . . . , δ d ∈ J . On the other hand, there exists a unique d -tuple( x , . . . , x d ) ∈ J ⊕ d satisfying the equation φ r ((ˆ e + x , . . . , ˆ e d + x d ) C ) = (ˆ e + x , . . . , ˆ e d + x d ) . Indeed, the d -tuple t X i =0 ( φ ir ( δ ) , . . . , φ ir ( δ d )) φ ( C ) · · · φ i − ( C ) φ i ( C )is stable for sufficiently large t by assumption and this limit gives a uniquesolution of the equation. Then we have( p (ˆ e + x ) , . . . , p (ˆ e d + x d )) = φ r ( p (ˆ e + x ) , . . . , p (ˆ e d + x d )) φ ( C ) . Since the d -tuple on the left-hand side is contained in J ⊕ d , we see that this d -tuple is zero and e i ˆ e i + x i defines a section M → E . We can prove thesecond assertion similarly. (cid:3) Next we show that the two categories Mod r,φ/ Σ ∞ and Mod r,φ/S ∞ are in factequivalent. For M ∈ Mod r,φ/ Σ ∞ , we associate to it an S -module M by setting M = S ⊗ Σ M . We also define its S -submodule Fil r M byFil r M = Ker( M = S ⊗ Σ M → S/ Fil r S ⊗ Σ M/ Fil r M ≃ M/ Fil r M ) , where the last isomorphism is induced by the natural isomorphisms of W -algebras W [ u ] / ( E ( u ) r ) → Σ / Fil r Σ → S/ Fil r S. These associations induce two functors from Mod r,φ/ Σ ∞ to the category of S -modules, M
7→ M and M Fil r M . Since the rings S and W [ u ] / ( E ( u ) r )are p -torsion free, we have Tor Σ1 (Σ , S ) = Tor Σ1 (Σ , Σ / Fil r Σ) = 0 and thusTor Σ1 ( M, S ) = Tor Σ1 ( M, Σ / Fil r Σ) = 0 for any M ∈ Mod r,φ/ Σ ∞ . Hence we seethat these two functors are exact.We define φ r : Fil r M → M as follows. Note that Fil r S ⊗ Σ M ⊆ M and Fil r M is equal to Fil r S ⊗ Σ M + Im( S ⊗ Σ Fil r M → M ). Set φ ′ r :Fil r S ⊗ Σ M → M to be φ ′ r = φ r ⊗ φ . Lemma 3.8.
The map φ ⊗ φ r : S ⊗ Σ Fil r M → M induces a φ -semilinearmap φ ′′ r : Im( S ⊗ Σ Fil r M → M ) → M .Proof. Let z = P i s i ⊗ m i be in S ⊗ Σ Fil r M with s i ∈ S and m i ∈ Fil r M . Let¯ z be its image in M and suppose that ¯ z = 0. Write s i = s ′ i + s ′′ i with s ′ i ∈ Σand s ′′ i ∈ Fil p S . Since we have an isomorphism M / Fil r S ·M ≃ M/ Fil r Σ · M , AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 13 we can find elements s ( j ) ∈ Fil r Σ and m ( j ) ∈ M such that the equality P i s ′ i m i = P j s ( j ) m ( j ) holds in M . Then we have0 = ¯ z = X i ⊗ s ′ i m i + X i s ′′ i ⊗ m i = X j s ( j ) ⊗ m ( j ) + X i s ′′ i ⊗ m i in M . On the other hand, the element ( φ ⊗ φ r )( z ) ∈ M is equal to X j ⊗ φ r ( s ( j ) m ( j ) ) + X i φ ( s ′′ i ) ⊗ φ r ( m i ) . Since φ = p r φ r , this equals φ ′ r ( P j s ( j ) ⊗ m ( j ) + P i s ′′ i ⊗ m i ) = 0. (cid:3) Lemma 3.9.
The maps φ ′ r and φ ′′ r patch together and define a φ -semilinearmap φ r : Fil r M → M .Proof.
Since φ ′ r and φ ′′ r coincide on Im(Fil r S ⊗ Σ Fil r M → M ), it is enoughto show that 1 ⊗ φ r ( m ) = φ ′ r ( P i s i ⊗ m i ) for any m ∈ Fil r M , s i ∈ Fil r S and m i ∈ M satisfying 1 ⊗ m = P i s i ⊗ m i in M . As in the proof of Lemma3.8, the element m can be written as m = P j s ( j ) m ( j ) for some s ( j ) ∈ Fil r Σand m ( j ) ∈ M . By assumption, we have P i s i ⊗ m i = P j s ( j ) ⊗ m ( j ) inFil r S ⊗ Σ M . Hence the lemma follows. (cid:3) Then we see that this construction defines a functor M Σ ∞ : Mod r,φ/ Σ ∞ → Mod r,φ/S ∞ . Lemma 3.10.
The functor M Σ ∞ induces an equivalence of categories Mod r,φ/ Σ → Mod r,φ/S .Proof. Consider the diagram of functorsMod r,φ/ Σ M Σ ∞ / / T , Σ $ $ IIIIIIIII
Mod r,φ/S T (cid:15) (cid:15) Mod r,φ/ ˜ S . From the definition, we see that this diagram is commutative. By Lemma3.1, the downward arrows are equivalences of categories. Thus the lemmafollows. (cid:3)
Then a devissage argument as in [15, Proposition 1.1.11] shows the fol-lowing corollary.
Corollary 3.11.
The functor M Σ ∞ : Mod r,φ/ Σ ∞ → Mod r,φ/S ∞ is fully faithful. To show the essential surjectivity of the functor M Σ ∞ , we define anotherfunctor M S ∞ : Mod r,φ/ S ∞ → Mod r,φ/ Σ ∞ which is defined in a similar way to thefunctor M S ∞ : Mod r,φ/ S ∞ → Mod r,φ/S ∞ . For an S -module M in Mod r,φ/ S ∞ , weassociate to it a Σ-module M ∈ ′ Mod r,φ/ Σ as follows: • M = Σ ⊗ φ, S M , • Fil r M = Ker( M ⊗ φ → Σ ⊗ S M → (Σ / Fil r Σ) ⊗ S M ), • φ r : Fil r M ⊗ φ → Fil r Σ ⊗ S M φ r ⊗ → Σ ⊗ φ, S M = M .We can check that this defines an exact functor Mod r,φ/ S ∞ → Mod r,φ/ Σ ∞ asin the proof of [15, Proposition 1.1.11]. We let this functor be denoted by M S ∞ . Lemma 3.12.
The diagram of functors
Mod r,φ/ S ∞ M S ∞ / / M S ∞ $ $ JJJJJJJJJ
Mod r,φ/ Σ ∞ M Σ ∞ (cid:15) (cid:15) Mod r,φ/S ∞ is commutative.Proof. For M ∈ Mod r,φ/ S ∞ , put M = M S ∞ ( M ) and M = M S ∞ ( M ). Then M = S ⊗ Σ M as an S -module. Let Fil r M and φ r : Fil r M → M denote thefiltration and Frobenius structure defined by the functor M S ∞ . We also letˆFil r M and ˆ φ r : ˆFil r M → M denote those defined by M Σ ∞ .The S -module Fil r M contains ˆFil r M . Conversely, let z be an element ofFil r M . Note that Fil p S · M ⊆ ˆFil r M . Thus, to show z ∈ ˆFil r M , we mayassume that z ∈ Im( M → M ). Then the commutative diagram whose rightvertical arrow is an isomorphism M = Σ ⊗ φ, S M ⊗ φ −−−−→ Σ ⊗ S M −−−−→ Σ / Fil r Σ ⊗ S M y y y M = S ⊗ φ, S M ⊗ φ −−−−→ S ⊗ S M −−−−→ S/ Fil r S ⊗ S M implies that z ∈ Im(Fil r M → Fil r M ) ⊆ ˆFil r M and hence Fil r M = ˆFil r M .From the definition, we also can show φ r = ˆ φ r . This implies the lemma. (cid:3) Proposition 3.13.
The functor M Σ ∞ : Mod r,φ/ Σ ∞ → Mod r,φ/S ∞ is an equiva-lence of categories.Proof. Since the functor M S ∞ is an equivalence of categories for p ≥ r = 0, put κ Σ = κ ∩ Σ, where κ = Ker( S → W ). Then, byusing a natural isomorphism Σ ≃ W [[ u, u ep /p ]], we can show that the functor M M/κ Σ M defines an equivalence of categories Mod ,φ/ Σ ∞ → Mod φ/W ∞ , as AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 15 in the case of the category Mod ,φ/S ∞ . Since the diagramMod ,φ/ Σ ∞ $ $ JJJJJJJJJ M Σ ∞ / / Mod ,φ/S ∞ (cid:15) (cid:15) Mod φ/W ∞ is commutative and the downward arrows are equivalences of categories, theproposition follows also for p = 2. (cid:3) Remark 3.14.
We can also define a fully faithful functor M Σ : Mod r,φ/ Σ → Mod r,φ/S in a similar way to M Σ ∞ and prove that this is an equivalence ofcategories. Indeed, the claim for p ≥ M be in Mod ,φ/S and e , . . . , e d be a basis of M over S . Let C ∈ GL d ( S ) bethe matrix such that φ ( e , . . . , e d ) = ( e , . . . , e d ) C. Then the elements φ ( e ) , . . . , φ ( e d ) also form a basis of M and φ ( φ ( e ) , . . . , φ ( e d )) = ( φ ( e ) , . . . , φ ( e d )) φ ( C ) . Since φ ( S ) ⊆ Σ, the Σ-module M defined by M = Σ φ ( e ) ⊕ · · · ⊕ Σ φ ( e d ) isstable under φ . Hence we see that M ∈ Mod ,φ/ Σ and M = M Σ ( M ). Proposition 3.15.
Let M be an object of Mod r,φ/ Σ ∞ and set M = M Σ ∞ ( M ) .Then there exists a natural isomorphism of G K ∞ -modules Hom Σ , Fil r ,φ r ( M, A crys , ∞ ) → Hom S, Fil r ,φ r ( M , A crys , ∞ ) . Moreover, this induces for any n an isomorphism of G K n -modules Hom Σ , Fil r ,φ r ( M, W PD n ( ˜ O ¯ K )) → Hom S, Fil r ,φ r ( M , W PD n ( ˜ O ¯ K )) . Proof.
By definition, M = S ⊗ Σ M and we have a natural isomorphismHom Σ ( M, A crys , ∞ ) → Hom S ( M , A crys , ∞ ) . From the definition, we can check that this isomorphism induces the map inthe proposition, which is injective. To prove the bijectivity, by devissage wemay assume that pM = 0. Then both sides of this injection have the samecardinality by Lemma 3.4 and the first assertion follows. Since the sequence0 / / W PD n ( ˜ O ¯ K ) / / A crys , ∞ p n / / A crys , ∞ / / ′ Mod r,φ/ Σ is exact, the first assertion implies the second one. (cid:3) A method of Abrashkin
In this section, we study the G K n -module Hom Σ , Fil r ,φ r ( M, W PD n ( ˜ O ¯ K ))following Abrashkin ([3]).Let p and 0 ≤ r < p − p -power roots ofunity { ζ p n } n ∈ Z ≥ in ¯ K such that ζ p = 1 and ζ p n = ζ pp n +1 for any n , and setan element ε of R to be ( ζ p n ) n ∈ Z ≥ . Then the elements [ ε ] − ε /p ] − W ( R ). The element of W ( R ) t = ([ ε ] − / ([ ε /p ] −
1) = 1 + [ ε /p ] + [ ε /p ] + · · · + [ ε /p ] p − is a generator of the principal ideal Ker( θ ). We define an element a ∈ W ( R )to be a = (cid:26) P p − k =1 p − (( − p − − kp − C k − ε /p ] k ( p ≥ − p = 2) , where p − C k = ( p − / ( k !( p − − k )!) is the binomial coefficient. Notethat the coefficient of [ ε /p ] k in each term is an integer. The element a isinvertible in the ring W ( R ), since θ ( a ) = ( ζ p − p − /p ∈ O × C and the idealKer( θ ) is topologically nilpotent in W ( R ).The element Z = ([ ε ] − p − /p of A crys is topologically nilpotent and wehave φ ( t ) = p ( Z − φ ( a )). Consider the formal power series ring W ( R )[[ u ′ ]]with the ( t, u ′ )-adic topology and the continuous ring homomorphism W ( R )[[ u ′ ]] → A crys which sends u ′ to Z . Let ˆ A denote the image of this homomorphism.Then we see that the ring ˆ A is ( t, Z )-adically complete. Since we have Z = at p − + t p /p , the element t p /p of A crys is contained in the subring ˆ A and topologically nilpotent in this subring. Hence we can consider the ring ˆ A as a Σ-algebra by u [ π ]. Put Fil i ˆ A = ( t i , Z ) for 0 ≤ i ≤ p −
1. The Frobe-nius endomorphism φ of A crys preserves ˆ A and satisfies φ (Fil i ˆ A ) ⊆ p i ˆ A for0 ≤ i ≤ p −
1. Set φ r = p − r φ | Fil r ˆ A . Then we can consider the ring ˆ A also asan object of the category ′ Mod r,φ/ Σ . Put ˆ A n = ˆ A/p n ˆ A and ˆ A ∞ = ˆ A ⊗ W K /W .We include here a proof of the following lemma stated in [3, Subsection 3.2]. Lemma 4.1.
The natural inclusion W ( R ) → ˆ A induces isomorphisms of W ( R ) -algebras W ( R ) / (([ ε ] − p − ) → ˆ A/ ( Z ) and W n ( R ) / (([ ε ] − p − ) → ˆ A n / ( Z ) .Proof. For a subring B of A crys , put I [ s ] B = { x ∈ B | φ i ( x ) ∈ Fil s A crys for any i } as in [12, Subsection 5.3]. Then we have I [ s ] W ( R ) = ([ ε ] − s W ( R ) and thenatural ring homomorphism W ( R ) /I [ s ] W ( R ) → A crys /I [ s ] A crysAMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 17 is an injection ([12, Proposition 5.1.3, Proposition 5.3.5]). Since the element Z is contained in the ideal I [ p − A crys , this injection factors as W ( R ) /I [ p − W ( R ) → ˆ A/ ( Z ) → A crys /I [ p − A crys . Hence the former arrow is an isomorphism and the lemma follows. (cid:3)
Therefore ˆ A/ Fil r ˆ A is p -torsion free and p n Fil r ˆ A = Fil r ˆ A ∩ p n ˆ A . Thuswe can also consider ˆ A n and ˆ A ∞ as objects of the category ′ Mod r,φ/ Σ . Theabsolute Galois group G K ∞ acts naturally on these Σ-modules. Lemma 4.2.
We have a natural decomposition as an R -module ˆ A = R/ ( t p ) ⊕ ( Z ) . Proof.
Consider the natural inclusion W ( R ) → ˆ A . We claim that this in-duces an injection R/ ( t p ) → ˆ A . Let x be in the ring R . If the element[ x ] ∈ W ( R ) is contained in p ˆ A , then its image in A crys /pA crys is zero. Wehave an isomorphism of R -algebras R [ Y , Y , . . . ] / ( t p , Y p , Y p , . . . ) → A crys /pA crys which sends Y i to the image of t p i /p i !. Thus the element x is contained inthe ideal ( t p ). Conversely, if v R ( x ) ≥ p , then we have[ x ] = w ([ ε ] − p − + pw ′ for some w, w ′ ∈ W ( R ) and this implies [ x ] ∈ p ˆ A . Now we have the com-mutative diagram of R -algebras R/ ( t p ) / / f $ $ IIIIIIIII ˆ A (cid:15) (cid:15) ˆ A / ( Z )and the map f : R/ ( t p ) → ˆ A / ( Z ) is an isomorphism by Lemma 4.1. Hencethe lemma follows. (cid:3) Since r < p −
1, from this lemma we can show the following lemma as inthe proof of [6, Lemme 2.3.1.3].
Lemma 4.3.
The functor M Hom Σ , Fil r ,φ r ( M, ˆ A ∞ ) from Mod r,φ/ Σ ∞ to the category of G K ∞ -modules is exact. Corollary 4.4.
For any M ∈ Mod r,φ/ Σ ∞ , the natural map Hom Σ , Fil r ,φ r ( M, ˆ A ∞ ) → Hom Σ , Fil r ,φ r ( M, A crys , ∞ ) is an isomorphism of G K ∞ -modules. Moreover, for any n , we have an iso-morphism of G K ∞ -modules Hom Σ , Fil r ,φ r ( M, ˆ A n ) → Hom Σ , Fil r ,φ r ( M, A crys /p n A crys ) . Proof.
Let us prove the first assertion. By Lemma 3.3 and Lemma 4.3, wemay assume pM = 0. Consider the commutative diagram of ringsˆ A / / % % JJJJJJJJJJJ A crys /pA crys (cid:15) (cid:15) R/ ( t p − )whose downward arrows are defined by modulo Fil p − of the rings ˆ A and A crys /pA crys , respectively. Since r < p −
1, we have φ r (Fil p − ˆ A ) = 0 andsimilarly for the ring A crys /pA crys . Thus these two surjections induce onthe ring R/ ( t p − ) the same structure of a filtered φ r -module over Σ. ByCorollary 3.7, we have a commutative diagramHom Σ , Fil r ,φ r ( M, ˆ A ) / / * * VVVVVVVVVVVVVVVVV
Hom Σ , Fil r ,φ r ( M, A crys /pA crys ) (cid:15) (cid:15) Hom Σ , Fil r ,φ r ( M, R/ ( t p − ))whose downward arrows are isomorphisms. This concludes the proof of thefirst assertion. Since we have an exact sequence0 / / ˆ A n / / ˆ A ∞ p n / / ˆ A ∞ / / ′ Mod r,φ/ Σ , the second assertion follows. (cid:3) Since the ideal ( Z ) of ˆ A n satisfies the condition of Corollary 3.7, the Σ-algebra ˆ A n / ( Z ) is naturally considered as an object of ′ Mod r,φ/ Σ . We also givethe ring W n ( R ) / (([ ε ] − p − ) the structures of a Σ-algebra and a filtered φ r -module over Σ induced from those of ˆ A n / ( Z ) by the isomorphism in Lemma4.1. The map Σ → W n ( R ) / (([ ε ] − p − )sends the element u ∈ Σ to the image of [ π ] in the ring on the right-handside. Put v = t/E ([ π ]) ∈ W ( R ) × . As for the element Y ∈ Σ, the equality Y = − av − E ([ π ]) p − + v − p Z holds in ˆ A . Hence the above homomorphism sends the element Y to theimage of − av − E ([ π ]) p − .Consider the surjective ring homomorphism R → ˜ O ¯ K x = ( x , x , . . . ) x n AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 19 and the induced surjection β n : W n ( R ) → W n ( ˜ O ¯ K ). Let J = { ( x , . . . , x n − ) ∈ W n ( R ) | v R ( x i ) ≥ p n for any i } be the kernel of the latter surjection. Lemma 4.5.
The ideal J is contained in the ideal (([ ε ] − p − ) of the ring W n ( R ) .Proof. Write the element ([ ε ] − p − also as x = ( x , . . . , x n − ) ∈ W n ( R )with v R ( x ) = p . Take an element z = ( z , . . . , z n − ) of the ideal J . We con-struct y ∈ W n ( R ) such that xy = z . By induction, it is enough to show thatif z = · · · = z i − = 0 for some 0 ≤ i ≤ n − x , . . . , x i )(0 , . . . , , y i ) =(0 , . . . , , z i ) in W i +1 ( R ), then x (0 , . . . , , y i , , . . . , ∈ J . Let us write thiselement as (0 , . . . , , w i , . . . , w n − ) with w i = z i . We have v R ( y i ) ≥ p n − p i +1 .In the ring of Witt vectors W n ( F p [ X , . . . , X n − , Y , . . . , Y n − ]), the k -th en-try of the vector ( X , . . . , X n − )(0 , . . . , , Y i , , . . . , X p i k − i Y p k − i i for any k ≥ i . Thus we have v R ( w k ) ≥ p n . (cid:3) Note that the elements [ ζ p n ] − ζ p n +1 ] − W n ( ˜ O ¯ K ).By the above lemma, we have an isomorphism of rings W n ( R ) / (([ ε ] − p − ) → W n ( ˜ O ¯ K ) / (([ ζ p n ] − p − ) . We let ¯ A n,p − denote the ring on the right-hand side and give the ring ¯ A n,p − the structure of a filtered φ r -module over Σ induced by this isomorphism.For an algebraic extension F of K , we put b F = { x ∈ O F | v K ( x ) > er/ ( p − } . Note that the ring O F / b F is killed by p . We consider the ring of Wittvectors W n ( O F / b F ) as a W n ( O F )-algebra by the natural ring surjection W n ( O F ) → W n ( O F / b F ) and as a W n -algebra by twisting the natural actionby σ − n , as before. For a ring B and its ideal I , we define an ideal W n ( I ) ofthe ring W n ( B ) to be W n ( I ) = { ( x , . . . , x n − ) ∈ W n ( B ) | x i ∈ I for any i } . Put F n = K n ( ζ p n +1 ). For an algebraic extension F of F n in ¯ K , theelements [ ζ p n ] − ζ p n +1 ] − W n ( m F ) are topologically nilpotentnon-zero divisors in W n ( O F ). Let the ring W n ( O F / b F ) / ([ ζ p n ] − r W n ( m F / b F )be denoted by ¯ A n,F,r + . We also put ¯ A n,r + = ¯ A n, ¯ K,r + . Lemma 4.6.
The ideal ([ ζ p n ] − r W n ( m F ) of W n ( O F ) contains the ideal W n ( b F ) for any r ∈ { , . . . , p − } . We also have (([ ζ p n ] − p − ) ⊇ W n ( p O F ) . Proof.
The proof is similar to the proof of Lemma 4.5. Let us show thefirst assertion. Since this is trivial for r = 0, we may assume r ≥
1. Put x = ( x , . . . , x n − ) = ([ ζ p n ] − r ∈ W n ( O F ). Then we have v p ( x ) = r/ ( p n − ( p − ≤ i ≤ n −
1, if( x , . . . , x i )(0 , . . . , , y i ) ∈ W i +1 ( b F ), then y i ∈ m F and x (0 , . . . , , y i , , . . . , ∈ W n ( b F ). By assumption, we have v p ( y i ) > rp − − p n − i − ) ≥ . Put (0 , . . . , , w i , . . . , w n − ) = x (0 , . . . , , y i , , . . . , w l ∈ b F forany l by induction. Indeed, let us suppose that w l ∈ b F for any i ≤ l ≤ k − i + 1 ≤ k ≤ n −
1. We have the equality p i y p k − i i ( x p k + px p k − + · · · + p k x k ) = ( p i w p k − i i + p i +1 w p k − i − i +1 + · · · + p k w k ) . Since r ≥
1, we have ( p k − l − r/ ( p − ≥ k − l for 0 ≤ l ≤ k −
1. Thisimplies v p ( p l w p k − l l ) > k + r/ ( p −
1) for 0 ≤ l ≤ k −
1. The valuation of theleft-hand side of the above equality also satisfies this inequality. Thus wehave v p ( w k ) > r/ ( p −
1) and the assertion follows. We can show the secondassertion similarly. (cid:3)
By this lemma, the natural surjections of rings W n ( O F ) / ([ ζ p n ] − r W n ( m F ) → W n ( O F /p O F ) / ([ ζ p n ] − r W n ( m F /p O F ) → ¯ A n,F,r + are isomorphisms. Then we see that the natural injection F → ¯ K inducesan injection of rings ¯ A n,F,r + → ¯ A n,r + .Write Z n for the image of the element Z of A crys in W PD n ( ˜ O ¯ K ). Then wehave a commutative diagram of Σ-algebrasˆ A n (cid:15) (cid:15) (cid:15) (cid:15) / / A crys /p n A crys ≀ (cid:15) (cid:15) W PD n ( ˜ O ¯ K ) (cid:15) (cid:15) (cid:15) (cid:15) W n ( R ) / (([ ε ] − p − ) ∼ / / ≀ (cid:15) (cid:15) ˆ A n / ( Z ) ' ' OOOOOOOOOOO ¯ A n,p − / / (cid:15) (cid:15) (cid:15) (cid:15) W PD n ( ˜ O ¯ K ) / ( Z n ) , ¯ A n,r + where all the vertical arrows are surjections satisfying the condition of Corol-lary 3.7. Hence this is also a commutative diagram in ′ Mod r,φ/ Σ . Note AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 21 that these rings and homomorphisms are independent of the choice of asystem { ζ p n } n ∈ Z ≥ . We also note that Fil r ¯ A n,r + = E ([ π n ]) r ¯ A n,r + and φ r ( E ([ π n ]) r y ) = c r φ ( y ) for any y ∈ ¯ A n,r + , where φ denotes the Frobeniusendomorphism of ¯ A n,r + induced from that of the ring W n ( O ¯ K / b ¯ K ). More-over, let M be an object of Mod r,φ/ Σ ∞ . Then, by Corollary 3.7 and Corollary4.4, we have a natural isomorphism of abelian groupsHom Σ , Fil r ,φ r ( M, W PD n ( ˜ O ¯ K )) → Hom Σ , Fil r ,φ r ( M, ¯ A n,r + ) . Next we investigate the module on the right-hand side of this isomor-phism, and prove this is in fact an isomorphism of G F n -modules. Considerthe element E ([ π n ]) ∈ W n ( O F n /p O F n ) and let us fix its lift ˆ γ ∈ W n ( O F n )by the natural surjection W n ( O F n ) → W n ( O F n /p O F n ). Let a ∈ W ( R ) × and v = t/E ([ π ]) ∈ W ( R ) × as before. We let a n , t n and v n denote the images of a , t and v by the surjection W ( R ) → W n ( ˜ O ¯ K ) induced by β n , respectively.The elements a n and t n of the ring W n ( ˜ O ¯ K ) are contained in the subring W n ( O F n /p O F n ). We abusively let them also denote their images by thenatural surjections W n ( ˜ O ¯ K ) → W n ( O ¯ K / b ¯ K ) → ¯ A n,r + . Lemma 4.7.
The element ˆ t n = 1 + [ ζ p n +1 ] + [ ζ p n +1 ] + · · · + [ ζ p n +1 ] p − = [ ζ p n ] − ζ p n +1 ] − is divisible by ˆ γ in the ring W n ( O F n ) . In particular, ˆ γ is a non-zero divisorof the ring W n ( O ¯ K ) .Proof. It is enough to show the divisibility in the ring W n ( O ¯ K ). Note thatthe element t n is also the image of ˆ t n by the natural map W n ( O F n ) → W n ( O F n /p O F n ). Let ˆ v n be a lift of v n by the natural surjection W n ( O ¯ K ) → W n ( ˜ O ¯ K ). Then we have ˆ t n − ˆ γ ˆ v n ∈ W n ( p O ¯ K ). By Lemma 4.6, there existsˆ y ∈ W n ( m ¯ K ) such that ˆ t n − ˆ γ ˆ v n = ˆ t n ˆ y . Hence we have ˆ t n (1 − ˆ y ) = ˆ γ ˆ v n .Since ˆ y is topologically nilpotent in the ring W n ( O ¯ K ), the element 1 − ˆ y isinvertible and the lemma follows. (cid:3) Lemma 4.8.
The image of Y ∈ Σ in the ring ¯ A n,r + ( resp. ¯ A n,p − ) iscontained in its subring ¯ A n,F n ,r + ( resp. W n ( O F n /p O F n ) / (([ ζ p n ] − p − )) .Proof. We have the equality E ([ π n ]) v n = t n = 1 + [ ζ p n +1 ] + [ ζ p n +1 ] + · · · + [ ζ p n +1 ] p − in the ring W n ( ˜ O ¯ K ). Note that any element v ′ n ∈ W n ( ˜ O ¯ K ) satisfying thesame equality is invertible and thus the elements ( v ′ n ) − E ([ π n ]) are equal toeach other. Since Y = − a n v − n E ([ π n ]) p − in the rings ¯ A n,r + and ¯ A n,p − , itsuffices to construct an element v ′ n of the ring W n ( O F n /p O F n ) such that theequality E ([ π n ]) v ′ n = t n holds. This follows from Lemma 4.7. (cid:3) From this lemma, we see that the natural G F n -actions on the rings ¯ A n,p − and ¯ A n,r + are compatible with the filtered φ r -module structures over Σ. In the big commutative diagram above, the lowest horizontal arrow and lowerright vertical arrow are G K -linear by definition. Hence we have shown thefollowing proposition. Proposition 4.9.
Let M be an object of Mod r,φ/ Σ ∞ . Then the map Hom Σ , Fil r ,φ r ( M, W PD n ( ˜ O ¯ K )) → Hom Σ , Fil r ,φ r ( M, ¯ A n,r + ) is an isomorphism of G F n -modules. Let M be as in the proposition. Let e , . . . , e d be a system of generators of M as in Lemma 3.5 and C = ( c i,j ) ∈ M d (Σ) be a matrix representing φ r as inCorollary 3.6. Consider the surjection Σ ⊕ d → M defined by ( s , . . . , s d ) s e + · · · + s d e d and let ( s , , . . . , s ,d ) , . . . , ( s q, , . . . , s q,d ) be a system ofgenerators of its kernel. Then the underlying G F n -set of the G F n -moduleHom Σ , Fil r ,φ r ( M, ¯ A n,r + )is identified with the set of d -tuples (¯ x , . . . , ¯ x d ) in ¯ A n,r + such that thefollowing three conditions hold: • s l, ¯ x + · · · + s l,d ¯ x d = 0 for any l , • c ,i ¯ x + · · · + c d,i ¯ x d ∈ Fil r ¯ A n,r + for any i , • the following equality holds: φ r ( c , ¯ x + · · · + c d, ¯ x d ) = ¯ x ... φ r ( c ,d ¯ x + · · · + c d,d ¯ x d ) = ¯ x d . We choose lifts ˆ c , ˆ c i,j and ˆ s i,j in W n ( O F n ) of the images of c , c i,j and s i,j in ¯ A n,r + by the natural ring homomorphism W n ( O ¯ K ) → W n ( ˜ O ¯ K ) → W n ( O ¯ K / b ¯ K ) → ¯ A n,r + , respectively. Recall that we have already chosen a lift ˆ γ ∈ W n ( O F n ) of E ([ π n ]) ∈ W n ( O F n /p O F n ).Fix a polynomial Φ i ∈ Z [ X , . . . , X n − ] such that Φ i ≡ X pi mod p . Thisinduces for any commutative ring B a map Φ = (Φ , . . . , Φ n − ) : W n ( B ) → W n ( B ) which is a lift of the Frobenius endomorphism on W n ( B/pB ). Inparticular, set B to be the polynomial ring Z [ X , . . . , X n − , Y , . . . , Y n − ].Put X = ( X , . . . , X n − ) and Y = ( Y , . . . , Y n − ) in the ring W n ( B ). Thenwe see that there exists elements U , . . . , U n − and U ′ , . . . , U ′ n − of the poly-nomial ring B such thatΦ( X + Y ) = Φ( X ) + Φ( Y ) + ( pU , . . . , pU n − ) , Φ( XY ) = Φ( X )Φ( Y ) + ( pU ′ , . . . , pU ′ n − )in the ring W n ( B ). Proposition 4.10.
Every d -tuple (¯ x , . . . , ¯ x d ) in ¯ A n,r + satisfying the abovethree conditions uniquely lifts to a d -tuple (ˆ x , . . . , ˆ x d ) in W n ( O ¯ K ) such that • ˆ s l, ˆ x + · · · + ˆ s l,d ˆ x d ∈ ([ ζ p n ] − r W n ( m ¯ K ) for any l , AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 23 • ˆ c ,i ˆ x + · · · + ˆ c d,i ˆ x d ∈ ˆ γ r W n ( O ¯ K ) for any i , • the following equality holds: ˆ c r Φ((ˆ c , ˆ x + · · · + ˆ c d, ˆ x d ) / ˆ γ r ) = ˆ x ... ˆ c r Φ((ˆ c ,d ˆ x + · · · + ˆ c d,d ˆ x d ) / ˆ γ r ) = ˆ x d . Proof.
Fix a lift ˆ x i of ¯ x i in W n ( O ¯ K ). Recall that the kernel of the surjection W n ( O ¯ K ) → ¯ A n,r + is equal to the ideal ([ ζ p n ] − r W n ( m ¯ K ). The first condi-tion in the proposition holds automatically for (ˆ x , . . . , ˆ x d ). By Lemma 4.7,the element ˆ c ,i ˆ x + · · · + ˆ c d,i ˆ x d is contained in ˆ γ r W n ( O ¯ K ) for any i . Sincethe map φ r : Fil r ¯ A n,r + → ¯ A n,r + satisfies φ r ( E ([ π n ]) r ¯ x ) = c r φ (¯ x ) for any¯ x ∈ ¯ A n,r + , we have ˆ c r Φ((ˆ c , ˆ x + · · · + ˆ c d, ˆ x d ) / ˆ γ r ) = ˆ x + ([ ζ p n ] − r ˆ δ ...ˆ c r Φ((ˆ c ,d ˆ x + · · · + ˆ c d,d ˆ x d ) / ˆ γ r ) = ˆ x d + ([ ζ p n ] − r ˆ δ d for some ˆ δ , . . . , ˆ δ d ∈ W n ( m ¯ K ). It suffices to show that there exists a unique d -tuple (ˆ y , . . . , ˆ y d ) in W n ( m ¯ K ) such thatˆ c r Φ((ˆ c ,i (ˆ x + ([ ζ p n ] − r ˆ y ) + · · · + ˆ c d,i (ˆ x d + ([ ζ p n ] − r ˆ y d )) / ˆ γ r )= ˆ x i + ([ ζ p n ] − r ˆ y i for any i . For this, we need the following lemma. Lemma 4.11.
Let N be a complete discrete valuation field and m N be themaximal ideal of N . Let ǫ , . . . , ǫ d be in m N . Let P , . . . , P d and P ′ . . . , P ′ d beelements of O N [[ Y , . . . , Y d ]] such that P i ∈ ( Y , . . . , Y d ) . Then the equation Y − P ( Y , . . . , Y d ) − ǫ P ′ ( Y , . . . , Y d ) = 0 ... Y d − P d ( Y , . . . , Y d ) − ǫ d P ′ d ( Y , . . . , Y d ) = 0 has a unique solution in m N .Proof. By assumption, we see that for any integer l ≥
1, a d -tuple ( y , . . . , y d )in m N /m lN satisfying the above equation lifts uniquely to a d -tuple in m N /m l +1 N satisfying the same equation. Thus the lemma follows. (cid:3) Let us write as ˆ y i = (ˆ y i, , . . . , ˆ y i,n − ). Since the image of Φ(([ ζ p n +1 ] − r )in ¯ A n,r + is equal to ([ ζ p n ] − r , we can find ˆ b ∈ W n ( O ¯ K ) such thatΦ(([ ζ p n ] − r / ˆ γ r ) = ([ ζ p n ] − r ˆ b. Then there exists polynomials U i,m over O ¯ K of the indeterminates Y =( Y i,m ) ≤ i ≤ d, ≤ m ≤ n − such that the equation we have to solve isˆ x i + ([ ζ p n ] − r ˆ y i = ˆ x i + ([ ζ p n ] − r ˆ δ i + ([ ζ p n ] − r ˆ b ˆ c r (Φ(ˆ c ,i )Φ(ˆ y ) + · · · + Φ(ˆ c d,i )Φ(ˆ y d ))+ ( pU i, (ˆ y ) , . . . , pU i,n − (ˆ y ))for any i , where we put ˆ y = (ˆ y i,m ) ≤ i ≤ d, ≤ m ≤ n − . As in the proof of Lemma4.6, we see that, for any elements P , . . . , P n − of the polynomial ring O ¯ K [ Y ],we can uniquely find elements Q , . . . , Q n − of this ring such that the coef-ficients of these polynomials are in the maximal ideal m ¯ K and the equality( pP , . . . , pP n − ) = ([ ζ p n ] − r ( Q , . . . , Q n − )holds in the ring of Witt vectors W n ( O ¯ K [ Y ]). Therefore, this equation isequivalent to the equationˆ y i = ˆ δ i + ˆ b ˆ c r (Φ(ˆ c ,i )Φ(ˆ y ) + · · · + Φ(ˆ c d,i )Φ(ˆ y d ))+ ( V i, (ˆ y ) , . . . , V i,n − (ˆ y )) , where V i,m is a polynomial of Y over O ¯ K whose coefficients are in the max-imal ideal m ¯ K . From the definition of Φ, we see that ˆ y = (ˆ y i,m ) i,m is asolution of a system of equations Y i,m − P i,m ( Y ) − ǫ i,m P ′ i,m ( Y ) = 0satisfying the condition of Lemma 4.11 for a sufficiently large finite extension N of K . Then, by this lemma, we can solve the equation uniquely in m ¯ K . (cid:3) Let F be an algebraic extension of F n in ¯ K and consider the ring ¯ A n,F,r + .By Lemma 4.8, we can consider this ring as a Σ-subalgebra of ¯ A n,r + . PutFil r ¯ A n,F,r + = E ([ π n ]) r ¯ A n,F,r + . Then Lemma 4.7 implies that¯ A n,F,r + ∩ Fil r ¯ A n,r + = Fil r ¯ A n,F,r + . Moreover, the Frobenius endomorphism φ of the ring ¯ A n,r + preserves thesubalgebra ¯ A n,F,r + and thus φ r : Fil r ¯ A n,r + → ¯ A n,r + induces a φ -semilinearmap φ r : Fil r ¯ A n,F,r + → ¯ A n,F,r + . Hence ¯ A n,F,r + is a subobject of ¯ A n,r + inthe category ′ Mod r,φ/ Σ . For M ∈ Mod r,φ/ Σ ∞ , let us set T ∗ crys ,π n ,F ( M ) = Hom Σ , Fil r ,φ r ( M, ¯ A n,F,r + ) . We see that ¯ A n,r + = ¯ A n, ¯ K,r + = [ F/F n ¯ A n,F,r + in ′ Mod r,φ/ Σ and thus we have a natural identification of abelian groups T ∗ crys ,π n , ¯ K ( M ) = [ F/F n T ∗ crys ,π n ,F ( M ) . The absolute Galois group G F n acts on the abelian group on the left-handside. AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 25
Lemma 4.12.
Let F be an algebraic extension of F n in ¯ K . Then the G F -fixed part T ∗ crys ,π n , ¯ K ( M ) G F is equal to T ∗ crys ,π n ,F ( M ) .Proof. From Proposition 4.10, we see that the elements of T ∗ crys ,π n , ¯ K ( M )correspond bijectively to the d -tuples in W n ( O ¯ K ) satisfying the three condi-tions in this proposition. The uniqueness assertion of the proposition showsthat g ∈ G F fixes such a d -tuple in W n ( O ¯ K ) if and only if g fixes its imagein ¯ A n,r + . Hence an element of T ∗ crys ,π n , ¯ K ( M ) is fixed by G F if and only if itis contained in the image of W n ( O F ). Thus the lemma follows. (cid:3) Corollary 4.13.
Let L n be the finite Galois extension of F n correspondingto the kernel of the map G F n → Aut( T ∗ crys ,π n , ¯ K ( M )) . Then an algebraic extension F of F n in ¯ K contains L n if and only if T ∗ crys ,π n ,F ( M ) = T ∗ crys ,π n , ¯ K ( M ) . Proof.
An algebraic extension F of F n contains L n if and only if the actionof G F on T ∗ crys ,π n , ¯ K ( M ) is trivial. By Lemma 4.12, this is equivalent to T ∗ crys ,π n ,F ( M ) = T ∗ crys ,π n , ¯ K ( M ). (cid:3) Ramification bound
In this section, we prove Theorem 1.1. Let M be an object of Mod r,φ,N/S ∞ which is killed by p n and let L be the finite Galois extension of K corre-sponding to the kernel of the map G K → Aut( T ∗ st ,π ( M )) . Then the theorem is equivalent to the inequality u L/K ≤ u ( K, r, n ), where u L/K denotes the greatest upper ramification break of the Galois extension
L/K ([10]). For r = 0, the G K -module T ∗ st ,π ( M ) is unramified and theassertion is trivial. Thus we may assume p ≥ r ≥ L n be the finite Galois extension of F n corresponding to the kernel ofthe map G F n → Aut( T ∗ st ,π ( M )) . Since F n is Galois over K , the extension L n = LF n is also a Galois extensionof K . Let M ∈ Mod r,φ/ Σ ∞ be the filtered φ r -module over Σ which correspondsto M by the equivalence M Σ ∞ of Proposition 3.13. Then Proposition 3.15and Proposition 4.9 show that L n is also the finite extension of F n cutout by the G F n -module T ∗ crys ,π n , ¯ K ( M ). It is enough to prove the inequality u L n /K ≤ u ( K, r, n ).Before proving this, we state some general lemmas to calculate the ram-ification bound. Let N be a complete discrete valuation field of positiveresidue characteristic, v N be its valuation normalized as v N ( N × ) = Z and N sep be its separable closure. We extend v N to any algebraic closure of N . Lemma 5.1.
Let f ( T ) ∈ O N [ T ] be a separable monic polynomial and z , . . . , z d be the zeros of f in O N sep . Suppose that the set { v N ( z k − z i ) | k = 1 , . . . , d, k = i } is independent of i . Put s f = X k =1 ,...,dk = i v N ( z k − z i ) and α f = sup k =1 ,...,dk = i v N ( z k − z i ) , which are independent of i by assumption. If j > s f + α f , then we have thedecomposition { x ∈ O N sep | v N ( f ( x )) ≥ j } = a i =1 ,...,d { x ∈ O N sep | v N ( x − z i ) ≥ j − s f } . Otherwise, the set on the left-hand side contains { x ∈ O N sep | v N ( x − z i ) ≥ α f } , which contains at least two zeros of f .Proof. A verbatim argument in the proof of [1, Lemma 6.6] shows the claim. (cid:3)
Corollary 5.2.
Let f ( T ) be as above and put B = O N [ T ] / ( f ( T )) . Letus write the N -algebra N ′ = B ⊗ O N N as the product N × · · · × N t offinite separable extensions N , . . . , N t of N . If j > s f + α f , then the j -thupper numbering ramification group ([1]) , which we let be denoted by G ( j ) N ,is contained in G N i for any i . Moreover, if N ′ is a field and B coincideswith O N ′ , then j > s f + α f if and only if G ( j ) N ⊆ G N ′ .Proof. Note that the algebra B is finite flat and of relative complete intersec-tion over O N . By the previous lemma, the conductor c ( B ) of the O N -algebra B ([1, Proposition 6.4]) is equal to s f + α f . Thus we have the inequality c ( O N × · · · × O N t ) ≤ c ( B ) = s f + α f by the definition of the conductor and a functoriality of the functor F j defined in [1]. This implies the corollary. (cid:3) Corollary 5.3.
We have the inequality u K ( ζ pn +1 ) /K ≤ − e ( K ( ζ p ) /K ) + e ( n + 1 p − , where e ( K ( ζ p ) /K ) denotes the relative ramification index of K ( ζ p ) over K .Proof. Since the Herbrand function is transitive and the finite extension K ( ζ p ) is tamely ramified over K , it is enough to show the inequality u K ( ζ pn +1 ) /K ( ζ p ) ≤ e ( K ( ζ p ))( n + 1 p − . Put N = K ( ζ p ) and f ( T ) = T p n − ζ p . These satisfy the assumptions ofCorollary 5.2. We have s f = ne ( K ( ζ p )) and α f = e ( K ( ζ p )) / ( p −
1) in thiscase. Hence the corollary follows. (cid:3)
AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 27
Corollary 5.4.
Consider the finite Galois extension F n = K n ( ζ p n +1 ) of K .Then we have the equality u F n /K = 1 + e ( n + 1 p − . Proof.
Applying Corollary 5.2 to the Eisenstein polynomial f ( T ) = T p n − π and N = K shows that j > e ( n + 1 / ( p − G ( j ) K ⊆ G K n . From Corollary 5.3, we see that if j > e ( n + 1 / ( p − G ( j ) K ⊆ G K ( ζ pn +1 ) . Since G F n = G K n ∩ G K ( ζ pn +1 ) , we conclude that j > e ( n + 1 / ( p − G ( j ) K ⊆ G F n . (cid:3) Remark 5.5.
Note that this argument also shows the equality u K n ( ζ pn ) /K = 1 + e ( n + 1 p − . Next we assume that the residue field of N is perfect. For an algebraicextension F of N , we put a jF/N = { x ∈ O F | v N ( x ) ≥ j } . Let Q be a finite Galois extension of N and consider the property( P j ) for any algebraic extension F of N , if there existsan O N -algebra homomorphism O Q → O F / a jF/N ,then there exists an N -algebra injection Q → F for j ∈ R ≥ , as in [10, Proposition 1.5]. Then we have the following propo-sition, which is due to Yoshida. Here we reproduce his proof for the conve-nience of the reader. Proposition 5.6 ([19]) . u Q/N = inf { j ∈ R ≥ | the property ( P j ) holds } . Proof.
By [10, Proposition 1.5 (i)], it is enough to show that the property( P j ) does not hold for j = u Q/N − ( e ′ ) − with an arbitrarily large e ′ > Q is totallyand wildly ramified over N . Take an arbitrarily large integer e ′′ > e ′′ , pe ( Q/N )) = 1. We may also assume that N contains a primitive e ′′ -th root of unity. Set N ′ = N ( π /e ′′ N ) and Q ′ = QN ′ . Note that we have u Q ′ /N = u Q/N by assumption. From this proposition in [10], we see that forsome algebraic extension F of N , there exists an O N -algebra homomorphism O Q ′ → O F / a jF/N for j = u Q/N − e ( Q ′ /N ) − but no N -algebra injection Q ′ → F . Since Q/N is wildly ramified, we see that e ( Q/N ) u Q/N − > e ( Q/N ).Hence we have u Q/N − e ( Q ′ /N ) − > ≥ u N ′ /N and there exists an N -algebra injection N ′ → F also by this proposition. Thus there exists no N -algebra injection Q → F and the property ( P j ) for Q/N does not hold.Since e ( Q ′ /N ) = e ′′ e ( Q/N ), the proposition follows. (cid:3)
We see from Proposition 5.6 that to bound the greatest upper ramificationbreak u L n /K , it is enough to show the following proposition. Proposition 5.7.
Let F be an algebraic extension of K . If j > u ( K, r, n ) and there exists an O K -algebra homomorphism η : O L n → O F / a jF/K , then there exists a K -algebra injection L n → F .Proof. We may assume that F is contained in ¯ K . By assumption, we have j > er/ ( p −
1) and we see that the ideal b F = { x ∈ O F | v K ( x ) > er/ ( p − } contains a jF/K . Thus η induces an O K -algebra homomorphism O L n → O F / b F . Since η also induces an O K -algebra homomorphism O F n → O F / a jF/K and r ≥
1, from Corollary 5.4 and [10, Proposition 1.5] we get a K -linearinjection F n → F . Thus we see that F contains π n and ζ p n +1 . More precisely,we have the following two lemmas. Lemma 5.8.
There exists i ∈ Z such that η ( π n ) ≡ π n ζ ip n mod b F .Proof. Since the map η is O K -linear, the equality η ( π n ) p n = π holds in O F / a jF/K . Set ˆ x to be a lift of η ( π n ) in O F . Then we have v K (ˆ x p n − π ) = p n − X i =0 v K (ˆ x − π n ζ ip n ) ≥ j. Let us apply Lemma 5.1 to f ( T ) = T p n − π ∈ O K [ T ]. Then, with thenotation of the lemma, we have s f = 1 − p n + ne and α f = 1 p n + ep − . Since j − s f > er/ ( p −
1) by assumption, we haveˆ x ≡ π n ζ ip n mod b F for some i . (cid:3) Lemma 5.9.
There exists g ′ ∈ G K such that η ( ζ p n +1 ) ≡ g ′ ( ζ p n +1 ) mod b F .Proof. Set N to be the maximal unramified subextension of K ( ζ p n +1 ) /K .Since the map O K → O N is etale, there exists a K -algebra injection g : N → F such that η ( x ) ≡ g ( x ) mod a jF/K for any x ∈ O N . Let ̟ be auniformizer of K ( ζ p n +1 ) and f ( T ) ∈ O N [ T ] be the Eisenstein polynomial of ̟ over O N . We let f g ( T ) ∈ O N [ T ] denote the conjugate of f by g . Then f g satisfies the conditions of Lemma 5.1. By definition we have s f g = s f and α f g = α f . Since the roots of f g ( T ) are conjugates of ̟ over K ,Lemma 5.1 implies as in the previous lemma that there exists g ′ ∈ G K such AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 29 that g ′ | N = g and η ( ̟ ) ≡ g ′ ( ̟ ) mod a j − s f F/K . Since O K ( ζ pn +1 ) is generatedby ̟ over O N , we see that η ( ζ p n +1 ) ≡ g ′ ( ζ p n +1 ) mod a j − s f F/K .Thus it is enough to check the inequality j − s f > er/ ( p − s f is equal to the valuation v K ( D K ( ζ pn +1 ) /N ) of the different of thetotally ramified Galois extension K ( ζ p n +1 ) /N . To bound this, put G =Gal( K ( ζ p n +1 ) /N ( ζ p )) and e ′ = e ( N ( ζ p ) /N ). We have v K ( τ ( ̟ ) − ̟ )) ≤ v K ( τ ( ζ p n +1 ) − ζ p n +1 )for any τ ∈ G and thus v K ( D K ( ζ pn +1 ) /N ( ζ p ) ) ≤ X τ =1 ∈ G v K ( τ ( ζ p n +1 ) − ζ p n +1 ) ≤ ne. We also have the equality v K ( D N ( ζ p ) /N ) = 1 − /e ′ and hence we get s f = v K ( D K ( ζ pn +1 ) /N ) ≤ − /e ′ + ne. Since e ′ ≤ p −
1, the inequality j − s f > er/ ( p −
1) holds. (cid:3)
Corollary 5.10.
There exists g ∈ G K such that η ( π n ) ≡ g ( π n ) mod b F and η ( ζ p n +1 ) ≡ g ( ζ p n +1 ) mod b F .Proof. Let i ∈ Z and g ′ ∈ G K be as in Lemma 5.8 and Lemma 5.9, re-spectively. Since K n ∩ K ( ζ p n +1 ) = K (see for example [17, Lemma 5.1.2]),we can find an element g ∈ G K such that g ( π n ) = π n ζ ip n and g ( ζ p n +1 ) = g ′ ( ζ p n +1 ). (cid:3) Lemma 5.11.
For m ∈ Z ≥ , set an ideal b ( m ) L n of O L n to be b ( m ) L n = { x ∈ O L n | v K ( x ) > erp m ( p − } and similarly for F . Then the O K -algebra homomorphism η induces an O K -algebra injection η ( m ) : O L n / b ( m ) L n → O F / b ( m ) F for any m .Proof. We may assume that L n is totally ramified over K . We write theEisenstein polynomial of a uniformizer π L n of L n over O K as P ( T ) = T e ′ + c T e ′ − + · · · + c e ′ − T + c e ′ , where e ′ = e ( L n /K ). Then z = η ( π L n ) satisfies P ( z ) = 0 in O F / a jF/K . Letˆ z be a lift of z in O F . Since j >
1, we have v K (ˆ z ) = 1 /e ′ . The condition i > e ( L n ) r/ ( p m ( p − v K (ˆ z i ) > e ( L n ) rp m ( p − · e ′ = erp m ( p − . Thus the claim follows. (cid:3)
Since L n contains F n , we can consider the ring¯ A n,L n ,r + = W n ( O L n / b L n ) / ([ ζ p n ] − r W n ( m L n / b L n )and similarly ¯ A n,F,r + for F . We give these rings structures of Σ-algebrasas follows. The ring ¯ A n,L n ,r + is considered as a Σ-algebra by using thesystem { π n } n ∈ Z ≥ we chose of p -power roots of π , as in the previous section.On the other hand, using g ∈ G K in Corollary 5.10, put ˜ π n = g ( π n ) and˜ ζ p n +1 = g ( ζ p n +1 ). Then we consider the ring ¯ A n,F,r + as a Σ-algebra by usinga system of p -power roots of π containing ˜ π n . We define Fil r and φ r of theserings in the same way as before. Lemma 5.12.
The induced ring homomorphism ¯ η : ¯ A n,L n ,r + → ¯ A n,F,r + is a morphism of the category ′ Mod r,φ/ Σ .Proof. Firstly, we check that ¯ η is Σ-linear. By definition, this homomor-phism commutes with the action of the element u ∈ Σ. To show the com-patibility with the element Y ∈ Σ, let us consider the commutative diagram W n ( O L n / b L n ) η n −−−−→ W n ( O F / b F ) y y ¯ A n,L n ,r + ¯ η −−−−→ ¯ A n,F,r + , where the horizontal arrows are induced by η . Note that we have η n ([ π n ]) =[˜ π n ] and η n ([ ζ p n +1 ]) = [˜ ζ p n +1 ]. Let a ∈ W ( R ) × and v = t/E ([ π ]) ∈ W ( R ) × be as in the previous section. Let a n and v n denote the images of a and v in W n ( O L n / b L n ), respectively. Then the element v n is a solution of theequation E ([ π n ]) v n = 1 + [ ζ p n +1 ] + · · · + [ ζ p n +1 ] p − . Similarly, we define elements ˜ a n and ˜ v n of W n ( O F / b F ) using ˜ π n and ˜ ζ p n +1 .By definition, the element ˜ v n is a solution of the equation E ([˜ π n ])˜ v n = 1 + [˜ ζ p n +1 ] + · · · + [˜ ζ p n +1 ] p − . Now what we have to show is the equality¯ η ( a n v − n E ([ π n ]) p − ) = ˜ a n ˜ v − n E ([˜ π n ]) p − in the ring ¯ A n,F,r + . Since the element a n of W n ( O L n / b L n ) is a linear com-bination of the elements 1 , [ ζ p n +1 ] , . . . , [ ζ p n +1 ] p − over Z , we have ¯ η ( a n ) = ˜ a n in ¯ A n,F,r + . The elements ˜ v n and ¯ η ( v n ) satisfy the same equation in ¯ A n,F,r + .Since these two elements are invertible, we get ¯ η ( v n ) − E ([˜ π n ]) = ˜ v − n E ([˜ π n ])and the equality holds. Since the diagram above is compatible with theFrobenius endomorphisms, we see from the definition that ¯ η also preservesFil r and commutes with φ r of both sides. (cid:3) AMIFICATION BOUND OF TORSION SEMI-STABLE REPRESENTATIONS 31
Thus we get a homomorphism of abelian groups T ∗ crys ,L n ,π n ( M ) → T ∗ crys ,F, ˜ π n ( M ) . Then the following lemma, whose proof is omitted in [3, Subsection 3.13],implies that this homomorphism is an injection. We insert here a proof ofthis lemma for the convenience of the reader.
Lemma 5.13.
The ring homomorphism ¯ η : ¯ A n,L n ,r + → ¯ A n,F,r + is an injec-tion.Proof. Let x = ( x , . . . , x n − ) be an element of W n ( O L n / b L n ) such that( η (0) ( x ) , . . . , η (0) ( x n − )) ∈ ([ ζ p n ] − r W n ( m F / b F ) , where η (0) is as in Lemma 5.11. Suppose that x = · · · = x m − = 0 for some0 ≤ m ≤ n −
1. Let ˆ z i ∈ O F be a lift of η (0) ( x i ). By Lemma 4.6, we have(0 , . . . , , ˆ z m , . . . , ˆ z n − ) = ([ ζ p n ] − r (ˆ y , . . . , ˆ y n − )for some ˆ y , . . . , ˆ y n − ∈ m F . Thus we get ˆ y = · · · = ˆ y m − = 0 and v K (ˆ z m ) >er/ ( p n − − m ( p − x m is contained in theideal b ( n − − m ) L n / b L n and x = ([ ζ p n ] − r (0 , . . . , , y, , . . . ,
0) + (0 , . . . , , x ′ m +1 , . . . , x ′ n − )for some y ∈ m L n / b L n and x ′ m +1 , . . . , x ′ n − ∈ O L n / b L n . Repeating this, wesee that x is zero in ¯ A n,L n ,r + and the lemma follows. (cid:3) Now Corollary 4.13 shows that the abelian group T ∗ crys ,L n ,π n ( M ) has thesame cardinality as T ∗ crys , ¯ K,π n ( M ). This implies that the abelian group T ∗ crys ,F, ˜ π n ( M ) has cardinality no less than T ∗ crys , ¯ K,π n ( M ). Let g ∈ G K be as in Corollary 5.10. Then we have the following lemma. Lemma 5.14.
The G F n -module T ∗ crys , ¯ K, ˜ π n ( M ) is isomorphic to the conju-gate of the G F n -module T ∗ crys , ¯ K,π n ( M ) by the element g .Proof. Let us consider the compositeΣ → ¯ A n,r + g → ¯ A n,r + of the ring homomorphism defined by u [ π n ] and the map induced by g . We can check that this is the natural ring homomorphism defined by u [˜ π n ] as in the proof of Lemma 5.12. Thus we have an isomorphism ofabelian groups Hom Σ ( M, ¯ A n,r + ) → Hom Σ ( M, ¯ A n,r + ) f g ◦ f, where we consider on the ring ¯ A n,r + on the right-hand side the filtered φ r -module structure over Σ defined by ˜ π n . As in the proof of Lemma 5.12, wecan check that this isomorphism induces an injectionHom Σ , Fil r ,φ r ( M, ¯ A n,r + ) → Hom Σ , Fil r ,φ r ( M, ¯ A n,r + ) . This is also an isomorphism, for the map f g − ◦ f defines its inverse. (cid:3) Thus we have T ∗ crys , ¯ K, ˜ π n ( M ) = T ∗ crys , ¯ K,π n ( M ). Since L n is Galois over K , this lemma also shows that the finite Galois extension of F n cut outby the action on T ∗ crys , ¯ K, ˜ π n ( M ) is L n . Hence we see from Corollary 4.13that F contains L n and Proposition 5.7 follows. This concludes the proof ofTheorem 1.1. (cid:3) Proof of Corollary 1.3.
The second assertion follows immediately from The-orem 1.1 and [8, Th´eor`eme 1.1]. As for the first assertion, note that if r = 0then V is unramified and the assertion is trivial. Thus we may assume p ≥ L /p n L → L / L ′ , we may also assume L ′ = p n L . For ˆ M ∈
Mod r,φ,N/S , let us consider the G K -module T ∗ st ,π ( ˆ M ) = Hom S, Fil r ,φ r ,N ( ˆ M , ˆ A st ) . By [17, Theorem 2.3.5], there exists ˆ
M ∈
Mod r,φ,N/S such that the G K -module L is isomorphic to T ∗ st ,π ( ˆ M ). Then we see that the G K -module L /p n L is isomorphic to T ∗ st ,π ( ˆ M /p n ˆ M ) and the assertion follows from Theorem1.1. (cid:3) Remark 5.15.
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