On logarithmic nonabelian Hodge theory of higher level in characteristic <span class="mathjax-formula">$p$</span>

On logarithmic nonabelian Hodge theory of higher level in characteristic p

SACHIOOHKAWA(*)

ABSTRACT- Given a natural numbermand a log smooth integral morphismX!Sof fine log schemes of characteristicp>0with a lifting of its Frobenius pull-back X⁰!S modulo p², we use indexed algebras A^gp_X, B^(m‡1)_X=S of Lorenzon-Montagnon and the sheaf D^(m)_X=S of log differential operators of level m of Berthelot-Montagnon to con- struct an equivalence between the category of certain indexed A^gp_X-modules with D^(m)_X=S-action and the category of certain indexed B^(m‡1)_X=S -modules with Higgs field.

Our result is regarded as a levelmversion of some results of Ogus-Vologodsky and Schepler.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 16H05; 14F30, 32C38.

KEYWORDS. Log geometry, LogD-module, Higgs module, Cartier transform.

1. Introduction

For a projective smooth complex algebraic variety, Simpson [12] estab- lished a correspondence, which is called the Simpson correspondence nowa- days, between local systems and Higgs bundles. In [10], Ogus and Vologodsky studied an analogue of the Simpson correspondence for certain integrable connections or equivalently certain D-modules in positive characteristic. As a natural generalization of their theory, Schepler [11] studied its log version and Gros, Le Stum and QuiroÂs [4] studied its higher level version. The aim of this article is to establish the log and higher level version of the theory of Ogus- Vologodsky.

Let us recall the Ogus-Vologodsky's analogue of the Simpson correspondence in positive characteristic, which is called the global Cartier transform (see Theorem 2.8 of [10]). Let X!S be a smooth morphism of schemes of char- acteristic p>0. Let us denote by X⁰ the pull-back of X!S via the absolute

(*) Indirizzo dell'A.: Sachio Ohkawa, Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan.

E-mail: ohkawa@ms.u-tokyo.ac.jp

Frobenius F_S of S. Denote the relative Frobenius morphism X!X⁰ by F_X=S. LetT_X⁰_=S be the tangent bundle ofX⁰overS,ST_X⁰_=S the symmetric algebra of T_X⁰_=S and Gthe nilpotent divided power envelope of the zero section of the co- tangent bundle ofX⁰=S, so thatO_GˆG:T^ _X⁰_=S. Assume that we are given a lifting of X⁰!Smodulop². Then there exists an equivalence between the category of OX-modules E with integrable connection r equipped with a horizontal OX- linear G-Higgs field u:OG!F_X=SEndOX(E;r) extending the horizontal map c:ST_X⁰_=S !F_X=SEnd_OX(E;r) given by the p-curvature and the category of O_X⁰-modulesE⁰ equipped with an O_X⁰-linearG-Higgs field u:O_G! End_OX0(E⁰).

There are two key technical results for the proof of the global Cartier transform.

One is the fact that the sheafD⁽⁰⁾_X=S of differential operators of level 0 onXis an Azumaya algebra over its center, which is isomorphic to ST_X⁰_=S via the p-cur- vature map. The other is a construction of the splitting module K_X=S for this Azumaya algebra over the scalar extension O_G of ST_X⁰_=S. This means an iso- morphism of O_X-algebras D⁽⁰⁾_X=SSTX0=SO_G!^ End_OG K_X=S

. Then the global Cartier transform can be obtained by the Morita equivalence. Ogus-Vologodsky also constructed a splitting module over the completionS^T_X⁰_=S ofS^:T_X⁰_=S under the assumption of an existence of a modp² lifting ofF_X=S and got an analogous equivalence called the local Cartier transform.

As is mentioned in the first paragraph, the theory of Ogus-Vologodsky has been generalized in (at least) two directions. First, Schepler [11] extended their theory to the case of log schemes. The difficulty for this generalization is that the Azumaya nature of the sheafD⁽⁰⁾_X=Sof the log differential operators of level 0 is no longer true in general. Schepler overcame this difficulty by using Lor- enzon's theory of indexed modules and indexed algebras A^gp_X and B_X=S asso- ciated to a log schemeXand its Frobenius pullbackX⁰!S. Roughly speaking, A^gp_X and B_X=S are the suitable scalar extensions of the structure sheaf OX and O_X⁰ respectively in the case of log schemes. He used the sheaf D~⁽⁰⁾_X=S:ˆ

A^gp_{X OX} D⁽⁰⁾_X=S in place ofD⁽⁰⁾_X=Sand proved the Azumaya nature of D~⁽⁰⁾_X=S over its center. Schepler also generalized the splitting module K_X=Sof Ogus-Vologodsky and got the log global Cartier transform. Second, in [4], Gros, Le Stum and QuiroÂs extended some results in [10] to the case of Berthelot's ring of differ- ential operators of higher level [2]. They proved the Azumaya nature of the sheaf D^(m)_X=S of differential operators of levelm, constructed a splitting module forD^(m)_X=S overS^T_X⁰_=S (hereX⁰denotes the pull-back ofX!Sby the (m‡1)-st iterate of the absolute Frobenius F_S :S!S) under the assumption of an existence of a good lifting of the (m‡1)-st relative Frobenius morphism F_X=S modp², which they call a strong lifting, and proved the local Cartier transform of higher level. They also constructed (but informally) a global splitting module by a gluing argument. But their construction is different from that of Ogus- Vologodsky. It should be remarked here that the sheaf D~⁽⁰⁾_X=S used by Schepler (or more generally the sheafD~^(m)_X=S of log differential operators of higher level) was introduced by Montagnon [9]. She established there the foundations of log

differential operators of higher level and especially obtained the log version of Berthelot's Frobenius descent by using the indexed algebras A^gp_X and B^(m)_X=S, where the latter denotes the higher level version of Lorenzon's B_X=S.

The purpose of this paper is to generalize Schepler's log global Cartier transform to the case of higher level by using the indexed algebras A^gp_X and

B^(m‡1)_X=S of Lorenzon and Montagnon. Our construction is a natural general-

ization of Ogus-Vologodsky and Schepler, but we also need some log differ- ential calculus of higher level which is based on Montagnon's result. We also prove the compatibility of the log global Cartier transform with Montagnon's log Frobenius descent.

Let us describe the content of each section. We work with a log smooth morphismX!Sof fine log schemes in positive characteristic. LetF_X=Sdenote the (m‡1)-st relative FrobeniusX!X⁰. In the second section, we review the theory of indexed modules. In the third section, we construct the log version of the higher curvature map b:T_X⁰_=S!F_X=SD^(m)_X=S, which we call the p^m‡1-curvature map, in Definition 3.10 after reviewing the theory of log differential operators of levelm. In the fourth section, after reviewing the construction and some basic results of in- dexed algebras associated to the log structure, we study the Azumaya nature of D~^(m)_X=S. We prove thatB^(m‡1)_{X=S OX}0 S^:T_X⁰_=Sis identified with the center ofD~^(m)_X=Svia the p^m‡1-curvature map (see Theorem 4.16) and D~^(m)_X=S is an Azumaya algebra over B^(m‡1)_X=S O_X0 S^:T_X⁰_=S (see Corollary 4.2-). We also prove the log Cartier descent theorem of higher level as an application (see Theorem 4.26). In the fifth section, we construct the splitting module K^(m);A_X=S forD~^(m)_X=S overB^(m‡1)_{X=S OX}0 G^_:T_X⁰_=S under the assumption of an existence of a modp²lifting ofX⁰!S(see (18)) and get the log global Cartier transform of higher level by using the indexed variant of the Morita equivalence (see Theorem 5.19). In the final section, we consider the compatibility of the log global Cartier transform with Montagnon's log Frobenius descent. Our new ingredient is to prove the behavior of the splitting moduleK^(m);A_X=S with respect to the Frobenius descent functor of Montagnon (see Theorem 6.10).

As a consequence of Theorem 6.10, we obtain the expected compatibility (see Theorem 6.8).

2. Indexed Azumaya Algebra

In this section, we give a review of the theory of indexed modules and indexed Azumaya algebras developed by ([5], see also [11]) which we will use to construct the log global Cartier transform of higher level. The general theory of indexed modules can be developed on a ringed topos but, for simplicity, we only consider the case of the ringed topos associated to the eÂtale site of a scheme and its structure sheaf. Also, we try to describe several notions more concretely than those given in [5] and [11]. We fix throughout this section a schemeXand an eÂtale sheaf of abelian groupsI.

2.1 ±Indexed module

Let us recall some notions on indexed modules.

DEFINITION2.1. (1) AnI-indexed sheaf onXis a sheaf of sets overI, namely, a map of sheaves F ! I. We denote the mapF ! I byp_F. AnI-indexed sheaf of abelian groups onXis anI-indexed sheafF ! I onXequipped with an addition mapF IF ! FoverI, a unit mapI ! FoverIand an inverse mapF ! Fover I satisfying the usual axioms of abelian groups.

(2) AnI-indexedO_X-module is anI-indexed sheaf of abelian groups equipped with a scalar multiplication map O_X F ! F over I satisfying the usual asso- ciativity, distributivity and unitarity conditions, where O_X F is regarded as a sheaf overI via the compositeO_X F ! F ! I.

(3) AnI-indexedOX-algebra is anI-indexedOX-module Aequipped with an OX-bilinear multiplication mapp:A A ! A over the addition map I I ! I and a global section 1_A of Aover the zero section 0:e! I satisfying the usual associativity and unitarity conditions. We say an I-indexed O_X-algebra A is commutative if the multiplication map p satisfies psˆp where s is the iso- morphismA A ! A Adefined by (a;b)7!(b;a).

(4) For an I-indexed OX-algebraA, anI-indexedA-algebra is anI-indexed O_X-algebraBequipped with a morphismA ! BofI-indexedO_X-algebras.

REMARK2.2. LetAbe anI-indexed sheaf onX. For an eÂtale openUofXand a sectioni2 I(U), we denote byAithe pullbackhUIAwherehU is an eÂtale sheaf onXrepresented byUandhU ! Iis the sectioni. We callA_ithe fiber ofA ! Iat i2 I(U). Note thatA_iis naturally considered as an eÂtale sheaf onU, and more- over, if A is an I-indexed O_X-module, then A_i has an O_U-module structure naturally induced by theI-indexedO_X-module structure onA. IfAis anI-indexed OX-algebra, the multiplication mappofAis equivalent to the following data: for each eÂtale open U of X and sections i;j2 I(U), a morphism of OU-modules p_ij:A_iOUA_j! A_i‡j functorial with respect toi;jsatisfying the obvious condi- tions of associativity and unitarity.

Now we recall the definition ofJ-indexedA-modules.

DEFINITION2.3. LetAbe anI-indexedO_X-algebra. LetJ be an eÂtale sheaf of I-sets, that is, an eÂtale sheaf of sets on X equipped with an I-action map I J ! J;(i;j)7!i‡j. AJ-indexed leftA-module is aJ-indexedOX-moduleE equipped with anO_X-bilinear mapr:A E ! Eover theI-action mapI J ! J satisfying the usual associativity and unitarity conditions. We can similarly define the notion ofJ-indexed rightA-module.

REMARK2.4. LetAbe anI-indexedOX-algebra andJ be an eÂtale sheaf ofI- sets on X. Let E be a J-indexed left A-module. Then the structure morphism

r:A E ! E over I J ! J is equivalent to the following data: for each eÂtale open U of X, and each section (i;j)2 I J, a morphism of O_U-modules r_ij:A_iOUE_j! E_i‡j;ae7!aefunctorial with respect toi;jsatisfying the ob- vious conditions of associativity and unitarity.

Next we recall the definition of tensor products and internal hom objects as an indexed module.

DEFINITION2.5. LetAbe anI-indexedO_X-algebra andJ,Kbe eÂtale sheaves of I-sets onX.

(1) LetEbe aJ-indexed rightA-module andFaK-indexed leftA-module. Let J IKbe theI-setJ K=, whereis the equivalence relation generated by the relation (i‡j;k)(j;i‡k) for i2 I;j2 J;k2 K. Then we define aJ IK- indexed sheaf of abelian groups E _AF (the tensor product of E andF) as the object representing the functor which sends J _IK-indexed sheaf of abelian groups M to the set of biadditive A-balanced morphisms E F ! Mover the natural projectionJ K ! J IK. Concretely this is the eÂtale sheaf onXasso- ciated to the presheaf

U7 ! G

l2J IK(U)

M

(j;k)ˆl

E_j(U)_OX(U)F_k(U)

!

=R

endowed with the natural projection toJ IK, whereRis theOX(U)-submodule generated by

xay xay

x2 E(U); y2 F(U) anda2 A(U) satisfying (p_E(x)‡p_A(a);p_F(y))ˆl

. When A is commutative, thenE _AF naturally forms aJ IK-indexedA-module.

(2) For aK-indexed leftA-module F andW2Hom_I(J;K), we define theJ- indexedA-moduleF(W) by the eÂtale sheafF K;WJwith the second projection and theA-action via the action onF.

(3) LetEbe aJ-indexed leftA-module andF aK-indexed leftA-module. We define the internal hom object ofEandF, which we denote byHom_A(E;F), as the eÂtale sheaf onX

U7 ! G

W2HomI…J;K†(U)

Hom_A Ej_U;Fj_U(W)

endowed with the natural projection toHomI(J;K), where HomAdenotes the set of homomorphism of J-indexed A-modules. When A is commutative, then Hom_A(E;F) naturally forms a HomI(J;K)-indexed A-module. Also, we denote Hom_A(E;E) simply byEnd_A(E).

Finally we recall the local freeness and faithful flatness as an I-indexed A- module.

DEFINITION2.6. LetAbe anI-indexedOX-algebra and letBbe aJ-indexed A-algebra.

(1) We say that anI-indexedA-moduleEis locally free of rankkif eÂtale locally onXthere exist sectionsn₁;. . .;n_kofI ˆ Hom_I(I;I) such thatEis isomorphic to L_k

iˆ1A(ni), whereA(ni) are as in Definition 2.5 (2).

(2) We say thatBis faithfully flat overAif the functorE 7! E ABis exact and faithful.

2.2 ±Indexed Azumaya algebra

The following proposition due to Schepler (see [11]) is an index version of the Morita equivalence.

PROPOSITION2.7. LetAbe a commutativeI-indexedOX-algebra. LetJ be an eÂtale sheaf of I-sets on X and M be a locally free I-indexedA-module of finite rank. We denoteEnd_A(M)byEwhich is anI-indexedO_X-algebra in natural way.

Then the functor E7 !M_AE is an equivalence of categories between the category ofJ-indexedA-modules and the category ofJ-indexed leftE-modules.

PROOF. The quasi-inverse ofE7 !M_AEis given byF7 ! Hom_A(M;F). For

more details, see Theorem 2.2 of [11]. p

Now let us recall the notion on indexed Azumaya algebra.

DEFINITION2.8. Let Abe a commutative I-indexedO_X-algebra andE anI- indexedA-algebra. Then, for a commutativeI-indexedA-algebraB,Esplits overB with splitting module M if there exists anI-indexed locally freeB-module M of finite rank such thatE _AB  EndB(M).Eis an Azumaya algebra overAof rankr² if there exists a faithfully flatI-indexedA-algebraBsuch thatEsplits overBwith splitting moduleMof rankr.

If we know thatEis an Azumaya algebra overA, then we can find a splitting module forEoverAin certain case by the following proposition.

PROPOSITION 2.9. Let A be a commutative I-indexed O_X-algebra and E an Azumaya algebra over Aof rank r². If there exists a locally freeI-indexedA- module M of rank r with a structure ofI-indexed leftE-module compatible with the given I-indexed A-module structure, then E splits over A with splitting module M.

PROOF. See Corollary 2.5 of [11]. p

3. Thep^m‡1-curvature map

From this section, we are mainly concerned with log schemes. Our aim of this section is to construct thep^m‡1-curvature map for a log smooth morphismX!S of fine log schemes defined over a field of positive characteristic, which generalizes the classicalp-curvature map.

3.1 ±Logarithmic differential operators of higher level

In this subsection, we briefly recall the log version of Berthelot's theory of differential operators of higher level which is studied by Montagnon. For more details, see [2] and [9].

Let us start with basics on log schemes [5]. A pre-log structure on a schemeXis a pair (MX;aX) whereMX is a sheaf of monoids on the eÂtale site ofXandaX is a homomorphism from MX to the multiplicative monoid OX. A pre-log structure (M_X;a_X) is a log structure ifa_Xinduces an isomorphism froma_X¹ O_X

toO_X. A log scheme is a pair of a schemeXand a log structure (M_X;a_X). We usually denote a log scheme by a single letter such asX and the log structure ofXbyM_X. For a schemeX, the natural forgetful functor from the category of log structures onXto that of pre-log structures onXhas a left adjoint functor (see (1.3) of [5]). We denote by (M^a_X;a^a_X) the log structure defined by the image of a pre-log structure (M_X;a_X) under this left adjoint functor. We call (M^a_X;a^a) the log structure associated to (M_X;a_X). A monoidP is integral if the natural map fromPto the groupP^gpas- sociated to P is injective. It is fine if it is finitely generated and integral. A log schemeXis fine if, eÂtale locally onX, there exists a fine monoidPwith a morphism P_X ! M_X of monoids such that the log structure of X is isomorphic to the log structure associated to a pre-log structureP_X ! M_X ! O_X. HereP_X denotes the constant eÂtale sheaf onX defined by P. Fine log schemes form a category in an obvious way. This category has all finite projective limits (see (2.8) of [5]). A morphism of log schemesf :X!Yis strict if the natural morphismfMY ! MX

induced by f is an isomorphism, wherefMY denotes the log structure onXas- sociated to a pre-log structuref ¹M_Y ! M_X ! O_X. It is an exact closed immer- sion if it is strict and the underlying morphism of schemes is a closed immersion.

One can define a log smooth (resp. log eÂtale) morphism of fine log schemes in terms of local infinitesimal liftings in the category of fine log schemes (see Subsection 3 of [5]). Let (X;MX)!(S;M_S) be a morphism of log schemes. Then we define the module of log differentialsV¹_{(X;MX)=(S;MS)}by the quotientV¹_X=S O_XZM^gp_X

=N.

Here V¹_X=S is the module of differentials of the underlying morphism of schemes X!SandN is theO_X-submodule locally generated by the sections of the form (daX(a);0) (0;aX(a)a) witha2 MX and (0;1a) witha2Im(fMS! MX).

From now on, we simply denote the module of log differentials ofX!SbyV¹_X=S. If X!S is a log smooth morphism of fine log schemes, then its module of log dif- ferentialsV¹_X=S is a locally freeO_X-module of finite rank.

From now on, all log schemes are assumed to be defined overZ_(p). Let us recall the definition of them-PD structure.

DEFINITION3.1. LetX be a log scheme andIa quasi-coherent ideal ofO_X. A divided power structure of levelm(m-PD structure) onIis a divided power ideal (J;g) ofOX such that

I^(pm)‡pIJI

and the divided power structuregonJis compatible with the unique one onpZ_(p). HereI^(pm)denotes the ideal ofO_Xgenerated byp^m-th powers of all sections ofI. If (J;g) is an m-PD structure on I, we call (I;J;g) an m-PD ideal of O_X and call (X;I;J;g) anm-PD log scheme.

Let (X;I;J;g) be anm-PD log scheme. For each natural numberk, we define the map I! O_X; f7!f^fkg(m) by f^fkg(m):ˆf^rg_q(f^pm) where kˆp^mq‡r and 0r<p^m. These maps satisfy the following formulas (see p. 13 of [9]).

PROPOSITION 3.2. Let(X;I;J;g) be an m-PD log scheme and k;l be positive integers.

(1) For any x2I;x^f0g(m)ˆ1;x^f1g(m)ˆx, and x^fkg(m) 2I:Moreover if kp^m, then x^fkg(m) 2J:

(2) For any x2I and a2 O_X,(ax)^fkg(m)ˆa^kx^fkg(m). (3) For any x;y2I,(x‡y)^fkg(m)= P

k⁰‡k⁰⁰ˆk

kk⁰ x^fk0g(m)y^fk00g(m): (4) For any x2I, q_k!x^fkg(m)ˆx^k.

(5) For any x2I,(x^fkg(m))^flg(m)ˆ q_kl!

(q_k!)^lq_l!x^fklg(m).

In the following, we sometimes denote an element f^fkg(m) simply by f^fkg, if there will be no confusions. For a while, we fix an m-PD fine log scheme (S;a;b;g) (i.e. S is a fine log scheme) on which p is locally nilpotent and a log smooth morphism X!S of fine log schemes. We assume that g extends to X (for definition, see [2] DeÂfinition 1.3.2(1)). Note that g always extends to X in the case bˆ(p) (see [2] DeÂfinition 1.3.2(1)), which is the case of our interest.

To recall the sheaf of log differential operators of higher level, we need the log m-PD envelope. The construction of the logm-PD envelope is the same as the classical case of level 0, which we explain now: Let i:X,!Y be an immersion of fine log schemes overS. EÂtale locally onX, we have a factorizationiˆgi⁰with an exact closed immersion i⁰:X,!Z and a log eÂtale morphism g:Z!Y. Let i⁰⁰:X,!Dbe the usualm-PD envelope ofi⁰(for definition, see [2]), and endowD with the inverse image log structure of Z. Then, since i⁰⁰ satisfies the obvious universal property, it descents to the exact closed immersionX,!P_X;(m)(Y) with

the m-PD structure globally on X. P_X;(m)(Y) is called the logm-PD envelope of i:X,!Y.

Let us consider the diagonal immersionX!X_SX. We simply denote its log m-PD envelope byP_X=S;(m)and the defining ideal ofX,!P_X=S;(m)byI. Then there exists the m-PD-adic filtration I^fng

n2N associated toI (for definition, see [3]

DeÂfinition A.3) which satisfies the following property.

If xis a local section ofI^fng; then x^fkg is inI^fnkg: …1†

LetP_X=S;(m)denote the structure sheaf ofP_X=S;(m). For each natural numbern, we denote by Pⁿ_X=S;(m) the quotient sheaf P_X=S;(m)I^fn‡1g and by Pⁿ_X=S;(m) the closed subscheme ofP_X=S;(m) defined byI^fn‡1g. We have a sequence of surjective homo- morphisms of sheaves

! Pⁿ_X=S;(m) ! Pⁿ_X=S;(m)¹ ! ! P¹_X=S;(m)! P⁰_X=S;(m):

Let p0 and p1 (resp. pⁿ₀ and pⁿ₁) denote the first and second projection P_X=S;(m)!X(resp.Pⁿ_X=S;(m)!X) respectively.

DEFINITION3.3. Letn;mbe natural numbers. The sheaf of differential opera- tors of levelmof ordernis defined by

D^(m)_X=S;n:ˆ Hom_OX(pⁿ₀Pⁿ_X=S;(m);O_X):

The sheaf of differential operators of levelmis defined by D^(m)_X=S:ˆ [

n2N

D^(m)_X=S;n:

REMARK3.4. Since, for any m⁰m, an m-PD ideal can be considered as an m⁰-PD ideal,

D^(m)_{X=S m0} naturally forms an inductive system.

D^(m)_X=S has the (non commutative) ring structure as follows. By using the uni- versality of m-PD envelope, we obtain the canonical homomorphism of OX-alge- bras

d^n;n_m ⁰:P^n‡n_X=S;(m)⁰ ! Pⁿ_X=S;(m)OXPⁿ_X=S;(m)⁰

for each natural numbern;n⁰, which is induced by the projectionX_SX_SX! X_SX to the first and the third factors (for precise definition ofd^n;n_m ⁰, see Sub- section 2.3.2 of [9]). For each F2 D^(m)_X=S;n andC2 D^(m)_X=S;n0, we define the product FC2 D^(m)_X=S;n‡n0by

P^n‡n_X=S;(m)⁰ ƒƒƒ!^dn;n

0

m Pⁿ_X=S;(m)OXPⁿ_X=S;(m)⁰ ƒƒƒ!^idC Pⁿ_X=S;(m)ƒ!^F O_X:

This is well-defined andD^(m)_X=Sforms a sheaf of non commutativeO_X-algebras onX.

REMARK3.5. LetEbe anOX-module. Then a logm-PD stratification onEis a family ofPⁿ_X=S;(m)-linear isomorphismse_n :pⁿ₀ E !^ pⁿ₁ Esatisfying the usual cocycle conditions. As is the same with the classical case, giving a D^(m)_X=S-action on E extending itsO_X-module structure is equivalent to giving a logm-PD stratification onE.

Finally we recall the local description of D^(m)_X=S. Letjdenote the logm-PD en- velopeX,!PX=S;(m)of the diagonalX!XSX. We have an exact sequence

0!j ¹(1‡I)ƒ!^l j ¹MPX=S;(m) ƒ!^j MX !0;

…2†

where l is the restriction of the log structure j ¹(a_PX=S;(m)¹ ):j ¹(P_X=S;(m) )! j ¹(M_PX=S;(m)). For any sectiona2 M_X,p₀(a) andp₁(a) have the same image inM_X. Thus, from the exact sequence (2), there exists a unique sectionm_(m)(a)2j ¹(1‡I) such that p₁(a)ˆp₀(a)l m_(m)(a)

. Log smoothness ofX!Simplies that, eÂtale locally onX, there is a logarithmic system of coordinatesm₁;. . .;m_r2 M^gp_X, that is, a system of sections such that the setfdlogm₁;. . .;dlogm_rgforms a basis of the log differential module V¹_X=S of X over S. We define h^(m)_i :ˆm_(m)(m_i) 1 and h^fkg(m):ˆQ_r

i:ˆ1h^fk_i ^ig(m)for each multi-indexk2N^r.

PROPOSITION-DEFINITION 3.6. We regard Pⁿ_X=S;(m) as an O_X-module via pⁿ₀. Then the setfh^fkg(m)jkj ngforms a local basis ofPⁿ_X=S;(m)overO_X. We denote the dual basis of

h^fkg(m)jkj n by

@_hki(m)jkj n . We also denote h^fkg(m) (resp.

@_hki(m))byh^fkg (resp.@_hki)simply, if there will be no confusions.

PROOF. See Proposition 2.2.1 of [9]. p

PROPOSITION 3.7. Let X!S be a log smooth morphism of fine log schemes.

Assume that we are given a logarithmic system of coordinates m₁;. . .;m_r2 M^gp_X. (1) D^(m)_X=S is locally generated by

@_heii; @_hpeii;. . .; @_hpme_ii1ir as an O_X- algebra.

(2) We have

@_hk⁰_i@_hk⁰⁰_iˆ ^kX^0‡k00

kˆsupfk⁰;k⁰⁰g

k!

(k⁰‡k⁰⁰ k)!(k k⁰)!(k k⁰⁰)!

q_k0!q_k00! q_k! @_hki: In particular,@_hki@_hk⁰_iˆ@_hk⁰_i@_hkiholds.

(3) For any x2 O_X, we have

@_hki:xˆX

ik

ki @_{hk ii}(x)@_hii inD^(m)_X=S:

(4) The natural mapD^(m)_X=S! D^(m_X=S⁰⁾ sends@_hki(m)to_q^q!0!@_hki(m0), where kˆp^mq‡r , k⁰ˆp^m0q⁰‡r⁰with0r<p^mand0r⁰<p^m0.

PROOF. (1) See Proposition 2.3.1 of [9]. (2) See Lemme 2.3.4 of [9]. (3) See (2.5)

of [9]. (4) See (2.6) of [9]. p

We prove the following lemma needed later.

LEMMA3.8. Let X!S be a log smooth morphism of fine log schemes defined overZ=pZ. For any k2N^r, l2Nand1ir, we have

@_hpm‡1ki@_hle

iiˆ@_hpm‡1k‡le_ii:

PROOF. We may assume 1lp^m‡1. Whenkiˆ0, the assertion follows easily from Proposition 3.7 (2). Thus we may also assumeki1. By Proposition 3.7 (2), we have

(|) @_hpm‡1ki@_hle

iiˆX^l

sˆ0

(p^m‡1k_i‡s)!

(l s)!s!(p^m‡1k_i l‡s)!

(pk_i)!q_l!

(pk_i‡q_s)!@_hpm‡1k‡se_ii: We put

A:ˆ (p^m‡1k_i‡s)!

(l s)!s!(p^m‡1k_i l‡s)!2Z andB:ˆ (pk_i)!q_l! (pk_i‡q_s)!2Q:

First, we consider the casesˆl. Then, we have ABˆ(p^m‡1ki‡l)!

l!(p^m‡1k_i)!

(pki)!ql! (pk_i‡q_l)!

ˆY^l

jˆ1

1‡p^m‡1k_i j

Y^ql

jˆ1

1‡pk_i

j ₁

:

Since 1‡^pm‡1_j ^ki21‡pZ_(p) if 1jp^m‡1 1,1‡^p_p^m‡1m‡1^kiˆ1‡k_i if jˆp^m‡1, (1‡^pk_jⁱ) ¹21‡pZ_(p) if 1jp 1 and (1‡^pk_pⁱ) ¹ˆ(1‡k_i) ¹ if jˆp, we have AB21‡pZ_(p) and thus AB1 mod p. Next, we consider the case 0sl 1;lˆp^m‡1. Then, we have

Bˆ (pk_i)!q_l! (pk_i‡q_s)!ˆY^qs

jˆ1

1

(pk_i‡j)p!2pZ_(p):

HenceAB2pZ(p). We thus haveAB0 modp. Finally, we consider the case 0sl 1;0lp^m‡1 1. Then, we have

BˆY^qs

jˆ1

1

(pki‡j)ql!2Z(p):

Let v:Q !Z denote the normalized p-adic valuation. For any n2N, it is

known that (p 1)v(n!)ˆn s(n), wheres(n):ˆP

ja_jifnˆP

ja_jp^j. Thus, we have

(p 1)v(A)ˆs(l s)‡s(s)‡s(p^m‡1ki l‡s) s(p^m‡1ki‡s)

ˆs(l s)‡s(p^m‡1k_i l‡s) s(p^m‡1k_i)

>0:

Hence AB2pZ(p). We thus have AB0 modp. The assertion follows from

these calculations and (|). p

3.2 ±The p^m‡1-curvature map

Throughout this subsection, all the log schemes are assumed to be defined over Z=pZ. Let us introduce some notations. For a log scheme X, F_X denotes the (m‡1)-st iterate of its absolute Frobenius. For a morphism X!S of fine log schemes, we consider the following commutative diagram:

whereX⁰⁰is the fiber product in the category of fine log schemes, and the morphism X!X⁰(denoted byF_X=Sand called the ((m‡1)-st) relative Frobenius morphism) is uniquely determined by the requirement that the morphismF_X=Sis purely in- separable and X⁰!X⁰⁰ is log eÂtale (see Proposition 4.10 of [5]). We denote the compositionX⁰!X⁰⁰!Xbyp_X=S or simply byp. We also denoteX⁰byX^(m‡1), if there is a risk of confusion.

First we prove the log levelmversion of Mochizuki's theorem which is used to construct thep^m‡1-curvature map (see also Proposition 3.2 of [4] and Proposition A.7 of [11]).

THEOREM3.9. Let X!S be a log smooth morphism of fine log schemes. Let P_X=S;(m)(resp. Y) be the log m-PD envelope (resp. the log formal neighborhood) of the diagonal immersion X !X_SX andI (resp. I) its defining ideal. LetP_X=S;(m) denote the structure sheaf of P_X=S;(m). Then there is an isomorphism ofO_X-modules

a:F_X=SV¹_X0=S!I=I^fpm‡1‡1g‡IP_X=S;(m) such that, for anyj2I with imagev2I=I²V¹_X=S,

a(1pv)ˆj^fpm‡1g: …3†

PROOF. First we show that the mapa⁰:I!I=IP_X=S;(m) defined by I ! I=IP_X=S;(m)

j 7 ! j^fpm‡1g

isF_X-linear and zero onI². Ifjandtare local sections ofI, by Proposition 3.2 (3) we have

(j‡t)^fpm‡1gˆj^fpm‡1g‡t^fpm‡1g‡ X

i‡jˆpm‡1 i;j>0

p^m‡1 i

j^figt^fjg:

Since 0<i;j<p^m‡1, we have q_i;q_j <p, where q_i;q_j are as in Subsection 1.3.

Therefore q_i! and q_j! are invertible. From Proposition 3.2 (4) we have j^figt^fjg ˆ(q_i!q_j!) ¹jⁱt^j2IP_X=S;(m). It follows that the last term in the sum is in IP_X=S;(m)and we see the additivity ofa⁰. Similarly, the fact thata⁰isF_X-linear and zero onI²follows from Proposition 3.2 (2), (5). We thus obtain theOX-linear map

a:F_X=SV¹_X0=SF_XV¹_X=S !I=I^fpm‡1‡1g‡IP_X=S;(m)

which satisfies (3). Let us show thatais an isomorphism. Since the assertion is eÂtale local on X, we may assume that we have a logarithmic system of coordinates m₁;. . .;m_r2 M^gp_X. Then the left hand side is isomorphic toL_r

iˆ1O_X…1pdlogm_i†.

On the other hand, by Proposition-Definition 3.6,I=I^fpm‡1‡1g is freely generated by n

h^figj1 jij p^m‡1o

as an O_X-module and the image of IP_X=S;(m) under the mapIP_X=S;(m)!I=I^fpm‡1‡1g is generated byfh_jh^figj0 jij p^m‡1 1;1jrg as an OX-module. Actually I=(I^fpm‡1‡1g‡IP_X=S;(m)) is freely generated by nh^fp_i ^m‡1g1iro

as an OX-module. So the right hand side is isomorphic to L_r

iˆ1O_Xh^fp_i ^m‡1g and, by construction,a sends 1pdlogm_i toh^fp_i ^m‡1g. This com-

pletes the proof. p

Letadenote the map defined by the composite Pⁿ_X=S;(m) ! O_X ! Pⁿ_X=S;(m);

where the first map is the natural projection and the second one is the structural morphismpⁿ₀ . Now, we are ready to define thep^m‡1-curvature map.

DEFINITION3.10. LetX!Sbe a log smooth morphism of fine log schemes. Let TX⁰=S:ˆ HomO_X0(V¹_X0=S;OX⁰) denote the log tangent bundle onX⁰. We define the mapb:T_X⁰_=S!F_X=SD^(m)_X=Sby sendingD2 TX⁰Sto the composition of maps

P_X=S;(m)^pm‡1 ƒƒƒƒƒƒ^{y7!y a(y)}!I=I^fpm‡1g!I=I^fpm‡1‡1g‡IP_X;m

 F_X=SV¹_X0=S ƒƒƒƒƒ!^F

X=S D

O_X; where the second map is the natural projection and the third one is the isomorphism in Theorem 3.9. We call it thep^m‡1-curvature map.

The local description of thep^m‡1-curvature map is the following.

PROPOSITION3.11. Let X!S be a log smooth morphism of fine log schemes.

Assume that we are given a logarithmic system of coordinates m₁;. . .;m_r2 M^gp_X . Letnj⁰_i1iro

denote the dual basis ofpdlogm_i1ir

. Thenbsendsj⁰_i to@_hpm‡1e_ii.

PROOF. We calculate thatb(j⁰_i) sendsh^fkgto 1 ifkˆp^m‡1e_iand 0 otherwise by

construction ofb, thereby completing the proof. p

REMARK3.12. Whenm is equal to 0, ourp^m‡1-curvature map is the usual p- curvature map ([10] Proposition 1.7). If the log structure ofX is trivial, then our p^m‡1-curvature map coincides with thep^m-curvature map studied in [4] section 3.

4. Azumaya algebra property

The goal of this section is the Azumaya algebra property of the indexed version of the sheaf of log differential operators D~^(m)_X=Sˆ A^gp_X OXD^(m)_X=S defined by Mon- tagnon [9]. We also study the Azumaya nature of D~⁽⁰⁾_X(m)=Sˆ B^(m)_X=SOX(m) D⁽⁰⁾_X(m)=S, which is also introduced by Montagnon. At first we give a review of the canonical indexed algebraA^gp_X associated to the log structure ofXintroduced by Lorenzon [5] and Montagnon'sD~^(m)_X=SandD~⁽⁰⁾_X(m)=S.

4.1 ±Indexed algebra associated to a log structure 4.1.1 ± TheI^gp_X-indexed algebraD~^(m)_X=S

First we recall the definition ofA^gp_X (I, 3.1 of [5]). LetXbe a fine log scheme. We consider the extension of sheaves of abelian groups

0 ! O_X ! M^gp_X !^d I^gp_X !0:

HereI^gp_X is the quotient sheafM^gp_X=O_X. We defineA^gp_X as the contracted product M^gp_X ^_OX OX which is the quotient of M^gp_X OX by the equivalence relation (ax;y)(x;ay) wherea;x;yare local sections ofO_X;M^gp_X andOXrespectively. The projectionM^gp_X O_X ! M^gp_X ! I^gp_X induces a mapA^gp_X ! I^gp_X which makesA^gp_X an I^gp_X-indexedO_X-module. For a local sectioniofI^gp_X, the fiberM^gp_X;iofM^gp_X ! I^gp_X ati is anO_X-torsor. This implies that the fiberA^gp_X;iofA^gp_X ! I^gp_X atiis an invertibleO_X- module.A^gp_X has a multiplication map induced by the addition mapM^gp_X M^gp_X ! M^gp_X overI^gp_X I^gp_X ! I^gp_X. HenceA^gp_X forms anI^gp_X-indexedOX-algebra, called the I^gp_X-indexed algebra associated to the log structure.

Next let us recall the definition of the sectione_sofA^gp_X associated to a sections ofM^gp_X. For each eÂtale openUofXand a sections2 M^gp_X(U),strivializes theO_X-

torsorM^gp_X;d(s). Thus it gives a basises:ˆ(s;1) of the invertibleOU-moduleA^gp_X;d(s). Then

e₀ˆ1; e_se_tˆe_s‡t; ae_s ˆe_as fors;t2 M^gp_X(U) anda2 O_X(U).

The construction of A^gp_X is functorial in the following sense. For a morphism f :X!Yof fine log schemes, we have a commutative diagram

where vertical arrows are isomorphisms iff is strict. This induces a commutative diagram

where horizontal arrows are isomorphisms if f is strict. So we get the following proposition.

PROPOSITION4.1. If f :X!Y is a strict morphism of fine log schemes, then A^gp_f :fA^gp_Y ! A^gp_X is an isomorphism ofI^gp_X-indexed algebras.

Next we recall the definition ofI^gp_X-indexed leftD^(m)_X=S-module structure onA^gp_X (Subsection 4.1.1 of [9]). Let be the trivial sheaf of abelian groups on X. W e naturally regardD^(m)_X=Sas a-indexedO_X-algebra andI^gp_X as a sheaf of-sets. Since the natural projections pⁿ_i :Pⁿ_X;(m)!X with iˆ0;1 are strict, we have the iso- morphism ofI^gp_X-indexedO_X-algebras

e_n :pⁿ₁ A^gp_X !^ A_Pⁿ_X=S;(m) ^ pⁿ₀ A^gp_X:

These isomorphisms are compatible with respect tonby construction and satisfy the cocycle condition. Hence we can define an I^gp_X-indexed left D^(m)_X=S-module structure onA^gp_X by

D^(m)_X=S A^gp_X ,! D^(m)_X=Sp₁A^gp_X ƒƒƒ^ide!D^(m)_X=Sp₀A^gp_X ,! HomOX PX=S;(m);OX

A^gp_X OXPX=S;(m)! A^gp_X: By calculation with the sectiones s2 I^gp_X

, one can see that the action ofD^(m)_X=S onA^gp_X satisfies the Leibniz formula (for more details, see Subsection 4.1 of [9]).

Therefore we have the following nontrivial ring structure on D~^(m)_X=S which is a central object in this article.

PROPOSITION-DEFINITION4.2. Let X!S be a log smooth morphism of fine log schemes. Let D~^(m)_X=S denote theI^gp_X-indexedOX-moduleA^gp_X OXD^(m)_X=S. Then there exists a unique I^gp_X-indexed O_X-algebra structure on D~^(m)_X=S such that the maps A^gp_X !D~^(m)_X=S; a7!a1 and D^(m)_X=S!D~^(m)_X=S; P7!1P are homomorphisms and that for any a2 A^gp_X, P2 D^(m)_X=S and k2N^r, we have the relations aPˆ (a1)(1P)and

(1@_hki)(a1)ˆX

ik

k i ( )

(@_{hk ii}:a)@_hii: Heref@_hkigis as in Proposition-Definition3.6.

PROOF. See Subsection 4.1 of [9]. p

REMARK4.3. In [9], Montagnon definesD~^(m)_X=S byA^gp_{X OX;I} D^(m)_X=S;I, whereO_X;I resp:D^(m)_X=S;I

denotes an I-indexed OX-module OX I resp:D^(m)_X=S I with the second projection. So, in Montagnon's definition of D~^(m)_X=S, if we take a section aP2D~^(m)_X=S, thenPmay have a nontrivial index. It seems to us thatPmust have the trivial index. For example, equation (4) in the following Proposition 4.4 should actually be considered as an equation inD~^(m)_X=S. If@_hkihas a nontrivial index, then (4) does not make sense because both sides of the equation must have the same index. If we defineD~^(m)_X=S:ˆ A^gp_X OXD^(m)_X=Sby regardingD^(m)_X=Sas a-indexedO_X-algebra in a natural way, then equation (4) makes sense (both sides have the same indexQ

im_i^ji).

Therefore, it seems to us that the natural definition of D~^(m)_X=S is A^gp_{X OX}D^(m)_X=S. However, this is not serious. Actually, if we replace in [9] A^gp_X OX;ID^(m)_X=S;I by A^gp_X OX D^(m)_X=S (and also modify the definition of B^(m)_X=S andF(E), see Proposition- Definition 4.7 and Subsection 6.1 respectively), then all equations in [9] make sense and all the proofs are correct without any essential changes.

Finally we recall the following formula needed later. Letm₁;. . .;m_r2 M^gp_X be a logarithmic system of coordinates. Fori2 f1;2;. . .;rgand a multi-indexj, we put ui:ˆemi andu^j:ˆQ

iu_i^ji. PROPOSITION4.4. For k2N^r,

@_hki:u^j ˆq_k! j k u^j: …4†

PROOF. See Lemme 4.2.3 of [9]. p

4.1.2 ± TheI^gp_X-indexed algebraD~⁽⁰⁾_X(m)=S

We start with a general theory of logD-modules of higher level. For details in more general settings, see Chapitre 3 of [9]. LetX!Sbe a log smooth morphism of fine log schemes defined overZ=pZ. We consider the following commutative diagram:

Here the vertical two arrows are them-th relative Frobenius ofX!Sdefined in Subsection 3.2 and q0 andq1 (resp. p0 and p1) are the first and second pro- jections. We denote by F the m-th relative Frobenius of X!S by abuse of notation. Then, by the universal property of log 0-PD envelopes, there exists a unique morphism of 0-PD fine log schemes F₄:P_X=S;(0)!P_X(m)=S;(0) which fits into the above diagram.

PROPOSITION4.5. Let X!S be a log smooth morphism of fine log schemes.

(1) There exists F:P_X=S;(m) !P_X^(m)_=S;(0) such that F4 uniquely factors as P_X=S;(0)!P_X=S;(m)!^F P_X(m)=S, where the first map is the natural homo- morphism.

(2) Assume that we are given a logarithmic system of coordinates fm_ig of X!S. Let

h^fkg(m) (resp.

h^0fkg(0) ) denote a basis of P_X=S;(m) (resp.

P_X^(m)_=S;(0)) associated to the basis fdlogmig (resp.fdlogpmig) (see Pro- position3.11), wherepdenotes the natural projection X^(m)!X explained in the beginning of Subsection 3.2. Then F:P_X^(m)_=S;(0)! P_X=S;(m) sends h^0fkg(0) toh^fpmkg(m).

PROOF. (1) see Proposition 3.3.1 of [9]. (2) See (i) of Proposition 3.4.1 of [9]. p Let E be a left D⁽⁰⁾_X(m)=S-module and feng the log 0-stratification on E via the equivalence in Remark 3.5. By endowingFEwith a leftD^(m)_X=S-module structure by pulling backfe_ngviaF, we have a functor

F:

left D⁽⁰⁾_X(m)=S-modules onX^(m)

!

left D^(m)_X=S-modules on X : …5†

REMARK4.6. Let Ebe a leftD⁽⁰⁾_X(m)=S-module. Let

h^fkg(m) and

h^0fkg(0) be as in Proposition 4.5 (2). Let

@_hki (resp.

@⁰_hki ) denote the dual of

h^fkg(m) (resp.

h^0fkg(0) ). Then the D^(m)_X=S-action on FE is characterized by the following for-