Notes on Generalizations of Local Ogus-Vologodsky Correspondence

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1 J. Math. Sc. Uv. Tokyo 22 (2015), Notes o Geeralzatos of Local Ogus-Vologodsky Correspodece By Atsush Shho Abstract. Gve a smooth scheme over Z/p Z wth a lft of relatve Frobeus to Z/p +1 Z, we costruct a fuctor from the category of Hggs modules to that of modules wth tegrable coecto as the composte of the level rasg verse mage fuctors from the category of modules wth tegrable p m -coecto to that of modules wth tegrable p m 1 -coecto for 1 m. I the case m =1, we prove that the level rasg verse mage fuctor s a equvalece whe restrcted to quas-lpotet objects, whch geeralzes a local result of Ogus-Vologodsky. We also prove that the above level rasg verse mage fuctor for a smooth p-adc formal scheme duces a equvalece of Q-learzed categores for geeral m whe restrcted to lpotet objects ( strog sese), uder a strog codto o Frobeus lft. We also prove a smlar result for the category of modules wth tegrable p m -Wtt-coecto. Cotets 1. Modules wth Itegrable p m -Coecto p-adc Dfferetal Operators of Negatve Level The case of p-adc formal schemes The case of schemes over Z/p Z Crystalle property of tegrable p m -coectos Frobeus Descet to the Level Mus Oe A Comparso of de Rham Cohomologes Modules wth Itegrable p m -Wtt-Coecto Mathematcs Subject Classfcato. 12H25, 14F30, 14F40. Key words: Module wth coecto, Hggs module, Frobeus descet, de Rham comohology, module wth Wtt-coecto. 793

2 794 Atsush Shho Itroducto For a proper smooth algebrac varety X over C, the equvalece of the category of modules edowed wth tegrable coecto o X ad the category of Hggs modules o X (wth semstablty ad vashg Cher umber codto) s establshed by Smpso [10]. I search for the aalogue of t chatacterstc p>0, Ogus ad Vologodsky proved [9] smlar equvaleces for a smooth scheme X 1 over a scheme S 1 of characterstc p> 0. Oe of ther results [9, 2.11] s descrbed as follows: Deote the structure morphsm X 1 S 1 by f 1, let F S1 : S 1 S 1 be the absolute Frobeus morphsm, put X (1) 1 := S 1 FS1,S 1 X 1, deote the projecto X (1) 1 S 1 by f (1) 1 ad let F X1 /S 1 : X 1 X (1) 1 be the relatve Frobeus morphsm. Assume that we are gve smooth lfts f 2 : X 2 S 2,f (1) 2 : X (1) 2 S 2 of f 1,f (1) 1 to morphsms of flat Z/p 2 Z-schemes ad a lft F 2 : X 2 X (1) 2 of the morphsm F X1 /S 1 whch s a morphsm over S 2. The there exsts a equvalece betwee the category of quas-lpotet Hggs modules o X (1) 1 ad that of modules wth quas-lpotet tegrable coecto o X 1. (There s also a verso [9, 2.8] whch does ot assume the exstece of f 2 ad F 2. I ths case, the categores they treat are a lttle more restrcted some sese.) The key of ther proof s the Azumaya algebra property of the sheaf D (0) X 1 /S 1 of dfferetal operators of level 0 o X 1 over S 1. Also, the above stuato, oe ca gve a explct descrpto of ths fuctor as the verse mage by dvded Frobeus. (See [9, ], [4, 5.9, 6.5] or Remark 1.12 ths paper.) The purpose of ths paper s to costruct a fuctor from Hggs modules to modules wth tegrable coecto for smooth schemes over some flat Z/p Z-schemes ad study the propertes of ths fuctor ad related fuctors. Let us fx N ad let S +1 be a scheme flat over Z/p +1 Z. Let us put S j := S +1 Z/p j Z (j N, +1), let f 1,F S1 be as above ad for 0 m, let us put X (m) 1 := S 1 F m S1,S 1 X 1, deote the projecto X (m) 1 S 1 by f (m) 1 ad for 1 m, let F (m) X 1 /S 1 : X (m 1) 1 X (m) 1 be the relatve Frobeus morphsm for f (m) 1. Moreover, assume that we are gve a smooth lft f +1 : X +1 S +1 of f 1, smooth lfts f (m) +1 : X(m) +1 S +1 of f (m) 1 (0 m ) wth f (0) 1 = f 1 ad lfts F (m) +1 : X(m 1) +1 X (m) +1 of the morphsms

3 Ogus-Vologodsky Correspodece 795 F (m) X 1 /S 1 (1 m ) whch are morphsms over S +1. f : X S, f (m) : X (m) S (0 m ), F (m) Furthermore, let : X (m 1) X (m) be f +1 Z/p Z,f (m) +1 Z/p Z,F (m) +1 Z/p Z, respectvely. Uder ths assumpto, frst we costruct the fuctor Ψ q :HIG(X () from the category HIG(X () ) q MIC(X ) q ) q of quas-lpotet Hggs modules o X () to the category MIC(X ) q of modules wth quas-lpotet tegrable coecto o X. The costructo of the fuctor Ψ q s doe the followg way: For 0 m, let MIC (m) (X (m) ) q be the category of modules wth quas-lpotet tegrable p m -coecto (also called quas-lpotet tegrable coecto of level m ) o X (m), that s, the category of pars (E, ) cosstg of a O (m) X -module E edowed wth a addtve map : E E Ω 1 wth X (m) /S the property (fe)=f (e)+p m e df (e E,f O (m) X ), the tegrablty ad the quas-lpotece codto. (The oto of p m -coecto s the case λ = p m of the oto of λ-coecto of Smpso [11].) Note that we have HIG(X () ) q =MIC () (X () fuctor Ψ q as the composte of fuctors F (m),,q ) q sce p =0oX (). We costruct the +1 :MIC (m) (X (m) ) q MIC (m 1) (X (m 1) ) q for 1 m, whch are defed as the verse mage by dvded Frobeus (whch we call the level rasg verse mage) assocated to F (m) +1, as the case of [9, ], [4, 5.9]. The frst ave questo mght be whether the fuctor Ψ q s a equvalece or ot. (Note that, sce the sheaf of dfferetal operators of level 0 o X over S does ot seem to have Azumaya algebra property, t would be hard to geeralze the method of Ogus-Vologodsky ths case.) Ufortuately, the fuctors F (m),,q are ot equvaleces (ot full, ot essetally surjectve) for m 2 ad so the fuctor Ψ q s ot a equvalece ether. So a terestg questo would be to costruct ce fuctors from the fuctors F (m),,q +1. As a frst aswer to ths questo, we prove that the fuctor F (1),,q +1 s a equvalece, uder the assumpto that there exsts a closed mmerso S +1 S of S +1 to a p-adc formal scheme S flat over Z p. Ths geeralzes [9, 2.11] uder the exstece of S. To prove ths, we may work locally ad so we may assume the exstece of a smooth lft f : X S of f +1, a smooth lft f (1) : X (1) S of f (1) 1 ad a lft F (1) : X X (1) of the morphsm F (1) X 1 /S 1. I ths stuato, we troduce

4 796 Atsush Shho the oto of the sheaf of p-adc dfferetal operators D ( 1) X (1) /S of level 1 of X (1) over S, whch s a level 1 verso of the sheaf of p-adc dfferetal operators defed ad studed by Berthelot [1], [2]. The we prove that the category MIC (1) (X (1) ) q s equvalet to the category of p -torso quas-lpotet left D ( 1) X (1) /S -modules ad re-defe the fuctor F (1),,q as the level rasg verse mage fuctor from the category of p -torso quaslpotet left D ( 1) X (1) /S -modules to the category of p -torso quas-lpotet left D (0) X/S-modules. Ths s a egatve level verso of the level rasg verse mage fuctor defed [2, 2.2]. (We also gve a defto of the sheaf of p-adc dfferetal operators of level m ad gve the terpretato of (quas-lpotet) modules wth tegrable p m -coecto ad (level rasg) verse mage fuctors of them terms of D-modules for geeral m N.) The, oe ca prove the equvalece of the fuctor F (1),,q by followg the proof of Frobeus descet by Berthelot [2, 2.3]. To expla our ext result, let us fx m N ad assume that we are gve smooth lfts f : X S of f 1, f (1) : X (1) S of f (1) 1 ad a ce lft F : X X (1) of F X1 /S 1 whch s a morphsm over S. The, uder certa assumpto o lpotece codto, we prove that the -th de Rham cohomology of a object the category MIC (m) (X (1) ) of modules wth tegrable p m -coecto o X (1) s somorphc to the -th de Rham cohomology of ts level rasg verse mage by F (whch s a object the category MIC (m 1) (X) of modules wth tegrable p m 1 -coecto o X) whe = 0, ad somorphc modulo torso for geeral. Ths mples that the level rasg verse mage by F duces a fully fathful fuctor betwee a full subcategory of MIC (m) (X (1) ) ad that of MIC (m 1) (X) satsfyg certa lpotece codto ad that t duces a equvalece of Q-learzed categores (aga uder lpotece codto), whch gves a secod aswer to the questo we rased the prevous paragraph. Also, we prove a Wtt verso of the result the prevous paragraph: We troduce the category of modules wth tegrable p m -Wtt coecto MIWC (m) (X 1 )ox 1 for a smooth morphsm X 1 S 1 of chatacterstc p > 0 wth S 1 perfect ad the level rasg verse mage fuctor F : MIWC (m) (X (1) 1 ) MIWC(m 1) (X 1 ) for X 1 S 1 as above ad X (1) 1 := S 1 FS1,S 1 X 1. We prove that the fuctor F duces a fully fathful fuctor betwee a full subcategory of MIWC (m) (X (1) 1 ) ad that of MIWC(m 1) (X 1 )

5 Ogus-Vologodsky Correspodece 797 satsfyg certa lpotece codto ad that t duces a equvalece of Q-learzed categores (aga uder lpotece codto). Ths result has a advatage that we eed o assumpto o the lftablty of objects ad Frobeus morphsms. The cotet of each secto s as follows: I Secto 1, we troduce the oto of modules wth tegrable p m -coecto, (level rasg) verse mage fuctors ad defe the fuctor Ψ (Ψ q ) from the category of (quaslpotet) Hggs modules to the category of modules wth (quas-lpotet) tegrable coecto as the terato of level rasg verse mage fuctors. We also gve a example whch shows that the fuctor Ψ, Ψ q are ot equvaleces. I Secto 2, we troduce the sheaf of p-adc dfferetal operators of egatve level ad prove basc propertes of t. I partcular, we prove the equvalece betwee the category of (quas-lpotet) left D-modules of level m ad the category of (quas-lpotet) modules wth tegrable p m - coecto (m N). I the case of schemes over Z/p Z, we also troduce certa crystalzed categores to descrbe the categores of (quas-lpotet) modules wth tegrable p m -coecto, ad prove certa crystalle property of them. I Secto 3, we prove that the level rasg verse mage fuctor from the category of modules wth tegrable p-coecto to that of modules wth tegrable coecto defed Secto 1 s a equvalece of categores whe restrcted to quas-lpotet objects. I Secto 4, we compare the de Rham cohomology of certa modules wth tegrable p m -coecto over smooth p-adc formal schemes ad that of the pull-back of them by the level rasg verse mage fuctor, ad deduce the fullfathfuless (resp. the equvalece) of the fuctor from the category (resp. the Q-learzed category) of modules wth tegrable p m -coecto to that of modules wth tegrable p m 1 -coecto satsfyg lpotet codtos. I Secto 4, we troduce the oto of modules wth tegrable p m -Wtt coecto ad the level rasg verse mage fuctor from the category of modules wth tegrable p m -Wtt coecto to that of modules wth tegrable p m 1 -Wtt coecto. We compare the de Rham cohomology of certa modules wth tegrable p m -Wtt-coecto ad the de Rham cohomology of the pull-back of them by the level rasg verse mage fuctor, ad deduce the full-fathfuless (resp. the equvalece) of the fuctor from the category (resp. the Q-learzed category) of modules wth te-

6 798 Atsush Shho grable p m -Wtt-coecto to that of modules wth tegrable p m 1 -Wttcoecto satsfyg lpotet codtos. Whle the ma part of ths work s doe, the author s partly supported by Grat--Ad for Youg Scetsts (B) from the Mstry of Educato, Culture, Sports, Scece ad Techology, Japa ad Grat--Ad for Scetfc Research (B) from Japa Socety for the Promoto of Scece. Curretly, he s partly supported by Grat--Ad for Scetfc Research (C) ad Grat--Ad for Scetfc Research (B) Coveto Throughout ths paper, p s a fxed prme umber. Fber products ad tesor products ths paper are p-adcally completed oes, uless otherwse stated. A local secto of a sheaf E o a (formal) scheme meas a secto of E o a affe ope (formal) subscheme. 1. Modules wth Itegrable p m -Coecto I ths secto, we defe the oto of modules wth tegrable p m - coecto. Also, we defe the verse mage fuctor of the categores of modules wth tegrable p m -coecto, ad the level rasg verse mage fuctor from the category of modules wth tegrable p m -coecto to that of modules wth tegrable p m 1 -coecto for a lft of Frobeus morphsm. As a composte of the level rasg verse mage fuctors, we defe the fuctor from the category of Hggs modules to the category of modules wth tegrable coecto, whch s a geeralzato of the verse of local Carter trasform of Ogus-Vologodsky [9, 2.11]. Frst we gve a defto of p m -coecto, whch s a specal case (the case λ = p m ) of the oto of λ-coecto of Smpso [11]. Defto 1.1. Let X S be a smooth morphsm of schemes over Z/p Z or p-adc formal schemes ad let m N. A p m -coecto o a O X -module E s a addtve map : E E OX Ω 1 X/S satsfyg (fe)= f (e) +p m e df for ay local sectos e E,f O X. We refer to a p m -coecto also as a coecto of level m.

7 Ogus-Vologodsky Correspodece 799 Whe we are gve a O X -module wth p m -coecto (E, ), we ca defe the addtve map k : E OX Ω k X/S E O X Ω k+1 X/S whch s characterzed by k (e ω) = (e) ω + p m e dω. Defto 1.2. Wth the otato above, we call (E, ) tegrable f we have 1 = 0. We deote the category of O X -modules wth tegrable p m -coecto by MIC (m) (X). Whe m = 0, the oto of modules wth tegrable p m -coecto s othg but that of modules wth tegrable coecto. I ths case, we deote the category MIC (m) (X) also by MIC(X). Also, whe X, S are schemes over Z/p Z wth m, the oto of modules wth tegrable p m - coecto s othg but that of Hggs modules. I ths case, we deote the category MIC (m) (X) also by HIG(X). Remark 1.3. For a smooth morphsm f : X S of p-adc formal schemes ad N, we deote the full subcategory of MIC (m) (X) cosstg of p -torso objects by MIC (m) (X). If we deote the morphsm f Z/p Z by X S, the drect mage by the caocal closed mmerso X X duces the equvalece of categores MIC (m) = (X ) MIC (m) (X). (1.1) Let us assume gve a commutatve dagram g X Y S T of schemes over Z/p Z or p-adc formal schemes wth smooth vertcal arrows ad ad a object (E, ) MIC (m) (Y ) (where MIC (m) (Y ) s defed for the morphsm Y T ). The we ca edow a structure of a tegrable p m -coecto g o g E by g (fg (e)) = fg ( (e)) + p m g (e) df (e E,f O X ). So we have the verse mage fuctor g :MIC (m) (Y ) MIC (m) (X); (E, ) g (E, ) :=(g E,g ). Remark 1.4. Let us assume gve the commutatve dagram (1.1) of p- adc formal schemes wth smooth vertcal arrows ad let us deote the morphsm g Z/p Z by g : X Y. The the verse mage fuctor g above

8 800 Atsush Shho duces the fuctor g :MIC (m) (Y ) MIC (m) (X), ad t cocdes wth the verse mage fuctor g :MIC (m) (Y ) MIC (m) (X ) assocated to g va the equvaleces MIC (m) = (X ) MIC (m) (X), MIC (m) = (Y ) MIC (m) (Y ) of Remark 1.3. Next we troduce the oto of quas-lpotece. Let X S be a smooth morphsm of schemes over Z/p Z whch admts a local coordate t 1,..., t d. The, for (E, ) MIC (m) (X), we ca wrte as (e) = d =1 θ (e)dt for some addtve maps θ : E E(1 d). The we have 0=( 1 )(e) = <j (θ θ j θ j θ )(e)dt dt j. So we have θ θ j = θ j θ. Therefore, for a = (a 1,..., a d ) N d, the map θ a := d =1 θa s well-defed. Defto 1.5. Wth the above stuato, we call (E, ) quas-lpotet wth respect to (t 1,..., t d ) f, for ay local secto e E, there exsts some N N such that θ a (e) = 0 for ay a N d wth a N. Lemma 1.6. The above defto of quas-lpotece does ot deped o the local coordate (t 1,..., t d ). Proof. Whe m = 0, ths s classcal [3]. Here we prove the lemma the case m>0. (The proof s easer ths case.) Frst, let us ote that, for f O X, we have the equalty θ (fe)dt = (fe)=f (e)+p m edf = (fθ (e)+p m f )dt. t So we have the equalty (1.2) θ f = fθ + p m f. t Now let us take aother local coordate t 1,..., t d, ad wrte as (e) = d =1 θ (e)dt. The we have θ (e)dt = θ (e) t t dt j = ( t,j j t θ (e))dt j. j j

9 Ogus-Vologodsky Correspodece 801 Hece we have θ j = t θ. t j Let us prove that, for ay local secto e E ad for ay a N d, there exst some f a,b O X (b N d, b a ) wth θ a (1.3) (e) = p m( a b ) f a,b θ b (e), b a by ducto o a: Ideed, ths s trvally true whe a = 0. If ths s true for a, wehave θ jθ a t (e) =( θ )( p m( a b ) f a,b θ b )(e) =,b t j b a (p m( a b ) t t f a,b θ b+e (e) j + p m( a b +1) f a,b θ b (e)) (by (1.2)) t ad from ths equato, we see that the clam s true for a + e j. Now let us assume that (E, ) s quas-lpotet wth respect to (t 1,..., t d ), ad take a local secto e E. The there exsts some N N such that θ b (e) = 0 for ay b N d, b N. The, for ay a N d, a N +, we have ether b N or a b for ay b N d. Hece we have ether p m( a b ) =0orθ b (e) = 0 o the rght had sde of (1.3) ad so we have θ a (e) = 0. So we have show that (E, ) s quas-lpotet wth respect to (t 1,..., t d ) ad so we are doe. Remark 1.7. By (1.2), we have θ a (fe)= p m b b f t b θa b (e) 0 b a for e E,f O X, ad we have p m b b f t b pm b b!o X. Hece, f we have θ a (e) = 0 for ay a N d, a N, we have θ a (fe) = 0 for ay a N d, a N + p d. Therefore, to check the quas-lpotece of (E, ) (wth respect to some local coordate t 1,..., t d ), t suffces to check that, for some local geerator e 1,..., e r of E, there exsts some N N such that θ a (e ) = 0 for

10 802 Atsush Shho ay a N d, a N ad 1 r. Also, we ca take N N such that θ a (e) = 0 for ay a N d, a N ad ay local secto e E. Whe a gve morphsm does ot admt a local coordate globally, we defe the oto of quas-lpotece as follows: Defto 1.8. (1) Let X S be a smooth morphsm of schemes over Z/p Z. The a object (E, ) MIC (m) (X) s called quas-lpotet f, locally o X, there exsts a local coordate t 1,..., t d of X over S such that (E, ) s quas-lpotet wth respect to (t 1,..., t d ). (Note that, by Lemma 1.6, ths defto s depedet of the choce of t 1,..., t d.) (2) Let X S be a smooth morphsm of p-adc formal schemes. The a object (E, ) MIC (m) (X) s called quas-lpotet f t s cotaed MIC (m) (X) for some ad the object MIC (m) (X ) (where X := X Z/p Z) correspodg to (E, ) va the equvalece Remark 1.3 s quas-lpotet. We deote the full subcategory of MIC (m) (X) cosstg of quas-lpotet objects by MIC (m) (X) q, ad the case of (2), we deote the category MIC (m) (X) MIC (m) (X) q by MIC (m) (X) q. Next we prove the fuctoralty of quas-lpotece. Proposto 1.9. Let us assume gve a commutatve dagram (1.1) of smooth morphsm of p-adc formal schemes or schemes over Z/p Z wth smooth vertcal arrows. The the verse mage fuctor g :MIC (m) (Y ) MIC (m) (X) duces the fuctor g,q :MIC (m) (Y ) q MIC (m) (X) q, that s, g seds quas-lpotet objects to quas-lpotet objects. Proof. I vew of Remark 1.4, t suffces to prove the case of schemes over Z/p Z. Whe m = 0, the proposto s classcal ([3], [2]). So we may assume m>0. Sce the quas-lpotece s a local property, we may assume that there exsts a local coordate t 1,..., t d (resp. t 1,..., t d )ofx over S (resp. Y over T ). Let us take a object (E, ) MIC (m) (Y ) q ad wrte

11 Ogus-Vologodsky Correspodece 803 the map,g as (e) = j θ j (e)dt j, g (fg (e)) = θ (fg (e))dt. Let us wrte g (dt j )= a jdt. The we have g (fg (e)) = fg ( (e)) + p m g (e) df =,j (a j fg (θ j(e)) + p m g (e) f t )dt ad so we have θ (fg (e)) = j a jfg (θ j f (e)) + pm g (e). Let us prove t that, for ay local sectos e E,f O X ad a N d, there exst some f a,b O X (b N d, b a ) (whch depeds o e, f) wth θ a (fg (e)) = p m( a b ) f a,b g (θ b (1.4) (e)), b a by ducto o a: Ideed, ths s trvally true whe a = 0. If ths s true for a, wehave θ θ a (fg (e)) = θ ( p m( a b ) f a,b g (θ b (e))) b a = (a j p m( a b ) f a,b g (θ b+e j (e)) j,b + p m( a b +1) f a,b g (θ b (e))) t ad from ths equato, we see that the clam s true for a + e. From (1.4), we ca prove the quas-lpotece of (g E,g ) as the proof of Lemma 1.6. So we are doe. Before we defe the level rasg verse mage fuctor, we gve the followg defto to fx the stuato. Defto I ths defto, a scheme flat over Z/p Z meas a p-adc formal scheme flat over Z p. For a, b, c N { } wth a b c, we mea by Hyp(a, b, c) the followg hypothess: We are gve a scheme S a flat over Z/p a Z, ad for j N,j a, S j deotes the scheme S a Z/p j Z. We are also gve a smooth morphsm of fte type f 1 : X 1 S 1, ad let F X1 : X 1 X 1,F S1 :

12 804 Atsush Shho S 1 S 1 be the Frobeus edomorphsm (p-th power edomorphsm). Let us put X (1) 1 := S 1 FS1,S 1 X 1 ad deote the projecto X (1) 1 S 1 by f (1) 1. The the morphsm F X 1 duces the relatve Frobeus morphsm F X1 /S 1 : X 1 X (1) 1. We assume that we are gve a smooth lft f b : X b S b of f 1, a smooth lft f (1) b : X (1) b S b of f (1) 1, ad for j N,j b, deote the morphsm f b Z/p j Z, f (1) b Z/p j Z by f j : X j S j, f (1) j : X (1) j S j, respectvely. Also, we assume that we are gve a lft F c : X c X c (1) of the morphsm F X1 /S 1 whch s a morphsm over S c. For j N,j c, let F j : X j X (1) j be F c Z/p j Z. Fally, whe a = (resp. b =, c = ), we deote S a (resp. f b : X b S b ad f (1) b : X (1) b S b, F c : X c X c (1) ) smply by S (resp. f : X S ad f (1) : X (1) S, F : X X (1) ). Roughly speakg, Hyp(a, b, c) meas that S 1 s lfted to a scheme S a flat over Z/p a Z, f 1 : X 1 S 1 ad f (1) 1 : X (1) 1 S 1 are lfted to morphsms over S b = S a Z/p b Z ad the relatve Frobeus F X1 /S 1 : X 1 X (1) 1 s lfted to a morphsm over S c = S a Z/p c Z. Now we defe the level rasg verse mage fuctor for a lft of Frobeus. Let N ad assume that we are the stuato of Hyp( + 1, + 1, + 1). Whe we work locally, we ca take a local coordate t 1,..., t d of X +1 over S +1 ad a local coordate t 1,..., t d of X(1) +1 over S +1 such that F+1 (t )=tp + pa for some a O X+1. Hece we have F +1 (dt ) = p(tp 1 F +1 :Ω1 X (1) +1 /S +1 dt + da ), that s, the mage of the homomorphsm Ω 1 X +1 /S +1 s cotaed pω 1 X +1 /S +1. So there exsts a uque morphsm F +1 :Ω 1 X (1) /S Ω 1 X /S whch makes the followg dagram commutatve Ω 1 X (1) proj. +1 /S +1 F +1 Ω 1 F X (1) /S Ω 1 X +1 /S +1 p +1 Ω 1 X /S, where proj. deotes the atural projecto ad p deotes the map aturally duced by the multplcato by p o Ω 1 X +1 /S +1. Usg ths, we defe the

13 Ogus-Vologodsky Correspodece 805 level rasg verse mage fuctor (1.5) F +1 :MIC (m) (X (1) ) MIC (m 1) (X ) as follows: a object (E, ) MI C (m) (X (1) ) s set by F+1 to the object (FE,F ), where F s the addtve map characterzed by (F )(ff (e)) = ff +1( (e)) + p m 1 F(e) df for e E,f O X. (Here, by abuse of otato, we deoted the map E Ω 1 F X (1) /S E Ω 1 X /S ; e ω F(e) F +1(ω) also by F +1.) We eed to check that the above defto of fuctor (1.5) s welldefed. Frst ote that the map F s well-defed, because we have for e E,f O X,g O (1) X the equalty (F )(ff (ge)) = ff +1( (ge)) + p m 1 F(ge) df = ff +1(g (e)+p m e dg)+p m 1 F(ge) df = ff(g)f +1( (e)) + p m 1 (ff(e) pf +1(dg)+F(ge) df ) = ff(g)f +1( (e)) + p m 1 F(e) d(ff(g)) =(F )((ff (g))f (e)). Next we check that (F E,F ) sap m 1 -coecto. Ths follows from the fact that, for e F E wth e = f F (e )(f O X,e E) ad f O X, we have the equalty (F )(fe)=(f )( ff (e )) = (ff F +1( (e )) + p m 1 F (e ) d(ff )) = f f F +1( (e )) + f p m 1 F (e ) df + p m 1 f F (e ) df = ff (e)+p m 1 e df. Fally, we eed to check the tegrablty of (F E,F ). Ths s doe as follows: Let us take the local coordate t 1,..., t d of X +1 over S +1 ad the

14 806 Atsush Shho local coordate t 1,..., t d of X(1) +1 over S +1 as above (so that F+1 (t )= t p + pa ), take a local secto e E ad wrte (e) = e dt, (e )= j e j dt j. The tegrablty of (E, ) mples that e j = e j. Let (F ) 1 : FE Ω 1 X /S FE Ω 2 X /S be the morphsm defed by (F ) 1 (g ω) = (F )(g) ω + p m 1 g dω. The our task s to show ((F ) 1 (F ))(ff (e)) = 0 for f O X. Ths s actually calculated as follows: ((F ) 1 (F ))(ff (e)) =(F ) 1 (ff +1( (e)) + p m 1 F(e)df ) =(F ) 1 ( ff(e ) F +1(dt )+p m 1 F(e)df ) = ff +1( (e )) F +1(dt )+ p m 1 F (e )df F +1(dt ) + p m 1 ff (e ) df +1(dt )+p m 1 F +1( (e)) df = f = f,j F +1( (e )) F +1(dt )+p m 1 f F (e j ) (F +1(dt j) F +1(dt )) F (e ) df +1(dt ) + p m 1 f F(e ) d(t p 1 dt + a dt j ) t j j = p m 1 f F(e ) 2 a dt k dt j =0. t k t j j,k So we have checked the tegrablty of (FE,F ) ad so the fuctor (1.5) s well-defed. Also, the stuato of Hyp(,, ), we have the homomorphsm

15 Ogus-Vologodsky Correspodece 807 F :Ω 1 X (1) /S Ω1 X/S whch makes the dagram Ω 1 X (1) /S Ω 1 X (1) /S F Ω 1 X/S p F Ω 1 X/S commutatve, ad usg ths, we ca defe the level rasg verse mage fuctor (1.6) F :MIC (m) (X (1) ) MIC (m 1) (X) the same way. Remark Assume we are the stuato of Hyp(,, ) ad put X (1) := X (1) Z/p Z,X := X Z/p Z. The the fuctor (1.6) duces the fuctor F : MI C (m) (X (1) ) MIC (m 1) (X), ad ths cocdes wth the fuctor (1.5) va the equvaleces MIC (m) (X (1) = ) MIC (m) (X (1) ), MIC (m 1) = (X ) MIC (m 1) (X) of Remark 1.3. Remark Assume that we are the stuato of Hyp(2, 2, 2). The the level rasg verse mage fuctor for m = 1 s wrtte as F2 : HIG(X (1) 1 ) MIC(X 1). Let us see how t s calculated locally. Let us take a local coordate t 1,..., t d of X 2 ad a local coordate t 1,..., t d of X (1) 2 wth F2 (t )=tp + pa. The F 2 :Ω 1 Ω 1 X (1) 1 /S X 1 1 /S 1 s wrtte as F 2(dt )=tp 1 dt + da ad the fuctor F2 s defed by usg t. So we obta the followg expresso of the fuctor F2 : A Hggs module (E,θ)o X (1) 1 of the form θ(e) = d =1 θ (e) dt s set to the tegrable coecto (FX 1 /S 1 E, ) such that, f we wrte = d =1 dt,wehave (1 e) =t p 1 θ (e)+ d j=1 a j t θ j (e). Let ι :HIG(X (1) 1 ) HIG(X(1) 1 ) be the fuctor (E,θ) (E, θ). The, by the above expresso, we see that the fuctor F2 ι cocdes wth a specal case of the fuctor defed [4, 5.8] (the case m = 0 the otato of [4])

16 808 Atsush Shho for quas-lpotet objects. (The uderlyg sheaf F1 E s globally the same as the mage of the fuctor [4, 5.8], ad the coectos cocde because they cocde locally.) Hece, by [4, 6.5], t cocdes wth the fuctor [9, 2.11] for quas-lpotet objects. We have the fuctoralty of quas-lpotece wth respect to level rasg verse mage fuctors, as follows: Proposto Assume that we are the stuato of Hyp( + 1,+1,+1)( N) (resp. Hyp(,, )). The the level rasg verse mage fuctor F+1 : MI C (m) (X (1) ) MIC (m 1) (X ) (resp. F : MI C (m) (X (1) ) MIC (m 1) (X)) duces the fuctor F,q +1 : MIC (m) (X (1) ) q MIC (m 1) (X ) q (resp. F,q :MIC (m) (X (1) ) q MIC (m 1) (X) q ), that s, F+1 (resp. F ) seds quas-lpotet objects to quas-lpotet objects. Proof. I vew of Remark 1.11, t suffces to prove the proposto for F+1. I the case m = = 1, the fuctor F 2 ι (ι s as Remark 1.12) cocdes wth the fuctor [9, 2.11]. Hece t seds quas-lpotet objects to quas-lpotet objects. Sce ι duces a auto-equvalece of MIC (1) (X (1) 1 )q, we see that F2 seds quas-lpotet objects to quaslpotet objects. Next, let us prove the proposto the case m = 1 ad geeral, by ducto o. Let us take a object (E, ) MIC (1) (X (1) ). The we have the exact sequece 0 (pe, pe ) (E, ) (E/pE, ) 0, where s the p-coecto o E/pE duced by. Sce F : X X (1) s fte flat, the above exact sequece duces the followg exact sequece: 0 F +1(pE, pe ) F +1(E, ) F +1(E/pE, ) 0. The, sce F+1 (pe, pe) = F(pE, pe ) ad F+1 (E/pE, ) = F2 (E/pE, ), they are quas-lpotet by ducto hypothess. The, f we work locally, take a local coordate t 1,..., t d of X +1 ad wrte the coecto o F+1 (E, ) ase θ(e) dt (e F+1 E), there exsts some N N such that θ a (e) s zero F+1 (E/pE) for ay a Nd wth a N

17 Ogus-Vologodsky Correspodece 809 ad ay local secto e F+1 E, by Remark 1.7. The, sce θa (e) s cotaed F+1 (pe), there exsts some M N such that θa+b (e) = 0 for ay b N d wth b M ad ay local secto e F+1 E, aga by Remark 1.7. Hece F+1 (E, ) s also quas-lpotet, as desred. Fally we prove the proposto the case m 2. Let us take a local coordate t 1,..., t d of X +1, a local coordate t 1,..., t d of X(1) +1 wth F+1 (t ) = tp + pa. Take a object (E, ) MIC (m) (X (1) ) ad wrte (e) := θ (e)dt, F (ff (e)) = θ(ff (e))dt. The we ca prove that, for ay local sectos e E,f O X ad a N d, there exst some f a,b O X (b N d, b a ) (whch depeds o e, f) wth θ a (ff +1(e)) = p (m 1)( a b ) f a,b F +1(θ b (1.7) (e)), b a the same way as the proof of Proposto 1.9. From ths we see the quaslpotece of F +1 (E, ) =(F E,F ) aga the same way as the proof of Proposto 1.9. Remark I the above proof, we used the results [9]. Later, we gve aother proof of Proposto 1.13 whch does ot use ay results [9] uder a slghtly stroger hypothess Hyp(,+1,+1) or Hyp(,, ). Now we defe a fuctor from the category of (quas-lpotet) Hggs modules to the category of modules wth (quas-lpotet) tegrable coecto as a composte of level rasg verse mage fuctors. Let us cosder the followg hypothess. Hypothess Let us fx N ad let S +1 be a scheme flat over Z/p +1 Z. For j N,j + 1, let us put S j := S +1 Z/p j Z. Let f 1 : X 1 S 1 be a smooth morphsm ad let F S1 : S 1 S 1 be the Frobeus edomorphsm. For 0 m, let us put X (m) 1 := S 1 F m S1,S 1 X 1, deote the projecto X (m) 1 S 1 by f (m) 1 ad for 1 m, let F (m) X 1 /S 1 : X (m 1) 1 X (m) 1 be the relatve Frobeus morphsm for f (m 1) 1. Assume that we are gve a smooth lft f +1 : X +1 S +1 of f 1, smooth lfts f (m) +1 : X(m) +1 S +1 of f (m) 1 (0 m ) wth f (0) +1 = f +1 ad lfts F (m) +1 : X(m 1) +1 X (m) (m) +1 of the morphsm F X 1 /S 1 (1

18 810 Atsush Shho m ) whch are morphsms over S +1. Fally, let f : X S, : X (m) S, F (m) : X (m 1) X (m) f (m) Z/p Z,F (m) +1 Z/p Z, respectvely. The we defe the fuctor as follows: be f +1 Z/p Z,f (m) +1 Defto Assume that we are the stuato of Hypothess The we defe the fuctors Ψ:HIG(X () ) MIC(X ), Ψ q :HIG(X () ) q MIC(X ) q as the composte F (1), +1 F (2), (), +1 F +1,F(1),,q +1 F (2),,q +1 F (),,q +1 of level rasg verse mage fuctors F (m), +1 :MIC (m) (X (m) +1 :MIC (m) (X (m) F (m),,q respectvely. ) MIC (k) (X (m 1) ) (1 m ), ) q MIC (m 1) (X (m 1) ) q (1 m ), Sce the morphsms F (m) are fte flat, we see that the fuctors Ψ, Ψ q are exact ad fathful. However, we see the followg example that the fuctors Ψ, Ψ q are ot so good as oe mght expect. Example I ths example, let us put S +1 =SpecZ/p +1 Z ad let S j := S +1 Z/p j Z =SpecZ/p j Z,X j := Spec (Z/p j Z)[t ±1 ] for j N,j + 1. Also, put X (m) j := Spec (Z/p j Z)[t ±1 ] for all m N,j N,j + 1 ad let F (m) j : X (m 1) j X (m) j be the morphsm defed by t t p. The F (m), +1 : Ω 1 Ω 1 seds f(t)t 1 dt to X (m) /S X (m 1) /S f(t p )t p t p 1 dt = f(t p )t 1 dt ad the level rasg verse mage fuctor F (m), +1 :MIC (m) (X (m) ) MIC (m 1) (X (m 1) ) s defed as the pull-back by F (m), +1. For m N ad f(t) Γ(X (m), O (m) X ), we defe the p m -coecto, f(t) )by f(t) = p m d + f(t)t 1 dt. It s locally free of rak 1. Sce (O X (m) ay locally free sheaf of rak 1 o X (m) whch s locally free of rak 1 has the form (O X (m) s free, ay p m -coecto o X (m), f(t) ) for some f(t). For

19 Ogus-Vologodsky Correspodece 811 a p m -coecto (O (m) X, f(t) ), the p m 1 -coecto F (m), s equal to (O (m 1) X prevous paragraph. +1 (O X (m), f(t) ), f(t p )) thaks to the descrpto of F (m), +1 gve the Let us make some more calculato o the p m -coecto (O (m) X It s easy to see that we have a somorphsm (O (m) X f ad oly f (O (m) X elemet. Sce we have, f(t) ) s geerated as O X (m), f(t) )., f(t) ) = (O (m) X, 0 ) -module by a horzotal f(t) (g(t)) = p m dg dt dt + gft 1 dt = g(p m 1 dg tg dt + f)t 1 dt, we see that (O X (m), f(t) ) s somorphc to (O (m) X, 0 ) f ad oly f there exsts a elemet g Γ(X (m) 1 dg, O ) wth f = p m tg. I f g s a X (m) dt ), t has the form g = c(t N + ph 1 ) for some c elemet Γ(X (m), O X (m) (Z/p Z), N Z ad h 1 Γ(X (m), O X (m)), ad ths case, g 1 has the form c 1 (t N + ph 2 ) for some elemet h 2 Γ(X (m) p m 1 dg tg dt = pm t(t N + ph 2 )(Nt N 1 + p dh 1 dt ) = p m N + p m+1 h(t) for some h Γ(X (m), O (m) X f(t) ) s somorphc to (O (m) X, O (m) X ). The we have ). Therefore, we have show that f (O (m) X,, 0 ), f has the form p m N + p m+1 h(t). Now, to vestgate the fuctor F (1), +1 : MI C (1) (X (1) frst let us cosder the p-coecto (O (1) X ot exst N N,h Γ(X (1) ) MIC(X ),, 1 ). The, sce there does, O (1) X ) wth 1 = pn + p 2 h, t s ot somorphc to (O (1) X, 0 ). O the other had, we see that the coecto F (1), +1 (O X (1), 1 ) = (O X, 1 ) s somorphc to F (1), +1 (O X (1), 0 ) = (O X, 0 ) because we have 1 = tg 1 dg dt whe g = t 1. So we ca coclude that the fuctor F (1), +1 s ot full. Secodly, let us cosder the coecto (O X, t ). If t s cotaed the essetal mage of F (1), +1, we should have (O X, t ) = F (1), +1 (O X (1), f(t) )=(O X, f(t p )) for some f(t). The we have f(t p ) t = N + ph for some N Z ad h Γ(X, O X ), but t s mpossble. Hece we see that the fuctor F (1), +1 s ot essetally surjectve.

20 812 Atsush Shho Next, let us vestgate the fuctors F (m), +1 : MI C (m) (X (m) ) MIC (m 1) (X (m 1) ), F (m),,q +1 :MIC (m) (X (m) ) q MIC (m 1) (X (m 1) ) q for m 2. Frst let us cosder the p m -coecto (O X (m), p m 1). We, 0 ), see as the prevous paragraph that t s ot somorphc to (O (m) X ad that F (m), +1 (O X (m), p m 1) = (O (m 1) X, p m 1) s somorphc to F (m), +1 (O X (m), 0 ) = (O (m 1) X, 0 ). If we put p m 1(e) = (e)dt, we ca see easly by ducto that l (1)=( l 1 =0 (pm 1 p m ))/t l. By ths ad Remark 1.7, we see that (O (m) X, p m 1) s quas-lpotet. So the fuctors F (m), +1, F (m),,q +1 are ot full. Secodly, let us cosder the co-, pt ). If t s cotaed the essetal mage of F (m), +1,, pt ) = F (m), ecto (O X (m 1) we should have (O (m 1) X for some f(t). +1 (O X (m), f(t) )=(O (m 1) X, f(t p )) The we have f(t p ) pt = p m 1 N + p m h for some N N ad h Γ(X (m 1), O (m 1) X ), but t s mpossble. Also, f we put pt (e) = (e)dt, we ca see easly by ducto that l (1) = p l. By ths ad Remark 1.7, we see that (O (m 1) X, pt ) s quas-lpotet. Hece the fuctors F (m), +1, F (m),,q +1 are ot essetally surjectve. I cocluso, Ψ s ot full, ot essetally surjectve for ay m 1, ad Ψ q s ot full, ot essetally surjectve for ay m 2. I vew of the above example, we would lke to ask the followg questo. Questo Is t possble to costruct some ce fuctor (a fully fathful fuctor or a equvalece) from the fuctors F (m), +1,F(m),,q +1, possbly uder some more assumpto? Several aswers to ths questo wll be gve Sectos 3, 4 ad p-adc Dfferetal Operators of Negatve Level I ths secto, frst we troduce the sheaf of p-adc dfferetal operators of level m (m N), whch s a egatve level verso of the sheaf of p- adc dfferetal operators of level m defed by Berthelot, for a smooth morphsm of p-adc formal schemes flat over Z p. We prove the equvalece of the oto of left D-modules ths sese ad that of modules wth

21 Ogus-Vologodsky Correspodece 813 tegrable p m -coecto. We also defe the verse mage fuctors ad the level rasg verse mage fuctors for left D-modules, whch are compatble wth the correspodg oto for modules wth tegrable p m -coecto over p-adc formal schemes. The defto of the sheaf of p-adc dfferetal operators of level m (m N) s possble oly for smooth morphsms of p-adc formal schemes, because we use the formal blow-up wth respect to a deal cotag p m the defto. I the case of smooth morphsms X S of schemes flat over Z/p Z, we gve a smlar descrpto by cosderg all the local lfts of X to smooth p-adc formal scheme ad cosder the crystalzed category of D-modules. We also cosder a varat of the crystalzed category of D-modules, whch s also related to the category of modules wth tegrable p m -coecto. As a cosequece, we prove certa crystalle property for the category of modules wth tegrable p m -coecto: Whe f : X S s a smooth morphsm of flat Z/p Z-schemes ad f we deote the morphsm f Z/pZ by X 1 S 1, we kow that the category MIC(X ) q, whch s equvalet to the category of crystals o the crystalle ste (X 1 /S ) crys, depeds oly o the dagram X 1 S 1 S. We prove here smlar results for the categores of modules wth tegrable p m -coecto, although the result the case m>0 s weaker tha that the case m = The case of p-adc formal schemes Let S be a p-adc formal scheme flat over Spf Z p ad let X be a p-adc formal scheme smooth over S. For a postve teger r, we deote the r-fold fber product of X over S by X r. For postve tegers m, r, let T X,( m) (r) be the formal blow-up of X r+1 alog the deal p m O X r+1 + Ker (r), where (r) : O X r+1 O X deotes the homomorphsm duced by the dagoal map (r) :X X r+1. Let T X,( m) (r) be the ope formal subscheme of T X,( m) (r) defed by T X,( m) (r) :={x T X,( m) (r) p m O T X,( m) (r),x =((pm O X r+1 + Ker (r) )O T X,( m) (r) ) x}. The, sce we have (p m O X r+1 + Ker (r) ) X = p m O X, the dagoal map (r) factors through a morphsm (r) : X T X,( m) (r) by the uversalty of formal blow-up. Let us put I X,( m) (r) := Ker (r). Let

22 814 Atsush Shho (P X,( m) (r), I X,( m) (r)) be the PD-evelope of O TX,( m) (r) wth respect to the deal I X,( m) (r), ad let us put P X,( m) (r) :=Spf OX P X,( m) (r). Also, for k N, let PX,( m) k (r),pk X,( m) (r) bep X,( m)(r),p X,( m) (r) modulo I X,( m) (r) [k+1]. I the case r = 1, we drop the symbol (r) from the otato. Note that P X,( m) admts two O X -algebra (O X -module) structure duced by the 0-th ad 1-st projecto X 2 X, whch we call the left O X -algebra (O X -module) structure ad the rght O X -algebra (O X -module) structure, respectvely. Note also that, for m m, we have the caocal morphsm P X,( m ) P X,( m). Locally, P X,( m) (r) s descrbed the followg way. Assume that X admts a local parameter t 1,..., t d over S. The, f we deote the q-th projecto X r+1 X by π (0 q r) ad f we put τ,q := πq+1 t πqt, Ker (r) s geerated by τ,q s (1 d, 0 q r 1) ad we have T X,( m) (r) = Spf OX O X {τ,q /p m },q, where { } meas the p-adcally completed polyomal algebra. So we have P X,( m) (r) =O X τ,q /p m,q, where meas the p-adcally completed PD-polyomal algebra. We see easly that the detty map X r+r +1 X r+1 X X r +1 aturally duces the somorphsm P X,( m) (r) OX P X,( m) (r ) P X,( m) (r + r ) ad the local stuato, the elemet τ,q /p m 1 (resp. 1 τ,q /p m )o the left had sde correspods to the elemet τ,q /p m (resp. τ,q+r /p m )o the rght had sde. The, the projecto X 3 X 2 to the (0, 2)-th factor duces the homomorphsm δ : P X,( m) P X,( m) (2) = P X,( m) OX P X,( m) wth δ(τ /p m )=τ /p m 1+1 τ /p m (here we deoted τ,0 smply by τ ) ad so t duces the homomorphsm δ k,k : P k+k X,( m) P k X,( m) O X P k X,( m). Usg these, we defe the sheaf of p-adc dfferetal operators of egatve level as follows: Defto 2.1. Let X, S be as above. The we defe the sheaf of p-adc dfferetal operators of level m ad order k by D ( m) X/S,k

23 Ogus-Vologodsky Correspodece 815 D ( m) X/S,k := Hom O X (PX,( m) k, O X) ad the sheaf D ( m) X/S of p-adc dfferetal operators of level m by D ( m) X/S := k=0 D( m) X/S,k. We defe the product D ( m) X/S,k D( m) X/S,k D ( m) X/S,k+k by sedg (P, P ) to the homomorphsm P k+k δ k,k X,( m) PX,( m) k O X PX,( m) k d P PX,( m) k P O X. By defto, D ( m) X/S also admts two O X-module structures, whch are defed as the multplcato by the elemets D ( m) X/S,0 = O X from left ad from rght. We call these the left ad the rght O X -module strucrure of D ( m) X/S. Note that P D( m) X/S,k acts o O X as the composte O X P k X,( m) P O X (where the frst map s defed by f 1 f), ad ths defes the acto of D ( m) X/S o O X. For m m, the caocal map P X,( m ) P X,( m) duces the homomorphsm of rgs ρ m, m : D ( m) X/S D ( m ) X/S. Assume that X admts a local parameter t 1,..., t d over S ad put τ := 1 t t 1 O X 2. The, as we saw before, we have P X,( m) = O X τ /p m ad so PX,( m) k admts a bass {(τ/pm ) [l] } l k as O X -module. (Here ad after, we use mult-dex otato.) We deote the dual bass of t D ( m) X/S,k by { l } l k. Whe l = (0,..., 1,..., 0) (1 s placed the -th etry), l s deoted also by. Whe we would lke to clarfy the level, we deote the elemet l by l m. Sce the caocal map P X,( m ) P X,( m) seds (τ/p m ) [l] to p (m m ) l (τ/p m ) [l], we have ρ m, m( l m )= p (m m ) l l m. We prove some formulas whch are the aalogues of the oes [1, 2.2.4]: Proposto 2.2. Wth the above otato, we have the followg: (1) For f O X, 1 f = l k l (f)(τ/p m ) [l] P k X,( m).

24 816 Atsush Shho (2) l (t )=l! (3) l l = l+l. ( ) p m l t l. l (4) For f O X, k f = k +k =k ( ) k k k (f) k. Proof. (1) s mmedate from defto. By lookg at the coeffcet of (τ/p m ) [l] of 1 t =(t + p m (τ/p m )), we obta (2). From the equalty ( l l )((τ/p m ) [] ) =( l (d l )δ l, l )((τ/p m ) [] ) =( l (d l ))( (τ/p m ) [a] (τ/p m ) [b] )= a+b= we see the asserto (3). From the equalty ( k f)((τ/p m ) [] )= k ((1 f)(τ/p m ) [] ) we see the asserto (4). = k ( l = k ( l { 1, f = l + l, 0, otherwse, l (f)(τ/p m ) [l] (τ/p m ) [] ) ( ) l + l (f)(τ/p m ) [l+] ), l Remark 2.3. Let D X/S be the formal scheme verso of the sheaf of usual dfferetal operators ad let us take a local bass { [l] } l N d of D X/S, whch ca be defed the same way as { l } above. The O X admts the atural acto of D X/S ad we see, for l N d, m N ad f O X, the equaltes l m (f) =p m l l 0 (f) =l!p m l [l] (f). I partcular, we have l m (f) 0as l. Next we defe the oto of ( m)-pd-stratfcato ad compare t wth the oto of left D ( m) X/S -module.

25 Ogus-Vologodsky Correspodece 817 Defto 2.4. A ( m)-pd-stratfcato o a O X -module E s a compatble famly of PX,( m) k -lear somorphsms {ɛ k : PX,( m) k O X E = E OX PX,( m) k } k wth ɛ 0 = d such that the followg dagram s commutatve for ay k, k N : PX,( m) k O X PX,( m) k O X E d ɛk δ k,k, (ɛ k+k ) P k X,( m) O X E OX P k X,( m) ɛ k d E OX P k X,( m) O X P k X,( m). The codtos put o {ɛ k } k the above defto s called the cocycle codto. It s easy to see that the cocycle codto s equvalet to the codto q02 k (ɛ k)=q01 k (ɛ k) q12 k (ɛ k) for k N, where qj k deotes the homomorphsm PX,( m) k P X,( m) k (2) duced by the (, j)-th projecto X 3 X 2. We have the followg equvalece, whch s a aalogue of [1, 2.3.2]: Proposto 2.5. equvalet. For a O X -module E, the followg three data are (a) A left D ( m) X/S -module structure o E whch exteds the gve O X- module structure. (b) A compatble famly of O X -lear homomorphsms {θ k : E E OX PX,( m) k } k (where we regard E OX PX,( m) k as O X-module by usg the rght O X -module structure of PX,( m) k ) wth θ 0 =dsuch that the followg dagram s commutatve for ay k, k N : (2.1) E OX P k+k θ k+k E X,( m) d δ k,k E OX PX,( m) k O X P k θ k d θ k E OX PX,( m) k. X,( m) (c) A ( m)-pd-stratfcato {ɛ k } k o E.

26 818 Atsush Shho Proof. Sce the proof s detcal wth the classcal case, we oly gve a bref sketch. The data (a) s equvalet to a compatble famly of homomorphsms µ k : D ( m) X/S,k O X E Esatsfyg the codto comg from the product structure of D ( m) X/S, ad µ k s duce the homomorphsms θ k : E Hom OX (D ( m) X/S,k, E) =E O X PX,( m) k (k N) whch satsfy the codtos (b). So the data (a) gves the data (b), ad we see easly that they are fact equvalet. Whe we are gve the data (b), we obta the PX,( m) k -lear homomorphsm ɛ k : PX,( m) k O X E E OX PX,( m) k by takg Pk X,( m) -learzato of θ k. Sce ɛ k (1 x) =θ k (x) s wrtte locally as l k l (x) (τ/p m ) [l], we see that ɛ k s actually a somorphsm because the verse of t s gve locally by x 1 l k ( 1) l (τ/p m ) [l] l (x). The cocycle codto for {ɛ k } k follows from the commutatve dagram (2.1) for {θ k } k ad so the data (b) gves the data (c). Aga we see easly that they are fact equvalet. Next we relate the oto of left D ( m) X/S -modules ad that of modules wth p m -coecto. Let X S be as above. Recall that a p m -coecto o a O X -module E s a addtve map : E E OX Ω 1 X/S satsfyg (fe)=f (e) +p m e df (e E,f O X ). To gve aother descrpto of p m -coecto, let us put JX/S 1 := Ker(P1 X,( m) O X). The we have a atural map α :Ω 1 X/S J X/S 1 duced by the map P1 X,(0) P X,( m) 1, ad locally α s gve by dt = τ p m (τ /p m ). So α s jectve ad the mage s equal to p m JX/S 1. Hece we have the uque somorphsm β :Ω 1 = X/S JX/S 1 satsfyg pm β = α. Va the detfcato by β, ap m - coecto o E s equvalet to a addtve map : E E OX JX/S 1 satsfyg (fe)=f (e)+e df for ay e E,f O X. (Atteto: the elemet df JX/S 1 here s the elemet 1 f f 1 P1 X,( m), ot the elemet β(1 f f 1).) The followg proposto s the aalogue of [3, 2.9]. Proposto 2.6. Let X S be as above. For a O X -module E, the followg data are equvalet: (a) A p m -coecto : E E OX JX/S 1 o E.

27 Ogus-Vologodsky Correspodece 819 (b) A P 1 X,( m) -lear somorphsm ɛ 1 : P 1 X,( m) O X E = E OX P 1 X,( m) whch s equal to detty modulo J 1 X/S. Proof. Sce the proof s aga detcal wth [3, 2.9], we oly gve a bref sketch. Frst assume that we are gve the somorphsm ɛ 1 as (b). The, f we defe : E E OX JX/S 1 by (e) :=ɛ(1 e) e 1, t gves a p m -coecto. Coversely, f we are gve a p m -coecto : E E OX JX/S 1, let us defe the P1 X,( m) -lear homomorphsm ɛ 1 : PX,( m) 1 O X E E OX PX,( m) 1 by ɛ(1 e) = (e)+e 1. The t s easy to see that ɛ 1 s equal to detty modulo JX/S 1. To show that ɛ 1 s a somorphsm, let us cosder the somomorphsm t : PX,( m) 1 P X,( m) 1 duced by the morphsm X 2 X 2 ;(x, y) (y, x) ad let s : E OX PX,( m) 1 P X,( m) 1 O X E be the somorphsm x ξ t(ξ) x. The we see that (s ɛ 1 ) 2 : PX,( m) 1 O X E PX,( m) 1 O X E s a PX,( m) 1 -lear edomorphsm whch s equal to the detty modulo JX/S 1. Hece t s a somorphsm ad we see from ths that ɛ 1 s also a somorphsm. Ω 1 X/S As for the tegrablty, we have the followg proposto. Proposto 2.7. d β,= Let E be a O X -module ad let : E E OX E OX J 1 X/S be a pm -coecto. Let ɛ 1 : P 1 X,( m) O X E E OX PX,( m) 1 be the P1 X,( m) -lear somorphsm correspodg to by the equvalece Proposto 2.6 ad let µ 1 : D ( m) X/S,1 O X E Ebe the homomorphsm duced by the composte E PX,( 1) 1 O X E ɛ 1 E OX PX,( 1) 1 = Hom OX (D ( m) X/S,1, E). The the followg codtos are equvalet. (1) (E, ) s tegrable. (2) µ 1 s (uquely) extedable to a D ( m) X/S -module structure o E whch exteds the gve O X -module structure. Proof. We may work locally. So we ca wrte (e) = θ d β (e)dt θ (e)(τ /p m ), usg local coordate. The we have µ 1 ( e) =θ (e).

28 820 Atsush Shho Frst assume the codto (2). The, sce [, j ](e) = 0 for ay e E, we have [θ,θ j ](e) = 0 for ay e ad so (E, ) s tegrable. So the codto (1) s satsfed. O the other had, let us assume the codto (1). The, we defe the acto of k D ( m) X/S o e E by k (e) := d =1 θk (e), where k =(k 1,..., k d ). To see that ths acto acturally defes a D ( m) X/S -module structure o E, we have to check the followg equaltes for local sectos e E ad f O X (see (3), (4) Proposto 2.2): (2.2) (2.3) k (fe)= k k (e) = k+k (e), k +k =k ( ) k k k (f) k (e). By the defto of the acto of k s o e gve above, the equalty (2.2) s reduced to the equalty θ θ j (e) =θ j θ (e), that s, the tegrablty of (E, ). I vew of the equalty (2.2), the proof of the equalty (2.3) s reduced to the case k = 1, ad ths case, t s rewrtte as θ (fe)=fθ (e)+ (f)e (1 d). Ths s equlvalet to the equalty (fe)=f (e)+e df E OX JX/S 1, whch s true by the defto of p m -coecto. So we have the well-defed D ( m) X/S -module structure o E ad hece the codto (2) s satsfed. So we are doe. Corollary 2.8. equvalet. For a O X -module E, the followg three data are (a) A tegrable p m -coecto o E. (b) A D (m) X/S -module structure o E whch exteds the gve O X-module structure. (c) A ( m)-pd-stratfcato {ɛ k } k o E. I partcular, we have the equvalece MIC (m) (X) = (left D ( m) X/S -modules)

29 Ogus-Vologodsky Correspodece 821 ad t duces the equvalece MIC (m) = ( m) (X) (left D X/S Z/p Z-modules). Proof. It suffces to prove the equvalece of (a) ad (b). Whe we are gve a tegrable p m -coecto o E, we have the desred D (m) X/S -module structure o E thaks to Proposto 2.7. Coversely, whe we are gve a D (m) X/S -module structure o E whch exteds the gve O X-module structure, we have the duced homomorphsm µ 1 : D ( m) X/S,1 O X E E. It gves the O X -lear homomorphsm E Hom OX (D ( m) X/S,1, E) = E OX P 1 X,( m) (where the O X -module structure o the target s duced by the rght O X - module structure o PX,( m) 1 ), ad by takg the P1 X,( m) -learzato of t, we obta the homomorphsm ɛ 1 : PX,( m) 1 O X E E OX PX,( m) 1 whch s equal to the detty modulo JX/S 1. It s automatcally a somorphsm by the last argumet the proof of Proposto 2.6, ad t gves a p m -coecto by Proposto 2.6. The, gves rse to the homomorphsm µ 1 by the recpe gve the statemet of Proposto 2.7. Sce µ 1 s extedable to the D ( m) X/S -module structure by assumpto, we see by Proposto 2.7 that s tegrable. So we obta the tegrable p m -coecto ad so we are doe. Next we gve a D-module theoretc terpretato of the quas-lpotece for objects MIC (m) (X). The followg proposto s the aalogue of [1, 2.3.7]. Proposto 2.9. Let f : X S be a smooth morphsm of p-adc formal schemes flat over Z p.letm Nad let E := (E, ) be a object MIC (m) (X), regarded as a left D ( m) X/S Z/p Z-module. The the followg codtos are equvalet. (a) (E, ) s quas-lpotet as a object MIC (m) (X). (b) Locally o X, f admts a local coordate such that the followg codto s satsfed: For ay local secto e E, there exsts some N N

30 822 Atsush Shho such that k (e) =0for ay k wth k N, where k s the elemet D ( m) X/S defed by usg the fxed local coordate. (c) The codto gve (b) s satsfed for ay local coordate. (d) There exsts (uquely) a P X,( m) -lear somorphsm ɛ : P X,( m) OX E = E OX P X,( m) satsfyg the cocycle codto o P X,( m) (2) whch duces the ( m)-pd stratfcato {ɛ k } k o E assocated to the D ( m) X/S -module structure o E va Proposto 2.5. (We call the somorphsm ɛ (d) the ( m)-hpd-stratfcato assocated to E.) Proof. The proof s smlar to that of [1, 2.3.7]. Frst, let us work locally o X, take a local coordate t 1,..., t d of f ad wrte as (e) = θ (e)dt. The, the otato (b), we have θ k = k for ay k N d. Hece we have the equvalece of (a) ad (b). Whe the codto (b) s satsfed, we ca defe the morphsm θ : E E OX P X,( m) by θ(e) = k k (e) (τ/p m ) [k] ad by P X,( m) -learzg t, we obta the homomorphsm ɛ : P X,( m) OX E E OX P X,( m) whch duces the stratfcato {ɛ k } k. The cocycle codto for ɛ follows from that for {ɛ k } k ad the uqueess s clear. Also, f we defe θ : E P X,( m) OX E by θ (e) := k ( 1) k (τ/p m ) [k] k (e), we see that the P X,( m) -learzato of t gves the verse of ɛ. Soɛ s a somorphsm ad thus defes a ( m)- HPD-stratfcato. Coversely, f we are gve a ( m)-hpd-stratfcato ɛ assocated to E, the coeffcet of (τ/p m ) [k] of the elememt ɛ(1 e) E OX P X,( m) = k E(τ/pm ) [k] s equal to k (e), by Proposto 2.5. Hece the codto (b) s satsfed. Fally, sce the codto (d) s depedet of the choce of the local coordate, we have the equvalece of the codtos (c) ad (d). Defto Let f : X S be as above. The, a left D ( m) X/S - module E s sad to be quas-lpotet f t s p -torso for some ad that t satsfes the codto (d) of Proposto 2.9. By Proposto 2.9, we have the equvalece MIC (m) (X) q = (quas-lpotet left D ( m) X/S -modules),

31 Ogus-Vologodsky Correspodece 823 whch s duced by MIC (m) (X) q = (quas-lpotet left D ( m) X/S Z/p Z-modules) ( N). Next we gve a defto of the verse mage fuctor for left D ( m) / - modules. Let (2.4) X S f X S be a commutatve dagram of p-adc formal schemes flat over Spf Z p such that the vertcal arrows are smooth. The, for ay m, k N, t duces the commutatve dagram (2.5) P k X,( m) g k P k X,( m) p P X,( m) X g f P X,( m) p X for =0, 1, where p,p deotes the morphsm duced by the -th projecto X 2 X,X 2 X, respectvely. So, f E s a O X -module edowed wth a( m)-pd-stratfcato {ɛ k } k, f E s aturally edowed wth the ( m)- PD-stratfcato {g k ɛ k } k. Hece, vew of Proposto 2.5, we have the fuctor (2.6) f :(left D ( m) X/S -modules) (left D( m) X /S -modules); (E, {ɛ k } k ) (f E, {g k ɛ k } k ), ad ths duces also the fuctor (2.7) f : (left D ( m) X/S Z/p Z-modules) (left D ( m) X /S Z/p Z-modules). As for the quas-lpotece, we have the followg: