aa r X i v : . [ m a t h . AG ] F e b Prismatic and q -crystalline sites of higher level Kimihiko Li
Abstract
In this article, we define the m -prismatic site and the m - q -crystallinesite, which is a higher level analogue of the prismatic site and the q -crystalline site respectively. We prove a certain equivalence between thecategory of crystals on the m -prismatic site (resp. the m - q -crystallinesite) and that on the prismatic site (resp. the q -crystalline site), whichcan be regarded as the prismatic (resp. the q -crystalline) analogue ofthe Frobenius descent due to Berthelot and the Cartier transform due toOgus-Vologodsky, Oyama and Xu. We also prove the equivalence betweenthe category of crystals on the m -prismatic site and that on the ( m − q -crystalline site. Contents
Introduction 11 m -prismatic site 42 m - q -crystalline site 173 m -prismatic site and ( m − q -crystalline site 264 Relation with Frobenius descent 285 Relation with the results of Xu, Gros-Le Stum-Quir´os andMorrow-Tsuji 41References 47 Introduction
Let p be a prime. In [BS19], Bhatt and Scholze defined two new p -adic cohomol-ogy theories generalizing crystalline cohomology, called prismatic cohomologyand q -crystalline cohomology. They are defined as the cohomology of corre-sponding sites called the prismatic site and the q -crystalline site respectively.The notion of prismatic site ( X/A ) ∆ is defined for a p -completely smooth formalscheme X over A/I , where (
A, I ) is a bounded prism. Also, the q -crystalline site1 X/D ) q -crys is defined for a p -completely smooth formal scheme X over D/I ,where (
D, I ) is a q -PD pair. (The assumptions imply that A, D are endowedwith a lift φ of Frobenius on A/pA, D/pD respectively. Also, D is a Z p J q − K -algebra such that ( D, [ p ] q D ), where [ p ] q = q p − q − , is a bounded prism and that I ⊆ φ − ([ p ] q D ). Moreover, when q = 1 in D , the q -PD pair ( D, I ) is a PDring.) Among other things, they proved the following comparison theorems:(1) If (
D, I ) is a q -PD pair and X is a p -completely smooth formal scheme over D/I , the q -crystalline cohomology of X over D is isomorphic to the pris-matic cohomology of its Frobenius pullback X ′ := X b × Spf(
D/I ) ,φ ∗ Spf( D/ [ p ] q D )over D (Theorem 16.17 of [BS19]).(2) If ( D, I ) is a q -PD pair with q = 1 in D and X is a p -completely smoothformal scheme over D/I , the q -crystalline cohomology of X over D isisomorphic to the crystalline cohomology of X over D (special case ofTheorem 16.14 of [BS19]).On the other hand, for m ≥
0, Berthelot ([Ber90], see also the works of LeStum-Quir´os [LSQ01] and Miyatani [Miy15]) had defined the notion of crys-talline cohomology of level m as the cohomology of the m -crystalline site. Thenotion of m -crystalline site ( X/D ) m -crys is defined for a smooth scheme X over D/I , where (
D, J, I ) is a p -torsion free p -complete m -PD ring or an m -PD ringin which p is nilpotent with p ∈ I , and when m = 0, it coincides with the crys-talline site. He proved the following results which are called Frobenius descent:(3) The category of crystals on the m -crystalline site of X over D is equivalentto that on the crystalline site of its m -times iterated Frobenius pullback X ′ := X × Spec(
D/I ) , ( φ m ) ∗ Spec(
D/I ) over D (Corollaire 2.3.7, Th´eor`eme4.1.3 of [Ber00] in local situation). (See also the equivalence e σ ∗ in Section4.)(4) The level m crystalline cohomology of X over D is isomorphic to thecrystalline cohomology of X ′ over D , where X ′ is as above (Proposition5.4 of [Miy15], Proposition 5.5 of [LSQ01]).The purpose of the present article is to introduce the notion of m -prismaticsite and m - q -crystalline site which are the level m version of the prismatic siteand the q -crystalline site respectively, and prove a prismatic and a q -crystallineversion of the equivalence of (3) in Theorems 1.23,2.16, which are actually com-patible with the equivalence in (3) when q = 1 and p ∈ I . We also prove theequivalence of categories of crystals corresponding to the higher level version of(1) and (2) in Theorem 3.1 and Proposition 4.19 respectively.Let us explain the content of each section. In Section 1, for a boundedprism ( A, I ) and a p -completely smooth and separated formal scheme X over A/J with J := ( φ m ) − ( I ), we define the m -prismatic site ( X/A ) m − ∆ (Def-inition 1.5), and we prove that the category of crystals on the m -prismatic2ite ( X/A ) m − ∆ is equivalent to that on the prismatic site ( X ′ /A ) ∆ , where X ′ := X b × Spf(
A/J ) , ( φ m ) ∗ Spf(
A/I ) (Theorem 1.23). We mainly concentrate on thecategory of crystals with respect to categories of modules with some technicalconditions (Definition 1.1) so that our argument works. Our proof of the equiv-alence is based on the idea developed by Oyama [Oya17] and Xu [Xu19] in theirstudy of Cartier transform in positive characteristic or in the case modulo p n :Namely, we define a functor ρ : ( X/A ) m − ∆ → ( X ′ /A ) ∆ of sites and prove thatit induces the equivalence of topoi ^ ( X/A ) m − ∆ → ^ ( X ′ /A ) ∆ (Theorem 1.17) andmoreover that this equivalence preserves the category of crystals on both handsides.In Section 2, for a q -PD pair ( D, I ) and a p -completely smooth and separatedformal scheme X over D/J with J := ( φ m ) − ( I ), we define the m - q -crystallinesite ( X/D ) m - q -crys (Definition 2.3), and we prove that the category of crystalson ( X/D ) m - q -crys is equivalent to that on the q -crystalline site ( X ′ /D ) q -crys ,where X ′ := X b × Spf(
D/J ) , ( φ m ) ∗ Spf(
D/I ) (Theorem 2.16). The proofs are donein a parallel way to those in Section 1. We also introduce the category ofstratifications which is equivalent to the category of crystals on (
X/D ) m - q -crys (Definition 2.18, Proposition 2.19), which is mainly used in the sections later toprove the equivalences of the categories of crystals.By the comparison of cohomologies (1) and (4) above, it would be natural toregard the prismatic cohomology as a kind of ‘level − q -crystalline cohomology’,and so it would be natural to compare the m -prismatic site and ( m − q -crystalline site. Based on this observation, in Section 3, for a q -PD pair ( D, I )with J q := ( φ m − ) − ( I ) and J ∆ := ( φ m ) − ([ p ] q D ) and a p -completely smoothand separated formal scheme X over D/J q , we prove that the category of crystalson the ( m − q -crystalline site ( X/D ) ( m − q -crys is equivalent to that on the m -prismatic site ( X ′ /D ) m − ∆ , where X ′ := X b × Spf(
D/J q ) Spf(
D/J ∆ ) (Theorem 3.1).The category of stratifications we introduced in Section 2 plays an importantrole for the proof.In Section 4, we relate our equivalence in Section 2 with the Frobenius de-scent (3). First, for a p -torsion free p -complete m -PD ring ( D, J, I ) with p ∈ I and a smooth and separated scheme e X over D/I , we give an alternative site-theoretic proof (a proof based on the idea of Oyama [Oya17] and Xu [Xu19]) ofthe equivalence in (3), namely, the equivalence between the category of crystalson the m -crystalline site ( e X/D ) m -crys and that on the crystalline site ( X ′ /D ) crys ,where X ′ := e X × Spec(
D/I ) , ( φ m ) ∗ Spec(
D/I ) (Corollary 4.13). We note that theoriginal definition of ( m -)crystalline site is not suitable to perform the site-theoretic argument of Oyama and Xu. Our strategy is to introduce variants( X/D ) m -crys,new (where X := e X × Spec(
D/I ) Spec(
D/J )), ( X ′ /D ) crys,new of the( m -)crystalline site (Definition 4.7) which do not change the category of crys-tals and for which the site-theoretic argument work. Furthermore, by comparingthe category of crystals on these variants and that on the ( m -) q -crystalline site3n Section 2, we see the compatibility of the equivalence in Section 2 with theFrobenius descent.In Section 5, we explain relations between our result and results in the worksof Xu [Xu19], Gros-Le Stum-Quir´os [GLSQ20b] and Morrow-Tsuji [MT20]. Wewill see that our equivalences between the category of crystals on the prismaticsite, that on the 1-prismatic site and that on the q -crystalline site fit naturallyinto the equivalence of Cartier transform in the case modulo p n by Xu, theequivalence between the category of twisted hyper-stratified modules of level − q -Higgs field and that of modules with flat q -connectionsappearing in the work of Morrow-Tsuji.Original motivation of introducing m -crystalline cohomology would be todevelop a p -adic cohomology theory over a ramified base, for example, whenthe base is a complete discrete valuation ring V of mixed characteristic (0 , p ) inwhich p is not a uniformizer. However, our definition of m -prismatic site and m - q -crystalline site is not enough for this purpose, because our base have tobe a δ -ring and there is no δ -ring structure on the ring V above. We hope togeneralize our definition to cover also the case of ramified base in a future.Finally, the author would like to express his sincere gratitude to his supervi-sor Atsushi Shiho who patiently answered many questions and had discussionson the constructions of this paper. The author is partly supported by WINGS-FMSP (World-leading Innovative Graduate Study for Frontiers of Mathemat-ical Sciences and Physics) program at the Graduate School of MathematicalSciences, the University of Tokyo. m -prismatic site In this section, we define the m -prismatic site which is a higher level analogueof the prismatic site defined in [BS19] and prove an equivalence between thecategory of crystals on the m -prismatic site of a smooth formal scheme X andthat on the prismatic site of the pullback of X by the m -th iteration φ m of theFrobenius lift φ on the base prism.To define a suitable category of crystals for which our argument works,first we need to impose some technical condition to our category of modules asfollows: Definition 1.1
Let (
E, I ) be a bounded prism.1. Let f M ∆ ( E, I ) be the category of E -modules M such that, for any map( E, I ) → ( E , I ) of bounded prisms and for any faithfully flat map( E , I ) → ( E ′ , I ′ ) of bounded prisms, the sequence0 → M b ⊗ E E → M b ⊗ E E ′ → M b ⊗ E ( E ′ b ⊗ E E ′ )is exact, where the completion is classical ( p, I )-completion.4. Let {M ∆ ( E, I ) ⊆ f M ∆ ( E, I ) } ( E,I ) be the largest family of full subcate-gories such that, for any M ∈ M ∆ ( E, I ), any (
E, I ) → ( E ′ , I ′ ) of boundedprisms and any faithfully flat map ( E , I ) → ( E ′ , I ′ ) of bounded prisms,any descent data ǫ on M b ⊗ E E ′ (an isomorphism( E ′ b ⊗ E E ′ ) b ⊗ E ′ ( M b ⊗ E E ′ ) ∼ = ( M b ⊗ E E ′ ) b ⊗ E ′ ( E ′ b ⊗ E E ′ )satisfying the cocycle condition on E ′ b ⊗ E E ′ b ⊗ E E ′ ) descends uniquely to M ∈ M ∆ ( E , I ).The above definition allows us to think only of the category M ∆ ( E, I ) of mod-ules with suitable “sheaf” property and descent property.We will also need the following variant of Definition 1.1 in which E does notadmit a δ -structure and that I = ( p ). Definition 1.2
Let E be a p -torsion free p -complete ring. Then1. Let f M ( E ) be the category of E -modules M such that, for any map E → E of p -torsion free p -complete rings and for any p -completely faithfullyflat map E → E ′ of p -torsion free p -complete rings, the sequence0 → M b ⊗ E E → M b ⊗ E E ′ → M b ⊗ E ( E ′ b ⊗ E E ′ )is exact, where the completion is classical p -completion.2. Let {M ( E ) ⊆ f M ( E ) } E be the largest family of full subcategories suchthat, for any M ∈ M ( E ), any E → E ′ of p -torsion free p -complete ringsand any p -completely faithfully flat map E → E ′ of p -torsion free p -complete rings, any descent data ǫ on M b ⊗ E E ′ (an isomorphism( E ′ b ⊗ E E ′ ) b ⊗ E ′ ( M b ⊗ E E ′ ) ∼ = ( M b ⊗ E E ′ ) b ⊗ E ′ ( E ′ b ⊗ E E ′ )satisfying the cocycle condition on E ′ b ⊗ E E ′ b ⊗ E E ′ ) descends uniquely to M ∈ M ( E ).The categories introduced above have the following properties. Proposition 1.3
1. For all M ∈ f M ∆ ( E, I ) and all maps ( E, I ) → ( E ′ , I ′ ) of bounded prisms, M b ⊗ E E ′ belongs to f M ∆ ( E ′ , I ′ ) . Similar property holdsalso for the three other categories M ∆ ( E, I ) , f M ( E ) and M ( E ) .2. If f : M → M ′ is a morphism in f M ∆ ( E, I ) , ( E, I ) → ( E ′ , IE ′ ) is afaithfully flat map of bounded prisms and M b ⊗ E E ′ → M ′ b ⊗ E E ′ is an iso-morphism, then f is an isomorphism. Similar property holds also for f M ( E ) . roof. The first property follows immediately from the definition. For the sec-ond property, we have a commutative diagram of exact sequences:0
M M b ⊗ E E ′ M b ⊗ E ( E ′ b ⊗ E E ′ )0 M ′ M ′ b ⊗ E E ′ M ′ b ⊗ E ( E ′ b ⊗ E E ′ ) . f f b ⊗ id f b ⊗ (id b ⊗ id) By assumption f b ⊗ id is an isomorphism and it follows that f b ⊗ (id b ⊗ id) is also anisomorphism. Then it is immediate that f is an isomorphism. (cid:3) Remark 1.4
1. If we define M fp ( E ) to be the category of finite projective E -modules, M p n -tors ( E ) to be the category ( { ( p, I ) n -torsion E -module } (when ( E, I ) is a bounded prism) { p n -torsion E -module } (when E is a p -torsion free p -complete ring)and M tors ( E ) to be [ n M p n -tors ( E ), we can show that M fp ( E ) ⊆ M ∆ ( E, I ) , M ( E )by Proposition A.12 of [AB19] and M tors ( E ) ⊆ M ∆ ( E, I ) , M ( E )by usual descent property.2. If E is a p -torsion free p -complete δ -ring, we have the inclusion f M ( E ) ⊆ f M ∆ ( E, pE ) by definition. The same holds for M and M ∆ .We will often denote f M ∆ ( E, I ) (resp. M ∆ ( E, I )) simply by f M ∆ ( E ) (resp. M ∆ ( E )).Now we define the m -prismatic site, which is a higher level analogue of theprismatic site as well as a prismatic analogue of the level m crystalline site. Wefix a non-negative integer m and a bounded prism ( A, I ). Definition 1.5
Let J = ( φ m ) − ( I ). Let X be a p -adic formal scheme seperatedand p -completely smooth over A/J . We define the m - prismatic site ( X/A ) m − ∆ of X over A as follows. Objects are the maps ( A, I ) → ( E, I E ) of boundedprisms together with a map Spf( E/J E ) → X over A/J satisfying the followingcondition, where J E = ( φ m ) − ( I E ):(*) Spf( E/J E ) → X factors through some open affine Spf( R ) ⊆ X .6e shall often denote such an object by(Spf( E ) ← Spf(
E/J E ) → X ) ∈ ( X/A ) m − ∆ or ( E, I E ) if no confusion arises. A morphism (Spf( E ′ ) ← Spf( E ′ /J E ′ ) → X ) → (Spf( E ) ← Spf(
E/J E ) → X ) is a map of formal schemes Spf( E ′ ) → Spf( E ) over A which is compatible with the morphisms Spf( E ′ /J E ′ ) → X , Spf( E/J E ) → X .When we denote an object by ( E, I E ), we shall write a morphism from ( E ′ , I E ′ )to ( E, I E ) as ( E, I E ) → ( E ′ , I E ′ ), not ( E ′ , I E ′ ) → ( E, I E ). A morphism ( E, I E ) → ( E ′ , I E ′ ) in ( X/A ) m − ∆ is a cover if it is a faithfully flat map of prisms, i.e., E ′ is ( p, I E )-completely faithfully flat over E . Remark 1.6
By Lemma 3.5 of [BS19], for an object (
E, I E ) in ( X/A ) m − ∆ , I E is always equal to IE .We need to check that ( X/A ) m − ∆ with the topology as defined above is indeeda site. Thanks to Lemma 3.8 of [MT20], we have the following lemma: Lemma 1.7
Let ( E , IE ) f ←− ( E, IE ) g −→ ( E , IE ) be maps in ( X/A ) m − ∆ such that f is a covering. Let E := E b ⊗ E E . Then ( E , IE ) is the objectthat represents the coproduct ( E , IE ) ⊔ ( E,IE ) ( E , IE ) in ( X/A ) m − ∆ , and thecanonical map ( E , IE ) → ( E , IE ) is a covering. By Lemma 1.7, the covers defined in (
X/A ) m − ∆ is actually a pretopology. Remark 1.8
1. Applying to the case m = 0, we see that the topology inprismatic site as defined above is actually a pretopology. This is thetopology defined in Definition 4.1 of [BS19].2. The condition (*), which we imposed due to technical reason, is not as-sumed in [BS19]. One can check easily that the topos is unchanged evenif we drop the condition (*).In the situation in Definition 1.5, let X ′ be X b × Spf(
A/J ) , ( φ m ) ∗ Spf(
A/I ). Then wehave the following diagram:Spf( A ) Spf( A/J ) X Spf( A ) Spf( A/I ) X ′ . (cid:3) ( φ m ) ∗ ( φ m ) ∗ We define a natural functor ρ from the m -prismatic site of X over A to theusual prismatic site of X ′ over A . For an object (Spf( E ) ← Spf(
E/J E ) → X )7f ( X/A ) m − ∆ , define the object ρ (Spf( E ) ← Spf(
E/J E ) → X ) of ( X ′ /A ) ∆ by(Spf( E ) ← Spf(
E/I E ) → X ′ ) where the right map is defined as follows:Spf( E/I E ) → Spf(
E/J E ) b × Spf(
A/J ) , ( φ m ) ∗ Spf(
A/I ) → X b × Spf(
A/J ) , ( φ m ) ∗ Spf(
A/I ) = X ′ . Here the first map is the one induced by a map of rings
E/J E b ⊗ A/J,φ m A/I → E/I E ; e ⊗ a φ m ( e ) a. This defines the functor ρ : ( X/A ) m − ∆ → ( X ′ /A ) ∆ .Next we want to show that ρ induces an equivalence of topoi. We follow theproof of Theorem 9.2 of[Xu19] where the author proves the analogous resultsfor the Oyama topoi. For this, we prepare following propositions. Proposition 1.9 (cf. [Xu19] 9.3 (i))
The functor ρ is fully faithful.Proof. The functor ρ is clearly faithful. We prove its fullness. Suppose that α : ρ (Spf( E ′ ) ← Spf( E ′ /J E ′ ) → X ) → ρ (Spf( E ) ← Spf(
E/J E ) → X ) is a mapin ( X ′ /A ) ∆ . We consider the diagramSpf( E ′ /I E ′ ) Spf( E ′ /J E ′ ) X ′ X Spf(
E/I E ) Spf( E/J E ) ( φ m ) ∗ α α ( φ m ) ∗ where the morphism α are the ones induced by α and the morphisms ( φ m ) ∗ arethe ones induced by φ m on E ′ and E . To prove the fullness of ρ , it suffices toprove the commutativity of the right triangle. Since the left triangle, the outersquare and the two trapezoids are commutative, it suffices to prove the followingclaim: claim. Let f i : Spf( E ′ /J E ′ ) → X ( i = 1 ,
2) be morphisms which factorthrough some affine Spf( R i ) ⊆ X ( i = 1 ,
2) respectively such that f ◦ ( φ m ) ∗ = f ◦ ( φ m ) ∗ : Spf( E ′ /I E ′ ) → X . Then f = f .We prove the claim. Let Spf( R ) = Spf( R ) ∩ Spf( R ). (Note that, since X is separated, the intersection on the right hand side is affine.) Then the map f ◦ ( φ m ) ∗ = f ◦ ( φ m ) ∗ factors through Spf( R ). Since ( φ m ) ∗ is a homeomorphismas a map of topological spaces, it implies that both f and f factor throughSpf( R ). If we denote the map R → E ′ /J E ′ corresponding to f i ( i = 1 ,
2) by f ∗ i ,we have the equality of the map of rings φ m ◦ f ∗ = φ m ◦ f ∗ : R → E ′ /I E ′ . Then,since the map φ m : E ′ /J E ′ → E ′ /I E ′ is injective, we conclude that f ∗ = f ∗ ,hence f = f . So we have proved the claim and we are done. (cid:3) roposition 1.10 (cf. [Xu19] 9.3 (ii)) The functor ρ is continuous.Proof. By definition, α : ( E, I E ) → ( E ′ , I E ′ ) in ( X/A ) m − ∆ is a covering if andonly if ρ ( α ) : ρ ( E, I E ) → ρ ( E ′ , I E ′ ) in ( X ′ /A ) ∆ is a covering (in this article,the definition of coverings in the prismatic site comes from the m = 0 casein Definition 1.5). Also, if ( E , IE ) f ←− ( E, IE ) g −→ ( E , IE ) is a diagramin ( X/A ) m − ∆ such that f is a covering, then by Lemma 1.7 ρ ( E ⊔ E E ) → ρ ( E ) ⊔ ρ ( E ) ρ ( E ) is an isomorphism. Hence ρ is continuous. (cid:3) Proposition 1.11 (cf. [Xu19] 9.3 (ii))
The functor ρ is cocontinuous.Proof. Suppose that α : (Spf( E ′ ) ← Spf( E ′ /I E ′ ) → X ′ ) → ρ (Spf( E ) ← Spf(
E/J E ) → X ) is a covering in ( X ′ /A ) ∆ . For Spf( E/J E ) → X , take an affine open formalsubscheme Spf( R ) ⊆ X in the condition (*). Then there exists a commutativediagram: R ′ = R b ⊗ A/J,φ m A/I E/J E b ⊗ A/J,φ m A/I E/I E EE ′ /I E ′ E ′ g α α where g is defined by the composition of the left triangle. By the commutativityabove, we have the following diagram R E/J E E/I E EE ′ /J E ′ E ′ /I E ′ E ′ f α φ m α αφ m where the solid arrows are commutative. If we define the dotted arrow to be thecomposite α ◦ f of the left triangle, we can make the whole diagram commutativeand it defines an object (Spf( E ′ ) ← Spf( E ′ /J E ′ ) → X ) in ( X/A ) m − ∆ and amorphism (Spf( E ′ ) ← Spf( E ′ /J E ′ ) → X ) → (Spf( E ) ← Spf(
E/J E ) → X ) in( X/A ) m − ∆ , which we also denote by α . We see that ρ ( α ) = α and by assumption α as a morphism in ( X/A ) m − ∆ is also a covering. Hence ρ is cocontinuous. (cid:3) Proposition 1.12 (cf. [Xu19] 9.8 (i))
Let ( E, I E ) be an object in ( X ′ /A ) ∆ .Then there exists an object ( E ′ , I E ′ ) in ( X/A ) m − ∆ and a covering of the form ( E, I E ) → ρ ( E ′ , I E ′ ) .Proof. Take an open covering X = S i Spf( R i ) of X by finitely many affine openformal subschemes, and let Spf( R ′ i ) = Spf( R i ) b × X X ′ , Spf( E i ) = Spf( R ′ i ) b × X ′ Spf(
E/I E ).Then, by Lemma 1.13 below, there exists uniquely an open affine formal sub-scheme Spf( E i ) of Spf( E ) which lifts Spf( E i ), namely, E i /I E E i = E i . By con-sidering the covering ( E, I E ) → Q i ( E i , I E E i ) in ( X ′ /A ) ∆ , we see that we may9eplace ( E, I E ) by ( E i .I E E i ) to prove the proposition. Thus we may assumethat the structure morphism Spf( E/I E ) → X ′ of the object ( E, I E ) in ( X ′ /A ) ∆ factors through an open affine formal subscheme Spf( R ′ ) ⊆ X ′ such that R ′ isof the form R b ⊗ A/J,φ m A/I for some open affine subscheme Spf( R ) ⊆ X .As the map A/J → R is p -completely smooth, it is topologically finitelygenerated. So there exists a surjection A [ x , . . . , x n ] ∧ → R whose kernel isthe ideal ( J, y , . . . , y r ) such that y , . . . , y r ∈ A/ ( p, J )[ x , . . . , x n ] is a regu-lar sequence. On the other hand, there is a natural map A [ x , . . . , x n ] ∧ → A { x , . . . , x n } ∧ , where the symbol {} denote the adjoining of elements in thetheory of δ -rings. From Corollary 1.15 below about regular sequence, we seethat φ m ( y ) , . . . , φ m ( y r ) is a ( p, I )-completely regular sequence relative to A .Then we can construct a map A { x , . . . , x n } ∧ → S def = (( A { x , . . . , x n } ) ∧ { KI } ) ∧ to the prismatic envelope by Proposition 3.13 of [BS19], where K denotes theideal ( φ m ( y ) , . . . , φ m ( y r ) , I ) in A { x , . . . , x n } ∧ . From the property of prismaticenvelope, for all i , φ m ( y i ) ∈ I S and so y i ∈ J S = ( φ m ) − ( I S ). This gives a map R → S/J S .Next, let A [ x , . . . , x n ] ∧ → R ′ be the morphism induced by the base change A [ x , . . . , x n ] ∧ → R b ⊗ A,φ m A of the map A [ x , . . . , x n ] ∧ → R in the previous para-graph by φ m : A → A . Then the kernel of this map is the ideal ( I, y ′ , . . . , y ′ r ),where y ′ i is the image of y i by the map A { x , . . . , x n } ∧ → A { x , . . . , x n } ∧ send-ing X j α j x j to X j φ m ( α j ) x j . (Here we used multi-index notation.) By defini-tion, y ′ , . . . , y ′ r ∈ A/ ( p, I )[ x , . . . , x n ] is a regular sequence. By Corollary 1.15below, we see that y ′ , . . . , y ′ r is a ( p, I )-completely regular sequence relative to A . Then we can construct a map A { x , . . . , x n } ∧ → S ′ def = (( A { x , . . . , x n } ) ∧ { K ′ I } ) ∧ to the prismatic envelope, where K ′ denotes the ideal ( y ′ , . . . , y ′ r , I ) in A { x , . . . , x n } ∧ .So we get the following diagram: A/I R ′ A A [ x , . . . , x n ] ∧ A { x , . . . , x n } ∧ S ′ def = (( A { x , . . . , x n } ) ∧ { K ′ I } ) ∧ . From the relation between y i and y ′ i , there exists a map ( φ m ) ′ : S ′ → S , where( φ m ) ′ is an A - δ -ring map sending x i to φ m ( x i ). From Proposition 1.16 below,( φ m ) ′ is ( p, I )-completely faithfully flat.Let ( E, I E ) be an object in ( X ′ /A ) ∆ as above. Then we have a map f : A [ x , . . . , x n ] ∧ → E/I E by composing A [ x , . . . , x n ] ∧ → R ′ in the abovediagram and R ′ → E/I E . As A [ x , . . . , x n ] ∧ is the completion of a polynomialring, there exists a lifting f : A [ x , . . . , x n ] ∧ → E of f . Since E is a δ -ring,we can extend the map f to the map f : A { x , . . . , x n } ∧ → E . As the image10f K ′ in E/I E equals to zero and S ′ is the prismatic envelope with respect to K ′ , we can extend the map f to the map g : S ′ → E . As the base change ofthe map ( φ m ) ′ : S ′ → S along g , we obtain a map h : E → E b ⊗ S ′ S which is( p, I )-completely faithfully flat by the property of ( φ m ) ′ shown above.It remains to show that this covering is actually a morphism of the form( E, I E ) → ρ ( E b ⊗ S ′ S, I E b ⊗ S ′ S )in ( X ′ /A ) ∆ . It is sufficient to prove the commutativity of the rectangle at theright below of the diagram in which all the other rectangles are all commutative A [ x , . . . , x n ] ∧ S E b ⊗ S ′ SA [ x , . . . , x n ] ∧ R ′ = R b ⊗ A/J,φ m A/I S/J S b ⊗ A/J,φ m A/I E b ⊗ S ′ S/J E b ⊗ S ′ S b ⊗ A/J,φ m A/IE E/I E E b ⊗ S ′ S/I E b ⊗ S ′ Si h where i is the map induced by the map to the coproduct in Lemma 1.7. Thiscan be checked by corresponding the elements x i : x i x i ⊗ x i x i x i ⊗ x i ⊗ f ( x i ) f ( x i ) 1 ⊗ φ m ( x i ) = f ( x i ) ⊗ . So we have the commutativity. (cid:3)
We prove the claims used in the proof above.
Lemma 1.13
Let A be an I -adically complete ring where I is a finitely gener-ated ideal. Let J ⊆ A be a closed ideal such that the image of J in A/I is anil ideal, and let f : U → Spf(
A/J ) be an I -completely smooth morphism ( resp.an open immersion ) with U affine. Then there exists uniquely an I -completelysmooth morphism ( resp. an open immersion ) f : U → Spf( A ) which lifts f .Also, U is affine.Proof. We may replace A by A/I n and Spf by Spec to prove the lemma. Then J is a nil ideal of A and the morphism f : U → Spec(
A/J ) is a smooth morphism(resp. an open immersion) of schemes. Then, when f is a smooth morphism, the11laim follows from Lemma 2.3.14 of [NS08]. When f is an open immersion, theexistence and the uniqueness of f is clear because Spec( A/J ) is homeomorphicto Spec( A ), and the affinity of U can be proven in the same way as the proof ofLemma 2.3.14 of [NS08]. (cid:3) To prove the other claims, we review several notions appeared in Section 2.6in [BS19]. A map A → B of simplicial commutative rings is called flat if π ( A ) → π ( B ) is flat and π i ( A ) ⊗ π ( A ) π ( B ) → π i ( B ) is an isomorphism forall i . This can be checked after we apply − ⊗ LA π ( A ). For a commutativering A , a finitely generated ideal I = ( f , . . . , f n ) ⊆ A and an I -completely flatcommutative A -algebra B , a sequence x , . . . , x r ∈ B is an I -completely regularsequence relative to A if the map of simplicial commutative rings A ⊗ L Z [ f ,...,f n ] Z → B ⊗ L Z [ f ,...,f n ,x ,...,x r ] Z is flat, where the maps Z [ f , . . . , f n ] → Z , Z [ f , . . . , f n , x , . . . , x r ] → Z used todefine both hand sides send each f i , x i to 0. Lemma 1.14
Let
A, I = ( f , . . . , f n ) ⊆ A and B as above and let x , . . . , x r ∈ B be a sequence of elements satisfying the following:1. The images x , . . . , x r of x , . . . , x r in B/IB is a regular sequence.2. B/ ( I, x , . . . , x r ) is flat over A/I .Then x , . . . , x r is an I -completely regular sequence relative to A .Proof. We prove the flatness of the map A ⊗ L Z [ f ,...,f n ] Z → B ⊗ L Z [ f ,...,f n ,x ,...,x r ] Z , and we can check it after we take the derived tensor product with the map A ⊗ L Z [ f ,...,f n ] Z → A/I ⊗ L Z [ f ,...,f n ] Z ∼ = A/I.
The map we obtain is
A/I → ( A/I ⊗ LA B ) ⊗ L Z [ f ,...,f n ,x ,...,x r ] Z ∼ = B/IB ⊗ L Z [ f ,...,f n ,x ,...,x r ] Z ∼ = B/IB ⊗ L Z [ x ,...,x r ] Z , where the first isomorphism follows from the I -complete flatness of B over A .Since x , . . . , x r is a regular sequence in B/IB , the complex
B/IB ⊗ L Z [ x ,...,x r ] Z ,which is equal to the Koszul complex of B/IB with respect to x , . . . , x r , is equalto B/ ( I, x , . . . , x r ) and since it is assumed to be flat over A/I , we see that themap we obtained is flat. So we are done. (cid:3)
Corollary 1.15
In the notation of the proof of Proposition 1.12, the sequence φ m ( y ) , . . . , φ m ( y r ) ∈ A { x , . . . , x n } ∧ and the sequence y ′ , . . . , y ′ r ∈ A { x , . . . , x n } ∧ are ( p, I ) -completely regular sequences relative to A . roof. The claim for the sequence y ′ , . . . , y ′ r is immediate from the previouslemma because y ′ , . . . , y ′ r ∈ A/ ( p, I )[ x , . . . , x n ] is a regular sequence, the ring A [ x , . . . , x n ] / ( p, I, y ′ , . . . , y ′ r ) = R ′ /pR ′ is smooth (hence flat) over A/ ( p, I )and the map A/ ( p, I )[ x , . . . , x n ] → A/ ( p, I ) { x , . . . , x n } is flat.We prove the claim for the sequence φ m ( y ) , . . . , φ m ( y r ). Recall that y , . . . , y r ∈ A/ ( p, J )[ x , . . . , x n ] is a regular sequence and ( A/ ( p, J )[ x , . . . , x n ]) / ( y , . . . , y r ) = R/ ( p, J ) is smooth over A/ ( p, J ). If we denote the ideal generated by x p m ( x ∈ J ) by J ( p m ) , we have the inclusion ( p, J ( p m ) ) ⊆ ( p, I ). Since ( p, J ) is a nilideal in A/ ( p, J ( p m ) )[ x , . . . , x n ], Jacobian criterion of smoothness implies that y , . . . , y r ∈ A/ ( p, J ( p m ) )[ x , . . . , x n ] is a regular sequence and that the quo-tient ( A/ ( p, J ( p m ) )[ x , . . . , x n ]) / ( y , . . . , y r ) is smooth over A/ ( p, J ( p m ) ). Thenit implies that y , . . . , y r ∈ A/ ( p, I )[ x , . . . , x n ] is a regular sequence and that( A/ ( p, I )[ x , . . . , x n ]) / ( y , . . . , y r ) is smooth over A/ ( p, I ). Moreover, since φ m ( y i ) = y ip m in A/ ( p, I )[ x , . . . , x n ], we see that φ m ( y ) , . . . , φ m ( y r ) ∈ A/ ( p, I )[ x , . . . , x n ]is a regular sequence and that ( A/ ( p, I )[ x , . . . , x n ]) / ( φ m ( y ) , . . . , φ m ( y r )) isflat over A/ ( p, I ). So, be the previous lemma, we conclude that the sequence φ m ( y ) , . . . , φ m ( y r ) ∈ A { x , . . . , x n } ∧ is a ( p, I )-completely regular sequencerelative to A . (cid:3) Proposition 1.16
In the notation of the proof of Proposition 1.12, the map ( φ m ) ′ : S ′ → S is ( p, I ) -completely faithfully flat.Proof. Since the map ( φ m ) ′ is the ( p, I )-completed base change of the A - δ -ringmap ( φ m ) ′′ : A { x , . . . , x n } ∧ → A { x , . . . , x n } ∧ , x i φ m ( x i ) , it suffices to prove that this map is ( p, I )-completely faithfully flat. By Lemma2.11 of [BS19], the A - δ -ring map( φ m ) ′′′ : A { x , . . . , x n } → A { x , . . . , x n } , x i φ m ( x i )is faithfully flat. Since the map ( φ m ) ′′ is obtained by taking the base changeof ( φ m ) ′′′ with respect to A { x , . . . , x n } → A { x , . . . , x n } ∧ and then taking thederived ( p, I )-completion, we conclude that ( φ m ) ′′ is ( p, I )-completely faithfullyflat. (cid:3) The functor ρ : ( X/A ) m − ∆ → ( X ′ /A ) ∆ induces a functor between the categoriesof presheaves of sets b ρ ∗ : \ ( X ′ /A ) ∆ → \ ( X/A ) m − ∆ G G ◦ ρ, and it admits a right adjoint b ρ ∗ : \ ( X/A ) m − ∆ → \ ( X ′ /A ) ∆ . By Proposition 1.10 and 1.11, we obtain a morphism between the topoi of thesheaves of sets C : ^ ( X/A ) m − ∆ → ^ ( X ′ /A ) ∆ C ∗ = b ρ ∗ , C ∗ = b ρ ∗ . heorem 1.17 The morphism
C : ^ ( X/A ) m − ∆ → ^ ( X ′ /A ) ∆ is an equivalence oftopoi.Proof. This follows from Propositions 1.9, 1.10, 1.11, 1.12 and Proposition 4.2.1of [Oya17] (see also Proposition 9.10 of [Xu19]). (cid:3)
Next we define the category of crystals with respect to the categories of modulesdefined in Definition 1.1.
Definition 1.18
Let C ∆ (( X/A ) m − ∆ ) (resp. C (( X/A ) m − ∆ ) , C fp (( X/A ) m − ∆ ) , C tors (( X/A ) m − ∆ )) be the category of presheaves F on ( X/A ) m − ∆ such that, forany object ( E, I E ) in ( X/A ) m − ∆ , F ( E ) ∈ M ∆ ( E ) (resp. M ( E ) , M fp ( E ) , M tors ( E ))and for any morphism ( E, I E ) → ( E ′ , I E ′ ), the induced map F ( E ) b ⊗ E E ′ → F ( E ′ ) is an isomorphism of E ′ -modules. This definition means that C ∆ (( X/A ) m − ∆ )(resp. C (( X/A ) m − ∆ ) , C fp (( X/A ) m − ∆ ) , C tors (( X/A ) m − ∆ )) is the category ofcrystals with respect to M ∆ (resp. M , M fp , M tors ). Remark 1.19
1. Presheaves F in the above definition is automatically sheavesby definition of the category of modules in Definition 1.1.2. We can prove that the category C ∆ (( X/A ) m − ∆ ) is unchanged even if wedrop the condition (*) in Definition 1.5.We want to prove that the morphism C induces an equivalence of categories ofcrystals. To do this, we need the following propositions. We follow the proof ofTheorem 9.12 of [Xu19]. Proposition 1.20 (cf. [Xu19] 9.5)
Let ( E, I E ) be an object of ( X/A ) m − ∆ and g : ρ ( E, I E ) → ( E ′ , I E ′ ) a morphism of ( X ′ /A ) ∆ . Then there exist an object ( E ′′ , I E ′′ ) of ( X/A ) m − ∆ and a morphism f : ( E, I E ) → ( E ′′ , I E ′′ ) of ( X/A ) m − ∆ such that g = ρ ( f ) . If g is a covering, so is f .Proof. This is what we have shown in the proof of Proposition 1.11. (cid:3)
Proposition 1.21 (cf. [Xu19] 9.8(ii))
Let g : ( E, I E ) → ( E , I E ) be a mor-phism of ( X ′ /A ) ∆ . Then there exists a morphism h : ( E ′ , I E ′ ) → ( E ′ , I E ′ ) of ( X/A ) m − ∆ and coverings f : ( E, I E ) → ρ ( E ′ , I E ′ ) , f : ( E , I E ) → ρ ( E ′ , I E ′ ) such that the following diagram is a pushout diagram: ( E , I E ) ρ ( E ′ , I E ′ )( E, I E ) ρ ( E ′ , I E ′ ) . f q g f ρ ( h ) roof. f is the covering constructed in Proposition 1.12. Let ( E ′′ , I E ′′ ) be thepushout of the diagram ( E , I E ) g ←− ( E, I E ) f −→ ρ ( E ′ , I E ′ ). Applying Propo-sition 1.20 to the map ρ ( E ′ , I E ′ ) → ( E ′′ , I E ′′ ), we obtain the required dia-gram. (cid:3) Proposition 1.22 (cf. [Xu19] 9.9)
Let ( E ′ , I E ′ ) be an object of ( X ′ /A ) ∆ , ( E, I E ) an object of ( X/A ) m − ∆ and ( E ′ , I E ′ ) → ρ ( E, I E ) a covering. Then thereexists an object ( E , I E ) of ( X/A ) m − ∆ and two morphisms p , p : ( E, I E ) → ( E , I E ) such that ρ ( E , I E ) = ρ ( E, I E ) b ⊗ ( E ′ ,I E ′ ) ρ ( E, I E ) and that ρ ( p ) ( resp. ρ ( p )) is the map ρ ( E, I E ) → ρ ( E, I E ) b ⊗ ( E ′ ,I E ′ ) ρ ( E, I E ) to the first ( resp. second ) component.Proof. Applying Proposition 1.20 to the map ρ ( E, I E ) → ρ ( E, I E ) b ⊗ ( E ′ ,I E ′ ) ρ ( E, I E )to the first component, we obtain the morphism p : ( E, I E ) → ( E , I E ) satis-fying the conditions. Existence of p follows from the fullness of ρ . (cid:3) Theorem 1.23
The functors C ∗ , C ∗ induce equivalences of categories quasi-inverse to each other C ∆ (( X/A ) m − ∆ ) ⇄ C ∆ (( X ′ /A ) ∆ ) . Proof.
We want to prove that the functors C ∗ , C ∗ preserve the property of crys-tals. The claim that the functor C ∗ preserves crystals follows easily from theequality C ∗ ( F )( E, I E ) = F ( ρ ( E, I E )) . On the other hand, as C ∗ C ∗ ≃ id, F ( E, I E ) = C ∗ C ∗ ( F )( E, I E ) = C ∗ ( F )( ρ ( E, I E )) . (1)Let F be an object in C ∆ (( X/A ) m − ∆ ) and g : ( E, I E ) → ( E , I E ) be a mor-phism in ( X ′ /A ) ∆ . We want to prove that C ∗ ( F )( E, I E ) ∈ M ∆ ( E ) , C ∗ ( F )( E , I E ) ∈M ∆ ( E ) and that the map E b ⊗ E C ∗ ( F )( E, I E ) → C ∗ ( F )( E , I E )is an isomorphism. By Proposition 1.21, we have a diagram( E , I E ) ρ ( E ′ , I E ′ )( E, I E ) ρ ( E ′ , I E ′ ) f q g f ρ ( h ) f and f are coverings. We consider the following diagram : E ′ b ⊗ E E b ⊗ E C ∗ ( F )( E, I E ) E ′ b ⊗ E ′ E ′ b ⊗ E C ∗ ( F )( E, I E ) E ′ b ⊗ E ′ C ∗ ( F )( ρ ( E ′ , I E ′ )) E ′ b ⊗ E C ∗ ( F )( E , I E ) C ∗ ( F )( ρ ( E ′ , I E ′ ))We want to prove that C ∗ ( F )( E, I E ) ∈ M ∆ ( E ) , C ∗ ( F )( E , I E ) ∈ M ∆ ( E )and that the left vertical arrow is an isomorphism by checking that the remainingarrows are all isomorphisms. If this holds, then by Proposition 1.3 we see that E b ⊗ E C ∗ ( F )( E, I E ) → C ∗ ( F )( E , I E )is an isomorphism and so we are done. The isomorphism of right column followsfrom (1) and the assumption that F is a crystal. So it is enough to show that,if f : ( E, I E ) → ρ ( E ′ , I E ′ ) is a covering, then C ∗ ( F )( E, I E ) ∈ M ∆ ( E ) and that E ′ b ⊗ E C ∗ ( F )( E, I E ) → C ∗ ( F )( ρ ( E ′ , I E ′ ))is an isomorphism.In the following, we will denote ( E, I E ) , ( E ′ , I E ′ ) simply be E, E ′ respectivelyto lighten the notation. Consider the following sequence E ρ ( E ′ ) ρ ( E ′ ) b ⊗ E ρ ( E ′ ) ρ ( E ′ ) b ⊗ E ρ ( E ′ ) b ⊗ E ρ ( E ′ ) , f where the arrows except f are the maps into suitable factors. By Proposition1.22, the above sequence is rewritten as E ρ ( E ′ ) ρ ( E ′′ ) ρ ( E ′′′ ) f for some E ′′ , E ′′′ ∈ ( X/A ) m − ∆ and the arrows except f come from morphismsin ( X/A ) m − ∆ . Then the diagramC ∗ ( F )( ρ ( E ′ )) C ∗ ( F )( ρ ( E ′′ )) C ∗ ( F )( ρ ( E ′′′ )) , which is equal to F ( E ′ ) F ( E ′′ ) F ( E ′′′ ) , defines a descent data on F ( E ′ ) with respect to the map E → E ′ of boundedprisms because F is a crystal. Hence it descents uniquely to an object M in16 ∆ ( E ). In particular, we have the isomorphism E ′ b ⊗ E M ∼ = F ( E ′ ) and thereexists an exact sequence0 M F ( E ′ ) F ( E ′′ ) . On the other hand, since C ∗ ( F ) is a sheaf, there exists an exact sequence0 C ∗ ( F )( E ) C ∗ ( F )( ρ ( E ′ )) C ∗ ( F )( ρ ( E ′′ )) . Comparing these two exact sequences, we see that C ∗ ( F )( E ) ∼ = M ∈ M ∆ ( E )and that the map E ′ b ⊗ E C ∗ ( F )( E, I E ) → C ∗ ( F )( ρ ( E ′ , I E ′ ))is an isomorphism. So we are done. (cid:3) We can apply the same proof as Theorem 1.23 to the other categories of crystals.
Corollary 1.24 C ∗ , C ∗ induce equivalences of categories quasi-inverse to eachother C (( X/A ) m − ∆ ) ⇄ C (( X ′ /A ) ∆ ) , C fp (( X/A ) m − ∆ ) ⇄ C fp (( X ′ /A ) ∆ ) , C tors (( X/A ) m − ∆ ) ⇄ C tors (( X ′ /A ) ∆ ) . m - q -crystalline site The purpose of this section is to prove a q -crystalline version of the theoremsin the previous section. Namely, we define the m - q -crystalline site which is ahigher level analogue of the q -crystalline site defined in Section 16 of [BS19], andprove an equivalence between the category of crystals on the m - q -crystalline siteof a smooth formal scheme X and that on the q -crystalline site of the pullbackof X by the m -th iteration φ m of the Frobenius lift φ on the base q -PD pair.First, we recall the definition of q -PD pair we want to consider. Set A = Z p J q − K with δ -structure given by δ ( q ) = 0. Definition 2.1 A q -PD pair is given by a derived ( p, [ p ] q )-complete δ -pair( D, I ) over ( A, ( q − f ∈ I , φ ( f ) − [ p ] q δ ( f ) ∈ [ p ] q I .2. The pair ( D, ([ p ] q )) is a bounded prism over ( A, ([ p ] q )), i.e., D is [ p ] q -torsion free and D/ ([ p ] q ) has bounded p ∞ -torsion.17. The ring D/ ( q −
1) is p -torsion free with finite ( p, [ p ] q )-complete Tor-amplitude over D .4. D/I is classically p -complete. Remark 2.2
1. For Definition 2.1, we adopt the one given in Section 7.1 of[Kos20]. The final condition is not required in Definition 16.2 of [BS19],but it makes possible to consider the affine p -adic formal scheme Spf( D/I ).In Definition 3.1 of [GLSQ20a], the condition (iii),(iv) in Definition 2.1and the condition that D/ ([ p ] q ) has bounded p ∞ -torsion are not imposed,and the condition of derived ( p, [ p ] q )-completeness of D is replaced bythe classical ( p, [ p ] q )-completeness. But we can see by condition (ii) inDefinition 2.1 that D is classically ( p, [ p ] q )-complete from Lemma 3.7 of[BS19].2. By definition, we see that for any x ∈ I , γ ( x ) def = φ ( x ) / [ p ] q − δ ( x ) ∈ I iswell defined.3. Let J = ( φ m ) − ( I ). As φ is continuous, J ⊆ D is closed in p -adic topology.So we see that D/J is also p -adically complete.Next, we define m - q -crystalline site, which is a higher level analogue of the q -crystalline site as well as a q -analogue of the level m crystalline site. We fix anon-negative integer m and a q -PD pair ( D, I ). Definition 2.3
Let J = ( φ m ) − ( I ). Let X be a p -adic formal scheme seper-ated and p -completely smooth over D/J . We define the m - q - crystalline site ( X/D ) m - q -crys of X over D as follows. Objects are the maps ( D, I ) → ( E, I E )of q -PD pairs together with a map Spf( E/J E ) → X over D/J satisfying thefollowing condition, where J E = ( φ m ) − ( I E ):(*) Spf( E/J E ) → X factors through some open affine Spf( R ) ⊆ X .We shall often denote such an object by(Spf( E ) ← Spf(
E/J E ) → X ) ∈ ( X/D ) m - q -crys or ( E, I E ) if no confusion arises. A morphism (Spf( E ′ ) ← Spf( E ′ /J E ′ ) → X ) → (Spf( E ) ← Spf(
E/J E ) → X ) is a map of formal schemes Spf( E ′ ) → Spf( E ) over D which is compatible with the morphisms Spf( E ′ /J E ′ ) → X , Spf( E/J E ) → X .When we denote an object by ( E, I E ), we shall write a morphism from ( E ′ , I E ′ )to ( E, I E ) as ( E, I E ) → ( E ′ , I E ′ ), not ( E ′ , I E ′ ) → ( E, I E ). A morphism ( E, I E ) → ( E ′ , I E ′ ) in ( X/D ) m - q -crys is a cover if it is a ( p, [ p ] q )-completely faithfullyflat map and [ I E E ′ = I E ′ , (2)where the completion is classical ( p, [ p ] q )-completion.18 emark 2.4 Note that the equality (2) does not immediately imply the equal-ity [ J E E ′ = J E ′ .We need to check that ( X/D ) m - q -crys with the topology as defined above isindeed a site. Thanks to Lemma 3.8 of [MT20], we have the following lemma: Lemma 2.5
Let ( E , I E ) f ←− ( E, I E ) g −→ ( E , I E ) be maps in ( X/D ) m - q - crys such that f is a covering. Let E := E b ⊗ E E . Then ( E , \ I E E ) is the objectthat represents the coproduct ( E , I E ) ⊔ ( E,I E ) ( E , I E ) in ( X/D ) m - q - crys , andthe canonical map ( E , I E ) → ( E , I E ) is a covering. By Lemma 2.5, the covers defined in (
X/D ) m - q -crys is actually a pretopology.Note that this topology differs from that in [BS19] and that in [Kos20]. Remark 2.6
Applying to the case m = 0, we see that the topology in q -crystalline site as defined above is actually a pretopology.In the situation in Definition 2.3 let X ′ be X b × Spf(
D/J ) , ( φ m ) ∗ Spf(
D/I ). Then wehave the following diagram:Spf( D ) Spf( D/J ) X Spf( D ) Spf( D/I ) X ′ . (cid:3) ( φ m ) ∗ ( φ m ) ∗ We can define a natural functor ρ from the m - q -crystalline site of X over D to the usual q -crystalline site of X ′ over D as in the case of m -prismatic site.For an object (Spf( E ) ← Spf(
E/J E ) → X ) of ( X/D ) m - q -crys , define the object ρ (Spf( E ) ← Spf(
E/J E ) → X ) of ( X ′ /D ) q -crys by (Spf( E ) ← Spf(
E/I E ) → X ′ )where the right map is defined as follows:Spf( E/I E ) → Spf(
E/J E ) b × Spf(
D/J ) , ( φ m ) ∗ Spf(
D/I ) → X b × Spf(
D/J ) , ( φ m ) ∗ Spf(
D/I ) = X ′ . Here the first map is the one induced by a map of rings
E/J E b ⊗ D/J,φ m D/I → E/I E ; e ⊗ d φ m ( e ) d. This defines the functor ρ : ( X/D ) m - q -crys → ( X ′ /D ) q -crys .Next we want to show that ρ induces an equivalence of topoi. As the m - q -crystalline site is defined in the same way as the m -prismatic site except sometechnical conditions, almost all the propositions can be proved in the same wayas the case of m -prismatic site by replacing the ring A with D . So we do notrepeat the proof. In particular, we have the following proposition.19 roposition 2.7 The functor ρ is fully faithful, continuous and cocontinuous. We make the proof of the next proposition in detail, as the construction ofthe ring S is different from that in the m -prismatic case. Proposition 2.8
Let ( E, I E ) be an object in ( X ′ /D ) q - crys . Then there ex-ists an object ( E ′ , I E ′ ) in ( X/D ) m - q - crys and a covering of the form ( E, I E ) → ρ ( E ′ , I E ′ ) .Proof. Take an open covering X = S i Spf( R i ) of X by finitely many affine openformal subschemes, and let Spf( R ′ i ) = Spf( R i ) b × X X ′ , Spf( E i ) = Spf( R ′ i ) b × X ′ Spf(
E/I E ).Then, by Lemma 1.13, there exists uniquely an open affine formal subschemeSpf( E i ) of Spf( E ) which lifts Spf( E i ), namely, E i / [ I E E i = E i . By consideringthe covering ( E, I E ) → Q i ( E i , [ I E E i ) in ( X ′ /D ) q -crys , we see that we may re-place ( E, I E ) by ( E i . [ I E E i ) to prove the proposition. Thus we may assume thatthe structure morphism Spf( E/I E ) → X ′ of the object ( E, I E ) in ( X ′ /D ) q -crys factors through an open affine formal subscheme Spf( R ′ ) ⊆ X ′ such that R ′ isof the form R b ⊗ D/J,φ m D/I for some open affine subscheme Spf( R ) ⊆ X .As the map D/J → R is smooth, it is topologically finitely generated.So there exists a surjection D [ x , . . . , x n ] ∧ → R whose kernel is the ideal( J, y , . . . , y r ) such that y , . . . , y r ∈ D/ ( p, J )[ x , . . . , x n ] is a regular sequence.On the other hand, there is a natural map D [ x , . . . , x n ] ∧ → D { x , . . . , x n } ∧ ,where the symbol {} denote the adjoining of elements in the theory of δ -rings. By the argument similar to the proof of Corollary 1.15, we see that φ m ( y ) , . . . , φ m ( y r ) is a ( p, [ p ] q )-completely regular sequence relative to D . Thenwe can construct a map D { x , . . . , x n } ∧ → S def = ( D { x , . . . , x n , φ m +1 ( y )[ p ] q , . . . , φ m +1 ( y r )[ p ] q } ) ∧ to the q -PD envelope with respect to ( I, φ m ( y ) , . . . , φ m ( y r )) by Lemma 16.10of [BS19]. From the property of q -PD envelope, for all i , φ m ( y i ) ∈ I S and so y i ∈ J S = ( φ m ) − ( I S ). This gives a map R → S/J S .Next, let D [ x , . . . , x n ] ∧ → R ′ be the morphism induced by the base change D [ x , . . . , x n ] ∧ → R b ⊗ D,φ m D of the map D [ x , . . . , x n ] ∧ → R in the previous para-graph by φ m : D → D . Then the kernel of this map is the ideal ( I, y ′ , . . . , y ′ r ),where y ′ i is the image of y i by the map D { x , . . . , x n } ∧ → D { x , . . . , x n } ∧ sending X j α j x j to X j φ m ( α j ) x j . (Here we used multi-index notation.) Bydefinition, y ′ , . . . , y ′ r ∈ D/ ( p, I )[ x , . . . , x n ] is a regular sequence. By the argu-ment similar to the proof of Corollary 1.15, we see that y ′ , . . . , y ′ r is a ( p, [ p ] q )-completely regular sequence relative to D . Then we can construct a map D { x , . . . , x n } ∧ → S ′ def = ( D { x , . . . , x n , φ ( y ′ )[ p ] q , . . . , φ ( y ′ r )[ p ] q } ) ∧ to the q -PD envelope with respect to ( I, y ′ , . . . , y ′ r ). So we get the following20iagram: D/I R ′ D D [ x , . . . , x n ] ∧ D { x , . . . , x n } ∧ S ′ def = ( D { x , . . . , x n , φ ( y ′ )[ p ] q , . . . , φ ( y ′ r )[ p ] q } ) ∧ . From the relation between y i and y ′ i , there exists a map ( φ m ) ′ : S ′ → S , where( φ m ) ′ is an D - δ -ring map sending x i to φ m ( x i ). From Proposition 2.9 below,( φ m ) ′ is ( p, [ p ] q )-completely faithfully flat.Let ( E, I E ) be an object in ( X ′ /D ) q -crys . Then we have a map f : D [ x , . . . , x n ] ∧ → E/I E by composing D [ x , . . . , x n ] ∧ → R ′ in the above diagram and R ′ → E/I E .As D [ x , . . . , x n ] ∧ is the completion of a polynomial ring, there exists a lifting f : D [ x , . . . , x n ] ∧ → E of f . Since E is a δ -ring, we can extend the map f tothe map f : D { x , . . . , x n } ∧ → E . As the image of ideal K def = ( I, y ′ , . . . , y ′ r )in E/I E equals to zero and S ′ is the q -PD envelope with respect to K , we canextend the map f to the map g : S ′ → E . As the base change of the map( φ m ) ′ : S ′ → S along g , we obtain a map h : E → E b ⊗ S ′ S which is ( p, [ p ] q )-completely faithfully flat by the property of ( φ m ) ′ shown above.It remains to show that this covering is actually a morphism of the form( E, I E ) → ρ ( E b ⊗ S ′ S, I E b ⊗ S ′ S )in ( X ′ /D ) q -crys . It is sufficient to prove the commutativity of the rectangleat the right below of the diagram, in which all the other rectangles are allcommutative: D [ x , . . . , x n ] ∧ S E b ⊗ S ′ SD [ x , . . . , x n ] ∧ R ′ = R b ⊗ D/J,φ m D/I S/J S b ⊗ D/J,φ m D/I E b ⊗ S ′ S/J E b ⊗ S ′ S b ⊗ D/J,φ m D/IE E/I E E b ⊗ S ′ S/I E b ⊗ S ′ Si h where i is the map induced by the map to the coproduct in Lemma 2.5. Thiscan be checked by corresponding the elements x i : x i x i ⊗ x i x i x i ⊗ x i ⊗ f ( x i ) f ( x i ) 1 ⊗ φ m ( x i ) = f ( x i ) ⊗ .
21o we have the commutativity. (cid:3)
We prove the claim used in the proof above.
Proposition 2.9
In the notation of the proof of Proposition 2.8, the map ( φ m ) ′ : S ′ → S is ( p, [ p ] q ) -completely faithfully flat.Proof. Since the map ( φ m ) ′ is the ( p, [ p ] q )-completed base change of the D - δ -ring map ( φ m ) ′′ : D { x , . . . , x n } ∧ → D { x , . . . , x n } ∧ , x i φ m ( x i ) , it suffices to prove that this map is ( p, [ p ] q )-completely faithfully flat. By Lemma2.11 of [BS19], the D - δ -ring map( φ m ) ′′′ : D { x , . . . , x n } → D { x , . . . , x n } , x i φ m ( x i )is faithfully flat. Since the map ( φ m ) ′′ is obtained by taking the base changeof ( φ m ) ′′′ with respect to D { x , . . . , x n } → D { x , . . . , x n } ∧ and then takingthe derived ( p, [ p ] q )-completion, we conclude that ( φ m ) ′′ is ( p, [ p ] q )-completelyfaithfully flat. (cid:3) The functor ρ : ( X/D ) m - q -crys → ( X ′ /D ) q -crys induces a functor between thecategories of presheaves of sets b ρ ∗ : \ ( X ′ /D ) q -crys → \ ( X/D ) m - q -crys G G ◦ ρ, and it admits a right adjoint b ρ ∗ : \ ( X/D ) m - q -crys → \ ( X ′ /D ) q -crys . By Proposition 2.7, we obtain a morphism between the topoi of the sheaves ofsets C : ^ ( X/D ) m - q -crys → ^ ( X ′ /D ) q -crys C ∗ = b ρ ∗ , C ∗ = b ρ ∗ . Theorem 2.10
The morphism
C : ^ ( X/D ) m - q - crys → ^ ( X ′ /D ) q - crys is an equiv-alence of topoi.Proof. This follows from Propositions 2.7,2.8 and Proposition 4.2.1 of [Oya17](see also Proposition 9.10 of [Xu19]). (cid:3)
Next we define the category of crystals with respect to the categories of modulesdefined in Definition 1.1. 22 efinition 2.11
Let C ∆ (( X/D ) m - q -crys ) (resp. C (( X/D ) m - q -crys ) , C fp (( X/D ) m - q -crys ) , C tors (( X/D ) m - q -crys )) be the category of presheaves F on ( X/D ) m - q -crys suchthat, for any object ( E, I E ) in ( X/D ) m - q -crys , F ( E ) ∈ M ∆ ( E ) (resp. M ( E ) , M fp ( E ) , M tors ( E )) and for any morphism ( E, I E ) → ( E ′ , I E ′ ), the induced map F ( E ) b ⊗ E E ′ → F ( E ′ ) is an isomorphism of E ′ -modules. Remark 2.12
As in Remark 1.19, we can prove that presheaves F in theabove definition is automatically sheaves and the category C ∆ (( X/D ) m - q -crys )is unchanged even if we drop the condition (*) in Definition 2.3.We can prove that the morphism C induces an equivalence of categories ofcrystals. Since we can prove the following three propositions, the theorem andthe corollary in the same way as the case of m -prismatic site, we do not repeatthe proof. Proposition 2.13
Let ( E, I E ) be an object of ( X/D ) m - q - crys and g : ρ ( E, I E ) → ( E ′ , I E ′ ) a morphism of ( X ′ /D ) q - crys . Then there exist an object ( E ′′ , I E ′′ ) of ( X/D ) m - q - crys and a morphism f : ( E, I E ) → ( E ′′ , I E ′′ ) of ( X/D ) m - q - crys suchthat g = ρ ( f ) . If g is a covering, so is f . Proposition 2.14
Let g : ( E, I E ) → ( E , I E ) be a morphism of ( X ′ /D ) q - crys .Then there exists a morphism h : ( E ′ , I E ′ ) → ( E ′ , I E ′ ) of ( X/D ) m - q - crys andcoverings f : ( E, I E ) → ρ ( E ′ , I E ′ ) and f : ( E , I E ) → ρ ( E ′ , I E ′ ) such that thefollowing diagram is a pushout diagram: ( E , I E ) ρ ( E ′ , I E ′ )( E, I E ) ρ ( E ′ , I E ′ ) . f q g f ρ ( h ) Proposition 2.15
Let ( E ′ , I E ′ ) be an object of ( X ′ /D ) q - crys , ( E, I E ) an objectof ( X/D ) m - q - crys and ( E ′ , I E ′ ) → ρ ( E, I E ) a covering. Then there exists an ob-ject ( E , I E ) of ( X/D ) m - q - crys and two morphisms p , p : ( E, I E ) → ( E , I E ) such that ρ ( E , I E ) = ρ ( E, I E ) b ⊗ ( E ′ ,I E ′ ) ρ ( E, I E ) and that ρ ( p ) ( resp. ρ ( p )) isthe map ρ ( E, I E ) → ρ ( E, I E ) b ⊗ ( E ′ ,I E ′ ) ρ ( E, I E ) to the first ( resp. second ) compo-nent. Theorem 2.16 C ∗ , C ∗ induce equivalences of categories quasi-inverse to eachother C ∆ (( X/D ) m - q - crys ) ⇄ C ∆ (( X ′ /D ) q - crys ) . Corollary 2.17 C ∗ , C ∗ induce equivalences of categories quasi-inverse to eachother C (( X/D ) m - q - crys ) ⇄ C (( X ′ /D ) q - crys ) , fp (( X/D ) m - q - crys ) ⇄ C fp (( X ′ /D ) q - crys ) , C tors (( X/D ) m - q - crys ) ⇄ C tors (( X ′ /D ) q - crys ) . Finally, we want to see the relationship between the category of crystals on the m - q -crystalline site and the category of certain stratifications. Definition 2.18
1. Let (
D, I ) be a q -PD pair, J = ( φ m ) − ( I ) and X be a p -adic formal scheme seperated and p -completely smooth over D/J . Wedefine the category S as follows. Objects are the maps ι : T → R suchthat T is a finite set, Spf( R ) ⊆ X is an open affine formal subscheme and ι is a map satisfying the following condition: D ι := D [ x t ] ∧ t ∈ T → R is surjective .x t ι ( t )A morphism from ι : T → R to ι ′ : T ′ → R ′ in S is a pair ( f, g ) of a mapof sets f : T → T ′ and a map of p -complete rings g : R → R ′ such thatthe diagram T RT ′ R ′ . ιf gι ′ is commutative and that the map g ∗ : Spf( R ′ ) → Spf( R ) induced by g iscompatible with open immersions Spf( R ′ ) ⊆ X, Spf( R ) ⊆ X .For objects ι : T → R and ι ′ : T ′ → R ′ in S , their sum ι ⊔ ι ′ is given bythe map T ⊔ T ′ → R b ⊗ D/J R ′ → R ′′ , where the first map is given by t ι ( t ) ⊗ t ∈ T ) , t ′ ⊗ ι ′ ( t ′ )( t ′ ∈ T ′ ), R ′′ is defined by the equality Spf( R ′′ ) = Spf( R ) ∩ Spf( R ′ ) ⊆ X (it is well-defined as X is separated) and the second map is the one induced by theopen immersions Spf( R ′′ ) ⊆ Spf( R ) , Spf( R ′′ ) ⊆ Spf( R ′ ).2. For ι : T → R ∈ S , we define S ι as in the proof of Proposition 2.8: Namely, we define S ι = D { x t , φ m +1 ( y w )[ p ] q } ∧ t ∈ T,w ∈ W to be the q -PD en-velope of D { x t } ∧ t ∈ T with respect to the ideal ( I, ( φ m ( y w )) w ∈ W ), where y w ( w ∈ W ) are defined so that ( J, ( y w ) w ∈ W ) is the kernel of the sur-jection D [ x t ] ∧ t ∈ T → R and that y , . . . , y r ∈ ( D/J )[ x t ] t ∈ T forms a regularsequence. Note that it is independent of the choice of the elements y w ( w ∈ W ): Indeed, the ideal ( J, ( y w ) w ∈ W ) is independent of the choice when re-garded as an ideal of D { x t } ∧ t ∈ T and so is the ideal ( I, ( φ m ( y w )) w ∈ W ) =( φ m ( J, ( y w ) w ∈ W ) , I ). Since the q -PD envelope depends only on the ideal( I, ( φ m ( y w )) w ∈ W ) by Lemma 16.10 of [BS19], we obtain the required in-dependence. In particular, we see that the construction of S ι is functorialwith respect to ι ∈ S . 24. A stratification with respect to S and M ∆ (resp. M ) is a pair (( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ),where M ι ∈ M ∆ ( S ι ) (resp. M ( S ι )) and ϕ ιι ′ : S ι ′ b ⊗ S ι M ι ≃ −→ M ι ′ is anisomorphism of S ι ′ -modules satisfying the cocycle condition. We denotethe category of stratifications with respect to S and M ∆ (resp. M ) byStr ∆ ( S ) (resp. Str( S )). Proposition 2.19
There exists an equivalence of the categories C ∆ (( X/D ) m - q - crys ) ≃ −→ Str ∆ ( S ) . Proof.
We have the functor C ∆ (( X/D ) m - q -crys ) → Str ∆ ( S ) defined by F (( F ( S ι )) ι ∈ S , ( S ι ′ b ⊗ S ι F ( S ι ) ≃ −→ F ( S ι ′ )) ι → ι ′ ) . The functor Str ∆ ( S ) → C ∆ (( X/D ) m - q -crys ) can be defined as follows: Given(( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ) ∈ Str ∆ ( S ) and ( E, I E ) ∈ ( X/D ) m - q -crys , choose an affineopen Spf( R ) ⊆ X such that Spf( E/J E ) → X factors through Spf( R ), and anobject in S of the form ι : T → R . Then we have a map f : D [ x t ] ∧ t ∈ T → E/J E by composing D [ x t ] ∧ t ∈ T → R and R → E/J E . As D [ x t ] ∧ t ∈ T is the completionof a polynomial ring, there exists a lifting f : D [ x t ] ∧ t ∈ T → E of f . Since E is a δ -ring, we can extend the map f to the map f : D { x t } ∧ t ∈ T → E . Sincethe image of y w ( w ∈ W ) by f belongs to J E , the image of φ m ( y w )( w ∈ W )belongs to I E . As the image of ideal K := ( I, ( φ m ( y w )) w ∈ W ) in E/I E equals tozero and S ι is the q -PD envelope with respect to K , we can extend the map f to the morphism g : S ι → E of the category ( X/D ) m - q -crys . So we can define apresheaf F by F ( E ) := E b ⊗ S ι M ι ∈ M ∆ ( E ) . We must show that the presheaf F is well-defined. If we choose another affineopen Spf( R ′ ), ι ′ : T ′ → R ′ which is an object in S and a morphism S ι ′ → E ,then we have a diagram S ι S ι ⊔ ι ′ E.S ι ′ So we have the isomorphism E b ⊗ S ι M ι = E b ⊗ S ι ⊔ ι ′ ( S ι ⊔ ι ′ b ⊗ S ι M ι ) ϕ ι,ι ⊔ ι ′ −−−−→ ≃ E b ⊗ S ι ⊔ ι ′ M ι ⊔ ι ′ ϕ ι ′ ,ι ⊔ ι ′ ←−−−−− ≃ E b ⊗ S ι ⊔ ι ′ ( S ι ⊔ ι ′ b ⊗ S ι ′ M ι ′ )= E b ⊗ S ι ′ M ι ′ . F ( E ) is independent of the choices. The map E ′ b ⊗ E F ( E ) → F ( E ′ )is also defined and is an isomorphism. So F ∈ C ∆ (( X/D ) m - q -crys ) and thefunctor Str ∆ ( S ) → C ∆ (( X/D ) m - q -crys ) can be defined by(( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ) F . These two functors we constructed are quasi-inverse to each other. (cid:3)
We can apply the same proof as Proposition 2.19 to C (( X/D ) m - q -crys ). Corollary 2.20
There exists an equivalence of the categories C (( X/D ) m - q - crys ) ≃ −→ Str( S ) . m -prismatic site and ( m − q -crystalline site In Theorem 16.17 of [BS19], it is shown that, in a certain setting, the q -crystalline cohomology of a smooth formal scheme X is isomorphic to the pris-matic cohomology of the pullback of X by the lift φ of Frobenius on the base.Because the latter is isomorphic to the 1-prismatic cohomology of X by our re-sult, it would be natural to expect that the m -prismatic site is naturally relatedto the ( m − q -crystalline site. In this section, we prove that the category ofcrystals on these two sites are equivalent.Let ( D, I ) be a q -PD pair, and let J q = ( φ m − ) − ( I ) and J ∆ = ( φ m ) − ([ p ] q D ).Note that by Corollary 16.8 of [BS19], I ⊆ φ − ([ p ] q D ) and in particular, J q ⊆ J ∆ . Let X ′ be X b × Spf(
D/J q ) Spf(
D/J ∆ ), where the completion is classi-cal ( p, [ p ] q )-completion. Then we have the following diagram:Spf( D ) Spf( D/J q ) X Spf( D ) Spf( D/J ∆ ) X ′ . (cid:3) We define a functor α from the ( m − q -crystalline site of X over D to the m -prismatic site of X ′ over D . For an object (Spf( E ) ← Spf(
E/J
E,q ) → X )of ( X/D ) ( m − q -crys , define the object α (Spf( E ) ← Spf(
E/J
E,q ) → X ) of( X ′ /D ) m − ∆ by (Spf( E ) ← Spf(
E/J E, ∆ ) → X ′ ) where the right map is de-fined as follows:Spf( E/J E, ∆ ) → Spf(
E/J
E,q ) b × Spf(
D/J q ) Spf(
D/J ∆ ) → X b × Spf(
D/J q ) Spf(
D/J ∆ ) = X ′ . α : ( X/D ) ( m − q -crys → ( X ′ /D ) m − ∆ . One can checkthat α is continuous and thus induces the morphism of topoi b α : ^ ( X ′ /D ) m − ∆ → ^ ( X/D ) ( m − q -crys . Theorem 3.1 b α induces an equivalence between the categories of crystals b α ∗ : C ∆ (( X ′ /D ) m − ∆ ) ≃ −→ C ∆ (( X/D ) ( m − - q - crys ) . The same holds true for C fp and C tors .Proof. Define I max = φ − ([ p ] q D ) , J q, max = ( φ m − ) − ( I max ) = ( φ m ) − ([ p ] q D ) = J ∆ . We claim that the category of crystals C ∆ (( X/D ) ( m − q -crys ) is unchanged when J q is replaced by J q, max and X is replaced by X max := X × Spf(
D/J q ) Spf(
D/J q, max ).If this is true, then we can replace J q by J q, max and then α is clearly the identity.The result follows.We show the claim above. We can see by the Proposition 2.19 that C ∆ (( X/D ) m - q -crys ) ≃ −→ Str ∆ ( S X ) , where S X is the category S constructed in Definition 2.18 with respect to X.It is enough to show that there exists a natural equivalence C ∆ (( X max / ( D, I max )) m - q -crys ) ≃ −→ Str ∆ ( S X ) . To do so, first we define a functor C ∆ (( X max / ( D, I max )) m - q -crys ) → Str ∆ ( S X ).For an object ( ι : T → R ) ∈ S X , put R ′ = R b ⊗ D/J q D/J q, max . Then, ifwe denote the kernel of the surjection D [ x t ] ∧ t ∈ T → R by ( J q , ( y w ) w ∈ W ) as inDefinition 2.18, the kernel of the surjection D [ x t ] ∧ t ∈ T → R → R ′ is equal to( J q, max , ( y w ) w ∈ W ). Hence, by definition, S ι is naturally regarded also as anobject of ( X max / ( D, I max )) m - q -crys . Thus we can define the functor C ∆ (( X max / ( D, I max )) m - q -crys ) → Str ∆ ( S X )by F (( F ( S ι )) ι ∈ S X , ( S ι ′ b ⊗ S ι F ( S ι ) ≃ −→ F ( S ι ′ )) ι → ι ′ ) . A functor in the opposite direction Str ∆ ( S X ) → C ∆ (( X max / ( D, I max )) m - q -crys )is defined as follows: Given (( M ι ) ι ∈ S X , ( ϕ ιι ′ ) ι → ι ′ ) ∈ Str ∆ ( S X ) and (Spf( E ) ← Spf(
E/J E ) → X max ) ∈ ( X max / ( D, I max )) m - q -crys , choose an affine open for-mal subscheme Spf( R ′ ) ⊆ X max such that Spf( E/J E ) → X max factors throughSpf( R ′ ). Since the map X max → X is a closed immersion defined by the ideal J q, max which is a nil ideal in D/J q , there exists an open formal subscheme U ⊆ X which lifts Spf( R ′ ). Since the morphism U → Spf(
D/J q ) is a ( p, [ p ] q )-completely27mooth morphism which lifts the morphism Spf( R ′ ) → Spf(
D/J q, max ), we seeby Lemma 1.13 that U is affine. So we write U = Spf( R ) and we have thefollowing diagram: Spf( R ) X Spf(
D/J q )Spf( E/J E ) Spf( R ′ ) X max Spf(
D/J q, max ) . We can take an object in S X of the form ι : T → R . Then we obtain S ι whichcan be regarded as an object in ( X max / ( D, I max )) m - q -crys and we can constructa morphism S ι → E in ( X max / ( D, I max )) m - q -crys , as in the proof of Proposition2.19. So we can define a presheaf F by F ( E ) := E b ⊗ S ι M ι ∈ M ∆ ( E ) . We can prove that the presheaf F is well-defined and that it defines an objectof C ∆ (( X max / ( D, I max )) m - q -crys ), in the same way as the proof in Proposition2.19. So the functor Str ∆ ( S X ) → C ∆ (( X max / ( D, I max )) m - q -crys ) is defined by(( M ι ) ι ∈ S X , ( ϕ ιι ′ ) ι → ι ′ ) F . Also, we see that the functors we constructed are quasi-inverse to each other.Hence C ∆ (( X max / ( D, I max )) m - q -crys ) is equivalent to Str ∆ ( S X ) and so the proofof the theorem is finished. (cid:3) The equivalences of categories of crystals proved in the previous sections aremodeled on the Frobenius descent of Berthelot, which gives an equivalence be-tween the category of crystals on m -crystalline site and that on crystalline site.However, Frobenius descent was proved through the description of crystals interms of stratifications and it did not follow from a certain equivalence of topoi,because the m -crystalline site is not defined so that such a site-theoretic proofwould work. In this section, first we give an alternative, site-theoretic proof ofFrobenius descent under a certain setting, by suitably modifying the definitionof m -crystalline site without changing the category of crystals and performingthe site-theoretic argument in the previous sections to the modified site. Also,using our alternative proof of Frobenius descent, we prove that the q = 1 case ofthe equivalence between the category of crystals on m - q -crystalline site and thaton q -crystalline site proved in Section 2 is compatible with Frobenius descent.First we recall the definition of the m -crystalline site. Note that the site we con-sider here is the ‘affine’, ‘big’, ‘flat topology’ and possibly ‘ p -adic base’ versionof the site and so it is not exactly the same as the original definition.28 efinition 4.1 Let (
D, J, I, γ ) be a p -torsion free p -complete m -PD ring or an m -PD ring on which p is nilpotent with p ∈ I . Let X be a scheme smooth andseperated over D/J . Then the m -crystalline site ( X/D ) m -crys of X over D isdefined as follows. Objects are the maps ( D, J, I, γ ) → ( E, J E , I E , γ E ) of m -PDrings together with a map Spec( E/J E ) → X over D/J satisfying the followingconditions:1. There exists some n ≥ p n E = 0.2. Spec( E/J E ) → X factors through some open affine Spec( R ) ⊆ X .We shall often denote such an object by(Spec( E ) ← Spec(
E/J E ) → X ) ∈ ( X/D ) m -crys or ( E, J E , I E , γ E ) if no confusion arises. A morphism (Spec( E ′ ) ← Spec( E ′ /J E ′ ) → X ) → (Spec( E ) ← Spec(
E/J E ) → X ) is a map of schemes Spec( E ′ ) → Spec( E ) over D which is compatible with the morphisms Spec( E ′ /J E ′ ) → X ,Spec( E/J E ) → X . When we denote an object by ( E, J E , I E , γ E ), we shallwrite a morphism from ( E ′ , J E ′ , I E ′ , γ E ′ ) to ( E, J E , I E , γ E ) as ( E, J E , I E , γ E ) → ( E ′ , J E ′ , I E ′ , γ E ′ ), not ( E ′ , J E ′ , I E ′ , γ E ′ ) → ( E, J E , I E , γ E ). A map ( E, J E , I E , γ E ) → ( E ′ , J E ′ , I E ′ , γ E ′ ) in ( X/D ) m -crys is a cover if it is a p -completely faithfully flatmap and J E ′ = J E E ′ , I E ′ = I E E ′ .We define the categories of crystals on ( X/D ) m -crys as follows. Definition 4.2
1. Let C qcoh (( X/D ) m -crys ) be the category of sheaves F such that, for any object ( E, J E , I E , γ E ) in ( X/D ) m -crys , F ( E ) is an E -module and for any morphism ( E ′ , J E , I E , γ E ) → ( E, J E ′ , I E ′ , γ E ′ ), theinduced map F ( E ) ⊗ E E ′ → F ( E ′ ) is an isomorphism of E ′ -modules.2. Let C tors (( X/D ) m -crys ) be the category of sheaves defined by C tors (( X/D ) m -crys ) = [ n C p n -tors (( X/D ) m -crys ) , where C p n -tors (( X/D ) m -crys ) is the category of p n -torsion objects in C qcoh (( X/D ) m -crys ).In particular, C tors (( X/D ) m -crys ) is a full subcategory of C qcoh (( X/D ) m -crys ).3. Let C fp (( X/D ) m -crys ) be the full subcategory of C qcoh (( X/D ) m -crys ) con-sisting of objects F such that, for all ( E, J E , I E , γ E ) in ( X/D ) m -crys , F ( E ) is a finite projective E -module. Remark 4.3
Our definition of m -crystalline site differs from the original defi-nition at the following points: 29. An object in our definition is ‘affine’ in the sense that the left two schemesappearing in the diagram(Spec( E ) ← Spec(
E/J E ) → X )are always affine, while in the usual definition, one allows an object of theform ( T ← U → X )(with the condition in our definition) such that T, U not necessarily affine.Also, in our definition, the morphism Spec(
E/J E ) → X in the diagramabove is assumed to factor through some open affine Spec( R ) ⊆ X , butno such condition is required in the usual definition.2. Our site is ‘big’ in the sense that, in the definition of an object(Spec( E ) ← Spec(
E/J E ) → X ) , no condition is imposed on the morphism Spec( E/J E ) → X , while, in theusual definition, the morphism U → X in the diagram in 1. is assumed tobe an open immersion.3. The topology on our site uses the flat topology on Spec( E ) while theZariski topology on T (in the diagram in 1.) is used in the usual definition.4. The base ring D can be a p -torsion free p -complete m -PD ring in ourdefinition, while D is an m -PD ring with p nilpotent in usual definition.By standard argument, we see that the change 1. does not change the associatedtopos and the category of crystals C qcoh (( X/D ) m -crys ), and the changes in 2.,3. do not change the category of crystals C qcoh (( X/D ) m -crys ). (Hence thecategories C p n -tors (( X/D ) m -crys ), C tors (( X/D ) m -crys ) and C fp (( X/D ) m -crys ) arealso unchanged.) As for the change in 4., when D is a p -torsion free p -complete m -PD ring, there exist equivalences C qcoh (( X/ ( D/p n D )) m -crys ) ≃ C p n -tors (( X/D ) m -crys ) , C qcoh (( X/D ) m -crys ) ≃ lim ←− n C p n -tors (( X/D ) m -crys ) ≃ lim ←− n C qcoh (( X/ ( D/p n D )) m -crys ) . Thus the category of crystals on (
X/D ) m -crys and that on ( X/ ( D/p n D )) m -crys ’s( n ∈ N ) recover each other.Next we review the Frobenius descent functor from the m -crystalline site to theusual crystalline site. Let ( D, I, γ ) be a p -torsion free p -complete PD ring or30 PD ring on which p is nilpotent such that p ∈ I and suppose there exists adiagram e X Spec(
D/I ) X ′ Spec(
D/I ) f (cid:3) ( φ m ) ∗ where f is smooth and separated and φ denotes the Frobenius on D/I . Thenwe have the functor of sites e σ : ( e X/D ) m -crys → ( X ′ /D ) crys that sends an ob-ject (Spec( E ) ← Spec(
E/J E ) → e X ) of ( e X/D ) m -crys to an object (Spec( E ) ← Spec(
E/I E ) → X ′ ) of ( X ′ /D ) crys , where the right map is defined as follows:Spec( E/I E ) g −→ Spec(
E/J E ) × Spec(
D/I ) , ( φ m ) ∗ Spec(
D/I ) → e X × Spec(
D/I ) , ( φ m ) ∗ Spec(
D/I ) . Here, the morphism g is induced by the map of rings E/J E ⊗ D/I,φ m D/I → E/I E ; e ⊗ d φ m ( e ) d. Note that (
D, I, I, γ ) is an m -PD ring and so the m -crystalline site ( e X/D ) m -crys is well-defined. Proposition 4.4
The functor e σ is cocontinuous.Proof. This can be shown as the proof of Proposition 1.11. (cid:3)
By Proposition 4.4, we can define the morphism of topoi ^ ( e X/D ) m -crys → ^ ( X ′ /D ) crys which we also denote by e σ . e σ induces the pullback functor of crystals e σ ∗ : C qcoh (( X ′ /D ) crys ) → C qcoh (( e X/D ) m -crys ) , which we call the Frobenius descent functor. Frobenius descent of Berthelotimplies that it is an equivalence: In fact, in the case where p is nilpotent in D ,we see the equivalence by gluing its local version proven in Corollaire 2.3.7 of[Ber00], and the last equivalence of Remark 4.3 implies the equivalence in thecase where D is a p -torsion free p -complete ring. (See also Th´eor`eme 4.1.3 of[Ber00].) The first purpose of this section is to give an alternative, site-theoreticproof of the equivalence e σ ∗ under a certain setting, which uses techniques similarto the ones in the previous sections. In the following, we will mainly work inthe case where D is a p -torsion free p -complete ring.For our purpose we need to modify suitably the m -crystalline site. To do so, firstwe recall the infinitesimal invariance of the category of crystals on m -crystalline31ite. Let ( D, J, I, γ ) be a p -torsion free p -complete m -PD ring or an m -PD ringin which p is nilpotent such that p ∈ I , ( J , I , γ ) be an m -PD subideal of( J, I, γ ) and suppose there exists a diagram X Spec(
D/J ) X Spec(
D/J ) f (cid:3) where f is a smooth and separated morphism. Then we have a cocontinuousfunctor of sites π : ( X / ( D, J , I , γ )) m -crys → ( X/ ( D, J, I, γ )) m -crys which isgiven by(Spec( E ) ← Spec(
E/J E ) → X ) (Spec( E ) ← X × X Spec(
E/J E ) → X ) . (It is well-defined by the proof of Proposition 2.11 in [Miy15].) Then the follow-ing holds, which is sometimes called the infinitesimal invariance of the categoryof crystals on m -crystalline site: Proposition 4.5 π ∗ induces equivalence of categories C qcoh (( X/D ) m - crys ) ≃ −→ C qcoh (( X /D ) m - crys ) . Proof.
In the case where p is nilpotent in D , the proposition follows from Corol-lary 2.14 of [Miy15]. In the case where D is a p -torsion free p -complete ring, theproposition follows from the previous case by the last equivalence of Remark4.3. (cid:3) Next, using the infinitesimal invariance of the category of crystals on m -crystallinesite, we relate the functor e σ to a functor which is in a more similar form to thefunctors studied in the previous two sections.Now, for ( D, I, γ ) be a p -torsion free p -complete PD ring such that p ∈ I , set J = Ker( D ։ D/I φ m −−→ D/I ) . Then (
D, J, I, γ ) is an m -PD ring. Suppose there exists a diagram e X Spec(
D/I ) X Spec(
D/J ) X ′ Spec(
D/I ) , f (cid:3)(cid:3) ( φ m ) ∗ f is smooth and separated. Then we can construct functors of sites π : ( e X/D ) m -crys → ( X/D ) m -crys e σ : ( e X/D ) m -crys → ( X ′ /D ) crys as above. We can also construct a functor of sites σ : ( X/D ) m -crys → ( X ′ /D ) crys that sends an object (Spec( E ) ← Spec(
E/J E ) → X ) of ( X/D ) m -crys to anobject (Spec( E ) ← Spec(
E/I E ) → X ′ ) of ( X ′ /D ) crys , where the right map isdefined as follows:Spec( E/I E ) g −→ Spec(
E/J E ) × Spec(
D/J ) , ( φ m ) ∗ Spec(
D/I ) → X × Spec(
D/J ) , ( φ m ) ∗ Spec(
D/I ) . Here, the morphism g is induced by the map of rings E/J E ⊗ D/J,φ m D/I → E/I E ; e ⊗ d φ m ( e ) d. Then we can easily check that σ ◦ π = e σ and the corresponding diagram ofpullback functors between the categories of crystals is commutative: C qcoh (( X ′ /D ) crys ) C qcoh (( X/D ) m -crys ) C qcoh (( e X/D ) m -crys ) . σ ∗ e σ ∗ ≃ π ∗ So the study of the Frobenius descent functor e σ ∗ can be reduced to studyingthe functor σ and this is more similar to our formulation of functor betweensites of level m and level 0.However, it is still difficult to apply the site-theoretic argument in the previoussections directly to the m -crystalline site because, for an object ( E, J E , I E , γ E ), E is killed by a power of p and the data ( J E , γ E ) is not determined by the pair( E, I E ). In order to overcome the difficulty, we introduce a variant of the m -crystalline site. We begin with some preparations. Fix a non-negative integer m . Definition 4.6
Let (
D, I, γ ) be a PD ring with p ∈ I . Define the ideal J max by J max = Ker( D ։ D/I φ m −−→ D/I ). Then ( J max , I, γ ) admits an m -PD structure,and for all m -PD structures of the form ( J, I, γ ), J ⊆ J max . We call an m -PDring ( D, J, I, γ ) satisfying J = J max a maximal m-PD ring .Note that, for a p -torsion free p -complete maximal m -PD ring ( D, J, I, γ ), thedata (
J, γ ) is determined by the pair (
D, I ): Indeed, J is determined by I and33he PD structure is uniquely determined by γ n ( x ) = x n n ! .Next we define a variant of m -crystalline site. Definition 4.7
Let (
D, J, I, γ ) be a p -torsion free p -complete maximal m -PDring with p ∈ I . Let X be a scheme smooth and seperated over D/J . Then the new m -crystalline site ( X/D ) m -crys,new is defined as follows. Objects are themaps ( D, J, I, γ ) → ( E, J E , I E , γ E ) of p -torsion free p -complete maximal m -PDrings together with a map Spec( E/J E ) → X over D/J satisfying the followingcondition:(*) Spec(
E/J E ) → X factors through some open affine Spec( R ) ⊆ X .We shall often denote such an object by(Spf( E ) ← Spec(
E/J E ) → X ) ∈ ( X/D ) m -crys,new or ( E, J E , I E , γ E ) if no confusion arises. A morphism (Spf( E ′ ) ← Spec( E ′ /J E ′ ) → X ) → (Spf( E ) ← Spec(
E/J E ) → X ) is a map of formal schemes Spf( E ′ ) → Spf( E ) over D which is compatible with the morphisms Spec( E ′ /J E ′ ) → X ,Spec( E/J E ) → X . When we denote an object by ( E, J E , I E , γ E ), we shallwrite a morphism from ( E ′ , J E ′ , I E ′ , γ E ′ ) to ( E, J E , I E , γ E ) as ( E, J E , I E , γ E ) → ( E ′ , J E ′ , I E ′ , γ E ′ ), not ( E ′ , J E ′ , I E ′ , γ E ′ ) → ( E, J E , I E , γ E ). A map ( E, J E , I E , γ E ) → ( E ′ , J E ′ , I E ′ , γ E ′ ) in ( X/D ) m -crys,new is a cover if it is a p -completely faithfullyflat map and I E ′ = [ I E E ′ (but we do not assume that J E ′ = [ J E E ′ ).Let X ′ be X × Spec(
D/J ) , ( φ m ) ∗ Spec(
D/I ). Then we have the functor of sites σ new : ( X/D ) m -crys,new → ( X ′ /D ) crys,new that sends an object (Spf( E ) ← Spec(
E/J E ) → X ) of ( X/D ) m -crys,new to an object (Spf( E ) ← Spec(
E/I E ) → X ′ ) of ( X ′ /D ) crys,new , where the right map is defined as follows:Spec( E/I E ) g −→ Spec(
E/J E ) × Spec(
D/J ) , ( φ m ) ∗ Spec(
D/I ) → X × Spec(
D/J ) , ( φ m ) ∗ Spec(
D/I ) . Here, the morphism g is induced by the map of rings E/J E ⊗ D/J,φ m D/I → E/I E ; e ⊗ d φ m ( e ) d. Proposition 4.8
The functor σ new is cocontinuous.Proof. This can be shown as the proof of Proposition 1.11. (cid:3)
By Proposition 4.8, the morphism of topoi can also be defined by σ new . Thefollowing statement follows. 34 roposition 4.9 The morphism σ new : ^ ( X/D ) m - crys , new → ^ ( X ′ /D ) crys , new isan equivalence of topoi.Proof. We can prove that the morphism of sites σ new : ( X/D ) m -crys,new → ( X ′ /D ) crys,new is fully faithful, continuous and cocontinuous in the same way asthe proofs in Section 1. The proposition follows from this fact, Proposition 4.10below and Proposition 4.2.1 of [Oya17] (see also Proposition 9.10 of [Xu19]). (cid:3) Proposition 4.10
Let ( E, I E ) be an object in ( X ′ /D ) crys , new . Then there existsan object ( E ′ , I E ′ ) in ( X/D ) m - crys , new and a covering of the form ( E, I E ) → σ new ( E ′ , I E ′ ) .Proof. By the argument in the proof of Proposition 1.12, we may assume thatthe structure morphism Spec(
E/I E ) → X ′ factors through some affine openSpec( R ′ ) ⊆ X ′ such that R ′ is of the form R ⊗ D/J,φ m D/I for some affine openSpec( R ) ⊆ X .Take a surjection D [ x , . . . , x n ] ∧ → R whose kernel is the ideal ( J, y , . . . , y r )such that y , . . . , y r ∈ D/J [ x , . . . , x n ] is a regular sequence. Since the imageof J in D/pD is a nil ideal, we see by Jacobian criterion that y , . . . , y r ∈ D/pD [ x , . . . , x n ] is also a regular sequence and e R := ( D/pD [ x , . . . , x n ]) / ( y , . . . , y r )is smooth over D/pD . Then, by the argument in Corollary 1.15, we see that y p m , . . . , y p m r is p -completely regular relative to D . Then, let D [ x , . . . , x n ] ∧ → S be the p -completed PD envelope with respect to the ideal ( I, y p m , . . . , y p m r ). Ifwe denote the PD ideal of S by I S and J S = Ker( S ։ S/I
S φ m −−→ S/I S ), wehave a map R → S/J S and thus ( S, I S ) is an object in ( X/D ) m -crys , new . Bydefinition, S = D [ x , . . . , x n ] ∧ [ y kpm k ! , . . . , y kpmr k ! ] ∧ k ∈ N .Next, let e R ′ be e R ⊗ D/pD,φ m D/pD . Then the natural surjection
D/pD [ x , . . . , x n ] → e R induces the surjection D/pD [ x , . . . , x n ] → e R ′ and the natural surjection e R → R induces the surjection e R ′ → R ′ via the base change by D/pD φ m −−→ D/pD .Composing these and the surjection D [ x , . . . , x n ] ∧ → D/pD [ x , . . . , x n ], we de-fine the surjection D [ x , . . . , x n ] ∧ → R ′ . Then the kernel of the map D [ x , . . . , x n ] ∧ → e R ′ is ( p, y ′ , . . . , y ′ r ), where y ′ i is the image of y i by the map D/pD [ x , . . . , x n ] → D/pD [ x , . . . , x n ] sending P j α j x j to P j α p m j x j . (Here we used multi-index no-tation.) So the kernel of the map D [ x , . . . , x n ] ∧ → R ′ is ( I, y ′ , . . . , y ′ r ). Thenlet D [ x , . . . , x n ] ∧ → S ′ be the p -completed PD envelope with respect to theideal ( I, y ′ , . . . , y ′ r ). By definition, S ′ = D [ x , . . . , x n ] ∧ [ y ′ k k ! , . . . , y ′ rk k ! ] ∧ k ∈ N .By the relation between y i and y ′ i , there exists a map ( φ m ) ′ : S ′ → S which is a D -PD-ring map sending x i to x p m i . The map ( φ m ) ′ is p -completelyfaithfully flat: Indeed, since S, S ′ are p -torsion free, it suffices to check that S ′ /pS ′ → S/pS is faithfully flat, and this map is the base change of the map
D/pD [ x , . . . , x n ] → D/pD [ x , . . . , x n ] sending x i to x p m i , which is clearly faith-fully flat. 35et ( E, I E ) be an object of ( X ′ /D ) crys , new as above. Then we have a map f : D [ x , . . . , x n ] ∧ → R ′ → E/I E and so there is a lift f : D [ x , . . . , x n ] ∧ → E of f . By the universality of PD envelope, we can extend the map f to the map g : S ′ → E . As the base change of the map ( φ m ) ′ : S ′ → S along g , we obtaina map E → E b ⊗ S ′ S which is p -completely faithfully flat.Then we can prove that this covering is actually a morphism of the form( E, I E ) → σ new ( E b ⊗ S ′ S, I E b ⊗ S ′ S ) in ( X ′ /D ) crys , new , in the same way as the proofof Proposition 1.12. So the proof of the proposition is finished. (cid:3) σ new also induces the equivalences of categories of crystals: Definition 4.11
Let C (( X/D ) m -crys,new ) (resp. C tors (( X/D ) m -crys,new ) , C p n -tors (( X/D ) m -crys,new ) , C fp (( X/D ) m -crys,new )) be the category of presheaves F on( X/D ) m -crys,new such that for any object ( E, J E , I E , γ E ) in ( X/D ) m -crys,new , F ( E ) ∈ M ( E ) (resp. M tors ( E ) , M p n -tors ( E ) , M fp ( E )) and for any morphism( E, J E , I E , γ E ) → ( E ′ , J E ′ , I E ′ , γ E ′ ), F ( E ) b ⊗ E E ′ → F ( E ′ ) is an isomorphism. Theorem 4.12 σ ∗ new induces the equivalences of categories of crystals C (( X ′ /D ) crys , new ) → C (( X/D ) m - crys , new ) . The same statement holds true for C tors , C p n - tors and C fp .Proof. We can prove the theorem in the same way as Theorem 1.23. (cid:3)
Corollary 4.13 σ ∗ induces the equivalence of categories of crystals C qcoh (( X ′ /D ) crys ) → C qcoh (( X/D ) m - crys )This corollary recovers the Frobenius descent, namely, we obtain a site-theoreticproof of the Frobenius descent in our setting. Proof.
For n ∈ N , let ν n : ( X/D ) m -crys,new → ( X/D ) m -crys be the functor ofsites which sends an object (Spf( E ) ← Spec(
E/J E ) → X ) to(Spec( E/p n E ) ← Spec((
E/p n E ) / ( J E /p n E )) → X ) . We define a functor ( X ′ /D ) crys,new → ( X ′ /D ) crys similarly, which we denotealso by ν n . Then one can check that the two functors ν n are continuous. If wedenote b ν n by the morphisms of topoi ^ ( X/D ) m -crys → ^ ( X/D ) m -crys,new , ^ ( X ′ /D ) crys → ^ ( X ′ /D ) crys,new , then b ν n, ∗ ’s induce the functors of the category of crystals36 p n -tors (( X/D ) m -crys ) → C p n -tors (( X/D ) m -crys,new ) , C p n -tors (( X ′ /D ) crys ) → C p n -tors (( X ′ /D ) crys,new ) . Also we have an equality σ ◦ ν n = ν n ◦ σ new as morphisms of sites and so itinduces the commutative diagram C p n -tors (( X ′ /D ) crys ) C p n -tors (( X/D ) m -crys ) C p n -tors (( X ′ /D ) crys,new ) C p n -tors (( X/D ) m -crys,new ) . σ ∗ b ν n, ∗ b ν n, ∗ σ ∗ new (Recall that the functors σ ∗ , σ ∗ new are the pullback functors induced by the mor-phism of topoi σ, σ new which are defined by the cocontinuity of the correspondingmorphism of sites.) Because σ ∗ new is an equivalence by Theorem 4.12 and b ν n, ∗ ’sare equivalences by Lemma 4.14 below, we conclude that the functor σ ∗ : C p n -tors (( X ′ /D ) crys ) → C p n -tors (( X/D ) m -crys )is an equivalence. By taking the projective limit with respect to n and usingthe last equivalence in Remark 4.3, we obtain the required equivalence σ ∗ : C qcoh (( X ′ /D ) crys ) → C qcoh (( X/D ) m -crys ) . (cid:3) Lemma 4.14
The functors b ν n, ∗ : C p n - tors (( X/D ) m - crys ) → C p n - tors (( X/D ) m - crys , new ) , b ν n, ∗ : C p n - tors (( X ′ /D ) crys ) → C p n - tors (( X ′ /D ) crys , new ) , in the proof of Corollary 4.13 are equivalences which are compatible with respectto n . To prove Lemma 4.14, we need to introduce the notion of stratifications for(
X/D ) m -crys , new , as in Section 2. Definition 4.15
Let (
D, I ) , J, X as above.1. We define the category S in the same way as in Definition 2.18. Namely,objects of S are the maps ι : T → R such that T is a finite set,Spec( R ) ⊆ X is an open affine subscheme and ι is a map of sets suchthat the induced map D [ x t ] ∧ t ∈ T → R ; x t ι ( t ) is surjective. For the def-inition of a morphism and the description of the sum ι ⊔ ι ′ for ι, ι ′ ∈ S ,see Definition 2.18. 37. For ι : T → R ∈ S , we define S ι as in the proof of Proposition 4.10: Namely, we define S ι to be the p -completed PD envelope of D [ x t ] ∧ t ∈ T with respect to ( I, ( y p m w ) w ∈ W ), where ( J, ( y w ) w ∈ W ) is the kernel of thesurjection D [ x t ] ∧ t ∈ T → R such that y , . . . , y r ∈ ( D/J )[ x t ] t ∈ T forms aregular sequence. Note that it is independent of the choice of the elements y w ( w ∈ W ) because the ideal ( I, ( y p m w ) w ∈ W ) = (( J, ( y w ) w ∈ W ) ( p m ) , I ) isdetermined independently of the choice. In particular, we see that theconstruction of S ι is functorial with respect to ι ∈ S .3. A stratification with respect to S and M (resp. M p n -tors ) is a pair(( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ), where M ι ∈ M ( S ι ) (resp. M p n -tors ( S ι )) and ϕ ιι ′ : S ι ′ b ⊗ S ι M ι ≃ −→ M ι ′ is an isomorphism of S ι ′ -modules satisfying the cocyclecondition. We denote the category of stratifications with respect to S and M (resp. M p n -tors ) by Str( S ) (resp. Str p n -tors ( S )).We can prove the following proposition in the same way as Proposition 2.19. Proposition 4.16
There exist equivalences of the categories C (( X/D ) m - crys , new ) ≃ −→ Str( S ) , C p n - tors (( X/D ) m - crys , new ) ≃ −→ Str p n - tors ( S ) . Using this, we prove Lemma 4.14.
Proof of Lemma 4.14 . Let S , Str p n -tors ( S ) be as above, so that the category C p n -tors (( X/D ) m -crys , new ) is equivalent to Str p n -tors ( S ). It suffices to prove thatthe category C p n -tors (( X/D ) m -crys ) is also naturally equivalent to Str p n -tors ( S ).First we define a functor C p n -tors (( X/D ) m -crys ) → Str p n -tors ( S ). For anobject ι : T → R in S , S ι /p n S ι (where S ι is as in Definition 4.15) is naturallyregarded also as an object of ( X/D ) m -crys . Thus we can define the requiredfunctor by F (( F ( S ι /p n S ι )) ι ∈ S , ( S ι ′ b ⊗ S ι F ( S ι /p n S ι ) ≃ −→ F ( S ι ′ /p n S ι ′ )) ι → ι ′ ) . A functor in the opposite direction Str p n -tors ( S ) → C p n -tors (( X/D ) m -crys ) isdefined as follows: Given (( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ) ∈ Str p n -tors ( S ) and (Spec( E ) ← Spec(
E/J E ) → X ) ∈ ( X/D ) m -crys , choose an affine open subscheme Spec( R ) ⊆ X such that Spec( E/J E ) → X factors through Spec( R ). If we take an object ι : T → R in S , we can construct a morphism S ι → E as in the proof ofProposition 2.19. So we can define a presheaf F by F ( E ) := E b ⊗ S ι M ι ∈ M ∆ ( E ) .
38e can prove that the presheaf F is well-defined and that it defines an objectof C p n -tors (( X/D ) m -crys ), in the same way as the proof in Proposition 2.19. Sothe required functor is defined by(( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ) F . Also, we see that the functors we constructed are quasi-inverse to each other.Hence C p n -tors (( X/D ) m -crys ) is equivalent to Str p n -tors ( S ) and so the proof ofthe lemma is finished. (cid:3) Finally, we compare the m - q -crystalline site and our variant of m -crystallinesite. Let ( D, I ) be a q -PD pair with q = 1 in D , i.e., a derived p -complete δ -pairover Z p satisfying the following conditions:1. For any f ∈ I , f p ∈ pI .2. The pair ( D, ( p )) is a bounded prism, i.e., D is p -torsion free. In particular, D is classically p -complete.3. D/I is classically p -complete.By Remark 16.3 of [BS19], there exists a canonical PD-structure on I .We suppose moreover that p ∈ I , and let J = ( φ m ) − ( I ). Then we have thefollowing: Lemma 4.17
With the notation above, J ( p m ) + pJ ⊆ I ⊆ J. Here, J ( p m ) denotes the ideal of D generated by x p m for all elements x of J . Inparticular, ( J, I ) is an m -PD ideal.Proof. Set J i = ( φ i ) − ( I ), then for all i , p ∈ J i . If x ∈ J i , then we see x p = φ ( x ) − pδ ( x ) ∈ J i − . So, for x ∈ J , x p m ∈ I . This implies J ( p m ) + pJ ⊆ I .On the other hand, for x ∈ I , φ m ( x ) = pφ m − ( φ ( x ) p ) ∈ I . So the statementfollows. (cid:3) Let X → Spec(
D/J ) be a smooth and seperated morphism of schemes. Wecan define a functor τ : ( X/D ) m - q -crys → ( X/D ) m -crys,new by sending ( E, I E ) to( E, J E , I E , γ E ), where J E = ( φ m ) − ( I E ) and γ E is the canonical PD-structureon I E by Remark 16.3 of [BS19]. Proposition 4.18
The functor τ is continuous.Proof. Noticing that ( p, [ p ] q )-completion is the same as p -completion since q = 1in D , we can prove the proposition in the same way as Proposition 1.10. (cid:3)
39e denote b τ by the morphism of topoi ^ ( X/D ) m -crys,new → ^ ( X/D ) m - q -crys . Then the following proposition comes true.
Proposition 4.19 b τ ∗ induces an equivalence of categories C (( X/D ) m - crys , new ) ≃ −→ C (( X/D ) m - q - crys ) . Proof.
Let S , Str( S ) be the categories defined in Definition 2.18, so that thecategory C (( X/D ) m - q -crys ) is equivalent to Str( S ). It suffices to prove that thecategory C (( X/D ) m -crys , new ) is also naturally equivalent to Str( S ).First we define a functor C (( X/D ) m -crys , new ) → Str( S ). For an object ι : T → R in S , S ι as defined in Definition 2.18 is naturally regarded also asan object of ( X/D ) m -crys , new . Thus we can define the required functor by F (( F ( S ι )) ι ∈ S , ( S ι ′ b ⊗ S ι F ( S ι ) ≃ −→ F ( S ι ′ )) ι → ι ′ ) . A functor in the opposite direction Str( S ) → C (( X/D ) m -crys , new ) is definedas follows: Given (( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ) ∈ Str( S ) and (Spf( E ) ← Spec(
E/J E ) → X ) ∈ ( X/D ) m -crys , new , choose an affine open subscheme Spec( R ) ⊆ X suchthat Spec( E/J E ) → X factors through Spec( R ). Take an object ι : T → R in S . Then we have a map f : D { x t } ∧ t ∈ T → R → E/J E . Since D { x t } ∧ t ∈ T is the completion of a polynomial ring by Lemma 2.11 of [BS19], there is alift f : D { x t } ∧ t ∈ T → E of f . Then, since S ι = D { x t , φ m +1 ( y w )[ p ] q } ∧ t ∈ T,w ∈ W isthe p -completed PD envelope of D { x t } ∧ t ∈ T with respect to ( I, ( φ m ( y w )) w ∈ W ) =( I, ( y p m w ) w ∈ W ) by Lemma 16.10 of [BS19], we can extend the map f to the map f : S ι → E . So we can define a presheaf F by F ( E ) := E b ⊗ S ι M ι ∈ M ( E ) . We can prove that the presheaf F is well-defined and that it defines an objectof C (( X/D ) m -crys , new ), in the same way as the proof in Proposition 2.19. Sothe required functor is defined by(( M ι ) ι ∈ S , ( ϕ ιι ′ ) ι → ι ′ ) F . Also, we see that the functors we constructed are quasi-inverse to each other.Hence C (( X/D ) m -crys , new ) is equivalent to Str( S ) and so the proof of the propo-sition is finished. (cid:3) We have the following commutative diagram:(
X/D ) m - q -crys ( X/D ) m -crys,new ( X ′ /D ) q -crys ( X ′ /D ) crys,new . ρ τ σ new τ
40n the other hand, we have the inclusion C (( X/D ) m - q -crys ) ⊆ C ∆ (( X/D ) m - q -crys ),so the above square induces the following diagram: C ∆ (( X/D ) m - q -crys ) C (( X/D ) m - q -crys ) C (( X/D ) m -crys,new ) C ∆ (( X ′ /D ) q -crys ) C (( X ′ /D ) q -crys ) C (( X ′ /D ) crys,new ) . b τ ∗ ≃≃ C ∗ ≃ b τ ∗ ≃ ≃ σ ∗ new In this sense, the equivalence of categories of crystals between the m - q -crystallinesite and the usual q -crystalline site in Section 2 is compatible with Frobeniusdescent. In this section, we explain the relationship of our equivalence in Sections 1,2 andthe results of Xu[Xu19], Gros-Le Stum-Quir´os[GLSQ18][GLSQ20a][GLSQ20b]and Morrow-Tsuji[MT20].First, we establish a relationship between our result and the result of Xu. Let k be a perfect field of characteristic p and W the Witt ring of k . Consider thefollowing diagram:Spf( W ) Spec( W/pW ) = Spec( k ) X Spf( W ) Spec( k ) X ′ , (cid:3) fφ ∗ φ ∗ where φ ∗ is the morphism induced by the lift φ : W → W of Frobenius and themorphism f is smooth and separated.We briefly recall some notations and results in [Xu19]. In [Xu19], Xu definedthe category E ′ (resp. E ) as follows: Objects are the diagrams ( T ← T → U )(resp. ( T ← T → U )), where T is a flat p -adic formal W -scheme, T → T (resp. T → T ) is the closed immersion defined by the ideal p O T (resp. theideal Ker( O T → O T /p O T φ −→ O T /p O T ) with φ the Frobenius) and T → U is an affine morphism over k to an open subscheme U of X ′ (resp. X ), andmorphisms are defined in obvious way. We endow E ′ , E with the topologyinduced by the fppf covering of T (see 7.13 of [Xu19]) and denote the resultingsites by E ′ fppf , E fppf , respectively. (Xu considers also the topology induced bythe Zariski topology on T in 7.9 of [Xu19], but we will not consider it here.)We define the categories of crystals C tors ( E ′ fppf ) = [ n C p n -tors ( E ′ fppf ) , C tors ( E fppf ) = [ n C p n -tors ( E fppf )41y setting C p n -tors ( E ′ fppf ) , C p n -tors ( E fppf ) to be the category of p n -torsion quasi-coherent crystals on E ′ fppf , E fppf respectively. (The categories C p n -tors ( E ′ fppf ) , C p n -tors ( E fppf )are written as C qcohfppf ( O E ′ ,n ) , C qcohfppf ( O E ,n ) respectively in [Xu19].)In 9.1 of [Xu19], Xu defined a functor ρ : E → E ′ by sending ( T ← T → U )to ( T ← T → U ′ ), where the right map T → U ′ is defined as follows: T → T × Spec( k ) ,φ ∗ Spec( k ) → U × Spec( k ) ,φ ∗ Spec( k ) =: U ′ . Here the first map is defined by the map of sheaves of rings O T ⊗ k,φ k → O T , t ⊗ a φ ( t ) a. Then, in Theorem 9.2 of [Xu19], he proved that the functor ρ induces theequivalence of topoi ρ : e E fppf ≃ −→ e E ′ fppf , and in Theorem 9.12 of [Xu19], he proved that the above equivalence inducesthe equivalence ρ ∗ : C p n -tors ( E ′ fppf ) ≃ −→ C p n -tors ( E fppf ) . We relate our result in Section 1 with his result. For a pair (
E, pE ) over(
W, pW ), define
E/pE = E/ Ker( E → E/pE φ −→ E/pE ) . (This is the ring-theoretic version of the notation T above.) Note that if( E, I E ) = ( E, pE ) is a δ -pair over ( W, pW ), then E/ Ker( E → E/pE φ −→ E/pE ) = E/ Ker( E φ −→ E → E/pE ) =
E/J E , where J E is the ideal considered in 1-prismatic site, i.e., J E = φ − ( I E ). Thenwe have the commutative diagram:( X/W ) − ∆ E fppf ( X ′ /W ) ∆ E ′ fppf , β ρ ′ ρβ where ρ is the functor defined by Xu, ρ ′ is the functor ρ we defined in Section1 and β , β are the functors defined as: β : (Spf( E ) ← ֓ Spec(
E/pE ) → X ′ ) (Spf( E ) ← ֓ Spec(
E/pE ) → X ′ ) ,β : (Spf( E ) ← ֓ Spec(
E/J E ) → X ) (Spf( E ) ← ֓ Spec(
E/J E = E/pE ) → X ) .
42t is easy to see that the functors β , β are continuous and so the functors aboveinduce the commutative diagram: C p n -tors (( X/W ) − ∆ ) C p n -tors ( E fppf ) C p n -tors (( X ′ /W ) ∆ ) C p n -tors ( E ′ fppf ) . b β , ∗ ≃ ρ ′∗ b β , ∗ ≃ ρ ∗ So our equivalence ρ ′∗ is compatible with Xu’s equivalence ρ ∗ .If X/k admits a lift e X/W and a lift of Frobenius to e X , then we have thefollowing diagram:MIC( e X/W ) qn C tors (( X/W ) crys ) C tors (( X/W ) − ∆ ) C tors ( E fppf ) p -MIC( e X ′ /W ) qn C tors (( X ′ /W ) ∆ ) C tors ( E ′ fppf ) . ≃ ψ ≃ λ b β , ∗ ≃ ϕ ≃ ̟ ≃ ρ ′∗ b β , ∗ ≃ ρ ∗ Here, we denote by MIC( e X/W ) (resp. p -MIC( e X ′ /W )) the category of p -powertorsion quasi-coherent O X -modules with quasi-nilpotent integrable connection(resp. p -connection) relative to W . The functor λ is obtained by composing thefunctor α ∗ : C tors (( X/W ) − ∆ ) ≃ −→ C tors (( X/W ) q -crys )in Theorem 3.1, the inverse of the functor b τ ∗ : C tors (( X/W ) crys , new ) ≃ −→ C tors (( X/W ) q -crys )constructed in Proposition 4.19 and the inverse of the functor b ν ∗ : C tors (( X/W ) crys ) ≃ −→ C tors (( X/W ) crys,new )in Lemma 4.14. The functor ϕ is constructed in Proposition 2.5 of [Shi12] andis an equivalence. By composing the functors that are equivalences, we obtainan equivalence ̟ from the category of crystals on the prismatic site to thecategory of modules with integrable p -connection. (Also, we have learned fromShiho that the equivalence ̟ has been obtained in more general situation byOgus.)Next, we consider the relationship between the functors we constructed and thetwisted Simpson correspondence by Gros-Le Stum-Quir´os. Let ( D, I ) be a q -PDpair with I = φ − ([ p ] q D ) and suppose we are given the diagram D d D [ x ] AD A ′ φ (cid:3) f f is a ( p, I )-completely ´etale morphism. We regard d D [ x ] as a δ -ringsuch that x is a rank one element, i.e., δ ( x ) = 0, and extend this δ -ring structureto A by using Lemma 2.18 of [BS19]. Set A = A ⊗ D D/I,A ′ = A ′ ⊗ D D/ [ p ] q D. Then, by Theorem 4.8 and Proposition 6.9 of [GLSQ20b], we have the followingcommutative diagram: C ∆ (( A/D ) − ∆ ) C ∆ (( A/D ) q -crys ) [ Strat (0) q ( A/D ) C ∆ (( A ′ /D ) ∆ ) [ Strat ( − q ( A ′ /D ) ≃ b α ∗ G ≃ ρ ∗ H ≃ F ∗ where b α ∗ and ρ ∗ are the functors we constructed, [ Strat (0) q ( A/D ) (resp. [ Strat ( − q ( A ′ /D ))is the category of twisted hyper-stratified modules of level 0 on A over D (resp. − A ′ over D ) defined in Definition 3.9 in [GLSQ20b], G,H and F ∗ are thefunctors which appear in Proposition 6.9 of [GLSQ20b] and F ∗ is an equivalenceof Theorem 4.8 in [GLSQ20b]. (Precisely speaking, it is not clear which categoryof modules they used in their definition of [ Strat (0) q ( A/D ) and [ Strat ( − q ( A ′ /D ).We guess that it would be reasonable to use the category M ∆ ( A ) and M ∆ ( A ′ )respectively, and so it would be related to our equivalences of the category ofcrystals with the symbol C ∆ .) So we see that our equivalence of functors fits intothe diagram in Proposition 6.9 of [GLSQ20b]. In particular, our argument givesa direct proof of the equivalence b α ∗ ◦ ρ ∗ , which they plan to prove indirectly in aforthcoming paper by showing the equivalence of functors G, H. (See Remark 2after Proposition 6.9 in [GLSQ20b].) Also, our proof of the equivalence b α ∗ ◦ ρ ∗ ,which is in the style of [Oya17], [Xu19], partially answers ‘the hope’ in Remark3 after Proposition 6.9 of [GLSQ20b].Finally, we explain a relationship of our result with a result of Morrow-Tsuji in[MT20]. To do so, we briefly recall some of the notations there. Let O be aring of integers of a characteristic 0 perfectoid field containing all p -power rootsof unity, and let A inf := W ( O ♭ ) , ǫ := (1 , ζ p , ζ p , . . . ) ∈ O ♭ , µ := [ ǫ ] − , ξ := µφ − ( µ ) , ˜ ξ := φ ( ξ ) , where φ is the Frobenius on A inf . ( φ is denoted by ϕ in[MT20].) Then the pair ( A inf , ( ˜ ξ )) is a bounded prism.Let R be a p -completely smooth O -algebra with a p -completely ´etale map Oh T ± , . . . , T ± d i → R called a framing. Using the framing, the ring R ∞ withGal( R ∞ /R ) =: Γ ∼ = Z dp and the rings A (cid:3) := A (cid:3) inf ( R ) , A (cid:3) ∞ := A inf ( R ∞ ) aredefined. For an A inf -algebra B , we put B (1) := A inf ⊗ φ,A inf B and denote the44elative Frobenius B (1) → B by F . (We use the same notation for an O -algebrabecause an O -algebra is regarded naturally as an A inf -algebra.)Let Rep µ Γ ( A (cid:3) ) (resp. Rep µ Γ ( A (cid:3) (1) )) be the category of generalized repre-sentations of Γ over A (cid:3) (resp. A (cid:3) (1) ) which is trivial modulo µ , and letqMIC( A (cid:3) ) (resp. qHIG( A (cid:3) (1) )) be the category of finite projective A (cid:3) -modules(resp. A (cid:3) (1) -modules) with flat q -connection (resp. flat q -Higgs field). We alsodefine qMIC( A (cid:3) ) qnilp (resp. qHIG( A (cid:3) (1) ) qnilp ) to be the full subcategory ofqMIC( A (cid:3) ) (resp. qHIG( A (cid:3) (1) )) consisting of ( p, µ )-adically quasi-nilpotent ob-jects. (This notation is not introduced in [MT20].)Then, in Section 3 in [MT20], they constructed the following commutativediagram C fp ((Spf( R (1) ) /A inf ) ∆ ) Rep µ Γ ( A (cid:3) (1) ) qHIG( A (cid:3) (1) )Rep µ Γ ( A (cid:3) ) qMIC( A (cid:3) ) , ev A (cid:3) (1) ev φA (cid:3) (1) ≃−⊗ A (cid:3) (1) ,F A (cid:3) ≃ where ev A (cid:3) (1) is the functor of evaluation at (Spf( A (cid:3) (1) ) ← Spf( R (1) ) = −→ Spf( R (1) )) and ev φA (cid:3) (1) is the composite of the functors ev A (cid:3) (1) and − ⊗ A (cid:3) (1) ,F A (cid:3) . Also, the arrows with ≃ are equivalences.By our result in Section 1, we see that the above diagram can be extendedas follows: C fp ((Spf( R (1) ) /A inf ) ∆ ) Rep µ Γ ( A (cid:3) (1) ) qHIG( A (cid:3) (1) ) C fp ((Spf( R ) /A inf ) − ∆ ) Rep µ Γ ( A (cid:3) ) qMIC( A (cid:3) ) , ev A (cid:3) (1) ≃ ≃−⊗ A (cid:3) (1) ,F A (cid:3) ev A (cid:3) ≃ where ev A (cid:3) is the functor of evaluation at (Spf( A (cid:3) ) ← Spf( R ) = −→ Spf( R )).In Theorem 3.2 in [MT20], they proved that the functors ev A (cid:3) (1) , ev φA (cid:3) (1) arefully faithful and they induce equivalences C fp ((Spf( R (1) ) /A inf ) ∆ ) ≃ −→ qHIG( A (cid:3) (1) ) qnilp , C fp ((Spf( R (1) ) /A inf ) ∆ ) ≃ −→ qMIC( A (cid:3) ) qnilp . Thus we see that the functor ev A (cid:3) is also fully faithful and it induces an equiv-alence C fp ((Spf( R ) /A inf ) − ∆ ) ≃ −→ qMIC( A (cid:3) ) qnilp . References [AB19] Johannes Ansch¨utz and Arthur-C´esar Le Bras. PrismaticDieudonn´e theory. arXiv: 1907.10525 , 2019.[Ber90] Pierre Berthelot. Letter to Illusie, 1990.45Ber00] Pierre Berthelot. D -modules arithm´etiques. II. Descente par Frobe-nius . M´em. Soc. Math. Fr. (N.S.), 2000. no. 81, p. vi+136.[BMS18] Bhargav Bhatt, Matthew Morrow, and Peter Scholze. Integral p -adic Hodge theory. Publ. Math. Inst. Hautes ´Etudes Sci. 128 , pages219–397, 2018.[BMS19] Bhargav Bhatt, Matthew Morrow, and Peter Scholze. TopologicalHochschild homology and integral p -adic Hodge theory. Publ. Math.Inst. Hautes ´Etudes Sci. 129 , pages 199–310, 2019.[BO78] Pierre Berthelot and Arthur Ogus.
Notes on crystalline cohomology .Princeton, N.J.: Princeton University Press, 1978. pages vi+243.[BS19] Bhargav Bhatt and Peter Scholze. Prisms and prismatic cohomol-ogy. arXiv: 1905.08229 , 2019.[GLSQ18] Michel Gros, Bernard Le Stum, and Adolfo Quir´os. Twisted dividedpowers and applications. arXiv: 1711.01907 , 2018.[GLSQ20a] Michel Gros, Bernard Le Stum, and Adolfo Quir´os. Twisted differ-ential operators and q -crystals. arXiv: 2004.14320 , 2020.[GLSQ20b] Michel Gros, Bernard Le Stum, and Adolfo Quir´os. Twisted dif-ferential operators of negative level and prismatic crystals. arXiv:2010.04433 , 2020.[Kos20] Teruhisa Koshikawa. Logarithmic prismatic cohomology I. arXiv:2007.14037 , 2020.[Miy15] Kazuaki Miyatani. Finiteness of crystalline cohomology of higherlevel . Annales de l’Institut Fourier Tome 65, no.3, 2015. p. 975-1004.[MT20] Matthew Morrow and Takeshi Tsuji. Generalised representations asq-connections in integral p -adic Hodge theory. arXiv: 2010.04059 ,2020.[NS08] Yukiyoshi Nakkajima and Atsushi Shiho. Weight filtrations on logcrystalline cohomologies of families of open smooth varieties . Lec-ture Notes in Mathematics 1959. Springer-Verlag, Berlin, 2008.[Oya17] Hidetoshi Oyama.
PD Higgs crystals and Higgs cohomology in char-acteristic p . J. Algebraic Geom. 26, 2017. 735-802.[Shi12] Atsushi Shiho. Notes on generalizations of local Ogus-Vologodskycorrespondence. arXiv: 1206.5907 , 2012.[LSQ97] Bernard Le Stum and Adolfo Quir´os. Transversal crystals of finitelevel . Ann. Inst. Fourier (Grenoble) , 1997. no. 1, p. 69-100.46LSQ01] Bernard Le Stum and Adolfo Quir´os. The exact Poincar´e lemmain crystalline cohomology of higher level . J. Algebra , 2001. no.2, p. 559-588.[Xu19] Daxin Xu. Lifting the Cartier transform of Ogus-Vologodsky mod-ulo p n . arXiv: 1705.06241arXiv: 1705.06241