Twisted differential operators of negative level and prismatic crystals

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Twisted differential operators of negative level and prismatic crystals

Michel Gros, Bernard Le Stum, Adolfo Quirós

To cite this version:

Michel Gros, Bernard Le Stum, Adolfo Quirós. Twisted differential operators of negative level and prismatic crystals. 2020. �hal-02968767�

Twisted differential operators of negative level and prismatic crystals

Michel Gros, Bernard Le Stum & Adolfo Quirós^∗ Version of October 12, 2020

Abstract

We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.

Contents

Introduction 1

1 Twisted divided powers of negative level 3

2 Twisted coordinate 9

3 Twisted calculus of negative level 12

4 Level raising and Frobenius descent 17

5 Prisms and twisted divided powers of negative level 22

6 Cartier descent and prismatic crystals 25

References 29

Introduction

Recently Barghav Bhatt and Peter Scholze have introduced in [BS19] the beautiful notion of a prism. It is built on the concept of a thickening which goes back to Alexander

∗Supported by grant PGC2018-0953092-B-I00 (MCTU/AEI/FEDER, UE).

Grothendieck in its modern form. A thickening of a ringB(always commutative) is simply another ring B together with a surjective map B Ñ B. Alternatively, we may consider the couplepB, Jq whereJ denotes the kernel of the surjection so that B“B{J. The idea behind any theory using this concept is to allow only some particular thickenings with some extra structure in order to get a better grasp onB, which is actually the main object of study even if it sometimes disappears from the picture. Grothendieck used infinitesimal thickenings (J nilpotent) in order to develop calculus overB and Berthelot extended the theory to positive characteristicp by using PD-thickenings (divided powers on J).

Aprism is a particular kind of a thickening. For example, if we start fromB “Z_prζswhere ζ is a primitivepth root of unity, we may consider the thickeningB “Zprrq´1ssobtained by sendingq toζ. The idealJ is then generated by theq-analogppq_q “1`q` ¨ ¨ ¨ `q^p´1 ofp. We will only consider here prisms over this base that we may callppq_q-prisms and we will stick tobounded prisms (a very light finiteness condition). More precisely, a prism (in this context) is appq_q-torsion freepp, q´1q-adically completeZ_prrq´1ss-algebraB endowed with a δ-structure¹ satisfying δpqq “0 (identifying q with its image inB) and such that B “B{ppqqB has bounded p⁸-torsion. An important example fromp-adic Hodge theory is Fontaine’s ring A_inf :“ WpO⁵_C

pq with q “ rζs (Teichmüller lifting of a sequence of pth roots of ζ) in which case A_inf “ O_Cp. This applies more generally to perfectoid integral rings which then correspond to what are calledperfect prisms.

Putting all infinitesimal thickenings together provides the infinitesimal site on which crystals are defined. Any crystal²(say onXsmooth overS) gives rise to a module endowed with an integrable connection (on X{S) and this defines an equivalence of categories.

Actually, in positive characteristic p, one uses PD-thickenings and obtain the crystalline site. A similar pattern occurs with prisms. In this article, we consider the following situation: we letR be appqq-prism andA a pp, q´1q-adically complete R-algebra which admits a coordinate x. There exists a unique δ-structure on A such that δpxq “ 0 and a unique endomorphism σ such that σpxq “ qx. Then, one may consider the prismatic site of A{R and we will show in section 6 that a prismatic crystal gives rise to an A- module endowed with what we call atwisted connection of level´1. In general, a twisted connection of level ´m on an A-module M is an R-linear map satisfying the following Leibniz rule:

@f PA,@sPM, ∇pf sq “ pp^mqqsbd_qpmf`σ^pmpfq∇psq.

We will denote the corresponding category by MIC^p´mq_q pA{Rq. There exists also the notion of a q-PD-thickening giving rise to theq-crystalline site and we showed in [GSQ20] that, ifais a q-PD-ideal inA, then aq-crystal onA{a gives rise to anA-module endowed with a twisted connection of level 0.

The main result of the present article is the following: we prove (theorem 4.8 and corollary 4.9) that, if A¹ denotes the frobenius pullback of A, then frobenius descent provides an equivalence of categories betweenA¹-modules endowed with a twisted connection of level

´1 andA-modules endowed with a twisted connection of level 0 (when we focus on finitely presented topologically quasi-nilpotent objects). Moreover, as we expected in [Gro20], section 6, this is related to prisms via a very general Cartier morphismC, whose definition is inspired by the proof of theorem 16.17 of [BS19], from theq-crystalline topos of a smooth formal schemeX to the prismatic topos of its frobenius pull backX¹. WhenX “SpfpA{aq,

1This is simply a stable version of a frobenius lift.

2We only consider crystals with quasi-coherent components.

in which caseX¹ “SpfpA¹{ppqqAq, there exists a commutative diagram tppq_q´prismatic crystals on X¹{Ru ^{C´1 //}

tq´crystals onX{Ru

MIC^p´1q_q pA¹{Rq ^{F˚ //}MIC^p0q_q pA{Rq.

As explained above, the bottom map is an equivalence (on finitely presented topologically quasi-nilpotent objects) and, introducing flat topology considerations, we will prove in a forthcoming article that this is also the case for the vertical maps. Of course, this implies that the upper map is also an equivalence and we expect this to be true more generally for any smooth formal scheme X. This would be a twisted arithmetic version of the results obtained by Oyama in [Oya17] and Xu in [Xu19].

Let us briefly describe the content of this article. In the first section, we introduce the ring of twisted divided polynomials of negative level and show in proposition 1.12 that it automatically inherits a δ-structure in the case of level ´1. In the second section, we explain the basics of twisted calculus on an adic ring and give a fundamental example. In section three, we develop twisted calculus of negative level. In section four, we introduce the level raising functor and show in theorem 4.8 that this is an equivalence when we move from level´1 to level 0 (Frobenius descent). In section five, we recall some basic notions on prisms and show in corollary 5.6 that the ring of twisted divided polynomials of level minus one is the prismatic envelope of the polynomial ring for thesymmetric δ-structure.

In the last section, we explain Cartier descent from prismatic crystals toq-PD-crystals and we show in proposition 6.9 that this is compatible with the raising level map of section four (which is an equivalence in this case).

1 Twisted divided powers of negative level

We recast here some results from [GLQ19] (see also section 2 of [GSQ20]) and extend them to negative level.

We letRbe any commutative ring. We fix someqPRand denote bypnq_q PRtheq-analog of an nPZě0 (see [LQ15] for example for a presentation of the theory ofq-analogs). We letA be any commutative R-algebra and fix some xPA. We also fix a prime pand some mPZ_ě0.

IfyPAand nPZě0, we will consider the twisted powers ξ^pnqq,y :“

n´1

ź

i“0

pξ` piq_qyq PArξs. (1)

They form an alternative basis for the polynomial ring Arξs as a free A-module. Note however that multiplication of twisted powers is kind of tricky because

ξ^pn1qq,yξ^pn2qq,y “

minpn1,n2q

ÿ

i“0

p´1qⁱpiqq!q^ipi´1q2 ˆn1

i

˙

q

ˆn2

i

˙

q

yⁱξ^{pn1`n2´iqq,y},

as shown in lemma 1.2 of [GLQ19]. We will need to understand how blowing-up and frobenius act on twisted powers and we can already notice the following:

Lemma 1.1. If y, z PA, then the blowing up

Arξs ÑArωs, ξÞÑzω sends ξ^pnqq,zy tozⁿω^pnqq,y.

Proof. We have śn´1

i“0 pzω` piqqzyq “zⁿśn´1

i“0 pω` piqqyq.

Later, we will assume thatR is aδ-ring (see section 1 of [GSQ20] for a short review) and in particular thatR is endowed with a lifting of frobenius given byφpfq “f^p`pδpfq. We will also assume that q has rank one which means that δpqq “0 (so that φpqq “q^p). We will denote³ by

A¹ :“R_φÔbRA

the frobenius pullback⁴ ofA. It is easy to see that the semilinear morphism of rings AÑA¹, f ÞÑf¹:“1_φÔbf

has the following effect on twisted polynomials:

Arξs ÑA¹rξs, ξ^pnqq,y ÞÑξ^pnqqp,y1 (withy PA andy¹ “1_φÔby).

Back to the general situation, the next result will follow from section 2 in [GLQ19]:

Proposition 1.2. Given y P A, there exists a unique natural multiplication on the free A-module Axξyq,y with basis tξ^rnsq,yunPZě0 turning the A-linear map

Arξs ÑAxξyq,y, ξ^pnqq,y ÞÑ pnqq!ξ^rnsq,y into a morphism of rings.

Proof. Since this is the first time that we use the term “natural”, we should make it precise (and leave it to the reader’s imagination in the future): it means that if we are given a commutative diagram of commutative rings

q_1

R1 //

A1

y_1

q2 R2 //A2 y2,

then the diagram ξ_

A1rξs ^//

A1xξyq1,y1

ξ^rns_{_}^q1,y1

ξ A₂rξs ^//A₂xξy_q2,y2 ξ^rnsq2,y2

is also commutative. Let us now prove our assertion. Existence follows from propositions 2.2 and 2.1 in [GLQ19]. For uniqueness, we may first replaceAwith some polynomial ring (in an infinite number of variables) overZrqs, then replaceZrqswith its fraction fieldQpqq and finally rely on proposition 2.1 of [GLQ19] again.

3Later we will writeA¹:“R_φÔbpRAbut the topology is discrete here.

4We always use the geometric vocabulary and say “pullback” instead of “scalar extension”.

Definition 1.3. The ringAxξyq,y is the ring of twisted divided polynomials.

Note that, whenpnqq! is invertible in R, we have ξ^rnsq,y“ ξ^pnqq,y

pnq_q!

so thatξ^rnsq,y deserves the name of atwisted divided power.

We shall now show that blowing up and frobenius both extend totwisted divided powers.

Lemma 1.4. If y, z PA, then the blowing up ξ ÞÑ zω extends naturally in a unique way to a morphism ofA-algebras

Axξy_q,zy ÑAxωy_q,y, ξ^rnsq,zy ÞÑzⁿω^rnsq,y.

Proof. First of all, the formula defines an A-linear map that extends the original blowing up. In order to show that it is a morphism of rings, we may actually assume thatR“Qpqq in which case the question becomes trivial. The same argument can be used to show uniqueness.

On proves exactly in the same way the following:

Lemma 1.5. If R is a δ-ring with δpqq “0 and A¹ denotes the frobenius pullback of A, then there exists a unique natural semilinear morphism of rings

Axξyq,yÑA¹xξyq^p,y¹, ξ^rnsq,y ÞÑξ^rnsqp,y1 (with yPA andy¹ “1_φÔby).

Note that we actually obtain an isomorphism

R_φÔbRAxξyq,y»A¹xξy_q^p_,y¹.

Recall that we fixed some parameter x PA (that we haven’t used yet). We will then set y:“ p1´qqxand simply writeξ^pnqq for the twisted powers so that

ξ^pnqq :“

n´1

ź

i“0

`ξ` p1´qⁱqx˘

PArξs.

In this situation, we will also denote byAxξyqthe ring of twisted divided polynomials and write ξ^rnsq for the corresponding twisted divided powers.

We can replaceq withq^pm in order to obtain a new ring Axξy_qpm buty needs then to be replaced byp1´q^pmqx“ pp^mqqy in order to fit the pattern (for fixedx) so that

Axξy_q:“Axξy_q,y and Axξy_qpm :“Axξy_qpm,pp^mqqy.

The new ingredient in our story is the ring obtained by changingq toq^pm but keeping the samey (in which case we’d use ω rather thanξ as indeterminate):

Definition 1.6. Given m P Zě0, the ring of twisted divided polynomials of level ´m with respect top is

Axωy_qp´mq:“Axωy_qpm,y

withy :“ p1´qqx.

We will then writeω^tnuq :“ω^rnsqpm,y and call them twisted divided powers of level ´m.

Remarks 1. The ring of twisted divided polynomials of level 0 is the same thing as the ring of twisted divided polynomials of definition 1.3 : Axωy_qp0q “Axξy_q (we will usually keepξ as indeterminate in this situation).

2. If we write explicitly the multiplication onAxωy_qp´mq, we get:

ω^tn1uqω^tn2uq “

ÿ

0ďiďmintn1,n2u

q^pmipi´1q2

ˆn1`n2´i n₁

˙

q^pm

ˆn1

i

˙

q^pm

pq´1qⁱxⁱω^tn1`n2´iuq.

We will now study blowing up and frobenius in this specific context.

Proposition 1.7. The blowing up ξ ÞÑ pp^mqqω extends uniquely to a natural morphism of A-algebras

Axξy_qpm ÑAxωy_qp´mq, ξ^rnsqpm ÞÑ pp^mqⁿ_qω^tnuq.

Proof. Apply lemma 1.4 with q replaced byq^pm and z replaced bypp^mqq.

Remarks 1. Whenpp^mqq (resp. and pnq_qpm!) is invertible inR, we may identify both rings in the proposition and we have

ω^tnuq “ ξ^rnsqpm pp^mqⁿ_q

˜

resp. “ ξ^pnqqpm pp^mqⁿ_qpnq_qpm!

¸

In particular, we will haveω “ξ{pp^mqqand this is why it is important to use different letters.

2. More generally, there exists a natural morphism ofA-algebras

Axωy_qprp´m`rq ÑAxωy_qp´mq, ω^tnuqpr ÞÑ pp^m´rqⁿ_qω^tnuq whenever rěm.

3. As a consequence, ifR is aδ-ring with δpqq “0, then there exists a unique natural semilinear morphism of rings

Axωy_qp´m`1qÑA¹xωy_qp´mq, ω^tnuq ÞÑ ppqⁿ_qω^tnuq.

(usingx¹ :“1_φÔbxPA¹ :“R_φÔbRAas fixed parameter on the right hand side).

AssumeR is a δ-ring and A is aδ-R-algebra with both q and x of rank one.

Definition 1.8. The symmetricδ-structure on Arξs is defined by δpξq “

p´1

ÿ

i“1

1 p

ˆp i

˙ x^p´iξⁱ.

It is easy to check that this is the unique δ-structure that extends the δ-structure of A and such that x`ξ also has rank one. We will also consider the corresponding relative frobenius

F :A¹rξs “R_φÔbRArξs ÑArξs which is then given byFpξq “ px`ξq^p´x^p.

We focus up to the end of the section on the case m“1.

Theorem 1.9. Assume R is a δ-ring and A is a δ-R-algebra with both q and x of rank one. Then the relative frobenius F :A¹rξs Ñ Arξs extends uniquely (through blowing up) to a natural morphism

rFs:A¹xωy_qp´1qÑAxξyq.

Proof. We have to prove that there exists a unique natural morphism making the diagram ξ_

A¹rξs ^{F //}

Arξs

ξ_{_}

ppqqω A¹xωy_qp´1q ^{rFs //}Axξyq ξ.

commutative. This requires some work but this is done in section 7 of [GLQ19] where we showed that rFsis explicitly given by

ω^tnuÞÑ

pn

ÿ

i“n

b_n,ix^pn´iξ^ris

with

b_n,i:“ piq_q!

pnq_q^p!ppqⁿ_qa_n,iPR (2) and

an,i:“

n

ÿ

j“0

p´1q^n´jqppn´jqpn´j´1q 2

ˆn j

˙

q^p

ˆpj i

˙

q

. (3)

Definition 1.10. The morphism rFs is the divided frobenius.

Corollary 1.11. The symmetric δ-structure of Arξsextends uniquely in a natural way to Axξy_q.

Proof. We define φas the composition ξ_

Axξyq

φ

((ppqqω A¹xωy_qp´1q ^{rFs //}Axξyq

and easily check that it solves the problem.

This last result also holds forAxωy_qp´1qbut we need to go back to our fancy formulas from [GLQ19] in order to prove it:

Proposition 1.12. Assume R is a δ-ring and A is a δ-R-algebra with both q and x of rank one. Then the symmetric δ-structure of Arξsextends uniquely (through blowing up) in a natural way to Axωy_qp´1q.

Proof. It is actually sufficient to show that there exists a unique natural φ-structure on Axωy_qp´1q: we may assume thatR“Zrqsand A“Rrxsarep-torsion free and the notions of δ and φ-structures are then equivalent. In order to avoid confusions, let us denote by φ_A : A Ñ A the frobenius of A. It follows from proposition 7.5 (and lemma 7.8 which shows that the sum actually starts at i“n) of [GLQ19] that the (absolute) frobenius on Arξsis theφ_A-linear morphism given by

φ:Arξs ÑArξs, ξ^pnqqp ÞÑ

pn

ÿ

i“n

φpan,iqx^pn´iξ^piqqp

where the an,i are defined in (3). We consider now the natural φA-linear morphism of A-modules

φ:Axωy_qp´1qÑAxωy_qp´1q, ω^tnuq ÞÑ

pn

ÿ

i“n

ppqⁱ_qφpb_n,iqx^pn´iω^tiuq

where the b_n,i are defined in (2). We want to show that this is a morphism of rings that reduces to the frobenius modulop. Actually, since the blowing up map

Arξs ÑAxωy_qp´1q, ξ^pnqqp ÞÑ pnq_q^p!ppqⁿ_qω^tnuq

becomes an isomorphism when allq-analogs are invertible, it is sufficient to prove that the diagram

ξ^pnq_{_}^qp

Arξs ^{φ //}

Arξs

ξ^pnq_{_}^qp

pnq_q^p!ppqⁿ_qω^tnuq Axωy_qp´1q ^{φ //}Axωy_qp´1q pnq_q^p!ppqⁿ_qω^tnuq

is commutative. We compute the image ofξ^pnqqp along both paths. On one hand, we find

pn

ÿ

i“n

piq_q^p!ppqⁱ_qφpa_n,iqx^pn´iω^tiuq and on the other hand

φ`

pnqq^p!ppqⁿ_q˘

pn

ÿ

i“n

ppqⁱ_qφpbn,iqx^pn´iω^tiuq.

In order to conclude, it is therefore sufficient to recall that φppiqq!q “ piqq^p! and formula (2) then provides an equality

piq_q^p!φpa_n,iq “φ`

pnq_q^p!ppqⁿ_q˘

φpb_n,iq.

Example Assume p“2. Then, we have

φpξq “ p1`qqξ<sup>r2s</sup>` p1`qqxξ and φpωq “ p1`qq²ω^t2u` p1`qqxω.

In particular, we see that there is nonatural δ-structure (i.e. compatible with blowing up) on the polynomial ringArωsin general: we do haveφpξq PArξsbutφpωq RArωs.

Remark The diagram

A¹xωy_qp´1q ^{rFs //}

φ

))

Axξyq

ξ_{_}

A¹xωy_qp´1q ppqqω

is commutative and may be used to define theδ-structure of Axωy_qp´1q when Aitself is a frobenius pullback (which will be the case in practice).

2 Twisted coordinate

This section is completely independent of the previous one. We recall some notions introduced in [LQ18] and give an important example of a situation where these notions apply. However, we work on adic rings as in [GSQ20], and not merely on usual rings.

We assume that R is an adic ring (not necessarily complete) and that A is a complete adic R-algebra (not necessarily R-adic at this point). Adic rings are always assumed to admit a finitely generated ideal of definition. Unless otherwise specified, completion is always meant relative to the adic topology. We will not need it here but we could even assume that R and A are Huber rings as long as P_A{R :“ AbRA is also a Huber ring (multiplication onP_A{R need not be continuous in general).

We denote by

p1 :AÑP_A{R, f ÞÑfb1 and p2 :AÑP_A{R, f ÞÑ1bf the “projections” and by

∆ :P_A{RÑA, fbgÞÑf g

the “diagonal map” (we use the - contravariant - geometric vocabulary and notations).

Unless otherwise specified, we will always considerP_A{R as anA-module through p₁ and callp2 the “Taylor map”.

Assume now that A is a twisted R-algebra, which simply means that the R-algebra A is endowed with acontinuousendomorphismσ. We extendσ toP_A{Rin an asymmetric way by the formulaσpfbgq “σpfq bg. We letI_A{R be the kernel of ∆ and define thetwisted powers and twisted principal parts of ordern:

I_A{R^{pn`1qσ</sup> :“I<sub>A{R</sub>σpI<sub>A{R</sub>q ¨ ¨ ¨σ<sup>n</sup>pI<sub>A{R</sub>q and P<sub>A{R,pnqσ</sub> :“Pp<sub>A{R</sub>{I<sub>A{R</sub><sup>pn`1qσ} (where the hat (resp. the bar) indicates the completion (resp. the closure)).

Definition 2.1. AssumeAis atwistedR-algebraand letxPA. Thenxis aσ-coordinate if the canonical map

Arξs_ďnÑP_A{R,pnqσ, ξÞÑ1bx´xb1 is bijective⁵ for all nPZ_ě0.

5This is theσ-analog ofdifferential smoothness (of relative dimension one).

It is actually convenient to extend σ to the polynomial ring Arξs by requiring that σpx`ξq “ x`ξ and to introduce the twisted powers ξ^pn`1qσ :“ ξσpξq. . . σⁿpξq P Arξs. One easily sees thatx is aσ-coordinate if and only if

Arξs{pξ^pn`1qσq »P_A{R,pnqσ (isomorphism ofA-algebras). In general, we also set

Ω_A{R,σ :“I_A{R{I^p2q_A{R^σ

and we denote by d_σ :AÑΩ_A{R,σ the map induced byp₂´p₁. ThisA-module represents theσ-derivations D:AÑM, i.e. the R-linear maps satisfying the twisted Leibniz rule

@f, gPA, Dpf gq “Dpfqg`σpfqDpgq.

Whenxis aσ-coordinate, Ω_A{R,σ is a free module on the generator d_σxand we denote by Bσ the corresponding derivation of A(so that d_σf “ Bσpfqd_σx).

In order to make clear the relation with the notions introduced in section 1, note that ifx is aσ-coordinate such thatσpxq “qxwithq PR, thenξ^{pn`1qq</sup> “ξ<sup>pn`1qσ}. In this situation, we will use q rather than σ as index or prefix in all our notations so that we will write I_A{R^pn`1qq,P_A{R,pnqq, Ω_A{R,q, d_qorBq, for example, and sayq-coordinate orq-derivation (even if everything actually depends on σ).

Let us give an important example where there exists aq-coordinate (we already used these results without proofs in [GSQ20]).

Definition 2.2. A topologically étale coordinatex onAwith respect to R is the image of an indeterminateX under some topologically étale (that is, formally étale and topologically finitely presented) map RrXs ÑA.

Proposition 2.3. Ifq´1is topologically nilpotent andxis a topologically étale coordinate on A, then there exists a unique continuous endomorphism σ of A such that σpxq “qx.

Moreover, x is a q-coordinate onA.

Proof. Since Ais complete adic andq´1 is topologically nilpotent,Ais complete for the pq ´1q-adic topology. We may therefore assume that q´1 is nilpotent. The existence and uniqueness ofσ are trivially true whenA“RrXsand x“X. In general, they follow from the commutativity of the diagram

RrXs ^{σ //}

RrXs ^//A.

A ^//

44

A{pq´1q

since RrXs Ñ A is formally étale. Now, since x is a topologically étale coordinate on A over R, it is also a topologically étale coordinate on A{pq´1q over R{pq´1q. Thus, moduloq´1, the canonical map

Arξs{ξ^pn`1qÑP_A{R,pnqq (4) reduces to an isomorphism

pA{pq´1qqrξs{ξ^n`1 »P_A{pq´1q,n.

Surjectivity of (4) therefore follows from Nakayama’s lemma. Moreover, we can build a section using the commutative diagram

RrX, ξs{ξ^{pn`1q //}

Arξs{ξ^pn`1q

P_A,pnqq ^//

33

P_A{pq´1q,n ^{oo »} pA{pq´1qqrξs{ξ^n`1 and obtain injectivity.

One may also produce a frobenius onA using the same methods:

Proposition 2.4. Assumep is a topologically nilpotent prime inAandR is aφ-ring. Ifx is a topologically étale coordinate onA, then there exists a unique structure ofφ-R-algebra onA with x of rank one (i.e. φpxq “x^p). Moreover, the relative frobenius

F :A¹ :“R_φÔbRAÑA is free of rank p.

Proof. We proceed exactly as above and it is sufficient to prove existence and uniqueness of the relative frobeniusF. First, we reduce to the case where pis nilpotent in A. In the case A “RrXsand x“X, there is nothing to do. In general, existence and uniqueness follow from the commutativity of the diagram

RrXs ^{F //}

RrXs ^//A

A^{1 ////}

44

A¹{p ^{F //}A{p,

in which the last map is the relative frobenius of A{p. It only remains to verify that F :A¹ÑAis free of rankp: actually, there exists an isomorphismA¹rTs{pT^p´1bxq »A which is obtained from the analog map when A “ RrXs after pulling back along RrXs ÑA¹.

We can improve a little bit on the previous result (see also [BS19], lemma 2.18):

Proposition 2.5. Assume pis a topologically nilpotent prime inAandRis aδ-ring. Ifx is a topologically étale coordinate onA, then there exists a unique structure of δ-R-algebra onA with x of rank one (i.e. δpxq “0).

Proof. There exists a unique structure ofδ-R-algebra onRrXssuch thatδpXq “0. Now, by definition (see section 1 of [GSQ20]), aδ-structure on A is a section of the projection W₁pAq Ñ A (where W1 denote the Witt vectors of length two). Note that the kernel V1

of this projection satisfiesV₁² ĂpV₁. By functoriality, there exists a commutative diagram W₁pRrXsq ^//

RrXs

xx

W₁pAq ^//A.

where the backward upper-map is the above trivial δ-structure of RrXs. Since V1 is nilpotent modulop and p is topologically nilpotent, the composite map

RrXs ÑW₁pRtXuq ÑW₁pAq

extends uniquely toA and defines aδ-structure onA withδpxq “0.

3 Twisted calculus of negative level

We keep the same hypotheses as before: R is an adic ring and A is a complete adic R- algebra, but we also fix⁶, as in section 1, a prime p and somem PZ_ě0. We assume that A is a twisted R-algebra and that x PA is a q^pm-coordinate (we will denote by σ^pm the corresponding endomorphism ofAeven if it needs not be a power).

Definition 3.1. Atwisted connection of level´m on anA-moduleM is an R-linear map

∇:M ÑMbΩ_A{R,qpm such that

@f PA,@sPM, ∇pf sq “ pp^mqq sbd_qpmf`σ^pmpfq∇psq.

Intuitively, this is a twisted connection with a vertical pole: a twisted connection on Mr1{pp^mqqswhich is defined over A.

We will denote⁷ by MIC^p´mq_q pA{Rq the category of A-modules endowed with a twisted connection of level ´m. Thetwisted de Rham cohomology of level ´m ofM is defined as

H_dR,qp´mq⁰ pMq “ker∇ and H_dR,qp´mq¹ pMq “coker∇ or in the derived category as

RΓ_dR,qp´mqpMq “ rM Ñ^∇ MbΩ_A{R,qpms.

Examples 1. Whenm“0, we fall back onto the notion of a twisted connection from our previous articles: MIC^p0q_q pA{Rq “MIC_qpA{Rq.

2. When q “1, we remove the word “twisted” and we obtain a p^m-connection in the sense of Deligne or Simpson as in definition 1.1 of [Shi15].

3. When pp^mqq “ 0 and σ^pm “ Id (which is then automatic in practice), a twisted connection of level´m is the same thing as aHiggs field and does not depend onq, p orm anymore. This condition obviously holds when p“0 but this also happens for example in the important case whenq is ap^mth root of unity.

4. The notion of a twisted connection of negative level is also related to the notion of a differential operator of finite radius that was studied in [LQ17] as well as, in the untwisted case, to the notion of a congruence level introduced in [HSS17]. See also corollary 1.3.15 in [Kis06], where Mark Kisin associates a λ-connection to a weakly admissible module.

6Actually, the whole story will only depend on the integerk:“p^mbut the terminology and the notations involvem.

7Be careful that our notation is different from Shiho’s [Shi15]. He use MIC^pmq for connections of level

´mbut we prefer to keep MIC^pmq for connections of positive level.

Definition 3.2. Atwisted derivation of level´m on anA-moduleM is anR-linear map θ:M ÑM such that

@f PA,@sPM, θpf sq “ pp^mqqB_q^pmpfqs`σ^pmpfqθpsq.

Of course, twisted connections and twisted derivations of level´m correspond bijectively via

∇psq “θpsqd_qpmx,

but the later notion is often more convenient for computations.

Definition 3.3. The ring of twisted differential operators of level´m onA{R is the free A-module D^p´mq_A{R,q on the generators B^xnyq for nPZ_ě0 with the commutation rules

@f PA, B^x1y_q ˝f “ pp^mqqB_q^pmpfq `σ^pmpfqB_q^x1y and

@n1, n2 PZě0, B^xn_q ^1y˝ B^xn_q ^2y“ B^xn_q ^1`n2y.

Remarks 1. If we assume that x is also a q-coordinate and if we denote by D_A{R,q the ring of twisted differential operators that we already considered in our previous articles, then there exists a naturalA-linear morphism of rings

D^p´mq_A{R,qÑD_A{R,q, B_q^xnyÞÑ pp^mqⁿ_qB_qⁿpm. (5) More generally, we have

D^p´mq_A{R,q ÑD^p´m`rq_A{R,qpr, B_q^xny ÞÑ pp^rqⁿ_qB^xny_qpr

for anyrďm ifx is also a q^pm´r-coordinate.

2. Giving a twisted connection (or a twisted derivation) of level ´m on an A-module M is equivalent to giving the structure of a D^p´mq_A{R,q-module. For the trivial twisted connection of level´m on A, we can check that

B^xny_q pfq “ pp^mqⁿ_qBⁿ_qpmpfq.

3. It is not difficult to see that

RΓ_dR,qp´mqpMq “RHom

D^p´mq_A{R,qpA, Mq.

Examples 1. In the casem“0, we fall back onto the ring D_A{R,q already considered in our previous articles.

2. In the case q “ 1, then D^p´mq_A{R,q is the same thing as the ring D^p´mq_A{R of differential operators of level´m introduced in definition 2.1 of [Shi15] (see also [HSS17]).

3. When A is ppq_q-torsion-free and x is also a q-coordinate, the map (5) is injective, and we could then as well define D^p´mq_A{R,q as the A-subalgebra of D_A{R,qpm generated byBq^x1y:“ pp^mqqB_q^pm.

Recall that we introduced in definition 1.6 the ringAxωy_qp´mq of twisted divided polyno- mials of level ´m. We will denote by I_A{R^tn`1uq the free A-module generated byω^tn1uq for n¹ąn. This defines an ideal filtration on Axωy_qp´mq.

Lemma 3.4. 1. The blowing up ξ ÞÑ pp^mqqω induces a canonical morphism of A- algebras

Arξs{pξ^{pn`1qqpm</sup>q ÑAxωy<sub>qp´mq</sub>{I<sub>A{R</sub><sup>tn`1uq}.

2. If pp^mqⁿ_qpnq_qpm! Ñ 0 in A, then the blowing up provides a canonical morphism of A-algebras

Arrξss_qpm :“lim

ÐÝArξs{pξ^pn`1qqpmq ÑAxωyz_qp´mq, ξ^pnqqpm ÞÑ pp^mqⁿ_qpnq_qpm!ω^tnuq (recall that completion is meant with respect to the adic topology of A).

Proof. The first assertion follows from the fact that pp^mqⁿ_qpnq_qpm!ω^tnuq P I_A{R^tnuq and the second one is obtained by taking the limit on both sides. More precisely, our hypothesis implies the existence of an ordered functionnÞÑrpnq and a canonical morphism

Arξs{pξ^pn`1qqpmq ÑAxωy_qp´mq{a^rpnqAxωy_qp´mq ifais an ideal of definition in A.

Definition 3.5. 1. The twisted Taylor map level ´m and order n of A{R is the composite

θ_n:A ^{p2 //}P_A{R ^//Pp_A{R{Ip_A{R^pn`1qqpm

»

Arξs{pξ^{pn`1qqpm</sup>q <sup>//</sup>Axωy<sub>qp´mq</sub>{I<sub>A{R</sub><sup>tn`1uq}.

2. If pp^mqⁿ_qpnq_qpm! Ñ 0 in A, then the twisted Taylor map of level ´m of A{R is the composite map

θ:A ^{p2 //}P_A{R ^//lim

ÐÝPp_A{R{Ip_A{R^{pn`1qqpm » //}Arrξss_qpm //Axωyz_qp´mq. Remarks 1. The twisted Taylor map of level ´m is a morphism of R-algebras that

endows Axωy_qp´mq{I_A{R^tn`1uq (resp. Axωyz_qp´mq) with a second structure of A-module commonly called the “right” structure (the usual one being the “left” structure).

2. It will follow from proposition 3.7 below that the twisted Taylor map of level´m is explicitly given by

f ÞÑÿ

B_q^xiypfqω^tiuq. This actually provides an alternative definition.

Definition 3.6. A twisted differential operator of level ´m and order at most n is an A-linear map

u:Axωy_qp´mq{I_A{R^tn`1uqb¹_AM ÑN

(in which the b¹ indicates that we are using the “right” structure given by the twisted Taylor map of level ´m on the left hand side).

One can compose twisted differential operators of level´m of different orders in the usual way using the diagonal maps

Axωy_qp´mq{I_A{R^{tn1`n2`1uq ∆n1,n2 //}Axωy_qp´mq{I_A{R^{tn1`1uq</sup>b<sup>1</sup><sub>A</sub>Axωy<sub>qp´mq</sub>{I<sub>A{R</sub><sup>tn2`1uq}

ω^{tiuq //}ř

i1`i2“iω^ti1uq bω^ti2uq.

(6)

More precisely, the composition of two differential operators u₁ and u₂ of order at most n1 and n2 respectively is the differential operator of order at most n1`n2 defined by

u₁u₂ :“u₁˝ pIdb¹u₂q ˝ p∆_n1,n2b¹Idq.

Proposition 3.7. The twisted differential operators of level ´m of all orders from A to itself form a ring which is isomorphic toD^p´mq_A{R,q. More precisely, the basistω^tnuqunPZě0 of Axωy_qp´mq is “topologically” dual to the basis tBq^xnyunPZě0 of D^p´mq_A{R,q.

Proof. The point is to check that the bilinear map

D^p´mq_A{R,qˆAxωy_qp´mq{I_A{R^tn`1uq ÑA, pB_q^xny, ω_q^tkuq ÞÑ

"

1 if n“k 0 otherwise provides an isomorphism of rings

D^p´mq_A{R,q »lim ÝÑ_n Hom_A

´

Axωy_qp´mq{I_A{R^tn`1uq, A

¯ .

Details are left to the reader.

Definition 3.8. Atwisted Taylor structure of level´mon anA-moduleM is a compatible family of A-linear maps

θ_n:M ÑMbAAxωy_qp´mq{I_A{R^tn`1uq such that

@n1, n2 PZě0, pθn1 bId_Axωy

qp´mq{I<sup>tn2`1uq</sup>q ˝θn2 “ pId<sub>M</sub> b∆<sub>n1,n2</sub>q ˝θn1`n2. Remarks 1. A twisted Taylor structure of level´m is equivalent to a D^p´mq_A{R,q-module

structure via the formula

θnpsq “

n

ÿ

i“0

B^xiy_q sbω^tiuq.

This is therefore also equivalent to a twisted connection (or derivation) of level´m.

2. One may define the Čech-Alexander cohomology of anA-module with a twisted Tay- lor structure of level´mand show that this is isomorphic to de Rham cohomology.

3. As in definition 3.9 below, whenppqⁿ_qpnq_q^p!Ñ0, there exists an “hyper” equivalent to the notion of a twisted Taylor structure of level´mwithAxωy_qp´mq{I_A{R^tn`1uq replaced with of Axωyz_qp´mq.

There remains one more notion to investigate (we denote by

∆_n:Axωy_qp´mq{I^tn`1uq ÑA the canonical map sendingω to 0).

Definition 3.9. 1. A twisted stratification of level ´m on an A-module M is a compatible family of Axωy_qp´mq{I^tn`1uq-linear isomorphisms

_n:Axωy_qp´mq{I^{tn`1uq</sup> b<sup>1</sup><sub>A</sub>M »MbAAxωy<sub>qp´mq</sub>{I<sup>tn`1uq} satisfying the normalization and cocycle conditions

∆^˚_np_nq “Id_M and ∆^˚_n,np_nq “p^˚₁p_nq ˝p^˚₂p_nq.

2. If ppqⁿ_qpnq_q^p! Ñ 0, then a twisted hyper-stratification of level ´m on an A-module M is an Axωyz_qp´mq-linear isomorphism

:Axωyz_qp´mqb¹_AM »MbAAxωyz_qp´mq satisfying the normalization and cocycle conditions.

We will denote by Strat^p´mq_q pA{Rq (resp. Stratz^p´mq_q pA{Rq) the category of twisted stratified (resp. hyper-stratified) modules of level´m.

Remarks 1. The following are equivalent:

• a twisted connection (or derivation) of level m,

• a structure of aD_A{R,q^p´mq-module,

• a twisted Taylor structure of levelm,

• a twisted stratification of levelm.

In particular, there exists an isomorphism of categories MIC^p´mq_q pA{Rq »Strat^p´mq_q pA{Rq.

2. Any twisted hyper-stratification of level ´m induces a twisted stratification of level

´m and we get a fully faithful functor

Stratz^p´mq_q pA{Rq ÑStrat^p´mq_q pA{Rq »MIC^p´mq_q pA{Rq.

Proposition 3.10. When M is finitely presented, a hyper-stratification of level ´m is equivalent to a topologically quasi-nilpotent twisted connection of level´m, meaning that

@sPM, B^xkypsq Ñ0.

Proof. Standard (see the proof of proposition 6.3 in [GSQ20], for example).