A Beilinson-Bernstein theorem for twisted arithmetic differential operators on the formal flag variety

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A Beilinson-Bernstein theorem for twisted arithmetic differential operators on the formal flag variety

Andrés Sarrazola Alzate

To cite this version:

Andrés Sarrazola Alzate. A Beilinson-Bernstein theorem for twisted arithmetic differential operators on the formal flag variety. 2019. �hal-02180190v2�

A Beilinson-Bernstein theorem for twisted arithmetic differential operators on the formal flag variety

Andrés Sarrazola Alzate

Abstract

Letℚ_𝑝be the field of𝑝-adic numbers and𝔾a split connected reductive group scheme overℤ_𝑝. In this work we will introduce a sheaf of twisted arithmetic differential operators on the formal flag variety of𝔾, associated to a general character. In particular, we will generalize the results of [23], concerning theD^†- affinity of the smooth formal flag variety of𝔾, of certain sheaves of𝑝-adically complete twisted arithmetic differential operators associated to an algebraic character, and the results of [27] concerning the calculation of the global sections.

Key words: Formal flag variety, twisted arithmetic differential operators, Beilinson-Bernstein correspondence.

1 Introduction 2

2 Arithmetic definitions 6

2.1 Partial divided power structures of level m . . . . 6

2.2 Arithmetic differential operators . . . . 7

2.3 Symmetric algebra of finite level . . . . 8

2.4 Arithmetic distribution algebra of finite level . . . . 9

2.5 Integral models . . . . 10

3 Integral twisted arithmetic differential operators 12 3.1 Torsors . . . . 12

3.2

𝕋

-equivariant sheaves and sheaves of

𝕋

-invariant sections . . . . 13

3.3 Relative enveloping algebras of finite level . . . . 18

3.4 Affine algebraic groups and homogeneous spaces . . . . 24

3.5 Relative enveloping algebras of finite level on homogeneous spaces . . . . 25

3.6 Finiteness properties . . . . 33

4 Passing to formal completions 35 4.1 Cohomological properties . . . . 35

4.2 Calculation of global sections . . . . 37

4.3 The localization functor . . . . 39

4.4 The arithmetic Beilinson-Bernstein theorem . . . . 39

5 The sheaves

D_𝔛,𝜆^†

40 5.1 The localization functor

L

𝑜𝑐

_𝔛,𝜆^†

. . . . 41 5.2 The arithmetic Beilinson-Bernstein theorem for the sheaves

D_𝔛,𝜆^†

. . . . 41 5.2.1 Calculation of global sections . . . . 44

References 45

1 Introduction

An important theorem in group theory is the so-called Beilinson-Bernstein theorem [3]. Let us briefly recall its statement. Let 𝐺 be a semi-simple complex algebraic group with Lie algebra

𝔤_ℂ

∶= Lie(𝐺). Let

𝔱_ℂ

⊂

𝔤_ℂ

be a Cartan subalgebra and

𝔷

⊂ U (𝔤

_ℂ

) the center of the universal enveloping algebra of

𝔤_ℂ

. For every character 𝜆 ∈

𝔱^∗_ℂ

we denote by

𝔪_𝜆

⊂

𝔷

the corresponding maximal ideal determined by the Harish-Chandra homomorphism.

We put U

_𝜆

∶= U (𝔤

_ℂ

)∕𝔪

_𝜆

. The theorem states that if 𝑋 is the flag variety associated to 𝐺 and D

_𝑋,𝜆

is the sheaf of 𝜆-twisted differential operators on 𝑋, then Mod

_𝑞𝑐

( D

_𝑋,𝜆

) ≃ Mod( U

_𝜆

), provided that 𝜆 is a dominant and regular character. Here Mod

_𝑞𝑐

( D

_𝑋,𝜆

) denotes the category of O

_𝑋

-quasi-coherent D

_𝑋,𝜆

-modules. Moreover, under this equivalence of categories coherent D

_𝑋,𝜆

-modules correspond to finitely generated U

_𝜆

-modules [3, théorème principal].

The Beilinson-Bernstein theorem has been proved independently by A. Beilinson and J. Bernstein in [3] and, by J-L. Brylinski and M. Kashiwara in [12]. This result is an essential tool in the proof of Kazhdan-Lusztig’s multiplicity conjecture [29]. In mixed characteristic, an important progress has been achieved by C. Huyghe in [22, 23] and Huyghe-Schmidt in [27]. They use Berthelot’s arithmetic differential operators [5] to prove an arithmetic version of the Beilinson-Bernstein theorem for the smooth formal flag variety over

ℤ_𝑝¹

. In this setting, the global sections of these operators equal a crystalline version of the classical distribution algebra Dist(𝐺) of the group scheme 𝐺.

Let

ℚ_𝑝

be the field of 𝑝-adic numbers. Throughout this paper

𝔾

will denote a split connected reductive group scheme over

ℤ_𝑝

,

𝔹

⊂

𝔾

a Borel subgroup and

𝕋

⊂

𝔹

a split maximal torus of

𝔾. We will also denote by

𝑋 =

𝔾∕𝔹

the smooth flag

ℤ_𝑝

-scheme associated to

𝔾. In this work, we introduce sheaves of

twisted differential operators on the formal flag scheme

𝔛

and we show an arithmetic Beilinson-Bernstein correspondence. Here the twist is respect to a morphism of

ℤ_𝑝

-algebras 𝜆 ∶ Dist(𝕋 )

→ ℤ_𝑝

, where Dist(𝕋 ) denotes the distribution algebra in the sense of [13]. Those sheaves are denoted by

D_𝔛,𝜆^†

. In particular, there exists a basis S of

𝔛

consisting of affine open subsets, such that for every

𝔘

∈ S we have

D_𝔛,𝜆^† |𝔘

≃

D_𝔘^†

.

In other words, locally we recover the sheaves of Berthelots’ differential operators, [5] (of course, this clarifies why they are called twisted differential operators).

To calculate their global sections, we remark for the reader that actually we dispose of a description of the distribution algebra Dist(𝔾) as an inductive limit of filtered noetherian

ℤ_𝑝

-algebras Dist(𝔾) = lim

←

←←←←←←←←←→^𝑚∈ℕ

𝐷

^(𝑚)

(𝔾),

1In reality over a discrete valuation ring.

such that for every 𝑚 ∈

ℕ

we have 𝐷

^(𝑚)

(𝔾) ⊗

_ℤ

𝑝ℚ_𝑝

= U (Lie(𝔾) ⊗

_ℤ

𝑝 ℚ_𝑝

) the universal enveloping algebra of Lie(𝔾) ⊗

_ℤ

𝑝 ℚ_𝑝

. For instance 𝐷

⁽⁰⁾

(𝔾) = U (Lie(𝔾)). In particular, every character 𝜆 ∶ Dist(𝕋 )

→ ℤ_𝑝

induces, via tensor product with

ℚ_𝑝

, an infinitesimal character 𝜒

_𝜆

. The last one allows us to define 𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

as the 𝑝-adic completion of the central reduction 𝐷

^(𝑚)

(𝔾)∕(𝐷

^(𝑚)

(𝔾) ∩ ker(𝜒

_𝜆+𝜌

)), and we denote by 𝐷

^†

(𝔾)

_𝜆

the limit of the inductive system 𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

⊗

_ℤ

𝑝 ℚ_𝑝→

𝐷 ̂

^(𝑚′)

(𝔾)

_𝜆

⊗

_ℤ

𝑝 ℚ_𝑝

. In this paper we show the following result

Theorem. Let us suppose that 𝜆 ∶ Dist(𝕋)

→ ℤ_𝑝

is a character of Dist(𝕋 ) such that 𝜆 + 𝜌 ∈

𝔱^∗_ℚ ²

(𝔱

_ℚ

∶= Lie(𝕋 ) ⊗

_ℤ

𝑝ℚ_𝑝

) is a dominant and regular character of

𝔱^∗_ℚ

. Then the global sections functor induces an equivalence between the categories of coherent

D_𝔛,𝜆^†

-modules and finitely presented 𝐷

^†

(𝔾)

_𝜆

-modules.

This theorem is based on a refined version for the sheaves (of level m twisted differential operators which we will define later)

D

̂

_{𝔛,𝜆,ℚ}^(𝑚)

. As in the classical case, the inverse functor is determined by the localization functor

L 𝑜𝑐

_𝔛,𝜆^†

(∙) ∶=

D_𝔛,𝜆^†

⊗

_𝐷†(𝔾)_𝜆

(∙), with a completely analogous definition for every 𝑚 ∈

ℕ.

The first section of this work is devoted to fix some important arithmetic definitions. They are necessary to define an integral model (in the sense of definition 2.5.1) of the usual sheaf of twisted differential operators on the smooth flag variety 𝑋

_ℚ

∶= 𝑋 ×

_Spec(ℤ

𝑝)

Spec(ℚ

_𝑝

), associated to the split connected reductive algebraic group

𝔾_ℚ

∶=

𝔾

×

_Spec(ℤ

𝑝)

Spec(ℚ

_𝑝

). One of the most important is the algebra of distributions of level 𝑚, which is denoted by 𝐷

^(𝑚)

(𝔾) and it is introduced in subsection 2.5. This is a filtered noetherian

ℤ_𝑝

-algebra which plays a fundamental roll in the rest of our work.

Inspired by the works [1], [8] and [11], in the first part of the third section we construct our level m twisted arithmetic differential operators on the formal flag scheme

𝔛. To do this, we denote by 𝔱

the commutative

ℤ_𝑝

-Lie algebra of the maximal torus

𝕋

and by

𝔱_ℚ

∶=

𝔱

⊗

_ℤ

𝑝 ℚ_𝑝

. They are Cartan subalgebras of

𝔤

and

𝔤_ℚ

, respectively. Let N be the unipotent radical subgroup of the Borel subgroup

𝔹

and let us define

𝑋 ̃ ∶=

𝔾∕N,

𝑋 ∶=

𝔾∕𝔹

the basic affine space and the flag scheme of

𝔾. These are smooth and separated schemes overℤ_𝑝

, and 𝑋 ̃ is endow with commuting (𝔾,

𝕋

)-actions making the canonical projection 𝜉 ∶ 𝑋 ̃

→

𝑋 a locally trivial

𝕋

-torsor for the Zariski topology on 𝑋. Following [11], the right

𝕋

-action on 𝑋 ̃ allows to define the level m relative enveloping algebra of the torsor 𝜉 as the sheaf of

𝕋

-invariants of 𝜉

_∗

D

^(𝑚)

𝑋̃

:

̃ D

^(𝑚)

∶=

(

𝜉

_∗

D

^(𝑚)

𝑋̃

)𝕋

.

As we will explain later this is a sheaf of 𝐷

^(𝑚)

(𝕋 )-modules and we will show that over an affine open subset 𝑈 ⊂ 𝑋 that trivialises the torsor, the sheaf ̃ D

^(𝑚)

may be described as the tensor product D

^(𝑚)_{𝑋 |𝑈}

⊗

_ℤ

𝑝

𝐷

^(𝑚)

(𝕋 ) (this is the arithmetic analogue of [11, page 180]).

2Here the character𝜆is induced via tensor product withℚ_𝑝and the correspondence (33).

Two fundamental properties of the distribution algebra Dist(𝕋 ) are: it constitutes an integral model of the uni- versal enveloping algebra U (𝔱

_ℚ

) and Dist(𝕋) = lim

←

←←←←←←←←←→^𝑚

𝐷

^(𝑚)

(𝕋 ). These properties allow us to introduce the central reduction as follows. First of all, we say that an

ℤ_𝑝

-algebra map 𝜆 ∶ Dist(𝕋 )

→ ℤ_𝑝

is a character of Dist(𝕋 ) (cf. 3.5.3). By the properties just stated, it induces a character of the Cartan subalgebra

𝔱_ℚ³

(𝜆 =) 𝜆 ⊗ 1 ∶

𝔱_ℚ→

U (𝔱

_ℚ

) = Dist(𝕋 ) ⊗

_ℤ

𝑝 ℚ_𝑝 →ℚ_𝑝

.

Now, let us consider the ring

ℤ_𝑝

as a 𝐷

^(𝑚)

(𝕋 )-module via 𝜆. We can define the sheaf of level m twisted arithmetic differential operators on the flag scheme 𝑋 by

D

^(𝑚)_𝑋,𝜆

∶= ̃ D

^(𝑚)

⊗

_𝐷(𝑚)(𝕋)ℤ_𝑝

. (1) This is a sheaf of

ℤ_𝑝

-algebras which is an integral model of D

_𝜆

.

⁴

The second part of the third section is dedicated to explore some finiteness properties of the cohomology of coherent D

_𝑋,𝜆^(𝑚)

-modules. Notably important is the case when the character 𝜆 + 𝜌 ∈

𝔱^∗_ℚ

is dominant and regular.

Under this assumption, the cohomology groups of every coherent D

^(𝑚)_𝑋,𝜆

-module have the nice property of being of finite 𝑝-torsion. This is one of the central results in this work.

In section 4 we will consider the 𝑝-adic completion of (1). It will be denoted by

D

̂

_𝔛,𝜆^(𝑚)

and the sheaf

D

̂

_{𝔛,𝜆,ℚ}^(𝑚)

∶=

D

̂

_𝔛,𝜆^(𝑚)

⊗

_ℤ

𝑝 ℚ_𝑝

is our sheaf of level m twisted arithmetic differential operators on the formal flag scheme

𝔛.

Let 𝑍(𝔤

_ℚ

) be the center of the universal enveloping algebra U (𝔤

_ℚ

). From now on, we will assume that 𝜆+𝜌 ∈

𝔱^∗_ℚ

is dominant and regular, and that 𝜒

_𝜆

∶ 𝑍(𝔤

_ℚ

)

→ ℚ_𝑝

is the central character induced by 𝜆 via the classical Harish-Chandra homomorphism. We will show that if Ker(𝜒

_𝜆+𝜌

)

_ℤ

𝑝

∶= 𝐷

^(𝑚)

(𝔾) ∩ Ker(𝜒

_𝜆+𝜌

) and 𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

denotes the 𝑝-adic completion of the central reduction 𝐷

^(𝑚)

(𝔾)

_𝜆

∶= 𝐷

^(𝑚)

(𝔾)∕𝐷

^(𝑚)

(𝔾)Ker(𝜒

_𝜆+𝜌

)

_ℤ

𝑝

, then we have an isomorphism of complete

ℤ_𝑝

-algebras

𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

⊗

_ℤ

𝑝ℚ_𝑝←←←←←←←→^≃

𝐻

⁰ (

𝔛, ̂D_{𝔛,𝜆,ℚ}^(𝑚) )

.

Finally, in subsection 4.3 we will introduce the localization functor

L

𝑜𝑐

^(𝑚)_𝔛,𝜆

, from the category of finitely gen- erated 𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

⊗

_ℤ

𝑝 ℚ_𝑝

-modules to the category of coherent

D

̂

_{𝔛,𝜆,ℚ}^(𝑚)

-modules as the sheaf associated to the presheaf defined by

U ⊆

𝔛→D

̂

_{𝔛,𝜆,ℚ}^(𝑚)

( U ) ⊗

_𝐷̂(𝑚)(𝔾)_𝜆⊗_ℤ

𝑝ℚ_𝑝

𝐸, where 𝐸 is a finitely generated 𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

⊗

_ℤ

𝑝 ℚ_𝑝

-module. We will show

theorem: Let us suppose that 𝜆 ∶ Dist(𝕋 )

→ ℤ_𝑝

is a character of Dist(𝕋 ) such that 𝜆 + 𝜌 ∈

𝔱^∗_ℚ

is a dominant and regular character of

𝔱_ℚ

.

3In order to soft the notation through this work we will always denote by𝜆the character𝜆⊗1_ℚ

𝑝. This should not cause any confusion to the reader.

4Tensoring withℚ_𝑝and restricting to the generic fiber𝑋_ℚ→𝑋equalsD_𝜆, [11, page 170].

(i) The functors

L

𝑜𝑐

_𝔛,𝜆^(𝑚)

and 𝐻

⁰

(𝔛, ∙) are quasi-inverse equivalences of categories between the abelian cat- egories of finitely generated (left) 𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

⊗

_ℤ

𝑝 ℚ_𝑝

-modules and coherent

D

̂

_{𝔛,𝜆,ℚ}^(𝑚)

-modules.

(ii) The functor

L

𝑜𝑐

_𝔛,𝜆^(𝑚)

is an exact functor.

Section 5 of this work is devoted to treat the problem of passing to the inductive limit. In fact, if 𝑚

≤

𝑚

^′

the sheaves

D

̂

_{𝔛,𝜆,ℚ}^(𝑚)

and

D

̂

_{𝔛,𝜆,ℚ}^(𝑚′)

are related via a natural map

D

̂

_{𝔛,𝜆,ℚ}^(𝑚) → D

̂

_{𝔛,𝜆,ℚ}^(𝑚′)

and we can define

D_𝔛,𝜆^†

to be the inductive limit. Similarly, we dispose of canonical maps 𝐷 ̂

^(𝑚)

(𝔾)

_𝜆

⊗

_ℤ

𝑝ℚ_𝑝→

𝐷 ̂

^(𝑚′)

(𝔾)

_𝜆

⊗

_ℤ

𝑝ℚ_𝑝

and we denote by 𝐷

^†

(𝔾)

_𝜆

its limit. Introducing the localization functor

L

𝑜𝑐

_𝔛,𝜆^†

exactly as we have made before, we get an analogue of the previous theorem for the sheaves

D_𝔛,𝜆^†

.

The work developed by Huyghe in [22] and by D. Patel, T. Schmidt and M. Strauch in [34], [35] and [24], shows that the Beilinson-Berstein theorem is an important tool in the following localization theorem [24, theorem 5.3.8]: if

𝔛

denotes the formal flag scheme of a split connected reductive group

𝔾, then the theorem provides

an equivalence of categories between the category of admissible locally analytic

𝔾(ℚ_𝑝

)- representations (with trivial character!) [33] and a category of coadmissible equivariant arithmetic

D-modules (on the family of

formal models of the rigid analytic flag variety of

𝔾). Our motivation is to study this localization in the twisted

case (in a work still in progress, we have proved the affinity of an admissible formal blow-up of

𝔛

for the sheaf of twisted arithmetic differential operators with a congruence level 𝑘

⁵

. This is a fundamental result to achieve the previous localization theorem [24]).

The case of a finite extension 𝐿 of

ℚ_𝑝

seems to present several technical problems (the reader can take a look to 3.5.3 to see one of this obstacles). We will treat this case in a future work.

Acknowledgements: The present article contains a part of the author PhD thesis written at the Universities of Strasbourg and Rennes 1 under the supervision of C. Huyghe and T. Schmidt. Both have always been very patient and attentive supervisors. For this, I express my deep gratitude to them.

Notation: Throughout this work 𝑝 will denote a prime number and

ℤ_𝑝

the ring of 𝑝-adic integers. Furthermore, if 𝑋 is an arbitrary noetherian scheme over

ℤ_𝑝

and 𝑗 ∈

ℕ, then we will denote by

𝑋

_𝑗

∶= 𝑋×

_Spec(ℤ

𝑝)

Spec(ℤ

_𝑝

∕𝑝

^𝑗+1

) the reduction modulo 𝑝

^𝑗+1

, and by

𝔛

= lim

←

←←←←←←←←←→

𝑗

𝑋

_𝑗

the formal completion of 𝑋 along the special fiber. Moreover, if E is a sheaf of

ℤ_𝑝

-modules on 𝑋 then its 𝑝-adic completion

E

∶= lim

←←←←←←←←←←←^𝑗

E ∕𝑝

^𝑗+1

E will be considered as a sheaf on

𝔛. Finally, the base change of a sheaf of ℤ_𝑝

-modules on 𝑋 (resp. on

𝔛) toℚ_𝑝

will always be denoted by the subscript

ℚ. For instance

E

_ℚ

∶= E ⊗

_ℤ

𝑝ℚ_𝑝

(resp.

Eℚ

∶=

E

⊗

_ℤ

𝑝ℚ_𝑝

).

5The point here is that an admissible formal blow-up is not necessarily smooth and therefore, we can not use immediately Berthelot’s differential operators. For the definition of these differential operators the reader is invited to visit [25].

2 Arithmetic definitions

In this section we will describe the arithmetic objects on which the definitions and constructions of our work are based. We will give their functorial constructions and we will enunciate their most remarkable properties.

For a more detailed approach, the reader is invited to take a look to the references [7], [22], [26] and [5].

2.1 Partial divided power structures of level m

Let 𝑝 ∈

ℤ

be a prime number. In this subsection

ℤ_(𝑝)

denotes the localization of

ℤ

with respect to the prime ideal (𝑝).

We start recalling the following definition [7, Definition 3.1].

Definition 2.1.1. Let 𝐴 be a commutative ring and 𝐼 ⊂ 𝐴 an ideal. By a structure of divided powers on I we mean a collection of maps 𝛾

_𝑖

∶ 𝐼

→

𝐴 for all integers 𝑖

≥

0, such that

(i) For all 𝑥 ∈ 𝐼 , 𝛾

₀

(𝑥) = 1, 𝛾

₁

(𝑥) = 𝑥 and 𝛾

_𝑖

(𝑥) ∈ 𝐼 if 𝑖

≥

2.

(ii) For 𝑥, 𝑦 ∈ 𝐼 and 𝑘

≥

1 we have 𝛾

_𝑘

(𝑥 + 𝑦) =

∑

𝑖+𝑗=𝑘

𝛾

_𝑖

(𝑥)𝛾

_𝑗

(𝑦).

(iii) For 𝑎 ∈ 𝐴 and 𝑥 ∈ 𝐼 we have 𝛾

_𝑘

(𝑎𝑥) = 𝑎

^𝑘

𝛾

_𝑘

(𝑥).

(iv) For 𝑥 ∈ 𝐼 we have 𝛾

_𝑖

(𝑥)𝛾

_𝑗

(𝑥) = ((𝑖, 𝑗))𝛾

_𝑖+𝑗

(𝑥), where ((𝑖, 𝑗 )) ∶= (𝑖 + 𝑗 )!(𝑖!)

⁻¹

(𝑗 !)

⁻¹

. (v) We have 𝛾

_𝑝

(𝛾

_𝑞

(𝑥)) = 𝐶

_𝑝,𝑞

𝛾

_𝑝𝑞

(𝑥), where 𝐶

_𝑝,𝑞

∶= (𝑝𝑞)!(𝑝!)

⁻¹

(𝑞!)

^−𝑝

.

Throughout this paper we will use the terminology: "(𝐼 , 𝛾) is a PD-ideal", "(𝐴, 𝐼 , 𝛾) is a PD-ring" and "𝛾 is a PD-structure on 𝐼 ". Moreover, we say that 𝜙 ∶ (𝐴, 𝐼 , 𝛾)

→

(𝐵, 𝐽 , 𝛿) is a PD-homomorphism if 𝜙 ∶ 𝐴

→

𝐵 is a homorphism of rings such that 𝜙(𝐼) ⊂ 𝐽 and 𝛿

_𝑘◦𝜙|𝐼

= 𝜙◦𝛾

_𝑘

, for every 𝑘

≥

0.

Example 2.1.2. [7, Section 3, Examples 3.2 (3)] Let 𝐿

|ℚ_𝑝

be a finite extension of

ℚ_𝑝

, such that

𝔬

is its valuation ring and 𝜛 is a uniformizing parameter. Let us write 𝑝 = 𝑢𝜛

^𝑒

, with 𝑢 a unit and 𝑒 a positive integer (called the absolute ramification index of

𝔬). Then

𝛾

_𝑘

(𝑥) ∶= 𝑥

^𝑘

∕𝑘! defines a PD-structure on (𝜛) if and only if 𝑒

≤

𝑝 − 1.

In particular, we dispose of a PD-structure on (𝑝) ⊂

ℤ_𝑝

. We let 𝑥

^[𝑘]

∶= 𝛾

_𝑘

(𝑥) and we denote by ((𝑝), [ ]) this PD-ideal.

Let us fix a positive integer 𝑚 ∈

ℤ. For the next terminology we will always suppose that

(𝐴, 𝐼 , 𝛾) is a

ℤ_(𝑝)

-PD- algebra whose PD-structure is compatible (in the sense of [5, subsection 1.2]) with the PD-structure induced by ((𝑝), [ ]) (we recall to the reader that the PD-structure ((𝑝), [ ]) always extends to a PD-structure on any

ℤ_(𝑝)

-algebra [7, proposition 3.15] ). We will also denote by 𝐼

^(𝑝𝑚)

the ideal generated by 𝑥

^𝑝𝑚

, with 𝑥 ∈ 𝐼 . Definition 2.1.3. Let 𝑚 be a positive integer. Let 𝐴 be a

ℤ_(𝑝)

-algebra and 𝐼 ⊂ 𝐴 an ideal. We call a m-PD- structure on 𝐼 a PD-ideal (𝐽 , 𝛾) ⊂ 𝐴 such that 𝐼

^(𝑝𝑚)

+ 𝑝𝐼 ⊂ 𝐽 .

We will say that (𝐼 , 𝐽 , 𝛾) is a m-PD-ideal of 𝐴. Moreover, we say that 𝜙 ∶ (𝐴, 𝐼 , 𝐽 , 𝛾)

→

(𝐴

^′

, 𝐼

^′

, 𝐽

^′

, 𝛾

^′

) is a m-

PD-morphism if 𝜙 ∶ 𝐴

→

𝐴

^′

is a ring morphism such that 𝜙(𝐼 ) ⊂ 𝐼

^′

, and such that 𝜙 ∶ (𝐴, 𝐽 , 𝛾 )

→

(𝐴

^′

, 𝐽

^′

, 𝛾

^′

)

is a PD-morphism.

For every 𝑘 ∈

ℕ

we denote by 𝑘 = 𝑝

^𝑚

𝑞 + 𝑟 the Euclidean division of 𝑘 by 𝑝

^𝑚

, and for every 𝑥 ∈ 𝐼 we define 𝑥

^{𝑘}(𝑚)

∶= 𝑥

^𝑟

(𝛾

_𝑞

(𝑥

^𝑝𝑚

)). We remark to the reader that the relation 𝑞!𝛾

_𝑞

(𝑥) = 𝑥

^𝑞

(which is an easy consequence of (i) and (iv) of definition 2.1.1) implies that 𝑞!𝑥

^{𝑘}(𝑚)

= 𝑥

^𝑘

.

On the other side, the m-PD-structure (𝐼 , 𝐽 , 𝛾) allows us to define an increasing filtration (𝐼

^{𝑛}

)

_𝑛∈ℕ

on the ring 𝐴 which is finer that the 𝐼-adic filtration and called the m-PD-filtration. It is characterized by the following conditions [6, 1.3]:

(i) 𝐼

^{0}

= 𝐴, 𝐼

^{1}

= 𝐼.

(ii) For every 𝑛

≥

1, 𝑥 ∈ 𝐼

^{𝑛}

and 𝑘

≥

0 we have 𝑥

^{𝑘}

∈ 𝐼

^{𝑘𝑛}

. (iii) For every 𝑛

≥

0, (𝐽 + 𝑝𝐴) ∩ 𝐼

^{𝑛}

is a PD-subideal of (𝐽 + 𝑝𝐴).

Proposition 2.1.4. [7, proposition 1.4.1] Let us suppose that 𝑅 is a

ℤ_(𝑝)

-algebra endowed with a m-PD-structure (𝔞,

𝔟, 𝛼). Let

𝐴 be a 𝑅-algebra and 𝐼 ⊂ 𝐴 an ideal. There exists an 𝑅-algebra 𝑃

_(𝑚)

(𝐼 ), an ideal 𝐼 ⊂ 𝑃

_(𝑚)

(𝐼 ) endowed with a m-PD-structure ( 𝐼 , ̃ [ ]) compatible with (𝔟, 𝛼), and a ring homomorphism 𝜙 ∶ 𝐴

→

𝑃

_(𝑚)

(𝐼 ) such that 𝜙(𝐼) ⊂ 𝐼 . Moreover, (𝑃

_(𝑚)

(𝐼 ), 𝐼 , ̃ 𝐼 , [ ], 𝜙) satisfies the following universal property: for every 𝑅- homomorphism 𝑓 ∶ 𝐴

→

𝐴

^′

sending 𝐼 to an ideal 𝐼

^′

which is endowed with a m-PD-structure (𝐽

^′

, 𝛾

^′

) com- patible with (𝔟, 𝛼), there exists a unique m-PD-morphism 𝑔 ∶ (𝑃

_(𝑚)

(𝐼), 𝐼 , ̃ 𝐼 , [ ])

→

(𝐴

^′

, 𝐼

^′

, 𝐽

^′

, 𝛾

^′

) such that 𝑔◦𝜙 = 𝑓 .

Definition 2.1.5. Under the hypothesis of the preceding proposition, we call the 𝑅-algebra 𝑃

_(𝑚)

(𝐼), endowed with the m-PD-ideal ( 𝐼 , ̃ ̄ 𝐼 , [ ]), the m-PD-envelope of (𝐴, 𝐼 ).

Finally, if we endow 𝑃

_(𝑚)^𝑛

(𝐼 ) ∶= 𝑃

_(𝑚)

(𝐼)∕𝐼

^{𝑛+1}

with the quotient m-PD-structure [5, 1.3.4] we have

Corollary 2.1.6. [5, Corollary 1.4.2] Under the hypothesis of the preceding proposition, there exists an 𝑅- algebra 𝑃

^𝑛

(𝑚)

(𝐼) endowed with a m-PD-structure (𝐼 , ̃ 𝐼 , [ ]) compatible with (𝔟, 𝛼) and such that 𝐼

^{𝑛+1}

= 0.

Moreover, there exists an 𝑅-homomorphism 𝜙

_𝑛

∶ 𝐴

→

𝑃

_(𝑚)^𝑛

(𝐼) such that 𝜙(𝐼) ⊂ 𝐼, and universal for the 𝑅-homomorphisms 𝐴

→

(𝐴

^′

, 𝐼

^′

, 𝐽

^′

, 𝛾

^′

) sending 𝐼 into a m-PD-ideal 𝐼

^′

compatible with (𝔟, 𝛼) and such that 𝐼

^′{𝑛+1}

= 0.

2.2 Arithmetic differential operators

Let us suppose that

ℤ_𝑝

is endowed with the m-PD-structure defined in example 2.1.2 (cf. [5, Subsection 1.3, example (i)]). Let 𝑋 be a smooth

ℤ_𝑝

-scheme, and I ⊂ O

_𝑋

a quasi-coherent ideal. The presheaves

𝑈

→

𝑃

_(𝑚)

(Γ(𝑈 , I )) and 𝑈

→

𝑃

_(𝑚)^𝑛

(Γ(𝑈 , I ))

are sheaves of quasi-coherent O

_𝑋

-modules which we denote by P

_(𝑚)

( I ) and P

_(𝑚)^𝑛

( I ). In a completely analogous way, we can define a canonical ideal I of P

_(𝑚)

( I ), a sub-PD-ideal ( I ̃ , [ ]) ⊂ ̄ I , and the sequence of ideals ( I ̄

^{𝑛}

)

_𝑛∈ℕ

defining the m-PD-filtration. Those are also quasi-coherent sheaves [5, subsection 1.4].

Now, let us consider the diagonal embedding Δ ∶ 𝑋

→

𝑋 ×

_ℤ

𝑝

𝑋 and let 𝑊 ⊂ 𝑋 ×

_ℤ

𝑝

𝑋 be an open subset

such that 𝑋 ⊂ 𝑊 is a closed subset, defined by a quasi-coherent sheaf I ⊂ O

_𝑊

. For every 𝑛 ∈

ℕ, the algebra

P

_𝑋,(𝑚)^𝑛

∶= P

_(𝑚)^𝑛

( I ) is quasi-coherent and its support is contained in 𝑋. In particular, it is independent of the open subset 𝑊 [5, 2.1]. Moreover, by proposition 2.1.4 the projections 𝑝

₁

, 𝑝

₂

∶ 𝑋 ×

_ℤ

𝑝

𝑋

→

𝑋 induce two morphisms 𝑑

₁

, 𝑑

₂

∶ O

_𝑋 →

P

_𝑋,(𝑚)^𝑛

endowing P

_𝑋,(𝑚)^𝑛

of a left and a right structure of O

_𝑋

-algebra, respectively.

Definition 2.2.1. Let 𝑚, 𝑛 be positive integers. The sheaf of differential operators of level 𝑚 and order less or equal to n on 𝑋 is defined by

D

_𝑋,𝑛^(𝑚)

∶=

H

𝑜𝑚

_O

𝑋

( P

_𝑋,(𝑚)^𝑛

, O

_𝑋

).

If 𝑛

≤

𝑛

^′

corollary 2.1.6 gives us a canonical surjection P

_𝑋,(𝑚)^𝑛′ →

P

_𝑋,(𝑚)^𝑛

which induces the injection D

^(𝑚)_𝑋,𝑛 →

D

^(𝑚)_𝑋,𝑛′

and the sheaf of differential operators of level 𝑚 is defined by D

^(𝑚)

𝑋

∶=

⋃

𝑛∈ℕ

D

^(𝑚)

𝑋,𝑛

. We remark for the reader that by definition D

^(𝑚)

𝑋

is endowed with a natural filtration called the order filtration, and like the sheaves P

_𝑋,(𝑚)^𝑛

, the sheaves D

^(𝑚)_𝑋,𝑛

are endowed with two natural structures of O

_𝑋

-modules. Moreover, the sheaf D

^(𝑚)_𝑋

acts on O

_𝑋

: if 𝑃 ∈ D

_𝑋,𝑛^(𝑚)

, then this action is given by the composition O

_𝑋←←←←←←←←←^𝑑→¹

P

_𝑋,(𝑚)^𝑛 ←←←←←←←→^𝑃

O

_𝑋

. Finally, let us give a local description of D

^(𝑚)_𝑋,𝑛

. Let 𝑈 be a smooth open affine subset of 𝑋 endowed with a family of local coordinates 𝑥

₁

, . . . , 𝑥

_𝑁

. Let 𝑑𝑥

₁

, . . . , 𝑑𝑥

_𝑁

be a basis of Ω

_𝑋

(𝑈 ) and 𝜕

_𝑥

1

, . . . , 𝜕

_𝑥

𝑁

the dual basis of T

_𝑋

(𝑈) (as usual, T

_𝑋

and Ω

_𝑋

denote the tangent and cotangent sheaf on 𝑋, respectively). Let 𝑘 ∈

ℕ^𝑁

. Let us denote by

|

𝑘

|

=

∑𝑁

𝑖=1

𝑘

_𝑖

and 𝜕

^[𝑘_𝑖 ^𝑖]

= 𝜕

_𝑥

𝑖

∕𝑘

_𝑖

! for every 1

≤

𝑖

≤

𝑁 . Then, using multi-index notation, we have 𝜕

^[𝑘]

=

∏𝑁

𝑖=1

𝜕

^[𝑘_𝑖 ^𝑖]

and 𝜕

^<𝑘>

= 𝑞

_𝑘

!𝜕

^[𝑘]

. In this case, the sheaf D

^(𝑚)_𝑋,𝑛

has the following description on 𝑈 D

^(𝑚)

𝑋,𝑛

(𝑈) =

{ ∑

|𝑘|≤𝑛

𝑎

_𝑘

𝜕

^<𝑘> |

𝑎

_𝑘

∈ O

_𝑋

(𝑈 ) and 𝑘 ∈

ℕ^𝑁 }

. (2)

2.3 Symmetric algebra of finite level

In this subsection we will focus on introducing the constructions in [22]. Let 𝑋 be a

ℤ_𝑝

-scheme, L a locally free O

_𝑋

-module of finite rank, S

_𝑋

( L ) the symmetric algebra associated to L and I the ideal of homogeneous elements of degree 1. Using the notation of section 2.1 we define

Γ

_𝑋,(𝑚)

( L ) ∶= P

_S

𝑋(L),(𝑚)

( I ) and Γ

^𝑛_𝑋,(𝑚)

( L ) ∶= Γ

_𝑋,(𝑚)

( L )∕ I ̄

^{𝑛+1}

. (3) Those algebras are graded [22, Proposition 1.3.3], and if 𝜂

₁

, ..., 𝜂

_𝑁

is a local basis of L , we have

Γ

^𝑛_𝑋,(𝑚)

( L ) =

⨁

|𝑙|≤𝑛

O

_𝑋

𝜂

^{𝑙}

.

As before 𝜂

^{𝑙}

=

∏𝑁

𝑖=1

𝜂

_𝑖^{𝑙𝑖}

and 𝑞

_𝑖

!𝜂

_𝑖^{𝑙𝑖}

= 𝜂

^𝑙𝑖

. We define by duality Sym

^(𝑚)

( L ) ∶=

⋃

𝑘∈ℕ

H

𝑜𝑚

_O

𝑋

(

Γ

^𝑘_𝑋,(𝑚)

( L

^∨

), O

_𝑋⁾

,

By [22, Propositions 1.3.1, 1.3.3 and 1.3.6] we know that Sym

^(𝑚)

( L ) = ⊕

_𝑛∈ℕ

Sym

^(𝑚)_𝑛

( L ) is a commutative graded algebra with noetherian sections over any open affine subset. Moreover, locally over a basis 𝜂

₁

, ..., 𝜂

_𝑁

we have the following description

Sym

^(𝑚)_𝑛

( L ) =

⨁

|𝑙|=𝑛

O

_𝑋

𝜂

^<𝑙>

, where 𝑙

_𝑖

!

𝑞

_𝑖

! 𝜂

^<𝑙_𝑖 ^𝑖>

= 𝜂

^𝑙_𝑖^𝑖

.

Remark 2.3.1. By [7, A.10] we have that

S⁽⁰⁾_𝑋

( L ) is the symmetric algebra of L , which justifies the terminology.

We end this subsection by remarking the following results from [22]. Let I be the kernel of the comor- phism Δ

^♯

of the diagonal embedding Δ ∶ 𝑋

→

𝑋 ×

_Spec(ℤ

𝑝)

𝑋. In [22, Proposition 1.3.7.3] Huyghe shows that the graded algebra associated to the m-PD-adic filtration of P

_𝑋,(𝑚)

it is identified with the graded m-PD- algebra Γ

_𝑋,(𝑚)

( I ∕ I

²

) = Γ

_𝑋,(𝑚)

(Ω

¹_𝑋

). More exactly, we have canonical isomorphisms Γ

^𝑛_𝑋,(𝑚)

∶= Γ

^𝑛_𝑋,(𝑚)

(Ω

¹_𝑋

)

←←←←←←←→^≃

𝑔𝑟

_∙

( P

_𝑋,(𝑚)^𝑛

) which, by definition, induce a graded isomorphism of algebras

Sym

^(𝑚)

( T

_𝑋

)

←←←←←←←→^≃

𝑔𝑟

_∙

D

_𝑋^(𝑚)

. (4)

2.4 Arithmetic distribution algebra of finite level

As in the introduction, let us consider

𝔾

a split connected reductive group scheme over

ℤ_𝑝

and 𝑚 ∈

ℕ

fixed. We propose to give a description of the algebra of distributions of level 𝑚 introduced in [26]. Let 𝐼 denote the kernel of the surjective morphism of

ℤ_𝑝

-algebras 𝜖

_𝔾

∶

ℤ_𝑝

[𝔾]

→ℤ_𝑝

, given by the identity element of

𝔾. We know that

𝐼 ∕𝐼

²

is a free

ℤ_𝑝

=

ℤ_𝑝

[𝔾]∕𝐼 -module of finite rank. Let 𝑡

₁

, . . . , 𝑡

_𝑙

∈ 𝐼 such that modulo 𝐼

²

, these elements form a basis of 𝐼 ∕𝐼

²

. The 𝑚-divided power enveloping of (ℤ

_𝑝

[𝔾], 𝐼) (proposition 2.1.4) denoted by 𝑃

_(𝑚)

(𝔾), is a free

ℤ_𝑝

-module with basis the elements 𝑡

^{𝑘}

= 𝑡

^{𝑘₁ ^1}

. . . 𝑡

^{𝑘_𝑙 ^𝑙}

, where 𝑞

_𝑖

!𝑡

^{𝑘_𝑖 ^𝑖}

= 𝑡

^𝑘_𝑖^𝑖

, for every 𝑘

_𝑖

= 𝑝

^𝑚

𝑞

_𝑖

+ 𝑟

_𝑖

and 0

≤

𝑟

_𝑖

< 𝑝

^𝑚

. These algebras are endowed with a decreasing filtration by ideals 𝐼

^{𝑛}

(subsection 2.1), such that 𝐼

^{𝑛}

= ⊕

_{|𝑘|≥𝑛}ℤ_𝑝

𝑡

^{𝑘}

. The quotients 𝑃

_(𝑚)^𝑛

(𝔾) ∶= 𝑃

_(𝑚)

(𝔾)∕𝐼

^{𝑛+1}

are therefore

ℤ_𝑝

-modules generated by the elements 𝑡

^{𝑘}

with

|

𝑘

|≤

𝑛 [5, Proposition 1.5.3 (ii)]. Moreover, there exists an isomorphism of

ℤ_𝑝

-modules

𝑃

_(𝑚)^𝑛

(𝔾) ≃

⨁

|𝑘|≤𝑛

ℤ_𝑝

𝑡

^{𝑘}

.

Corollary 2.1.6 gives us for any two integers 𝑛, 𝑛

^′

such that 𝑛

≤

𝑛

^′

a canonical surjection 𝜋

^𝑛′,𝑛

∶ 𝑃

_(𝑚)^𝑛′

(𝔾)

→

𝑃

_(𝑚)^𝑛

(𝔾). Moreover, for every 𝑚

^′ ≥

𝑚, the universal property of the divided powers gives us a unique morphism of filtered

ℤ_𝑝

-algebras 𝜓

_𝑚,𝑚′

∶ 𝑃

_(𝑚′)

(𝔾)

→

𝑃

_(𝑚)

(𝔾) which induces a homomorphism of

ℤ_𝑝

-algebras 𝜓

^𝑛

𝑚,𝑚^′

∶ 𝑃

_(𝑚^𝑛_′)

(𝔾)

→

𝑃

_(𝑚)^𝑛

(𝔾). The module of distributions of level 𝑚 and order 𝑛 is 𝐷

^(𝑚)_𝑛

(𝔾) ∶= Hom(𝑃

_(𝑚)^𝑛

(𝔾),

ℤ_𝑝

). The algebra of distributions of level m is

𝐷

^(𝑚)

(𝔾) ∶= lim

←

←←←←←←←←←→

𝑛

𝐷

_𝑛^(𝑚)

(𝔾),

where the limit is formed respect to the maps Hom

_ℤ

𝑝

(𝜋

^𝑛′,𝑛

,

ℤ_𝑝

). The multiplication is defined as follows. By universal property (Corollary 2.1.6) there exists a canonical application 𝛿

^𝑛,𝑛′

∶ 𝑃

^𝑛+𝑛′

(𝑚)

(𝔾)

→

𝑃

^𝑛

(𝑚)𝔾)

⊗

_ℤ

𝑝

𝑃

^𝑛′

(𝑚)

(𝔾).

If (𝑢, 𝑣) ∈ 𝐷

_𝑛^(𝑚)

(𝔾) × 𝐷

^(𝑚)_𝑛′

(𝔾), we define 𝑢.𝑣 as the composition 𝑢.𝑣 ∶ 𝑃

_(𝑚)^𝑛+𝑛′

(𝔾)

^𝛿

𝑛,𝑛′

←

←←←←←←←←←←←←←←←→

𝑃

_(𝑚)^𝑛

(𝔾) ⊗

_ℤ

𝑝

𝑃

_(𝑚)^𝑛′

(𝔾)

←←←←←←←←←←←←←←←←^𝑢⊗𝑣→ℤ_𝑝

. Let us denote by

𝔤

∶= Hom

_ℤ

𝑝

(𝐼 ∕𝐼

²

,

ℤ_𝑝

) the Lie algebra of

𝔾. This is a freeℤ_𝑝

-module with basis 𝜉

₁

, . . . , 𝜉

_𝑙

defined as the dual basis of the elements 𝑡

₁

, . . . , 𝑡

_𝑙

. Moreover, if for every multi-index 𝑘 ∈

ℕ^𝑙

,

|

𝑘

| ≤

𝑛, we denote by 𝜉

^<𝑘>

the dual of the element 𝑡

^{𝑘}

∈ 𝑃

_(𝑚)^𝑛

(𝔾), then 𝐷

_𝑛^(𝑚)

(𝔾) is a free

ℤ_𝑝

-module of finite rank with a basis given by the elements 𝜉

^<𝑘>

with

|

𝑘

|≤

𝑛 [26, proposition 4.1.6].

Remark 2.4.1.

⁶

Let A be an

ℤ_𝑝

-algebra and 𝐸 a free 𝐴-module of finite rank with base (𝑥

₁

, ..., 𝑥

_𝑁

). Let (𝑦

₁

, ..., 𝑦

_𝑁

) be the dual base of 𝐸

^∨

∶= Hom

_𝐴

(𝐸, 𝐴). As in the preceding subsection, let

S(𝐸^∨

) be the symmetric algebra and

I(𝐸^∨

) the augmentation ideal. Let Γ

_(𝑚)

(𝐸

^∨

) be the 𝑚-PD-envelope of (S(𝐸

^∨

),

I(𝐸^∨

)). We put Γ

^𝑛_(𝑚)

(𝐸

^∨

) ∶= Γ

_(𝑚)

(𝐸

^∨

)∕𝐼

^{𝑛+1}

. These are free 𝐴-modules with base 𝑦

^{𝑘₁ ^1}

...𝑦

^{𝑘_𝑁^𝑁}

with

∑

𝑘

_𝑖≤

𝑛 [22, 1.1.2]. Let {𝑥

^<𝑘>

}

_{|𝑘|≤𝑛}

be the dual base of Hom

_𝐴

(Γ

^𝑛_(𝑚)

(𝐸

^∨

), 𝐴). We define

𝑆𝑦𝑚

^(𝑚)

(𝐸) ∶=

⋃

𝑛∈ℕ

Hom

_𝐴 (

Γ

^𝑛_(𝑚)

(𝐸

^∨

), 𝐴

)

.

This is a free 𝐴-module with a base given by all the 𝑥

^<𝑘>

. The inclusion Sym

^(𝑚)

(𝐸) ⊆ Sym

^(𝑚)

(𝐸) ⊗

_ℤ

𝑝 ℚ_𝑝

gives the relation

𝑥

^<𝑘_𝑖 ^𝑖>

= 𝑘

_𝑖

!

𝑞

_𝑖

! 𝑥

^𝑘𝑖

. (5)

Moreover, it also has a structure of algebra defined as follows. By [22, Proposition 1.3.1] there exists an applica- tion Δ

_𝑛,𝑛′

∶ Γ

^𝑛+𝑛_(𝑚)^′

(𝐸

^∨

)

→

Γ

^𝑛_(𝑚)

(𝐸

^∨

) ⊗

_𝐴

Γ

^𝑛_(𝑚)^′

(𝐸

^∨

), which allows to define the product of 𝑢 ∈ 𝐻 𝑜𝑚

_𝐴

(Γ

^𝑛_(𝑚)

(𝐸

^∨

), 𝐴) and 𝑣 ∈ 𝐻 𝑜𝑚

_𝐴

(Γ

^𝑛_(𝑚)^′

(𝐸

^∨

), 𝐴) by the composition

𝑢.𝑣 ∶ Γ

^𝑛+𝑛_(𝑚)^′

(𝐸

^∨

)

←←←←←←←←←←←←←←←←←←^Δ𝑛,𝑛→^′

Γ

^𝑛_(𝑚)

(𝐸

^∨

) ⊗

_𝐴

Γ

^𝑛_(𝑚)^′

(𝐸

^∨

)

←←←←←←←←←←←←←←←←^𝑢⊗𝑣→

𝐴.

This maps endows 𝑆𝑦𝑚

^(𝑚)

(𝐸) of a structure of a graded noetherian

ℤ_𝑝

-algebra [22, Propositions 1.3.1, 1.3.3 and 1.3.6].

We have the following important properties [26, Proposition 4.1.15].

Proposition 2.4.2. (i) There exists a canonical isomorphism of graded

ℤ_𝑝

-algebras 𝑔𝑟

_∙

(𝐷

^(𝑚)

(𝔾)) ≃ 𝑆𝑦𝑚

^(𝑚)

(𝔤).

(ii) The

ℤ_𝑝

-algebras 𝑔𝑟

_∙

(𝐷

^(𝑚)

(𝔾)) and 𝐷

^(𝑚)

(𝔾) are noetherian.

2.5 Integral models

We start this subsection with the following definition [5, subsection 3.4].

Definition 2.5.1. Let 𝐴 be a

ℚ_𝑝

-algebra (resp. a sheaf of

ℚ_𝑝

-algebras). We say that a

ℤ_𝑝

-subalgebra 𝐴

₀

(resp.

a subsheaf of

ℤ_𝑝

-algebras) is an integral model of 𝐴 if 𝐴

₀

⊗

_ℤ

𝑝 ℚ_𝑝

= 𝐴.

6This remark exemplifies the local situation when𝑋=Spec(𝐴)with𝐴aℤ_𝑝-algebra [23, Subsection 1.3.1].