Spectrum of a linear differential equation with constant coefficients

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Spectrum of a linear differential equation with constant coefficients

Tinhinane Amina Azzouz

To cite this version:

Tinhinane Amina Azzouz. Spectrum of a linear differential equation with constant coefficients. 2018.

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Spectrum of a linear differential equation with constant coefficients

Tinhinane A. AZZOUZ February 22, 2018

Abstract

In this paper we compute the spectrum, in the sense of Berkovich, of an ultrametric linear differ- ential equation with constant coefficients, defined over an affinoid domain of the analytic affine line A¹_k^,an. We show that it is a finite union of either closed disks or topological closures of open disks and that it satisfies a continuity property.

1 Introduction

Differential equations constitute an important tool for the investigation of algebraic and analytic varieties, over the complex and the p-adic numbers. Notably, de Rham cohomology is one of the most powerful way to obtain algebraic and analytic informations. Besides, ultrametric phenomena appeared naturally studying formal Taylor solutions of the equation around singular and regular points.

As soon as the theory of ultrametric differential equations became a central topic of investigation around 1960, after the work of B. Dwork, P. Robba (et al.), the following intersting phenomena appeared.

In the ultrametric setting, the solutions of a linear differential equation may fail to converge as expected, even if the coefficients of the equation are entire functions. For example, over the ground field Qp

of p-adic numbers, the exponential power series exp(T) = P

n≥0Tⁿ

n! which is solution of the equation y^′ = y has radius of convergence equal to |p|^(p−1)1 even though the equation shows no singularities.

However, the behaviour of the radius of convergence is well controlled, and its knowledge permits to obtain several informations about the equation. Namely it controls the finite dimensionality of the de Rham cohomology. For more details, we refer the reader to the recent work of J. Poineau and A. Pulita [Pul15], [PP15], [PP13a] and [PP13b].

The starting point of this paper is an interesting relation between this radius and the notion ofspectrum (in the sense of Berkovich). Before discussing more in detail our results, we shall quickly explain this relation.

Consider a ringA together with a derivation d:A →A. A differential module on (A, d) is a finite freeA-moduleM together with aZ-linear map

∇:M →M

satisfying for allm∈M and allf ∈Athe relation∇(f m) =d(f)m+f∇(m). If an isomorphismM ∼=A^ν is given, then∇ coincides with an operator of the form

d+G:A^ν →A^ν

wheredacts onA^ν component by component and Gis a square matrix with coefficients inA.

IfAis moreover a Banach algebra with respect to a given normk.kandA^ν is endowed with the max norm, then we can endow the algebra of continuous Z-linear endomorphisms with the operator norm, that we still denotek.k. The spectral norm of∇is given by

k∇kSp= lim

n→∞k∇ⁿkⁿ¹.

The link between the radius of convergence of∇ and its spectrum is then the following: on the one hand, whenk.kis a Berkovich point of type other than1of the affine line andAis the field of this point, the spectral normk∇kSp coincides with the inverse of the radius of convergence ofM atk.k multiplied by some constant (cf. [CD94, p. 676] or [Ked10, Definition 9.4.4]). On the other hand the spectral norm k∇kSp is also equal to the radius of the smallest disk centred at zero and containing the spectrum of∇ in the sense of Berkovich (cf. [Ber90, Theorem 7.1.2]). We refer to the introduction of Section3.2for a more precise explanation.

The spectrum appears then as a new invariant of the connection∇generalizing and refining the radius of convergence. This have been our first motivation, however the study of the spectrum of an operator has its own interest and it deserves its own independent theory.

In this paper we focus on the computation of the spectrum of ∇ in the case where ∇ has constant coefficients, which means that there is a basis of M in which the matrixG has coefficients in the base field of constants (i.e. the field of elements killed by the derivations). We show that when we change the domain of definition of the equation the spectrum has a uniform behaviour: it is a finite union of either closed disks or topological closures (in the sense of Berkovich) of open disks.

Letkbe an arbitrary field andEbe ak-algebra with unit. Recall that classically, the spectrumΣf(E) of an element f of E (cf. [Bou07, §1.2. Defintion 1]) is the set of elements λ of k such that f −λ.1E

is not invertible in E. In the case wherek =Cand (E,k.k) is a C-Banach algebra, the spectrum of f satisfies the following properties:

• It is not empty and compact.

• The smallest disk centred at 0 and containingΣf(E)has radius equal tokfkSp= lim

n→+∞kfⁿkⁿ¹.

• The resolvent functionRf :C\Σf →E, λ7→(f −λ.1E)⁻¹ is an analytic function with values in E.

Unfortunately, this may fail in the ultrametric case. In [Vis85] M. Vishik provides an example of operator with empty spectrum and with a resolvent which is only locally analytic. We illustrate this pathology with an example in our context, where connections with empty classical spectrum abound.

Example 1.1. Consider the field k = Cendowed with the trivial absolute value. Let A :=C((S)) be the field of Laurent power series endowed with the S-adic absolute value given by|P

n≥n0anSⁿ|=rⁿ⁰, if an0 6= 0, where |S| =r < 1 is a nonzero real number¹. We consider a rank oneirregular differential module over A=C((S)) defined by the operator _dS^d +g :A→A, whereg ∈AhasS-adic valuation n0

that is less than or equal to−2. We consider the connection∇= _dS^d +g as an element of theC-Banach algebra E =LC(C((S))) of bounded C-linear maps of C((S)) with respect to the usual operator norm:

kϕk= sup

f∈A\{0}

kϕ(f)k

kfk , for allϕ∈ E. Then, the classical spectrum of the differentiel operator _dS^d +g is empty. Indeed, since k_dS^d k = ¹_r and |(g−a)⁻¹| ≤r² for all a ∈C, then k(g−a)⁻¹◦(_dS^d )k ≤r (resp.

k(_dS^d )◦(g−a)⁻¹k ≤r) and the seriesP

n≥0(−1)ⁿ·((g−a)⁻¹◦(_dS^d ))ⁿ(resp. P

n≥0(−1)ⁿ((_dS^d )◦(g−a)⁻¹)ⁿ) converges in Lk(C((S))). Hence, for all a ∈ C, (P

n≥0(−1)ⁿ ·((g−a)⁻¹◦(_dS^d ))ⁿ)◦(g−a)⁻¹ (resp.

(g−a)⁻¹◦(P

n≥0(−1)ⁿ·((_dS^d )◦(g−a)⁻¹)ⁿ) is a left (resp. right) inverse of _dS^d + (g−a)inLk(C((S))) andadoes not belong to the spectrum.²

To deal with this issue V. Berkovich understood that it was better not to define the spectrum as a subset of the base field k, but as a subset of the analytic line A^1,an_k , which is a bigger space³. His theory of analytic spaces (cf. [Ber90], [Ber93]) enjoys several good local topological properties such as

1In the language of Berkovich, this field can be naturally identified with the complete residual field of the pointx0,r∈ A^1,an_k (cf. section2.2).

2Notice that it is relatively easy to show that any non trivial rank one connection overC((S))is set theoretically bijective.

This follows from the classical index theorem of B. Malgrange [Mal74]. However, the set theoretical inverse of the connection may not be automatically bounded. This is due to the fact that the base fieldCis trivially valued and the Banach open mapping theorem does not hold in general. However, it is possible to prove that any such set theoretical inverse is bounded (cf. [Azz18]).

3This can be motivated by the fact that the resolvent is an analytic function on the complement of the spectrum.

compactness, arc connectedness ... In this setting Berkovich developed a spectral theory for ultrametric operators [Ber90, Chapter 7]. The definition of the spectrum given by Berkovich is the following: Let (k,|.|) be complete field with respect to an ultrametric absolute value and let A^1,an_k be the Berkovich affine line. For a pointx∈A^1,an_k we denote byH(x)the associated complete residual field. We fix on A^1,an_k a coordinate functionT. The spectrumΣf(E)of an elementf of ak-Banach algebraEis the set of pointsx∈A^1,an_k , such thatf⊗1−1⊗T(x)is not invertible inE⊗ˆkH(x). It can be proved (cf. [Ber90, Proposition 7.1.6]) that this is equivalent to say that there exists a complete valued field Ω containing isometricallykand a constantc∈Ωsuch that

• the image ofcby the canonical projectionA^1,an_Ω →A^1,an_k isx;

• f⊗1−1⊗c is not invertible as an element ofE⊗ˆkΩ.

In some senseΩ =H(x)andc=T(x)are the minimal possible choices. This spectrum is compact, non empty and it satisfies the properties listed above (cf. [Ber90, Theorem 7.12]). Notice that iff =∇ is a connection and ifE=Lk(M)is thek-Banach algebra of continuousk-linear endomorphisms ofM, then f ⊗1−1⊗c is no more a connection as an element of E⊗ˆkΩ. Indeed, Lk(M) ˆ⊗kΩ does not coincide with LΩ(M⊗ˆkΩ), unlessΩis a finite extension. In particular,no index theorem can be usedto test the set theoretical bijectivity of∇ ⊗1−1⊗c.

Coming back to the above example, using this definition it can be proved that the spectrum of _dS^d +g is now reduced to the individual non-rational pointx0,rⁿ⁰ ∈A^1,an_C (cf. [Azz18]). The dependence onr shows that the Berkovich spectrum depends on the chosen absolute value on k; whereas, instead, the classical spectrum is a completely algebraic notion.

Now, let(k,|.|)be an ultrametric complete field. LetX be an affinoid domain ofA^1,an_k andxa point of type (2), (3) or (4). To avoid confusion we fix another coordinate functionSoverX. We setA=O(X) or H(x) andd=g(S)_dS^d with g(S)∈A. We will distinguish the notion of differential equation in the sense of a differential polynomial P(d) acting on A from the notion of differential module (M,∇) over (A, d) associated to P(d) in a cyclic basis (cf. section 3.2). Indeed, it is not hard to prove that the spectrum of P(_dS^d ) = (_dS^d )ⁿ+an−1(_dS^d)ⁿ⁻¹+· · ·+a0as an element ofLk(A), whereai∈k, is given by the easy formula (cf. [Bou07, p. 2] and Lemma2.25)

Σ_P(^d

dS)(Lk(A)) =P(Σ^d

dS(Lk(A))). (1)

This set is always either a closed disk or the topological closure of an open disk (cf. Lemma 2.25, Remark4.16). On the other hand, surprisingly enough, this differs form the spectrumΣ∇(Lk(M))of the differential module(M,∇)associated toP(_dS^d )in a cyclic basis (i.e the spectrum of∇ as an element of Lk(M)). Indeed, this last is a finite union of either closed disks or topological closures of open disks (cf.

Theorem1.2) centered on the roots of the (commutative) polynomialQ=Xⁿ+an−1Xⁿ⁻¹+· · ·+a0∈k[X] associated toP.⁴ In order to introduce our next result, we denote byk˜the residual field ofk(cf. Section 2) and we set:

ω=

(|p|^p−11 if char(˜k) =p

1 if char(˜k) = 0. (2)

The main statement of this paper is then the following:

Theorem 1.2. We suppose thatkis algebraically closed. LetX be a connected affinoid domain of A^1,an_k and x∈A^1,an_k a point of type (2), (3) or (4). We setA=O(X)or H(x). Let(M,∇) be a differential module over (A,_dS^d )such that there exists a basis for which the associated matrixGhas constant entries (i.e. G ∈ Mn(k)). Let {a1, . . . , aN} ⊂ k be the set of eigenvalues of G. Then the behaviour of the spectrumΣ∇(Lk(M))of ∇as an element of Lk(M) is summarized in the following table:

4Notice thatQis the Fourier transform ofP and that its roots are related to the eigenvalues of the matrixGof the connection in the companion form (cf. Remark4.15).

A= O(X) H(x) X=D⁺(c, r) X=D⁺(c₀, r₀)\ S^µ

i=1

D⁻(c_i, r_i) xof type (2) or (3) xof type (4)

char(˜k) =p >0 S^N

i=1

D⁺(ai,^ω_r) ^NS

i=1

D⁺(ai,_min^ω

j rj) S^N

i=1

D⁺(ai,_r(x)^ω )

char(˜k) = 0 S^N

i=1

D⁻(ai,¹_r) ^NS

i=1

D⁺(ai,_min¹

j rj) S^N

i=1

D⁺(ai,_r(x)¹ ) S^N

i=1

D⁻(ai,_r(x)¹ )

Using this result we can prove that when x varies over Berkovich a segment(x1, x2) ⊂ A^1,an_k , the behaviour of the spectrum is left continuous, and continuous at the points of type (3) (cf. Section5).

The paper is organized as follows. Section2 is devoted to recalling the definitions and properties. It is divided into three parts: in the first one we recall some basic definitions and properties of k-Banach spaces, in the second we provide settings and notations related to the affine lineA^1,an_k and in the last one we recall the definition of the spectrum given by Berkovich and some properties.

In Section3, we introduce the spectrum associated to a differential module, and show how it behaves under exact sequences. The main result of this section is the following:

Proposition 1.3. We suppose that k is algebraically closed. Let A be as in Theorem 1.2, and let d=g(S)d/dS, withg∈A. Let(M,∇)be a differential module over(A, d) such that there exists a basis for which the associated matrix G has constant entries (i.e G ∈ Mn(k)). Then the spectrum of ∇ is Σ∇=

Sn i=1

(ai+ Σd(Lk(A))), where {a1, . . . , an} ⊂kis the multiset of the eigenvalues ofG.

In particular the spectrumhighly depends on the choice of the derivation d. In this paper we choose d= _dS^d . This claim shows the importance of computing the spectrum ofΣd(Lk(A)). Therefore, in Section 4, a large part is devoted to the computation of the spectrum of _dS^d acting on various rings. In the last part of this section, we state and prove the main result.

In the last section, we will explain, in the case of a differential equation with constant coefficients, that the spectrum associated to(M,∇)over(H(x), d)satisfies a continuity property, whenxvaries over a segment(x1, x2)⊂A^1,an_k .

Acknowledgments The author wish to express her gratitude to her advisors Andrea Pulita and Jérôme Poineau for their precious advice and suggestions, and for carful reading. We also thank F.

Beukers and F. Truc for useful occasional discussions.

2 Definitions and notations

All rings are with unit element. We denote byRthe field of real numbers, andR+={r∈R|r≥0}. In all the paper (k,|.|) will be a valued field of characteristic 0, complete with respect to an ultrametric absolute value|.|:k→R+(i.e. verifying|1|= 1,|a·b|=|a||b|,|a+b| ≤max(|a|,|b|)for alla,b∈k and

|a| = 0 if and only if a= 0). We set |k| :={r ∈R+|∃a∈ ksuch thatr=|a|}, k^◦ :={a∈ k||a| ≤ 1}, k^◦◦:={a∈k||a|<1} and˜k:=k^◦/k^◦◦. LetE(k)be the category whose objects are pairs(Ω,|.|), where Ω is a field extension of k complete with respect to |.|, and whose morphisms are the isometric rings morphisms. For (Ω,|.|)∈E(k), we setΩ^alg to be an algebraic closure of Ω, the absolute value extends uniquely to an absolute value defined onΩ^alg. We denote byΩd^alg the completion ofΩ^alg with respect to this absolute value.

2.1 Banach spaces

An ultrametric normon a k-vector spaceM is a mapk.k:M →R+ verifying:

• kmk= 0⇔m= 0.

• ∀m∈M,∀λ∈k;kλmk=|λ|kmk.

• ∀m, n∈M,km+nk ≤max(kmk,knk).

A normedk-vector spaceM is ak-vector space endowed with an ultrametric norm k.k. If moreover M is complete with respect to this norm, we say thatM isk-Banach space. Ak-linear mapϕ:M →N between two normedk-vector spaces is a boundedk-linear mapsatisfying the following condition:

∃C∈R+, ∀m∈M;kϕ(m)k ≤Ckmk.

IfC= 1we say that ϕis a contracting map.

LetM andN be two ultrametric normed k-vector spaces. We endow the tensor product M ⊗kN with the following norm:

k.k:M ⊗kN −→ R+

f 7→ inf{max

i kmikknik|f =P

i

mi⊗ni} . (3)

The completion ofM ⊗kN for this norm will be denoted byM⊗ˆkN.

An ultrametric norm on ak-algebraAis an ultrametric normk.k:A→R+ on thek-vector spaceA, satisfying the additional properties:

• k1k= 1ifAhas a unit element.

• ∀m, n∈A;kmnk ≤kmkknk.

A normedk-algebraAis ak-algebra endowed with an ultrametric normk.k. If moreoverAis complete with respect to this norm, we say thatAis ak-Banach algebra.

We setBan_kthe category whose objects arek-Banach vector spaces and arrows are boundedk-linear maps. An isomorphism in this category will be called bi-bounded isomorphism. We set BanALk the category whose objects arek-Banach algebras and arrows are bounded morphisms ofk-algebras.

Definition 2.1. LetA be ak-Banach algebra. Thespectral semi-norm associated to the norm of A is the map:

k.kSp,A:A −→ R+

f 7→ lim

n→+∞kfⁿkⁿ¹ = inf

n∈Nkfⁿkⁿ¹ . (4)

The existence of the limit and its equality with the infimum are well known (cf. [Bou07, Définition 1.3.2]).

Ifk.kSp,A=k.k, we say thatAis a uniform algebra.

Lemma 2.2. LetM be ak-vector space and letk.k:M →R+be a map verifying the following properties:

• kmk= 0⇔m= 0.

• ∀m∈M,∀λ∈k;kλmk ≤ |λ|kmk.

• ∀m, n∈M,km+nk ≤max(kmk,knk).

Then k.k is an ultrametric norm onM.

Proof. See [BGR84, Section 2.1.1, Proposition 4].

Lemma 2.3. Let Ω ∈ E(k) and let M be a k-Banach space. Then, the inclusion M ֒→M⊗ˆkΩ is an isometry. In particular, for allv∈M andc∈Ωwe havekv⊗ck=|c|kvk.

Proof. SinceΩ contains isometricallyk, the morphismM⊗ˆkΩ→Ω(resp. Ω→M⊗ˆkΩ) is an isometry (cf. [Poi13, Lemma 3.1]). We know thatkv⊗ck ≤ |c|kvk=|c|kv⊗1kfor allv∈M andc∈Ω. Therefore, by Lemma 2.2 the tensor norm is a norm on M⊗ˆkΩ as an Ω-vector space. Consequently, we obtain kv⊗ck=|c|kvkfor allv∈M andc∈Ω.

Proposition 2.4. Let M be ak-Banach space, andB be a uniform k-Banach algebra. Then inM⊗ˆkB we have:

∀m∈M, ∀f ∈B, km⊗fk=kmkkfk.

Proof. LetM(B)be the analytic spectrum ofB(see [Ber90, Chapter 1]). For allxinM(B)the canonical map B → H(x) is a contracting map, therefore the map M⊗ˆkB →M⊗ˆkH(x)is a contracting map too. Then, by Lemme 2.4we have:

∀x∈ M(B) :∀m∈M, ∀f ∈B, km⊗f(x)k=kmk|f(x)| ≤km⊗fk ≤kmkkfk thus,

∀m∈M, ∀f ∈B, kmk max

x∈M(B)|f(x)| ≤km⊗fk ≤kmkkfk.

Since B is uniform, we have max

x∈M(B)|f(x)| =kfkSp =kfk (cf. [Ber90, Theorem 1.3.1]), and kmkkfk ≤ km⊗fk ≤kmkkfk.

LetM and N be two k-Banach spaces. We denote byLk(M, N) thek-Banach algebra of bounded k-linear mapsM →N endowed with the operator norm:

Lk(M, N) −→ R+

ϕ 7→ sup

m∈M\{0}

kϕ(m)k

kmk . (5)

We setLk(M) :=Lk(M, M).

Lemma 2.5. LetΩ∈E(k). There exists an isometrick-linear mapLk(M)֒→ Lk(M⊗ˆkΩ)which extends to an Ω-linear contracting map Lk(M) ˆ⊗kΩ→ LΩ(M⊗ˆkΩ).

Proof. Letϕ∈ Lk(M), we have the bilinear map:

ϕ×1 :M×Ω −→ M⊗kΩ (x, a) 7→ ϕ(x)⊗a

where M×Ω is endowed with the product topology and M ⊗Ω with the topology induced by tensor norm (cf. (3)). The map ϕ×1is continuous see [Sch01, Chap. IV Lemma 17.1]. The diagram below:

M ×Ω

⊗

ϕ×1

//M⊗kΩ

M ⊗kΩ

∃!ϕ⊗1

88

r rr rr r rr rr

is commutative, in addition ϕ⊗1 is continuous see [Sch01, Chap. IV Lemma 17.1]. By the universal property of the completion of a metric space, the mapϕ⊗1extends to a continuous mapϕ⊗1 :ˆ M⊗ˆkΩ→ M⊗ˆkΩ. So we obtain ak-linear mapLk(M)֒→ LΩ(M⊗ˆkΩ). We prove now that it is an isometry. Indeed, letm∈M anda∈Ωthen by Lemma 2.3:

kϕ⊗1(m⊗a)k=kϕ(m)⊗ak=kϕ(m)k|a| ≤kϕkkmk|a|=kϕkkm⊗ak.

Now, letx=P

i

mi⊗ai∈M⊗ˆkΩ, then we have:

kϕ⊗1(x)k ≤ˆ inf{max

i (kϕ(mi)k|ai|)|x=X

i

mi⊗ai} ≤kϕkinf{max

i kmi⊗aik|x=X

i

mi⊗ai}.

Consequently,

kϕ⊗1k ≤kϕk.ˆ

On other hand, sinceM ֒→M⊗ˆkΩis an isometry andϕ⊗1ˆ |M =ϕ, we havekϕk ≤kϕ⊗1k.ˆ

The map Lk(M)֒→ LΩ(M⊗ˆkΩ)extends to a continuousΩ-linear map Lk(M) ˆ⊗kΩ֒→ LΩ(M⊗ˆkΩ). We need to prove now that it is a contracting map. Let ψ=P

iϕi⊗ai be an element of Lk(M) ˆ⊗kΩ, its image inLΩ(M⊗ˆkΩ)is the elementP

iaiϕi⊗1. we have:ˆ kX

i

aiϕi⊗1k ≤ˆ max

i kaiϕi⊗1kˆ = max

i kϕik|ai|.

Consequently,

kX

i

aiϕi⊗1k ≤ˆ inf{max

i kϕik|ai||ψ=X

i

ϕi⊗ai}=kψk.

Hence we obtain the result.

Lemma 2.6. Let M be ak-Banach space and Lbe a finite extension of k. Then we have a bi-bounded isomorphism:

Lk(M) ˆ⊗kL≃ LL(M⊗ˆkL).

Proof. As L is a sequence of finite intermediary extensions generated by one element, by induction we can assume thatL=k(α), whereαis an algebraic element overk. By Lemma2.5 we have a morphism Lk(M) ˆ⊗kL → LL(M⊗ˆkL). As L = k(α), we have a k-isomorphism M⊗ˆkL ≃ Ln−1

i=0 M ⊗(αⁱ ·k).

Letψ∈ LL(M⊗L). The restrictionˆ ψ|M⊗1:M⊗1→M⊗ˆkLis of the formm⊗17→Pn−1

i=0 ϕi(m)⊗αⁱ, whereϕi∈ Lk(M). AsψisL-linear, it is determined by theϕi. This gives rise to an inverseL-linear map LL(M⊗ˆkL)→ Lk(M) ˆ⊗kL. In the case where k is not trivially valued, by the open mapping theorem (see [BGR84, Section 2.8 Theorem of Banach]) the last map is bounded. Otherwise, the extensionL is trivially valued. Consequently, we have an isometrick-isomorphisms: L≃Ln−1

i=0 kequipped with the max norm. Therefore, we haveLk(M) ˆ⊗kL≃Ln−1

i=0 Lk(M) andM⊗ˆkL≃Ln−1

i=0 M with respect to the max norm. Then we have:

maxi kϕik ≤kψ|M⊗1k ≤kψk.

Hence, we obtain the result.

2.2 Berkovich line

In this paper, we will consider k-analytic spaces in the sense of Berkovich (see [Ber90]). We denote by A^1,an_k the affine analytic line over the ground fieldk, with coordinate T. We set k[T] to be the ring of polynomial with coefficients in kandk(T)its fractions field.

Recall that a point x∈A^1,an_k corresponds to a multiplicative semi-norm|.|x on k[T] (i.e. |0|x = 0,

|P +Q|x ≤max(|P|x,|Q|x)and |P ·Q|x =|P|x|Q|x for allP, Q ∈k[T]), that its restriction coincides with the absolute value ofk. The set px:={P ∈k[T]| |f|x= 0}is a prime ideal ofk[T]. Therefore, the semi-norm extends to a multiplicative norm on the fraction field Frac(A/px)).

Notation 2.7. We denote byH(x) the completion of Frac(A/px) with respect to |.|x, and by |.| the absolute value on H(x)induced by|.|x.

LetΩ∈E(k)andc∈Ω. Forr∈R^∗₊ we set

D⁺_Ω(c, r) ={x∈A^1,an_Ω ||T(x)−c| ≤r}

and

D⁻_Ω(c, r) ={x∈A^1,an_Ω ||T(x)−c|< r}

Denote byxc,rthe unique point in the Shilov boundary ofD_Ω⁺(c, r).

Forr1,r2∈R+, such that0< r1≤r2we set

C_Ω⁺(c, r1, r2) ={x∈A^1,an_Ω |r1≤ |T(x)−c| ≤r2} and forr1< r2 we set:

C_Ω⁻(c, r1, r2) ={x∈A^1,an_Ω |r1<|T(x)−c|< r2} We may delete the indexΩwhen it is obvious from the context.

Hypothesis 2.8. Until the end of this section we will suppose thatkis algebraically closed.

Each affinoid domainX ofA^1,an_k is a finite union of connected affinoid domain of the form:

D⁺(c0, r0)\ [µ i=1

D⁻(ci, ri) (6)

Wherec0, . . . , cµ∈D⁺(c0, r0)∩k and0< r1, . . . , rµ ≤r0 (the case whereµ= 0 is included).

LetX be an affinoid domain ofA^1,an_k , we denote byO(X)thek-Banach algebra of global sections of X. For a diskD⁺(c, r)we have:

O(D⁺(c, r)) ={X

i∈N

ai(T −c)ⁱ|ai∈k, lim

i→+∞|ai|rⁱ= 0}

equipped with the multiplicative norm:

kX

i∈N

ai(T−c)ⁱk= max

i∈N |ai|rⁱ

More generally, ifX is of the form (6), then by the Mittag-Leffler decomposition [FP04, Proposition 2.2.6], we have:

O(X) = Mµ

i=1

{X

j∈N^∗

aij

(T−ci)^j|aij∈k, lim

j→+∞|aij|r^−j_i = 0} ⊕ O(D⁺(c0, r0)).

wherekP

j∈N^∗ aij

(T−ci)^jk= maxj|aij|r^−j_i and the sum above is equipped with the maximum norm.

ForΩ∈E(k), we setXΩ=X⊗ˆkΩ. We have a canonical projection of analytic spaces:

π_Ω/k:XΩ→X (7)

Definition 2.9. Letx∈A^1,an_k . we define the radius ofxto be the value:

rk(x) = inf

a∈k|T(x)−a|.

We may delete kif it is obvious from the context.

We can describe the fieldH(x), wherexis a point ofA^1,an_k , in a more explicit way. In the case where xis of type (1), we haveH(x) =k. Ifxis of type (3) of the formx=xc,rwherec∈kandr /∈ |k|, then it is easy to see thatH(x) =O(C⁺(c, r, r)). But for the points of type (2) and (4), a description is not obvious, we have the following Propositions:

Proposition 2.10 (Mittag-Leffler Decomposition). Let x=xc,r be a point of type (2) of A^1,an_k (c∈k andr∈ |k^∗|). We have the decomposition:

H(x) =E⊕ O(D⁺(c, r))

whereE is the closure inH(x) of the ring of rational fractions ofk(T−c)whose poles are inD⁺(c, r).

i.e. for γ∈k with|γ|=r:

E:=dM

α∈˜k{X

i∈N^∗

aαi

(T−c+γα)ⁱ|aαi∈k, lim

i→+∞|aαi|r⁻ⁱ= 0}.

Proof. In the case where k is not trivially valued we refer to [Chr83, Theorem 2.1.6]. Otherwise, the only point of type (2) ofA^1,an_k isx0,1, which corresponds to the trivial norm onk[T]. Therefore, we have H(x) =k(T).

Lemma 2.11. Letx∈A^1,an_k be a point of type (4). The fieldH (x)is the completion ofk[T]with respect to the norm |.|x.

Proof. Recall that for a pointx∈A^1,an_k of type (4), the fieldH(x)is the completion ofk(T)with respect to |.|x. To prove that H(x) is the completion of k[T], it is enough to show thatk[T] is dense in k(T) with respect to |.|x. For alla∈kit is then enough to show that there exists a sequence(Pi)i∈N⊂k[T] which converges to _T¹_−a. Leta∈k. Sincexis of type (4), there existsc∈ksuch that|T−c|x<|T−a|x. Therefore we have|c−a|x=|T−a|x and we obtain:

1

T−a = 1

(T−c) + (c−a) = 1 c−a

X

i∈N

(T −c)ⁱ (a−c)ⁱ. So we conclude.

Proposition 2.12 ([Chr83]). Let x ∈ A^1,an_k be a point of type (4). Then there exists an isometric isomorphism ψ:O(DH⁺(x)(T(x), rk(x)))→H(x) ˆ⊗kH(x)of H(x)-Banach algebras.

Proof. Note that, for any element f ∈ H(x) ˆ⊗kH(x) with |f| ≤ rk(x), we can define a morphism of H(x)-Banach algebra ψ:O(D⁺H(x)(T(x), rk(x)))→H(x) ˆ⊗kH(x), that associatesf toT −T(x). To prove the statment we choosef =T(x)⊗1−1⊗T(x). Indeed, for alla∈kwe haveT(x)⊗1−1⊗T(x) = (T(x)−a)⊗1−1⊗(T(x)−a). Hence,

|T(x)⊗1−1⊗T(x)| ≤ inf

a∈k(max(|(T(x)−a)|,|(T(x)−a)|) = inf

a∈k|T(x)−a|=rk(x)

By constructionψis a contractingH(x)-linear map. In order to prove that it is an isometric isomorphism, we need to construct its inverse map and show that it is also a contracting map. For alla∈k,|T(x)−a|>

rk(x), henceT−ais invertible inO(D_H⁺_(x)(T(x), rk(x))). This means thatk(T)⊂ O(D⁺_H(x)(T(x), rk(x))) as k-vector space. As for alla∈k we have:

|T−a|= max(rk(x),|T(x)−a|) =|T(x)−a|

the restriction of the norm ofO(DH⁺(x)(T(x), rk(x)))tok(T)coincides with|.|x. Consequently, the closure of k(T)in O(D⁺H(x)(T(x), rk(x))) is exactly H(x), which means that we have an isometric embedding H(x)֒→ O(D⁺H(x)(T(x), rk(x))) of k-algebras which associatesT(x)to T. This map extends uniquely to a contracting morphism ofH(x)-algebras:

H(x) //

O(DH⁺(x)(T(x), rk(x)))

H(x) ˆ⊗kH(x)

∃!ϕ

55

❦❦

❦❦

❦❦

❦❦

❦❦

❦❦

❦

Then we have ϕ(T(x)⊗1−1⊗T(x)) = T −T(x). Since (T −T(x)) (resp. T(x)⊗1−1⊗T(x)) is a topological generator of the k-algebra DH⁺(x)(T(x), r(x)) (resp. H(x)⊗H(x)) and both of ϕ◦ψ and ψ ◦ϕ are bounded morphisms of k-Banach algebras, we have ϕ◦ψ=Id_O(D⁺

H(x)(T(x),rk(x))) and ψ◦ϕ=IdH(x) ˆ⊗_kH(x). Hence, we obtain the result.

For any point xofA^1,an_k and any extensionΩ∈E(k)the tensor norm on the algebraH(x) ˆ⊗kΩ is multiplicative (see [Poi13, Corollary 3.14.]). We denote this norm byσΩ/k(x).

Proposition 2.13. Let x∈A^1,an_k be a point of type (i), where i∈ {2,3,4}. Let Ω∈E(k)algebraically closed. If Ω∈/ E(H(x)), thenπ⁻¹_Ω/k{x}={σΩ/k(x)}.

Proof. Recall that ifπ⁻¹_Ω/k{x} \ {σΩ/k(x)}is not empty, then it is a union of disjoint open disks (cf. [PP15, Theorem 2.2.9]). Therefore, it contains points of type (1) which implies thatΩ∈E(H(x)). Hence, we obtain a contradiction.

2.3 Berkovich’s spectral theory

We recall here the definition of the sheaf of analytic functions with value in a k-Banach space over an analytic space and the definition of the spectrum given by V. Berkovich in [Ber90].

Definition 2.14. LetXbe ak-affinoid space andBbe ak-Banach space. We define the sheaf of analytic functions with values in B overX to be the sheaf:

OX(B)(U) = lim_←−

V⊂U

B⊗ˆkAV

whereU is an open subset ofX,V an affinoid domain andAV thek-affinoid algebra associated toV. As eachk-analytic space is obtained by gluingk-affinoid spaces (see [Ber90], [Ber93]), we can extend the definition to k-analytic spaces. Let U be an open subset of X. Every element f ∈ OX(B)(U) induces a fonction: f : U → `

x∈UB⊗ˆkH(x), x7→f(x), where f(x) is the image of f by the map OX(B)(U) → B⊗ˆkH(x). We will call an analytic function over U with value in B, any function ψ:U →`

x∈UB⊗ˆkH(x)induced by an elementf ∈ OX(B)(U).

Hypothesis 2.15. Until the end of the paper, we will assume that all Banach algebras are with unit element.

Definition 2.16. LetEbek-Banach algebra andf ∈E. The spectrum off is the setΣf,k(E)of points x∈A^1,an_k such that the elementf ⊗1−1⊗T(x)is not invertible in the k-Banach algebraE⊗ˆkH(x).

The resolvent off is the function:

Rf :A^1,an_k \Σf,k(E) −→ `

x∈A^1,an_k \Σf,k(A)

E⊗ˆkH(x)

x 7−→ (f⊗1−1⊗T(x))⁻¹

Remark 2.17. If there is no confusion we denote the spectrum off, as an element ofE, just by Σf. Theorem 2.18. Let E be a Banachk-algebra andf ∈E. Then:

1. The spectrumΣf is a non-empty compact subset of A^1,an_k .

2. The radius of the smallest (inclusion) closed disk with center at zero which containsΣf is equal to kfkSp.

3. The resolvent Rf is an analytic function onP^1,an_k \Σf which is equal to zero at infinity.

Proof. See [Ber90, Theorem 7.1.2].

Remark 2.19. LetE be ak-Banach algebra andr∈R^∗₊. Note that we have OA^1,an

k (E)(D⁺(a, r)) =E⊗ˆkO(D⁺(a, r)), i.e any element ofOA^1,an

k (E)(D⁺(a, r))has the formP

i∈Nfi⊗(T−a)ⁱwithfi∈E. Letϕ=P

i∈Nfi⊗(T− a)ⁱ be an element ofO_A^1,an

k (E)(D⁺(a, r)). Sincekfi⊗(T−a)ⁱk=kfikk(T−a)kⁱin O_A^1,an

k (E)(D⁺(a, r)) (cf. Proposition2.4), the radius of convergence ofϕwith respect toT−ais equal tolim inf

i→+∞kfik⁻¹ⁱ. Lemma 2.20. We maintain the same assumption as in Theorem2.18. Ifa∈(A^1,an_k \Σf)∩k, then the biggest open disk centred in acontained inA^1,an_k \Σf has radius R=k(f−a)⁻¹k⁻¹_Sp.

Proof. Since Σf is compact and not empty, the biggest disk D⁻(a, R) ⊂A^1,an_k \Σf has finite positive radiusR. In the neighbourhood of the pointa, we have:

Rf = (f⊗1−1⊗T)⁻¹= ((f−a)⊗1−1⊗(T−a))⁻¹= ((f −a)⁻¹⊗1)X

i∈N

1

(f−a)ⁱ ⊗(T−a)ⁱ. On the one hand the radius of convergence of the latter series with respect to (T −a) is equal to k(f−a)⁻¹k⁻¹_Sp (cf. Remark2.19). On the other hand, the analyticity ofRf onD⁻(a, R)implies that the radius of convergence with respect toT −ais equal toR. Hence, we haveR=k(f−a)⁻¹k⁻¹_Sp.

Proposition 2.21. Let E be a commutativek-Banach algebra element, andf ∈E. The spectrum Σf

of f coincides with the image of the analytic spectrum M(E)by the map induced by the ring morphism k[T]→E,T 7→f.

Proof. See [Ber90, Proposition 7.1.4].

Definition 2.22. LetE be a k-Banach algebra andB a commutativek-subalgebra of E. We say that B is a maximal subalgebra of E, if for any subalgebraB^′ of Ewe have the following property:

(B⊂B^′⊂E)⇔(B^′=B).

Remark 2.23. A maximal subalgebraB is necessarily closed inE, hence ak-Banach algebra.

Proposition 2.24. LetE be ak-Banach algebra. For any maximal commutative subalgebra B ofE, we have:

∀f ∈B; Σf(B) = Σf(E).

Proof. See [Ber90, Proposition 7.2.4].

Let P(T) ∈ k[T], let E be a Banach k-algebra and let f ∈ E. We set P(f) to be the image of P(T)by the morphismk[T]→E, T 7→f, andP :A^1,an_k →A^1,an_k to be the analytic map associated to k[T]→k[T],T 7→P(T).

Lemma 2.25. Let P(T) ∈k[T], let E be a Banach k-algebra and let f ∈ E. We have the equality of sets:

ΣP(f)=P(Σf)

Proof. LetB a maximal commutative k-subalgebra of E containingf (which exists by Zorn’s Lemma).

Then B contains alsoP(f). By Proposition 2.24we haveΣf(E) = Σf(B)andΣP(f)(E) = Σf(B). Let

∗f : M(B) → A^1,an_k (resp. ^∗P(f) :M(B)→ A^1,an_k ) be the map induced by k[T]→ E, T 7→ f (resp.

T 7→P(f)). By Proposition2.21 we haveΣf(B) =^∗ f(M(B))and ΣP(f)(B) =^∗ P(f)(M(B)). Since

∗P(f) =P◦^∗f, we obtain the equality.

Remark 2.26. Note that, we can imitate the proof provided in [Bou07, p.2] to prove the statement of Lemma 2.25.

Lemma 2.27. Let E and E^′ be two Banach k-algebras and ϕ : E → E^′ be a bounded morphism of k-algebras. Iff ∈E then we have:

Σϕ(f)(E^′)⊂Σf(E).

If moreover ϕis a bi-bounded isomorphism then we have the equality.

Proof. Consequence of the definition.

Let M1 and M2 be two k-Banach spaces, let M = M1 ⊕M2 endowed with the max norm (i.e.

∀m1∈M1 and∀m2∈M2km1+m2k= max(km1k,km2k)). We set:

M(M1, M2) =

L1 L2

L3 L4

|L1∈ Lk(M1), L2∈ Lk(M2, M1), L3∈ Lk(M1, M2), L4∈ Lk(M2)

We define the multiplication inM(M1, M2)as follows:

A1 A2

A3 A4

B1 B2

B3 B4

=

A1B1+A2B3 A1B2+A2B4

A3B1+A4B4 A3B2+A4B4

. ThenM(M1, M2)endowed with the max norm is ak-Banach algebra.

Lemma 2.28. We have a bi-bounded isomorphism of k-Banach algebras:

Lk(M)≃ M(M1, M2).

Proof. Letpj be the projection ofM ontoMj and ij be the inclusion ofMj into M, where j ∈ {1,2}.

We define the following twok-linear maps:

Ψ1:Lk(M) −→ M(M1, M2) ϕ 7→

p1ϕi1 p1ϕi2

p2ϕi1 p2ϕi2

Ψ2:M(M 1, M2) −→ Lk(M) L1 L2

L3 L4

7→ i1L1p1+i1L2p2+i2L3p1+i2L4p2

Since the projections and inclusions are bounded maps, then the mapsΨ1 andΨ2 are bounded too. It is easy to show thatΨ1◦Ψ2 =idM(M1,M2) andΨ2◦Ψ1 =idLk(M). Hence we have an isomorphism of k-Banach spaces.

We will need the following lemma for the computation of the spectrum:

Lemma 2.29. LetM1 andM2 bek-Banach spaces and letM =M1⊕M2 endowed with the max norm.

Let p1,p2 be the respective projections associated to M1 and M2 andi1,i2 be the respective inclusions.

Let ϕ∈ Lk(M)and setϕ1=p1ϕi1∈ Lk(M1)andϕ2=p2ϕi2∈ Lk(M2). Ifϕ(M1)⊂M1, then we have:

i) Σϕi(Lk(Mi))⊂Σϕ(Lk(M))∪Σϕj(Lk(Mj)), wherei, j∈ {1,2}andi6=j.

ii) Σϕ(Lk(M)) ⊂ Σϕ1(Lk(M1))∪Σϕ2(Lk(M2)). Furthermore, if ϕ(M2) ⊂ M2, then we have the equality.

iii) IfΣϕ1(Lk(M1))∩Σϕ2(Lk(M2)) =∅, thenΣϕ(Lk(M)) = Σϕ1(Lk(M1))∪Σϕ2(Lk(M2)).

Proof. By Lemma2.28, we can represent the elements ofLk(M)as follows:

Lk(M) =

L1 L2

L3 L4

|L1∈ Lk(M1), L2∈ Lk(M2, M1), L3∈ Lk(M1, M2), L4∈ Lk(M2)

andϕhas the form

ϕ1 L 0 ϕ2

, whereL∈ Lk(M2, M1).

Letx∈A^1,an_k . We have an isomorphism ofk−Banach algebras:

Lk(M) ˆ⊗kH(x) =

L1 L2

L3 L4

| L1∈ Lk(M1) ˆ⊗kH(x), L2∈ Lk(M2, M1) ˆ⊗kH(x), L3∈ Lk(M1, M2) ˆ⊗kH(x), L4∈ Lk(M2) ˆ⊗kH(x)

Consequently,

ϕ⊗1−1⊗T(x) =

ϕ1⊗1−1⊗T(x) L⊗1 0 ϕ2⊗1−1⊗T(x)

. We first prove i). Let

L1 C 0 L2

be an invertible element ofLk(M) ˆ⊗kH(x). We claim that if, for i ∈ {1,2}, Li is invertible inLk(Mi) ˆ⊗kH(x), then so is Lj, where j 6=i. Indeed, let

L^′₁ C^′ B L^′₂

such that we have:

L1 C 0 L2

L^′₁ C^′ B L^′₂

= 1 0

0 1

;

L^′₁ C^′ B L^′₂

L1 C 0 L2

= 1 0

0 1

.

Then we obtain: 









L1L^′₁+CB= 1 L1C^′+CL2= 0 L2B= 0

L2L^′₂= 1

;











L^′₁L1= 1 L^′₁C+C^′L2= 0 BL1= 0 BC+L^′₂L2= 1

.

We deduce that L1 is left invertible and L2 is right invertible. If L1 is invertible, then B = 0 which implies thatL2is left invertible, hence invertible. IfL2 is invertible, thenB= 0which implies thatL1 is right invertible, hence invertible. Therefore, ifϕ⊗1−1⊗T(x)andϕi⊗1−1⊗T(x)are invertible where i ∈ {1,2}, thenϕj⊗1−1⊗T(x)is invertible for j ∈ {1,2} \ {i}. We conclude thatΣϕj ⊂Σϕ∪Σϕi

wherei, j ∈ {1,2}and i6=j.

We now prove ii). Ifϕ1⊗1−1⊗T(x)andϕ2⊗1−1⊗T(x)are invertible, thenϕ⊗1−1⊗T(x) is invertible. This proves that Σϕ ⊂ Σϕ1 ∪Σϕ2. Ifϕ(M2) ⊂ M2, then L = 0 which implies that: if ϕ⊗1−1⊗T(x) is invertible, then ϕ1⊗1−1⊗T(x) andϕ2⊗1−1⊗T(x) are invertible. Hence we have the equality.

We now prove iii). IfΣϕ1∩Σϕ2 =∅, then by above we haveΣϕ1 ⊂Σϕ andΣϕ2 ⊂Σϕ. Therefore, Σϕ1∪Σϕ2 ⊂Σϕ.

Remark 2.30. Set notations as in Lemma2.29. In the proof above, we showed also: ifϕ⊗1−1⊗T(x) is invertible, then ϕ1⊗1−1⊗T(x)is left invertible andϕ2⊗1−1⊗T(x)is right invertible.

3 Differential modules and spectrum

3.1 Preliminaries

Recall that a differential k-algebra, denoted by (A, d), is a commutative k-algebra A endowed with a k-linear derivation d : A → A. A differential module (M,∇) over (A, d) is a finite free A-module M equipped with ak-linear map∇:M →M, called connection of M, satisfying∇(f m) =df.m+f.∇(m) for allf ∈Aandm∈M. If we fix a basis ofM, then we get an isomorphism ofA-modulesM→A˜ ⁿ, and the operator∇ is given in this basis by the rule:

∇



 f1

... fn



=



 df1

... dfn



+G



 f1

... fn



 (8)

where G ∈ Mn(A) is a matrix. Conversely the data of such a matrix defines a differential module structure onAⁿ by the rule (8).

A morphism between differential modules is a k-linear map M →N commuting with connections.

We setAhDi= L

i∈N

A.Dⁱ to be the ring of differential polynomials equipped with the non-commutative multiplication defined by the rule: D.f =df+f.D for all f ∈A. LetP(D) =g0+· · ·+gν−1D^ν−1+ D^ν be a monic differential polynomial. The quotient AhDi/AhDi.P(D) is a finite free A-module of rank ν. Equipped with the multiplication by D, it is a differential module over (A, d). In the basis {1, D, . . . , D^ν−1} the multiplication byD satisfies:

D









 f1

fν











=









 df1

dfν









 +













0 0 −g0

1 0 0

0

0

0 0 1 −gν−1





















 f1

fν











Theorem 3.1 (The cyclic vector theorem). Let (A, d) be a k-differential field (i.e A is a field), with d6= 0, and let(M,∇) be a differential module over(A, d)of rankn. Then there existsm∈M such that {m,∇(m), . . .∇ⁿ⁻¹(m)} is a basis ofM. In this case we say that mis cyclic vector.

Proof. See [Ked10, Theorem 5.7.2.].

Remark 3.2. The last theorem means that there exists an isomorphism of differential modules between (M,∇)and(AhDi/AhDi.P(D), D)for some monic differential polynomial P(D)of degreen.

Lemma 3.3. Let L,P and Q be differential polynomials, such that L=QP. Then we have an exact sequence of differential modules:

0 //AhDi/AhDiQ ⁱ //AhDi/AhDiL ^p //AhDi/AhDiP //0

where the mapsiandpare defined as follows: for a differential polynomialR,i( ¯R) =RP andp( ¯R) = ¯R.

Proof. See [Chr83, Section 3.5.6].

3.2 Spectrum associated to a differential module

Hypothesis 3.4. From now onAwill be either ak-affinoid algebra associated to an affinoid domain of A^1,an_k orH(x)for somex∈A^1,an_k not of type (1). Letdbe a bounded derivation onA. It is of the form d=g(T)_dT^d whereg(T)∈A.

Let (M,∇) be a differential module over (A, d). In order to associate to this differential module a spectrum we need to endow it with a structure of k-Banach space. For that, recall the following proposition:

Proposition 3.5. There exists an equivalence of category between the category of finite BanachA-modules with bounded A-linear maps as morphisms and the category of finite A-modules with A-linear maps as morphisms.

Proof. See [Ber90, Proposition 2.1.9].

This means that we can endow M with a structure of finite Banach A-module isomorphic to Aⁿ equipped with the maximum norm, and any other structure of finite BanachA-module onM is equivalent to the previous one. This induces a structure of Banachk-space onM. As∇ satisfies the rule (8) and d∈ Lk(A), we have∇ ∈ Lk(M). The spectrum associated to (M,∇)is denoted by Σ∇,k(Lk(M))⁵ (or just by Σ∇ if the dependence is obvious from the context).

Let ϕ : (M,∇) → (N,∇^′) be a morphism of differential modules. If we endow M and N with a structure of k-Banach space (as above) then ϕ is automatically an admissible⁶ bounded k-linear map (see [Ber90, Proposition 2.1.10]). In the caseϕis an isomorphism, then it induces a bi-boundedk-linear isomorphism and according to Lemma2.27we have:

Σ∇,k(Lk(M)) = Σ∇^′,k(Lk(N)). (9) This prove the following proposition:

Proposition 3.6. The spectrum of a connection is an invariant by isomorphisms of differential modules.

Proposition 3.7. Let 0 → (M1,∇1) → (M,∇) → (M2,∇2) → 0 be an exact sequence of differential modules over (A, d).

Then we have: Σ∇(Lk(M))⊂Σ∇1(Lk(M1))∪Σ∇2(Lk(M2)), with equality if Σ∇1(Lk(M1))∩Σ∇2(Lk(M2)) =∅.

Proof. AsM1,M2 andM are freeA-modules, the sequence:

0 //M1 f

//M ^g //M2 //0

splits. Hence, we haveM =M1⊕M2wheref is the inclusion ofM1intoM andgis the projection ofM ontoM2. Letp1 be the projection ofM ontoM1 andi2 be the inclusion ofM2 into M. As bothf and g are morphisms of differential modules, we have∇(M1)⊂M1,∇1=p1∇f and∇2=g∇i2. By Lemma 2.29and Remark2.30we obtain the result.

5Note that, since all the structures of finite BanachA-module onM are equivalente, the spectrum does not depend on the choice of such structure.

6Which means that: M/Kerϕendowed with quotient the topology is isomorphic ask-Banach space toImϕ.

Remark 3.8. We maintain the assumption of Lemma3.7. If in addition we have an other exact sequence of the form:

0→(M2,∇2)→(M,∇)→(M1,∇1)→0,

then the equality holds, this is a consequence of Remark2.30. Indeed, If∇ ⊗1−1⊗T(x)is invertible, then the first exact sequence shows that ∇1⊗1−1⊗T(x) is left invertible and ∇2⊗1−1⊗T(x) is right invertible, the second exact sequence to∇2⊗1−1⊗T(x)is left invertible and∇1⊗1−1⊗T(x) is right invertible. Therefore, both of∇1⊗1−1⊗T(x)and∇2⊗1−1⊗T(x)are invertible. Hence, we obtainΣ∇1∪Σ∇2 ⊂Σ∇.

Remark 3.9. If moreover we haveM =M1⊕M2as differential modules then we haveΣ∇= Σ∇1∪Σ∇2. Remark 3.10. Set notation as in Proposition3.7. We suppose thatA=H(x)for some pointx∈A^1,an_k not of type (1). For the spectral semi-norm it is know (see [Ked10, Lemma 6.2.8]) that we have:

k∇kSp= max{k∇1kSp,k∇2kSp}.

We say that a differential module(M,∇)over(A, d)of ranknis trivial if it isomorphic to(Aⁿ, d)as a differential module.

Lemma 3.11. Let(M,∇)be a differential module over a differential field(K, d)of rankn. If thek-vector spaceKer∇ has dimension equal ton, then(M,∇)is a trivial differential module.

Proof. See [Chr83, Proposition 3.5.3].

Corollary 3.12. We suppose that A = H(x) for some x ∈ A^1,an_k not of type (1). Let (M,∇) be a differential module over (A, d). If dim(Ker∇) =n, thenΣ∇(Lk(M)) = Σd(Lk(A)).

Proof. By Lemma 3.11there exists {e1, . . . , en} a basis of M as anA-module for which∇ satisfies the rule:

∇



 f1

... fn



=



 df1

... dfn



.

By induction and Remark3.9 we obtain the result.

Lemma 3.13. We suppose thatkis algebraically closed. Let (M,∇)be a differential module over(A, d) such that G ∈ Mn(k)(cf. (8)) and {a1,· · · , aN} is the set of the eigenvalues of G. Then we have an isomorphisme of differential modules:

(M,∇)≃( M

1≤i≤N

M

1≤j≤N_i

AhDi/(D−ai)^ni,j, D)

where theni,j are positive integers such thatPNi

j=1ni,j is the multiplicity ofai andP

i,jni,j=n.

Proof. Consequence of the Jordan reduction.

Lemma 3.14. Let (M,∇)be the differential module over(A, d)associated to the differential polynomial (D−a)ⁿ, where a∈ k. The spectrum of∇ is Σ∇(Lk(M)) = a+ Σd(Lk(A)) (the image of Σd by the polynomialT +a).

Proof. By Lemma3.3, we have the exact sequences:

0→(AhDi/(D−a)ⁿ⁻¹, D)→(AhDi/(D−a)ⁿ, D)→(AhDi/(D−a), D)→0 and

0→(AhDi/(D−a), D)→(AhDi/(D−a)ⁿ, D)→(AhDi/(D−a)ⁿ⁻¹, D)→0 By induction and Remark3.8, we haveΣD= Σd+a. By Lemma2.25, we obtainΣD=a+ Σd.