Deformations of solutions of differential equations

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Deformations of solutions of differential equations

Mercedes Haiech

To cite this version:

Mercedes Haiech. Deformations of solutions of differential equations. 2021. �hal-03230481�

DEFORMATIONS OF SOLUTIONS OF DIFFERENTIAL EQUATIONS

MERCEDES HAIECH

ABSTRACT. As in algebraic geometry where the formal neighborhood of a point of a scheme contains informations about the singularities of the object, we ex- tend this study to schemes where a point represents a solution of an algebraic differential equation. The obtained geometric object being of infinite dimension, a first step is to show that the formal neighborhood of a point not canceling the separant is noetherian, using considerations on the embedding dimension. We show that, in the neighborhood of points making the separant invertible, the em- bedding dimension is exactly the order of the considered differential equation. In a second step, we relate, for a certain type of differential equations of order two, the existence of essential singular components to the decrease of the embedding dimension, in the neighborhood of certain points.

INTRODUCTION

The study of the deformations of solutions of differentials equations can be seen as a natural generalization of the local study of the arc scheme. The arc scheme is an object introduced by J. NASHin the 60’s in an article published later. It is constructed as the arcs drawn on a given scheme. More precisely, ifXis a scheme over a fieldK-in this articleKis assumed to be of characteristic zero-, and ifAis a K-algebra, then anA-point of the arc scheme ofXcorrespond to anArrTss-point ofX.

On the other hand the arc scheme has a natural definition thanks to differential algebra. Let Kty₁,¨ ¨ ¨, y_nu :“ Kry_i,j | 1 ď i ď n, j P Ns be the ring of differential polynomials endowed with the derivation∆such that∆pyi,jq “yi,j`1. Then if X “ SpecpKry₁,¨ ¨ ¨ , y_ns{Iq whereI is some ideal of Kry₁,¨ ¨ ¨ , y_ns, then the arc scheme ofX, denoted byL8pXq, can be described by theK-scheme SpecpKty₁,¨ ¨ ¨ , y_nu{rIsq, whererIsstands for the differential ideal generated by I.

In the case of X “ SpecpKry₁,¨ ¨ ¨ , y_ns{Iq, the ideal that defines the arc scheme is -differentially- generated by elements of order 0. This idea can be generalized by studying geometrical objects likeSpecpKty1,¨ ¨ ¨ , ynu{JqwithJ a differential ideal of Kty₁,¨ ¨ ¨ , y_nu (not necessarily generated by differentials polynomials of order 0).

In addition to be some natural generalization of the arc scheme, it has also a meaning regarding differential algebra. The schemeSpecpKty1,¨ ¨ ¨ , ynu{Jqcan be understood as the the solutions -which are formal series- of the elements ofJ seen as differential equations. In order to underline the link between this object

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and the arc scheme, and since its related to differential equations, we are going to refer to schemes likeSpecpKty₁,¨ ¨ ¨ , y_nu{Jqasdifferential arc schemes.

Motivated by a result about the local structure of the arc scheme at the neigh- borhood of a rational point by M. GRINBERG& D. KAZHDANand V. DRINFELD

(see [GK00] and [Dri02]) D. BOURQUI& J. SEBAGbegan the study of the local structure of differential arc scheme.

Theorem 0.1(Drinfeld, Grinberg-Kazhdan). LetXbe a scheme of finite type over a fieldK, letγ: SpecKrrTss Ñ Xbe a non-degenerated rational arc,i.ean el- ement ofpL8pXqzL8pnSmpXqqqpkq. We denote byL8pXq_γthe formal neigh- borhood ofγinL8pXq. Assume thatdim_γp0qpXq ě1. There exists aK-scheme Sof finite type, a pointsPSpkqand an isomorphism of formalK-scheme

L8pXqγ–SsˆkSpfpKrrpTiqiPNssq.

The local structure theorem of the arc scheme states that all the informations about the formal neighborhood of a rational arc are encoded by a scheme of finite type although the arc scheme is, in general, not of finite type. The result of D.

BOURQUI& J. SEBAG(see [BS16]) states that for differential arc schemes defined by a differential equationFof order 1 the formal neighborhood of non degenerated points is a formal disk of dimension 1. More precisely:

Theorem 0.2(Bourqui-Sebag). Let K be a field of characteristic zero. LetF P Ktyube a nonconstant differential polynomial of order 1. LetX^B“SpecpKtyu{rFsq. Letγ PX^Bpkqbe a nonconstant differential arc. Assume thatγ does not conceal the separant ofF. Then the formal neighborhood ofγ, denotedX_γ^B, is isomorphic toSpfpKrrTssq.

A natural question is to ask if this theorem remains true for differential poly- nomials of order greater than 1. By looking at several examples, it turns out that it is not simple (see section 6.2). If, in general, for F a differential polynomial, the formal neighborhood of a solution ofF “ 0 is not easy to compute, we can show that, as in theorem 0.2, it remains noetherian by looking at the embedding dimension of the ring.

This article is organized as follow : the first section recalls some facts and theo- rems about differential algebra. In particular we will states the Low power theorem due to J. F. RITTwith gives a effective way, givenF a differential polynomial, to decide if a differential polynomialAgives rise to a a essential component ofF. In the second and third sections we recall the definitions of the functor of deforma- tions (definition 2.10) and embedding dimension (definition 3.1) and some useful properties to deal with it. Along the way, we prove the following theorem

Theorem A. LetpA,M_Aqbe a local ring. The following assertions are equivalent:

(1) The embedding dimension of the local ring A, denoted emb.dimpAq is finite.

(2) The completionAp:“lim

ÐÝnA{Mⁿ_AofAis noetherian.

(3) The embedding dimension of the local ring A, denotedp emb.dimpAqp is finite.

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Furthermore, if one of this equivalent conditions is verified, thenemb.dimpAq “ emb.dimpAqp and the ringApis complete for theMy_Apre-adic topology.

Note that the equivalence p1q ô p2q can already be found in [dFD, Lemma 10.12]. The rest is, up to my knowledge, new. Moreover, even if it is well known that forAa noetherian ring the topology onApis theMy_Apre-adic topology, this statement is, in general, false whenAis not noetherian (see for example [Yek11]

or [Hai20]).

In the 4th part, givenFa differential polynomial andγa solution ofF “0that does not conceal the separant ofF, we show that the problem of deciding if the formal neighborhood ofF atγis noetherian can be reduced to a problem of linear algebra, and we state that the embedding dimension of the formal neighborhood is smaller than the order ofF.

Theorem B. Let K be a field of characteristic zero. Let F P Ktyu be a non constant differential polynomial of ordern. Letγ P KrrTssbe a solution of the ODEF “0which does not cancel the separantS_F. LetX^B “Ktyu{rFs. So the formal neighborhoodL8{pX^Bqγis noetherian and its embedding dimension is less or equal ton.

This result gives a upper bound for the embedding dimension, but we can have a more precise result when the embedding dimension is at most one.

Theorem C. Let K be a field of characteritic zero. Let F P Ktyu be a non constant irreducible differential polynomial andX^B :“ SpecpKtyu{rFsqthe as- sociated differential scheme. Let γ P KrrTss be a non constant solution of the differential equationF “ 0. Assume that the embedding dimension of X^B at γ is at most 1. Then the formal neighborhood L{8pX^Bq_γ is isomorphic, as formal K-scheme, to the formal diskD“SpfpKrrTssqof dimension 1.

This statement applies for non constantFof any order, in particular it allows us to recover the statement of theorem 0.2.

Last section investigates the case of particular differentials equations of order two. For equationsF PKtyuof the formx^a₁´αx^b₂x^c₃withxi P ty0, y1, y2uztyj1, yj2u forj₁, j₂ ‰i, called binomomial, the embedding dimension seems to be linked to the existence of essential singular component. More precisely we have:

Theorem D. Let K be a field of characteristic zero. Let F P Kry₀, y₁, y₂s an irreductible polynomial. Assume that the ODEFpy, y¹, y²q “ 0is abimonomial equation of order two with constant coefficients. Letdě2be an integer such that γpTq “T^dis a solution of theF “0andX^B “SpecpKtyu{rFsq. If the perfect differential idealtFuhas a essential singular component, thenedimpO_L{

8pX^Bq,γq “ 1.

However the converse is false, there exist bimonomial equations without essen- tial singular components such that the embedding dimension in the neighborhood of a solution is 1.

To conclude, we present, at the end of the last section, various examples of equations and computations of embedding dimensions.

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1. RECOLLECTION ON THE LOW POWER THEOREM

LetK be a field of characteristic 0. Recall that ifpR,`,¨qis a ring, a deriva- tion ∆ on R is a linear application for + and which satisfies the Leibniz Rule

∆pabq “a∆pbq `∆paqb. A ring endowed with one, or many derivations is called a differential ring. An ideal of a differential ring is called differential if it is stable under the action of the derivations. A differential idealI of a differential ringRis say to be perfect of for everyaPR, the fact thataⁿPIimpliesaPI.

The notationKtyuwill stand for the differential ringpKry_i|iPNs,∆q, where the derivation sends an element ofK to 0 and satisfies∆pyiq “ yi`1. IfI is an ideal ofKtyu, thentIuwill denote the intersection of perfect differential ideals of KtyucontainingI. IfF is an element ofKtyuof order`, the separantSF ofF is

BF By`.

1.1. Decomposition of perfect differential ideals. Let F P Ktyu be a non- constant differential polynomial. We will study the decomposition of the ideal tFuas intersection of prime differential ideals.

We know that any perfect differential idealIdecomposes into an intersection of prime differential ideals (see [Rit50, Chapter 1, 16] or [Kap76, Chapter VII, Thm 7.5 & Thm 7.6]). SincetF, S_Fuis a perfect ideal ofKtyuthere exists irredundant differential prime idealsP₁, . . . ,P_ssuch that

tF, S_Fu “ X^s_i“1P_i. Furthermore, theP_iare unique up to reordering.

DefineptFu : SFq as the set of elementsA in Ktyu such thatASF P tFu.

A classical result of differential algebra can be stated as (see [Kap76, Theorem 7.10]):

Proposition 1.1. Let K be a field of characteristic 0. Let F P Ktyu be an ir- reducible differential polynomial. ThentFuhave the following decomposition as intersection of perfect ideals

tFu “ ptFu:S_Fq X tF, S_Fu.

Moreover ptFu : SFq is a prime differential ideal. If we denote tF, SFu “ X^s_i“1P_i, there exists a subset J Ă t1, . . . , su such that the non-redundant irre- ducible decomposition of the perfect differential idealJ Ă t1, . . . , suis given by

tFu “ ptFu:SFq X pXjPJP_jq.

Definition 1.2. The prime differential idealptFu:SFqis calledcomponent of the general solution ofF ¹; theP_j intervening in the formula, thecomponents of the essential singular solutions ofF ².

Remark1.3. Thanks to proposition 1.1 we can deal with cases where the polyno- mialF is not irreducible. The irreducible decomposition of the differential ideal

1Sometimes we also say general component.

2We sometimes also say essential singular components.

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tFuwill, indeed, be obtained from this proposition applied to the irreducible fac- tors ofF.

Let’s illustrate proposition 1.1 by an example from [Kap76, chap VII, 31].

Example 1.4. LetF “ y₁² ´4y0 P Ktyu. Thanks to proposition 1.1, we can decompose the perfect differential idealtFuunder the form

tFu “ ptFu:SFq X tF, SFu.

ButSF “2y1, hencetF, SFu “ ty0u. Since the idealty0uis prime, we have the decomposition oftF, S_Fuas an intersection of prime differential ideal. It remains to identify the idealptFu:SFq, but we already know that it is prime.

The derivative of F factors: ∆pFq “ 2y₁py₂ ´2q. We will prove that y₁ R ptFu :SFq. Assume thaty1 P ptFu : SFq. Sincey1 P tF, SFuand thattFuis the intersection of these two ideals we should have y₁ P tFu. In particular, this impliesSol_KpFq ĂSol_Kpy₁q “K. But the set

Sol_KpFq “ txptq PKrrtss |Fpxptqq “0u

containsxptq “t²which is not an element ofSol_Kpy1q. Hencey1 R ptFu:SFq, but since this ideal is prime and contain∆pFq, we deduce thatpy₂´2q P ptFu: SFq. We denoteQ “ ty²₁ ´4y0, y2 ´2u.This ideal is contained inptFu : SFq and is prime because the morphism

Ktyu{tF, y2´2u Ñ Kry0, y1s{xFy

y₀ ÞÑ y₀

y₁ ÞÑ y₁

y2 ÞÑ 2

y_i ÞÑ 0 foriě3

is a ring isomorphism. Butxy²₁´4y₀yis a prime ideal ofKry₀, y₁s, hence the ideal Qis prime. SincetFu ĂQ, we have

tFu “ ptFu:SFq X tF, SFu XQ.

ButQĂ ptFu:SFq, then the decomposition ofFas prime ideals is the following:

tFu “ ty₁²´4y0, y2´2u X ty0u and we deduce thatty₁²´4y₀, y₂´2u “ ptFu:S_Fq.

1.2. Low power theorem. Ritt’s low power theorem is one of the great success of the Rittian approach to differential algebra. This theorem is a simple algorithmic statement which fits into the problem of determining the irreducible decomposition of a perfect idealtFu ofKty1, . . . , ynu. Introduced in [Rit50, II] in the case of one-derivative differential fields, this theorem also finds a presentation in [Kol73, 13,15] in the case of arbitrary finite sets of derivations. In this section, we will limit ourselves to the case of a differential fieldK, equipped with a derivation∆ possibly trivial as proposed by Ritt.

This theorem exploits an algorithmic preparation procedure, which can be in- terpreted as a kind of pseudo-division. Here is how the algorithmic preparation procedure is expressed (see [Rit50, II, 17]).

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Proposition 1.5(Preparation process). LetAandF be two differential polynomi- als ofKty₁, . . . , y_nuof the same classh. We notelthe order ofAandmthe order ofF with respect toyh. One notesSAthe separant ofA. There exists two integers tPNandrP zt0usuch thatS_A^tF is of the form

S_A^tF “

r

ÿ

j“1

CjA^pi∆pAq^i1,j∆²pAq^i2,j¨ ¨ ¨∆^m´lpAq^im´l,j with

(1) thep_j andi_k,jare positive integers;

(2) thepm´lq-upletspi1,j,¨ ¨ ¨ , im´l,jqare all distinct

(3) the order of theC_j is smaller thanliny_h and are not divisible byA.

Example 1.6. (1) LetF “y₁²´4y₀ andA “y₀, thenF is already in "pre- pared" form witht “ 0, r “ 2, C1 “ 1, p1 “ 0, i1,1 “ 2,C2 “ ´4, p₂ “1, and all other integers are 0.

(2) Similarly, ifF “y²₁´4y0 andA “ y1, thenF is already in "prepared"

form witht “0,r “ 2,C₁ “ 1,p₁ “ 2andC₂ “ ´4y₀, and all other integers are zero.

The low power theorem can then be stated as follows (see [Rit50, II, 20]).

Theorem 1.7(Low power theorem). LetK be a field of characteristic zero. Let A, F PKtyutwo irreducible differential polynomials.

Let

(1.1) S_A^tF “

r

ÿ

j“1

CjA^piδpAq^i1,jδ²pAq^i2,j¨ ¨ ¨δ^m´lpAq^im´l,j

be a preparation ofF with respect toA. So the prime ideal differentialtAu :S_A is an irreducible component oftFuif and only if

(1) the right part of equation 1.1 contain a termC_kA^pk free of proper deriva- tion ofA

(2) for every integerj‰k, we havep_kăpj`i1,j` ¨ ¨ ¨ `i_m´l,j.³ Example 1.8. Let’s go back to the previous examples 1.6.

(1) Let F “ y²₁ ´4y0 andA “ y0, then there is a term of the form´4y0, where the derivative ofAdoes not appear. Andp₂ “1 ăp₁ `i_1,1 “ 2.

Soy₀is an irreducible component oftFu.

(2) IfF “y₁²´4y0andA“y1, then the first hypothesis of the lower power theorem is verified, but not the second, soy₁isn’t an irreducible component oftFu.

LetF P Ktyu. If we want to test if ty₀u is an essential singular component oftFu, the low power theorem takes the following simpler form: the differential idealty₀uis a component oftFuif and only if the expression ofF contains a term

3Ifm“land 1.1 contains a single term of the formCkA^pk, the condition will also be regarded as fulfilled.

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of the formαy0- withαPK- of -total- degree strictly inferior to the degree of any other (non-zero) monomial appearing in the expression ofF.

2. DEFORMATIONS OF A POINT

This section is a recollection about the space of deformations of a point. Nota- tions and definitions are introduced in order recall the proof that the functor that describes the space of deformation (definition 2.10) is representable.

2.1. Notations. In the followingKwill refer to an arbitrary field. We will define three categories:

(1) The categoryAlgloc_Kwhose objects are localK-algebras whose residual field isK-isomorphic toKand whose morphisms are morphisms of local rings.

(2) The categoryAlgLC_Kwhose objects are topological and localK-algebras which are a completion of an object ofAlgloc_Kand whose morphisms are the continuous morphisms ofK-algebras.

(3) The categoryQArt_K, which is a full subcategory ofAlgLC_K, whose the objects are the objects ofAlgLC_K whose maximal ideal is nilpotent.

Definition 2.1. An object of the categoryQArt_K will be called aquasi-artinian ring.

Remark2.2. LetpA,m_Aqbe an object ofAlgloc_K which is noetherian. Then it is known (see [Mat80, 23.L, Corollary 4]) that the completion ofAwith respect to its maximal ideal is complete for them_A(pre)adic topology. However this turn out to be false ifAis not noetherian (see [Hai20]).

Remark2.3. LetpA,M_Aqan object ofAlgloc_KandB“lim

ÐÝnA{Mⁿ_Aits comple- tion. In addition to the preceding remark let us insist on the fact that the maximum ideal of B is described by M_B “ My_A. The natural topology on B (resulting from its construction as projective limit) is described by the familypMyⁿ_AqnPN. In general, the familypMyⁿ_AqnPN does not coincide with the family of powers of the maximum ideal of B (unless A is Noetherian). If A is not Noetherian, usually Myⁿ_A‰My_Aⁿ. So we’ll take special care to distinguish these two topologies.

Remark2.4. The category of quasi-Artinian rings is referred to in some references astest rings(see for example [Dri02, BS17]).

Proposition 2.5. Let pA,m_Aq,pB,m_Bq be two objects of Algloc_K. Let Ap “ ÐÝlimnA{mⁿ_A, and Bp “ lim

ÐÝnB{mⁿ_B. By construction Ap and Bp are objects of AlgLC_K.

Letϕ:ApÑBpbe a ring morphism.

(1) If the morphismϕis continuous, then it’s local;

(2) If the morphismϕis local and if the family of ideals pMy_AⁿqnPN forms a base for the topology ofApthenϕis continuous.

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Proof. (1) Let’s assume thatϕis continuous, and let’s show thatϕ^´1pMy_Bq “My_A. Sinceϕis continuous, thenϕ^´1pMy_Bqis an open subset ofMy_A. In particular, since 0Pϕ^´1pMy_Bq, there is an integerjsuch as

My_A^j ĂMy^j_AĂϕ^´1pMy_Bq.

Now since ϕ^´1pMy_Bq is a prime ideal of A, the previous inclusions imply thatp My_AĂϕ^´1pMy_Bq. Soϕis local.

(2) Assume that ϕ is local. Let U be an open B. Letp x P ϕ^´1pUq. Since ϕpxq P U, there exists an integer j such that ϕpxq `My<sup>j</sup><sub>B</sub> Ă U. Furthermore My<sub>B</sub><sup>j</sup> Ă My<sup>j</sup><sub>B</sub>. So ϕpx`My_A^jq Ă ϕpxq `My_B^j Ă U. So ϕ^´1pUq is an open

subset, thusϕis continuous.

Proposition 2.6. LetpA,m_Aq,pB,m_Bq be two objects of Algloc_K . Let Ap “ ÐÝlimnA{mⁿ_A, and Bp “ lim

ÐÝnB{mⁿ_B. By construction Ap and Bp are objects of AlgLC_K.

Letϕ:AÑBbe a local ring morphism. So the morphism induced byϕp:ApÑ Bpis continuous.

Proof. Since the morphismϕis local then, for any integernPN, the morphismϕp satisfiesϕpp Myⁿ_Aq ĂMyⁿ_B.

LetU be an open subset ofB. Letp x Pϕp^´1pUq. Sinceϕpxq Pp U, there exists an integerjsuch thatϕpxq `p My<sup>j</sup><sub>B</sub> ĂU. Thusϕpxp `My^j_Aq Ă ϕpxq `p My^j_B ĂU. Soϕp^´1pUqis an open subset, henceϕpis continuous.

Proposition 2.7. Let pA,m_Aq,pB,m_Bq be two objets of Algloc_K. Let Ap “ ÐÝlimnA{mⁿ_A, andBp “lim

ÐÝnB{mⁿ_B. Letϕ:A Ñ Bpbe a morphism of local rings.

Then there exists an unique continuous morphism of local rings such that the fol- lowing diagram is commutative.

A ^{ϕ //}

Bp

A.p

ϕp

??

Proof. Letjbe an integer. Since the morphismϕis local and that My_B^j Ă My^j_B, thenϕinduces a morphism of local rings.

ϕj:A{M^j_AÑB{p My^j_B“B{M^j_B.

Letˆa“ panqnPN PA. We notep ϕpp ˆaq “ pϕnpanqqnPN. Since the morphism ϕis local, thenϕpp aq Pˆ Bpis well defined. And since everyϕ_j is local, thenϕpis local.

By constructing of the morphismϕ, we also havep ϕpp Myⁿ_Aq ĂMyⁿ_B, which implies that the morphismϕpis continuous (see previous proof for details).

Uniqueness is verified becauseAis dense inApandϕpis continuous.

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Remark2.8. In particular, the proposition 2.7 proves that, ifpA,m_Aq,pB,m_Bqare two objects ofAlgloc_K, then

Hom_Algloc

KpA,Bq “p Hom_AlgLC

KpA,p Bpq.

In particular, ifBis a quasi-Artinian ring (an object ofQArt_K), then we have Hom_Algloc

KpA, Bq “Hom_AlgLC

KpA, Bqp because, in this case,B “B.p

2.2. Functor of points. According to the Yoneda’s lemma, we have a fully faith- ful functor defined on the objects by :

α: AlgLC_K Ñ FuncpAlgLC_K,Setq

Bˆ ÞÑ hBˆ: AlgLC_K Ñ Set Aˆ ÞÑ Hom_AlgLC

KpB,ˆ Aqˆ

LetBˆandCˆbe two objects ofAlgLC_K. This fully faithful functor is described by the existence of an isomorphism betweenHom_AlgLC

KpC,ˆ Bqˆ andHomph_Cˆ, h_Bˆq.

In other words, a morphismϕPHom_AlgLC

KpC,ˆ Bqˆ is the same as considering a collection of morphisms

!

Hom_AlgLC

KpC,ˆ Aq ш Hom_AlgLC

KpB,ˆ Aqˆ )

APAlgLCˆ _K

functorials inA.ˆ

This functorαcan be restricted to the sub-categoryQArt_K. AlgLC_K ^{α //}

α¹ ((

FuncpAlgLC_K,Setq

FuncpQArt_K,Setq whereα¹pBq “ˆ h_Bˆ : ˆAPQArt_K ÞÑHom_AlgLCKpB,ˆ Aq.ˆ

The following proposition was used in [Dri02], but the reader can refer to [BS17, Section 2.1] for more details.

Proposition 2.9. The functorα¹:AlgLC_K ÑFuncpQArt_K,Setqis fully faith- ful.

2.3. Infinitesimal deformations of a rational point. This section is a recollec- tion of an important description of the deformation space of a rational point, which is the core of proposition 2.12. The same considerations can be found in [BS17, Section 2.2].

IfXis aK-scheme andxbe aK-point ofX, we denoteO_X,xthe stalk ofXat x.

Definition 2.10. LetX be aK-scheme and x be aK-point ofX. The space of deformations ofXatxis the functor from the categoryQArt_K toSet

AÞÑ tx_APXpAq |xĎ_A“xu wherexĎ_Ais the reduction ofx_Avia the mapXpAq ÑXpkq.

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Lemma 2.11. Let K be a field and R a K-algebra. Given two morphisms of K-algebrasf, g:RÑKsuch thatKerpfq “Kerpgq, thenf “g.

Proof. Sincef andgare morphisms ofK-algebras, note thatf andgare neces- sarily surjectives. Thus, the morphism deduced from the universal property of the quotientR{Kerpfq Ñ K is bijective. Since Kerpfq “ Kerpgq, we deduce the following commutative diagram by the universal property of the quotient.

R ^{g ////}

πf

K

R{Kerpfq

D!˜g

Now the only morphism ofK-algebra betweenKandKis the identity. Sog˜“id, andg“π_f. In the same way we show thatf “π_f, hencef “g.

Proposition 2.12. IfX is aK-affine scheme,xP Xpkqa rational point andAa quasi-Artinian ring, then

Hom_AlgLC

KpOz_X,x, Aq “Hom_Algloc

KpO_X,x, Aq “ tx_APXpAq |xĎ_A“xu wherexĎAis the reduction ofxAvia the mapXpAq ÑXpkq.

Remark2.13. IfXis a general scheme a proposition like the previous is also true, up to working with an affine open neighborhood ofx.

Proof. Remark 2.8 already proves the equality Hom_AlgLC

KpOz_X,x, Aq “Hom_Algloc

KpO_X,x, Aq.

We aim to define a bijective map fromtxAPXpAq |xĎA“xutoHom_Algloc

KpO_X,x, Aq.

Let’s considerx_APXpAq, which corresponds to a morphism ofK-algebrax_A:O_X Ñ Asuch thatxĎA“x. Let’s denoteKerpxq “p_x.

By universal property of the localization,x_A factorizes byO_X,x if and only if foryRp_xthenx_Apyqis invertible inA. SinceyRp_x, thenxĎ_Apyq “xpyq ‰0. So xAfactorizes uniquely :

O_X ^{xA //}

A

O_X,x

D!ϕ_xA

==

This defines an injective morphism

α: tx_APXpAq |xĎ_A“xu ãÑ Hom_AlglocKpO_X,x, Aq

x_A ÞÑ ϕx_A.

It remains to show thatαis surjective. LetϕP Hom_Algloc

KpO_X,x, Aqbe a mor- phism. We denotelx: O_X Ñ O_X,x the canonical morphism of localization. We denote againKerpxq “p_x.

We setxA “ ϕ˝lx. By definition ofϕ, ify Rp_x, then xApyq is invertible in A, in other words xĎ_Apyq ‰ 0 (in K). We deduce that y R Kerpxq implies that yRKerpxĎ_Aq. In other wordsKerpxĎ_Aq ĂKerpxq.

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But, we also know thatKerpxqandKerpxĎAqare maximum ideals ofO_X since O_X{KerpxqandO_X{KerpxĎ_AqareK-isomorphic toK. HenceKerpxĎ_Aq “Kerpxq.

Thanks to lemma 2.11, this implies thatxĎA“x. This proves thatαis surjective and thus the equality

Hom_Algloc

KpO_X,x, Aq “ tx_APXpAq |xĎ_A“xu.

3. TANGENT SPACE AND EMBEDDING DIMENSION

In this section, we will present and gather essentially known results on the notion of tangent space and embedding dimension for general schemes. The material in this chapter constitutes the key ingredients of the results we will obtain in the next chapters. Given the role that the statements will play in the following, we have chosen to present the most important proofs for the ease of reading. Even if most of the results presented in this chapter are known however, to our knowledge, corollary 3.9 and proposition 3.10 are new.

3.1. Definition.

Definition 3.1. LetpA,M_Aqbe a local ring. Theembedding dimensionofAis the dimension of theA{M_A-vector spaceM_A{M²_Aand is denotedemb.dimpAq.

The tangent space ofAis the dual of theK-vector space M_A{M²_A that is de- notedpM_A{M²_Aq^_.

Remark3.2. LetpA,M_Aq be a local ring and n P Nan integer. TheA-module Mⁿ_A{M^{n`1</sup><sub>A</sub> is endowed with a structure of A{M<sub>A</sub>-vector space. If a P A and bPM<sup>n</sup><sub>A</sub>thenpa`M_Aqpb`M<sup>n`1}_A q “ab`M<sup>n`1</sup>_A .

LetpA,M_Aqbe a localK-algebra which residual field isK isomorphic toK.

We denoteA{M_A “Kto mean that the structural morphism ofK-algebraK Ñ A{MA is an ismorphism. Let π: A Ñ A{MA “ K be the quotient morphism.

LetaPA. We denotea₀ “πpaq P K. Thenπpa´a₀q “ πpaq ´a₀ “0. Thus a´a₀ PM_A. In other words, for everyaPA, there existsa₀ PKanda₁ PM_A such thata“a0`a1. We also observe that this decomposition is unique.

Hence the setHom_Algloc

KpA, Krs{pq²qcan be endow with a structure ofK- vector space defined by:

(1) Ifϕ₁andϕ₂are two elements ofHom_Algloc

KpA, Krs{pq²qandλPK, we define

ϕ1`λϕ2: A Ñ Krs{pq<sup>2</sup> a“a<sub>0</sub>`a₁ ÞÑ a₀`ϕ<sub>1</sub>pa<sub>1</sub>q `λϕ₂pa₁q.

(2) The identity element is given by

0_Hom: A Ñ Krs{pq² a“a0`a1 ÞÑ a0

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Moreover, this structure ofK-vector space is fonctorial inA, i.e. ifψ:A ÑB is a morphism ofK-algebras, then the induced morphism

Hompψq: Hom_Algloc

KpA, Krs{pq²q Ñ Hom_Algloc

KpB, Krs{pq²q

ϕ ÞÑ ψ˝ϕ

is a morphism ofK-vector spaces.

Within this framework, we can therefore propose an equivalent definition of the embedding dimension.

Proposition 3.3. Let pA,M_Aq be a local K-algebra whose residual field isK- isomorphic to K. Then the vector spaceHom_Algloc

KpA, Krs{pq²q is isomor- phic, asK-vector space, topM_A{M²_Aq^_.

Proof. LetϕPHom_Algloc

KpA, Krs{pq²qandaPA. Thenacan be written in a unique way asa“a0`a1 aveca0PK anda1 PM<sub>A</sub>. Thenϕpaq “a0`ϕpa1q, where ϕpa₁q “ ϕ₁pa₁q. The application ϕ₁: M_A Ñ K send M²_A to zero, so we can quotient. We still noteϕ1:M_A{M²_A Ñ Kthe quotient application. This application isK-linear. So it is a morphism ofK-vector spaces.

Let’s define

ψ: Hom_Algloc

KpA, Krs{pq²q Ñ pM_A{M²_Aq^_

ϕ ÞÑ ϕ1.

Let’s check thatψis bijective.

Letφ, ϕ P Hom_Algloc

KpA, Krs{pq²q, verifyingψpφq “ ψpϕq. Leta P A.

We write, as before,ain the forma“a0`a1. Then

φpaq “a₀`φ<sub>1</sub>pa<sub>1</sub>q “a<sub>0</sub>`ϕ₁pa₁q “ϕpaq.

Henceψis injective.

LetθP pM_A{M²_Aq^_anda“a0`a1 PAWe defineϕpaq “a0`θpa1q. Let a, bPA. SinceθisK-linear, we verify thatϕpa`bq “ϕpaq `ϕpbq. Furthermore, with the notationsa“a₀`a<sub>1</sub>andb“b<sub>0</sub>`b₁, sinceθsendsM²_Ato zero, we have ϕpabq “ϕpa₀b₀`a<sub>0</sub>b<sub>1</sub>`b₀a₁`a<sub>1</sub>b<sub>1</sub>q “a<sub>0</sub>b<sub>0</sub>`pa₀θpb₁q`b₀θpa₁qq “ϕpaqϕpbq.

Henceψis surjective.

We also check thatψis aK-linear application.

Proposition 3.4. LetpA,M_Aqbe a localK-algebra which residual fieldK-isomorphic toK. Thenemb.dimpAqthe embedding dimension ofAis finite if and only if the dimension of theK-vector spaceHom_Algloc

KpA, Krs{pq²qis finite. In this case, these two dimensions are the same.

Proof. According to proposition 3.3, there exists an isomorphism of K-vector space between Hom_Algloc

KpA, Krs{pq²q and the dual of M_A{M²_A. We con- clude sinceM_A{M²_Ais of finite dimension if and only if its dual is of finite dimen-

sion.

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3.2. Properties and Characterizations.

Lemma 3.5. Let pA,M_Aq be a local ring and Ap “ lim

ÐÝnA{Mⁿ_A its comple- tion. Let n P N^˚ be an integer. Assume that there exists an integerdsuch that dim_A{MApMⁿ_A{M^n`1_A q “d, thenMyⁿ_Ais anA-module of finite type generated byp at mostdelements.

In particular, withn“1, ifemb.dimpAq “dthenMy_Ais aA-module of finitep type generated by at mostdelements.

Proof. Sincedim_A{M

ApMⁿ_A{M^n`1_A q “d, there exists elements a₁,¨ ¨ ¨ , a_dPMⁿ_A

which form a basis of the A{M_A-vector space Mⁿ_A{M^n`1_A . Let’s consider the morphism ofA-modulesψ: A^d Ñ Mⁿ_A, ei ÞÑ ai, where ei “ p0,¨ ¨ ¨ ,1,¨ ¨ ¨,0q with the unique 1 is at thei-th position.

According to [Sta, Tag 0315 (1)] ifRis a ring,I ĂRan ideal andϕ:M ÑN a morphism ofR-modules then, ifM{IM ÑN{IN is surjective, thenMxÑ Np is also surjective.

By applying this proposition withR “A,I “ M_A,M “A^d, N “Mⁿ_Aand ϕ“ψ, we deduce that the morphism

ψp:Ap^dÑMyⁿ_A is surjective (becauselim

ÐÝmA^d{M^n`m_A “lim

ÐÝmA^d{M^m_A).

The following lemma is an adaptation of the lemma [Sta, Tag 05GH], which does not apply directly in our context because, in general, if the local ringpA,M_Aq is not noetherian, then,Ap“ lim

ÐÝnA{Mⁿ_Aandx xA “lim

ÐÝnA{p My_Aⁿare not isomor- phic (see remark 2.2). However the proof of the lemma can be adjusted for our statement by noticing that if a sequence is Cauchy for theMy_Apre-adic topology onA, then this sequence is also Cauchy for the topology that makesp Apcomplete.

Lemma 3.6. Let pA,M_Aqbe a local ring and Ap “ lim

ÐÝnA{Mⁿ_A its completion.

Assume thatMy_Ais an ideal of finite type. ThenApis noetherian.

Proof. Let’s denotef1,¨ ¨ ¨, ft P My_Aa generating family of My_A. Consider the direct sum B “ ‘ně0My_Aⁿ{My_A^n`1 in the category of A-modules. This directp sum has an additional ring structure. Let’s consider the morphism of rings

A{p My_ArT₁,¨ ¨ ¨, T_ts ÑB

which sendTjonf¯j. This morphism is surjective, soBis a noetherian ring.

LetJ be an ideal ofA. Consider the idealp J_B “ ‘ně0JXMy_Aⁿ{JXMy_A^{n`1</sup> de B. Since B is noetherian, there exists a finite familyg¯1,¨ ¨ ¨g¯m of elements of B generating J<sub>B</sub>. Up to increasing the family size, we can assume that, for everyj, we haveg¯<sub>j</sub> PJXMy<sub>A</sub><sup>dj</sup>{JXMy<sub>A</sub><sup>dj`1}for some integerd_j. There exists

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gj P J XMy_A^dj which maps tog¯j in the quotient. We will show that the family g₁,¨ ¨ ¨ , g_mgeneratesJ.

LetxPJ. There exists an integernsuch thatxPJXMy_AⁿandxRJXMy_A^{n`1</sup>. If we considerx¯the image ofxinJXMy<sub>A</sub><sup>n</sup>{J XMy<sub>A</sub><sup>n`1}, we deduce that there exists a family aj P My_A^maxp0,n´djq such that x¯ “ ř_m

j“1a¯jg¯j since the family of the gj generate JB. Thusx ´řm

j“1ajgj P J XMy_A^{n`1</sup>. By iterating this process, for every integer N, we can find a family pa<sub>1,`</sub>,¨ ¨ ¨ , a<sub>m,`</sub>qnď`ďN with a<sub>j,`</sub>PMy<sub>A</sub><sup>maxp0,`´djq}such that

x“ ÿN

`“0

ÿm

j“1

a_j,`gj mod My_A^N

We set Aj “ ř

`ě0a<sub>j,`</sub>. The sequence př_N

`“0a<sub>j,`</sub>qNPN is a Cauchy sequence (for the topology for which Ap is complete) because, for every integer ` P N, we have My<sub>A</sub><sup>`</sup> Ă My^`_A hence A_j is well defined in A. Thus we can writep x “ ř_m

j“0Ajgj, since XNPNMy^N_A “ 0 (becauseAˆ is complete -and separate- for the filtrationpMy^N_AqNPN) which proves thatJ is of finite type.

Lemma 3.7. LetpA,M_Aqbe a local ring. The following assertions are equivalent:

(1) The embedding dimension of the local ringpA,M_Aqis finite.

(2) The completionApofA, with respect to its maximal ideal, is noetherian.

Proof. Assume that dim_A{MApM_A{M²_Aq ă 8, the by applying lemma 3.5 we deduce thatMy_Ais an ideal of finite type, and then, by appliying lemma 3.6, we deduce that the ringApis noetherian.

Conversely, if the ringApis noetherian, the inclusionMy_A² ĂMy²_Aimplies that the morphismMy_A{My_A² ÑM_A{M²_Ais surjective. Hence the embedding dimen-

sion ofAis finite andemb.dimpAq ďemb.dimpAq.p

The previous lemma is a result that can be found in [dFD, Lemma 10.12] in the following form.

Remark3.8. The previous statement can be reformulating as follow from the point of view of schemes. LetK be a field. LetX be aK-scheme. Letx P X. The following assertions are equivalent:

(1) The embedding dimension of the local ringO_X,x is finite.

(2) The completionOz_X,xof the local ring inxis noetherian.

Corollary 3.9. Let pA,M_Aq be a local ring. We denote Ap “ lim

ÐÝnA{Mⁿ_A the completion of A. Thenemb.dimpAqis finite if and only ifemb.dimpAqp is finite, and in this caseemb.dimpAq “emb.dimpAq.p

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Proof. The fact thatemb.dimpAqis finite if and only ifemb.dimpAqp is finite, can be deduced from lemma 3.7.

Let’s assume that these embedding dimensions are finite. Thanks to the inclu- sionMy_A²ĂMy²_A, we have a surjective morphism ofA{M_A-vector spaces

My_A{My_A²ÑMy_A{My²_A“M_A{M²_A. Henceemb.dimpAq ďemb.dimpAq.p

Furthermore, we know thanks to lemma 3.5 thatMy_Ais generated asA-modulep by at mostd:“ emb.dimpAqelements. Denote byu1,¨ ¨ ¨, ud P My_Aa generat- ing family of My_A. Let x P My_A, then there existspa1,¨ ¨ ¨, a_dq P Ap^d such that x “ ř_d

i“1a_iu_i. Letπ: Ap Ñ A{p My_A “ A{M_A the canonical morphism of pro- jection. SinceMy_A{My_A² is endowed with a structure ofA{M_A-vector space, we can consider the element y “ řd

i“0πpaiqu¯i ofMy_A{My_A². Hence x¯´y “ 0in My_A{My_A². Thus theA{M_A-vector space My_A{My_A² is of dimension at mostd.

Henceemb.dimpAq ďp emb.dimpAq.

The above propositions allow us to prove the following statement about ring completion:

Proposition 3.10. LetpA,M_Aqbe a local ring. Let’s denoteAp“lim

ÐÝnA{Mⁿ_Athe completion ofA. IfApis a noetherian ring thenApis complete for theMy_Apre-adic topology.

Proof. We are going to show by induction, that for everynPN, we haveMy_Aⁿ“ Myⁿ_A.

Forn“1there is nothing to prove.

Assume now that My_Aⁿ “ Myⁿ_A for some integer n. Thanks to the inclusion My_A^{n`1</sup> ĂM{<sup>n`1}_A , we deduce a surjective morphism

ϕn:My_Aⁿ{My_A^{n`1</sup>ÑMy<sup>n</sup><sub>A</sub>{M{<sup>n`1}_A “Mⁿ_A{M^n`1_A .

SinceApis noetherian, then theA{M_A-vector spaceMyⁿ_A{M{^{n`1</sup><sub>A</sub> is of finite dimen- sion and soMy<sup>n</sup><sub>A</sub>{M{<sup>n`1}_A is also of finite dimension.

We setd“dim_A{MApMⁿ_A{M^n`1_A q, then, thanks to lemma 3.5, we deduce that Myⁿ_Ais anA-module generated by at mostp delements. But, by hypothesisMyⁿ_A “ My_Aⁿ. Denoteu1,¨ ¨ ¨ , udPMy_Aⁿa generating family ofMy_Aⁿ. LetxPMy_Aⁿ, then there existspa₁,¨ ¨ ¨, a_dq PAp^dsuch thatx “ ř_d

i“1a_iu_i. Letπ:ApÑ A{p My_A “ A{M_Abe the canonical morphism of projection. SinceMy_Aⁿ{My_A^n`1is endowed with a structure ofA{MA-vector space, we consider the elementy“ř_d

i“0πpaiqu¯i

ofMy_Aⁿ{My_A^{n`1</sup>. Hence x¯´y “ 0 inMy<sub>A</sub><sup>n</sup>{My<sub>A</sub><sup>n`1}. Thus the A{M_A-vector spaceMy_Aⁿ{My_A^n`1 is of dimension at mostd.

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