the lipschitz saturation of a module

Instituto de Ciências Matemáticas e de Computação

UNIVERSIDADE DE SÃO PAULO

Bi-Lipschitz invariant geometry

Thiago Filipe da Silva Doctoral Dissertation of the Graduate Program in Mathematics (PPG-Mat)

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito: Assinatura: ______________________

Thiago Filipe da Silva

Bi-Lipschitz invariant geometry

Doctoral dissertation submitted to the Instituto de Ciências Matemáticas e de Computação – ICMC-USP, in partial fulfillment of the requirements for the degree of the Doctorate Program in Mathematics. FINAL VERSION Concentration Area: Mathematics Advisor: Prof. Dr. Nivaldo de Góes Grulha Júnior Co-advisor: Prof. Dr. Terence James Gaffney

USP – São Carlos January 2018

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados inseridos pelo(a) autor(a)

d111b

da Silva, Thiago Filipe Bi-Lipschitz invariant geometry / Thiago Filipe da Silva; orientador Nivaldo de Góes Grulha Júnior; coorientador Terence James Gaffney. -- São Carlos, 2018. 143 p. Tese (Doutorado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2018. 1. Double of a module. 2. Lipschitz saturation of a module. 3. Bi-Lipschitz equisingularity. 4. Determinantal Varieties. 5. Integral closure of ideals and modules. I. de Góes Grulha Júnior, Nivaldo, orient. II. Gaffney, Terence James, coorient. III. Título.

Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176

Thiago Filipe da Silva

Geometria bi-Lipschitz invariante

Tese apresentada ao Instituto de Ciências Matemáticas e de Computação – ICMC-USP, como parte dos requisitos para obtenção do título de Doutor em Ciências – Matemática. VERSÃO REVISADA Área de Concentração: Matemática Orientador: Prof. Dr. Nivaldo de Góes Grulha Júnior Coorientador: Prof. Dr. Terence James Gaffney

USP – São Carlos Janeiro de 2018

To Jesus, the Prince of Peace.

ACKNOWLEDGEMENTS

Primeiramente agradeço a Deus, por Jesus, pela vida e pelo privilégio de aprender um pouco mais de Matemática. Agradeço a minha esposa Mariana por todo o amor dedicado, pela compreensão nas horas difíceis e por ter feito parte deste tempo tão especial na minha vida. No início dessa jornada nós ainda éramos "apenas" namorados, separados por 1.140 km, mas nada que o aeroporto de Viracopos e as promoções da Azul não pudessem dar um jeitinho na saudade. Depois fiz a melhor escolha da minha vida: após aquele período turbulento do exame de qualificação, eu casei com você. Te amo! Agradeço meus pais Josué e Maria pelo amor, dedicação e orações. Foi muito bom ter vocês comigo aqui em São Carlos em 2014, com o "caçorro", o Lex! Agradeço minha irmã Jeane e ao meu cunhado Marcos pelas orações e pelas excelentes sugestões de boas músicas para ouvir durante o trabalho. Agradeço também minha irmã Samyra e ao meu cunhado José Antônio pelo carinho, por sempre nos receberem com tanto amor em sua casa em São Paulo. Foi um tempo precioso poder estar mais perto de vocês, da Clarice e Cecília, amores do titio. Além disso não poderia esquecer de agradecer a todos vocês especialmente pelo natal de 2014, onde tivemos a oportunidade de estarmos todos juntos comemorando o nascimento do nosso Salvador. Agradeço também minha segunda família em Vitória, Marlene e Luiz, pais da Mariana que me receberam tão carinhosamente em sua família, ao Higor, Heitor, a tia Carminha e a Laurinha, amor do titio. Agradeço o Prof. Nivaldo, meu orientador, pelos valiosos momentos de pesquisa, pelo incentivo, por ter me dado a oportunidade de ter contato com outros grandes pesquisadores e especialmente pela grande amizade construída ao longo desses anos. Com certeza você, a Suelen e o Lucca tornaram nosso tempo aqui em São Carlos muito mais agradável e sentiremos muitas saudades. I am very grateful to Professor Terence Gaffney, my co-advisor, for the valuable opportunity in work together. I am also grateful to Professor Gaffney and Mary Gaffney for the friendship and for the great support when I have been in Boston, for one year, for the care and kindness that you receive us. Agradeço a Profa. Miriam, da UFPB, por ter nos recebido de forma tão carinhosa quando estivemos em João Pessoa em outubro e novembro de 2016. Foi um tempo muito proveitoso para a minha pesquisa poder trabalhar com a profa. Miriam, além de ter sido muito bom poder

conhecer aquela cidade com um litoral tão lindo. Agradeço ao Prof. Marcelo Escudero, da UEM, pela hospitalidade quando estivemos em Maringá em fevereiro de 2017. Foi um enorme prazer poder apresentar alguns pontos da minha pesquisa ao professor. Minha visita em Maringá com certeza me motivou bastante a continuar trabalhando, tendo em vista as palavras de incentivo que o professor Marcelo direcionou a mim. Além disso foi um grande prazer conhecer uma cidade tão linda como Maringá. I am grateful to professor Anne Frühbis-Krüger for the valuable comments and suggestions in our work about Bi-Lipschitz equisingularity of Determinantal Surfaces. I am grateful to the members of the board examination for the careful reading and the valuable suggestions which certainly improved this work. Agradeço aos professores Behrooz, Irene, Ana Cláudia e Denise, meus professores nas disciplinas do programa, que tanto contribuíram para a minha formação. Agradeço à professora Maria Aparecida Soares Ruas pelo incentivo e pelas valiosas conversas a respeito da minha pesquisa. Agradeço a todos os professores do Departamento de Matemática da Universidade Federal do Espírito Santo pelo suporte e incentivo em realizar o doutorado. Em especial aos professores Leonardo e Alan, pelos conselhos e disposição em me auxiliar na escolha do ICMC como local de doutoramento, e ao professor Ricardo pelo grande auxílio com o fechamento das disciplinas sob minha responsabilidade antes do início do meu doutorado. Agradeço especialmente aos meus amigos Maico, Fernando, Leandro, Telau, Karlo e Giovani, colegas da turma de 2009 do mestrado/UFES, pela força e companheirismo. Também agradeço a Jovane, Elivelton e Jhonatan, amigos muito especiais que apesar da distância, a amizade continua a mesma. Agradeço aos irmãos da IPB Jardim Camburi, da IPB Central de São Carlos e da Segunda IPB de Nova Venécia pelas orações e acolhimento. Agradeço o suporte financeiro da Fundação de Amparo à Pesquisa do Estado de São Paulo para a realização deste doutorado, processo no. 2013/22411-2, e pela Bolsa Estágio de Pesquisa no Exterior, processo no. 2015/09529-0.

“Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room. . . ” (Andrew Wiles)

ABSTRACT DA SILVA, T. F.. Bi-Lipschitz invariant geometry. 2018. 143 f. Doctoral dissertation (Doctorate Candidate Program in Mathematics) – Instituto de Ciências Matemáticas e de Computação (ICMC/USP), São Carlos – SP.

The study about bi-Lipschitz equisingularity has been a very important subject in Singularity Theory in last decades. Many different approach have cooperated for a better understanding about. One can see that the bi-Lipschitz geometry is able to detect large local changes in curvature more accurately than other kinds of equisingularity. The aim of this thesis is to investigate the bi-Lipschitz geometry in an algebraic viewpoint. We define some algebraic tools developing classical properties. From these tools, we obtain algebraic criterions for the bi-Lipschitz equisingularity of some families of analytic varieties. We present a categorical and homological viewpoints of these algebraic structure developed before. Finally, we approach algebraically the bi-Lipschitz equisingularity of a family of Essentially Isolated Determinantal Singularities. Key-words: Double of a module, Lipschitz saturation of a module, bi-Lipschitz equisingularity, Determinantal varieties, Integral closure of ideals and modules.

RESUMO DA SILVA, T. F.. Geometria bi-Lipschitz Invariante. 2017. 143 p. Tese de doutorado (Candidato ao Programa de Doutorado em Matemática) – Instituto de Ciências Matemáticas e de Computação (ICMC/USP), São Carlos – SP. O estudo da equisingularidade bi-Lipschitz tem sido amplamente investigado nas últimas décadas. Diversas abordagens têm contribuído para uma melhor compreensão a respeito. Observa-se que a geometria bi-Lipschitz é capaz de detectar grandes alterações locais de curvatura com maior precisão quando comparada a outros padrões de equisingularidade. O objetivo desta tese é investigar a geometria bi-Lipschitz do ponto de vista algébrico. Definimos algumas estruturas algébricas desenvolvendo algumas propriedades clássicas. A partir de tais estruturas obtemos critérios algébricos para a equisingularidade bi-Lipschitz de algumas classes de famílias de variedades analíticas. Apresentamos uma visão categórica e homológica dos elementos desenvol- vidos. Finalmente abordamos algebricamente a equisingularidade de famílias de Singularidades Determinantais Essencialmente Isoladas. Palavras-chave: O double de um módulo, Saturação Lipschitz de um módulo, Equisingularidade biLipschitz, Variedades Determinantais, Fecho Integral de ideais e módulos.

LIST OF FIGURES

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Four moving lines . . . . X: y2 z2 = z3 + x2 . . . . Cone . . . . . . . . . . . {y = 0} ∪ {y = xz }, z ≥ 0 Cusp and Node . . . . . Family of Cusps . . . . .

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LIST OF TABLES

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Possible candidates of 1-jets with 4 or more variables . . . . . . . . . . . . . 104 Simple isolated Cohen-Macaulay in codimension 2 singularities in (C4 , 0) . . 106 Simple isolated Cohen-Macaulay in codimension 2 singularities in (C5 , 0) . . 107

LIST OF SYMBOLS

Z — Integer numbers C — Complex numbers R — Real numbers K — The field R or C PN — N-dimensional complex projective space OX — Analytic sheaf of rings of the analytic variety X ∆(X) — Diagonal of the set X in X × X `(M) — The length of a module M M — Integral closure of M R(M) — Rees algebra of the module M IS — Lipschitz saturation of the ideal I BI (X) — Blow-up of X with respect to the ideal I MD — Double of the module M Proj(G) — The projective spectrum of a graded ring G e(M) — The Buchsbaum-Rim multiplicity of a module M e(M, N) — The multiplicity of the pair of modules (M, N) Fx — The stalk of the sheaf F at the point x supp(F ) — The support of the sheaf F cosupp(F ) — The cosupport of the sheaf F

CONTENTS

1 1.1 1.2 1.3 1.4 1.5 1.6

PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . The integral closure of ideals . . . . . . . . . . . . . . . . . . . The integral closure of modules . . . . . . . . . . . . . . . . . Whitney equisingularity and the integral closure of modules . The Lipschitz saturation and the double of an ideal . . . . . Determinantal Varieties . . . . . . . . . . . . . . . . . . . . . . Bi-Lipschitz geometry and Lipschitz stratifications . . . . . .

2 2.1 2.2 2.3

THE DOUBLE OF A MODULE . . . . . . . The double of a module and basic properties The infinitesimal Lipschitz conditions iLA and The genericity theorem applied in a family of

3 3.1 3.2 3.3

THE LIPSCHITZ SATURATION OF A MODULE . . . . . The Lipschitz saturation of a module and basic properties . The generic equivalence among the Lipschitz saturations . Geometric applications . . . . . . . . . . . . . . . . . . . . . .

4

4.2

BI-LIPSCHITZ EQUISINGULARITY OF DETERMINANTAL SURFACES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Simple isolated Cohen-Macaulay of codimension 2 singularities in C4 and C5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 The bi-Lipschitz equisingularity on determinantal varieties . . . . . . 108

5 5.1 5.2 5.3 5.4 5.5

CATEGORICAL ASPECTS OF THE DOUBLE STRUCTURE The double homomorphism and basic properties . . . . . . . . Homological aspects of the double structure . . . . . . . . . . . The Double category . . . . . . . . . . . . . . . . . . . . . . . . . The double in a quotient of a free OX -module of finite rank . The double homomorphism relative to an analytic map germ .

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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Index

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23

INTRODUCTION

The study about bi-Lipschitz equisingularity was started by Pham and Teissier [84, 85] at 1969. In these papers they defined the notion of the Lipschitz saturation of a local ring on the context of analytic complex algebras. After, in 1971 Zariski [111] developed the Lipschitz saturation of a complete local domain of dimension one. In the same year, [112] he defined the saturation of a local ring of dimension one, and in 1975 [113] he generalized this notion for local rings with arbitrary dimension. For more about Lipschitz saturation of local rings one can see the important contributions of Lipman [65, 66]. As reported by Trotman in [101], while Thom [99] was describing Whitney’s work about stratifications of analytic varieties [109, 110] in 1964 at the Bourbaki seminar, he suggested that the vector fields yielding local topological triviality of Whitney stratifications (conjectured by Whitney in his contribution to the Morse Jubilee volume [109], although Whitney’s conjecture, still unproven, required a much stronger notion of local triviality) might be made Lipschitz, hence the local bi-Lipschitz triviality would follow as a corollary. In his 1985 dissertation, Mostowski [72] proved the Thom’s conjecture. Few years before in 1969, Pham and Teissier [84, 85] obtained a positive result for the special case of families of plane curves. Using a set of equisingularity conditions, Mostowski [72] defined a notion of Lipschitz stratification (see the section 1.6) stronger than Verdier’s (w)-regularity (see the section 1.3), and he proved that every complex analytic variety admits a Lipschitz stratification. Further, Mostowski proved that these Lipschitz stratifications are locally bi-Lipschitz trivial along strata by integrating stratified Lipschitz vector fields. Then, Parusiński generalized these theorems for every analytic real variety [78], after for all semianalytic sets [79], and finally in 1994 for every subanalytic set [82]. There are canonical Whitney (equivalently Verdier) stratifications of complex analytic varieties by the theory of Teissier [98] (see also [54]) which shows identifies canonical Whitney strata as precisely the loci of equimultiplicity of the different polar varieties. However, in general there is no canonical Lipschitz stratification attached to a given subanalytic set, even in the special cases of real algebraic varieties or complex algebraic varieties. For details, see Mostowski’s habilitation dissertation [72]. Nevertheless, Mostowski [71] gave equivalent algebraic criteria for Lipschitz regularity in the case of complex surfaces and in [70] he also explained the close relation of the polar varieties of a complex analytic variety X and the stratifications of X with locally trivializing Lipschitz vector fields. Many mathematicians worked or are still working on bi-Lipschitz geometry. We can

24

Introduction

cite, for instance, Birbrair [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], Comte [20], Costa [22, 23, 24], Fernandes [6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35], Gaffney [39, 40], Grandjean [8, 9], Grulha [26], Henry [55], Juniati [57, 58, 59], Kovalev [60], Lê [10], Mostowski [70, 71, 72, 73], Neumann [11, 12, 14, 74], O’Shea [9, 77], Parusiński [55, 78, 79, 80, 81, 82], Paunescu [61], Pham [84, 85], Pichon [74], Ruas [6, 33, 34, 91], Saia [23, 24], Sampaio [10, 35, 92], Teissier [85], Trotman [58, 59, 101], Valette [57, 59, 91, 102, 103, 104, 105, 106] and Zariski [111, 112, 113]. In [41], Gaffney used the integral closure of modules to describe the Whitney equisingularity. If X ⊆ Cn × Ck is a family of complex analytic varieties defined by an analytic map F : (Cn × Ck , 0) → (C p , 0), Y = 0 × Ck ≡ Ck ⊆ X the parameter space and the singular locus of a small enough representative of X, Gaffney showed that (X −Y,Y ) is Whitney equisingular if and only if all the partial derivatives ofnF withorespect to the parameter space are in the integral n closure of the submodule generated by zi ∂∂ zFj , where z1 , ..., zn are the coordinate functions on Cn . We denote this inclusion as

i, j=1

JM(X)Y ⊆ mY JMz (X). Since there is a close relation between the double structure and Lipschitz behavior, it is natural to hope that the above condition may desbribe the bi-Lipschitz equisingularity adding the double structure, i.e, (JM(X)Y )D ⊆ (mY JMz (X))D , or even a weaker condition as (JM(X)Y )D ⊆ (JMz (X))D . In the hypersurface case, Gaffney [39] called these the infinitesimal Lipschitz conditions mY and A, respectively. This thesis is divided into five chapters. In the first chapter we first remember the main definitions and results about the integral closure of ideals and modules. Then we recall the Whitney and Verdier regularity conditions and its algebraic characterization obtained by Gaffney in [41] using the integral closure of modules. We see the definition of the Lipschitz saturation of an ideal, following the ideas of Pham, Teissier [85] and Zariski [111, 112, 113]. We also remember the main about the double of an ideal and the infinitesimal Lipschitz conditions for hypersurfaces given by Gaffney in [39, 40]. After we recall the definition of Determinantal Varieties, which will be used on the fourth chapter. We end the first chapter presenting the classical approach about bi-Lipschitz geometry due to Mostowski and Parusiński. In the second chapter we start generalizing the concept of the double for modules, and we obtain several properties and results. Then we use the double structure to rephrase the infinitesimal Lipschitz conditions in any codimension, and we generalize some results of [39]. In the third chapter we define some different notions of Lipschitz saturations of a module, each one generalizing the original concept for ideals, and we get relations among them. At the end we apply the algebraic machinery developed to obtain conditions for bi-Lipschitz equisingularity of some families of analytic varieties.

Introduction

25

In the fourth chapter we investigate the bi-Lipschitz equisingularity for families of determinantal varieties. First we remember the classification of simple isolated singularities of codimension 2 in C4 and C5 , given by Frühbis-Krüger and Neumer in [37]. Then we get a canonical stratified vector field defined on a semi-universal 1-unfolding of a determinantal variety which is Lipschitz when the double of the partial derivative of the unfolding with respect to the parameter belongs to the integral closure of the ideal generated by the double of the another partial derivatives. We also see that for the special case that the matrix of deformation is constant, the last above condition always is true and then the canonical vector field is Lipschitz. Finally, in the fifth chapter we develop a categorical viewpoint of the double structure. We start defining the double of a homomorphism in a quite natural way. Then we look for homological properties of the double structure and we compare algebraic properties of the double homomorphism with the "single" homomorphism. We define the double category and we end the chapter generalizing the double homomorphism for two analytic varieties linked by an analytic map germ. Have a good reading!

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CHAPTER

1 PRELIMINARIES

1.1

The integral closure of ideals

In this section we present some classical results and definitions about the integral closure of an ideal following [95]. For more about this subject one can see the works of Bivià-Ausina [15, 16]. Definition 1.1.1. Let I be an ideal in a ring R. An element h ∈ R is said to be integral over I if there exists a positive integer n and elements ai ∈ I i , with i ∈ {1, ..., n}, such that hn + a1 hn−1 + ... + an−1 h + an = 0. Such an equation is called an equation of integral dependence of h over I. The set of all elements that are integral over I is called the integral closure of I, and ¯ If I¯ = I, then I is called integrally closed. If I ⊆ J are ideals, we say that J is is denoted by I. ¯ integral over I if J ⊆ I. A basic example is the following: Example 1.1.2. For arbitrary elements x, y ∈ R, xy ∈ (x2 , y2 ). In fact, take n = 2, a1 = 0 ∈ (x2 , y2 ) and a2 = −x2 y2 ∈ (x2 , y2 )2 , notice that (xy)2 + a1 (xy) + a2 = 0 is an equation of integral dependence of xy over (x2 , y2 ). Therefore, (x, y)2 is integral over (x2 , y2 ). Definition 1.1.3. Let R ⊆ S be an extension of rings. An element s ∈ S is called integral over R if f (s) = 0 for some monic polynomial f ∈ R[T ].

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Chapter 1. Preliminaries

The next lemma is a link between the notions of integral closure of extensions of rings and ideals, and it is an important tool to prove that the integral closure of an ideal is also an ideal. Lemma 1.1.4. ([64] Lemma 1.3) Let R be a ring and denote by R[T ] the ring of polynomials of one variable over R. Let h ∈ R and I an ideal of R. Then: h is integral over I if and only if hT is L n n integral over the subring I T of R[T ]. n∈Z+

Now, using the previous lemma and the fact that the set of the elements which are integral over a subring is also a subring, we have the next Proposition 1.1.5. ([64] Corollary 1.4) If I is an ideal of R then I¯ is an integrally closed ideal ¯ of R, i.e, I¯ = I. Remark 1.1.6. ([95] Remark 1.1.3) Basic properties of the integral closure of ideals: ¯ since for each r ∈ I, taking n = 1 and a1 = −r give an equation of integral depen∙ I ⊆ I, dence over I. ∙ If I ⊆ J are ideals then I¯ ⊆ J,¯ since every equation of integral dependence of r over I is also an equation of integral dependence of r over J. √ ∙ I¯ ⊆ I, since from the equation of integral dependence of some degree n for h ∈ I, we conclude that hn ∈ (a1 , ..., an ) ⊆ I. ∙ Radical, hence prime and maximal, ideals are integrally closed. √ ∙ The nilradical 0 of the ring is contained in I¯ for every ideal I, because for each nilpotent element r there exists an integer n such that rn = 0 and this is an equation of integral dependence over I. ∙ Intersections of integrally closed ideals are integrally closed. ∙ The following property is called persistence: if ϕ : R −→ S is a ring homomorphism, then ¯ ⊆ ϕ(I)S. This follows as by applying ϕ to an equation of integral dependence of an ϕ(I) element r over I to obtain an equation of integral dependence of ϕ(r) over ϕ(I)S. ∙ Another important property is contraction: if ϕ : R −→ S is a ring homomorphism and I is an integrally closed ideal of S, then ϕ −1 (I) is integrally closed in R. In fact, if r is integral over ϕ −1 (I), then applying ϕ to an equation of integral dependence of r over ϕ −1 (I) gives an equation of integral dependence of ϕ(r) over I, hence ϕ(r) ∈ I and r ∈ ϕ −1 (I). ∙ In particular, if R is a subring of S, and I is an integrally closed ideal of S, then I ∩ R is an integrally closed ideal in R. Integral closure behaves well under localization:

29

1.1. The integral closure of ideals

Proposition 1.1.7. ([95] Proposition 1.1.4) Let R be a ring and I an ideal in R. For any multiplicatively closed subset W of R, W −1 I¯ = W −1 I. Furthermore, the following are equivalent: a) I¯ = I; b) W −1 I = W −1 I, for all multiplicatively closed subsets W of R; c) IP = IP , for all prime ideals P of R; d) IM = IM , for all maximal ideals M of R. The following proposition reduces questions about the integral closure to questions about integral closure in integral domains. Proposition 1.1.8. ([95] Proposition 1.1.5) Let R be a ring and I an ideal of R. a) The image of the integral closure of I in Rred is the integral closure of the image of I in ¯ red = IRred . Thus I¯ equals the natural lift to R of the integral closure of I in the Rred : IR reduced ring Rred . b) An element r ∈ R is in the integral closure of I if and only if for every minimal prime ideal P of R, the image of r in RP is in the integral closure of I+P P . We next rephrase integral closure with ideal equalities: Proposition 1.1.9. ([95] Proposition 1.1.7) Let R be a ring, I an ideal of R and r ∈ R. Then: r ∈ I¯ if and only if there exists an integer n such that (I + (r))n = I(I + (r))n−1 . One can also use modules to express integral dependence, usually known as determinantal trick. Corollary 1.1.10. ([95] Corollary 1.1.8) Let I be an ideal of R and r ∈ R. Then, the following are equivalent: (a) r is integral over I; (b) There exists a finitely generated R-module M such that rM ⊆ IM and such that whenever √ aM = 0 for some a ∈ R, then ar ∈ 0. Moreover, if I is finitely generated and contains a non-zerodivisor, r is integral over I if and only if there exists a finitely generated faithful R-module M such that IM = (I + (r))M.

30


We introduce reductions, which are an extremely useful tool for integral closure in general. Definition 1.1.11. Let J ⊆ I be ideals in R. The ideal J is said to be a reduction of I if there exists a non-negative integer n such that I n+1 = JI n . As a consequence of Proposition 1.1.9 we have the following result. Corollary 1.1.12. ([95] Corollary 1.2.2) An element r ∈ R is integral over J if and only if J is a reduction of J + (r). Remark 1.1.13. Note that if JI n = I n+1 , then for all positive integers m, I m+n = JI m+n−1 = ... = J m I n . In particular, if J ⊆ I is a reduction, there exists an integer n such that for all m ≥ 1, I m+n ⊆ J m . The reduction property is transitive: Proposition 1.1.14. ([95] Proposition 1.2.4) Let K ⊆ J ⊆ I be ideals in R. a) If K is a reduction of J and J is a reduction of I, then K is a reduction of I; b) If K is a reduction of I then J is a reduction of I; c) If I is finitely generated, J = K + (r1 , ..., rk ), and K is a reduction of I, then K is a reduction of J. The next corollary gives a relation between the integral closure and reductions. Corollary 1.1.15. ([95] Corollary 1.2.5) Let J ⊆ I be ideals in R. Suppose that I is finitely generated. Then J is a reduction of I if and only if I ⊆ J.¯ In particular, if R is a noetherian ring, then reduction property in R is equivalent to the integral dependence. Notice that in Example 1.1.2 we have seen that (x2 , y2 ) is a reduction of (x, y)2 . The next result says about the behaviour of the integral closure with powers of an ideal. Proposition 1.1.16. ([64] Corollary 1.8) Let I be an ideal of R. ¯ n ⊆ In; a) (I) b) I · I n ⊆ I n+1 . As a corollary of the Corollary 1.1.10, we have the

31

1.1. The integral closure of ideals

Corollary 1.1.17. ([64] Corollary 1.11) If I and J are ideals in R then I¯ · J¯ ⊆ I · J. Proposition 1.1.18. ([64] Proposition 1.14) Let R be an excellent reduced ring and I an ideal in R containing a non-zerodivisor element of R. Then, there exists an integer N such that, if n ≥ N: a) I · I n = I n+1 ; ¯ n = (I) ¯ n+1 . b) I.(I) For the definition of excellent rings, see [52]. Every complete local ring is an excellent ring. For example, the local rings of the analytic geometry. Notation 1.1.19. Let I be an ideal of R. We denote by e(I) the Hilbert-Samuel multiplicity of I. Proposition 1.1.20. ([64] Proposition 1.18) Let (R, m) be a local noetherian ring. If I and J are m-primary ideals of R with the same integral closure, i.e., I¯ = J¯ then e(I) = e(J). For the definition of m-primary ideals and Hilbert-Samuel multiplicity see [31]. Remark 1.1.21. If (R, m) is an analytic equidimensional local algebra, David Rees [86] showed a reciprocal: if I and J are m-primary ideals of R such that e(I) = e(J) and I ⊆ J then I¯ = J.¯ Let us recall the definition of nowhere dense in a topological space. Definition 1.1.22. Let X be a topological space. A subset A ⊆ X is called nowhere dense if A has empty interior. Equivalently, a nowhere dense set is a set that is not dense in any non-empty open subset of X. The next theorem is one of the most important result to our work. It gives several points of view of the integral closure of an ideal when we are working in analytic geometry. Theorem 1.1.23. ([64] Theorem 2.1) Let (X, OX ) be a reduced complex analytic space, x ∈ X, and let I be a coherent sheaf of ideals of OX . Denote I = Ix the stalk of I on x, which is an ideal of OX,x . Let h ∈ OX,x . Suppose that I defines a nowhere dense closed subset of X. The following are equivalent: (a) h is integral over I; (b) There exist a neighborhood U of x, a positive real number C, representatives of the space germ X, the function germ h, and generators g1 , ..., gm of I on U, which we identify with the corresponding germs, so that ‖h(z)‖ ≤ C max {‖g1 (z)‖ , ..., ‖gm (z)‖} , for all z ∈ U;

32


(c) For all analytic path germs φ : (C, 0) → (X, x), the pullback φ * (h) is contained in the ideal generated by φ * (I) in the local ring OC,0 ; (d) Let NB denote the normalization of the blowup of X by I, D¯ the pullback of the exceptional divisor of the blowup of X by I. Then, for any irreducible component C of the underlying ¯ the order of vanishing of the pullback of h to NB along C is greater than or equal set of D, to the order of the divisor D¯ along C. We end this section with a theorem that allows us work with the integral closure of a sheaf of ideals. Theorem 1.1.24. ([64] Theorem 2.6) Let (X, OX ) be a reduced complex analytic space, and let I be a coherent sheaf of ideals of OX . Suppose that I defines a nowhere dense closed subset of X. Then there exists a coherent sheaf of ideals of OX , denoted I¯, such that (I¯)x = Ix , for all x ∈ X.

1.2

The integral closure of modules

In this section we develop the notion of the integral closure of a module and we develop analogous properties for this object. In the contexts considered in this work, it seems most convenient to define the integral closure of a module using the valuative criterion. We follow the Gaffney’s language for the integral closure of a module following [41]. Here we denote On := OCn ,0 and mn its maximal ideal. p Definition 1.2.1. Suppose (X, x) is a complex analytic germ and that M is a submodule of OX,x . p Let h ∈ OX,x .

∙ We say that h is integral over M if h ∘ ϕ ∈ (ϕ * (M))O1 , for all analytic path ϕ : (C, 0) → (X, x). ∙ We say that h is strictly integral over M if h ∘ ϕ ∈ m1 (ϕ * (M))O1 , for all analytic path ϕ : (C, 0) → (X, x). p The integral closure of M is defined as M := {h ∈ OX,x | h is integral over M} and the strict p integral closure of M is defined as M † := {h ∈ OX,x | h is strictly integral over M}. Using the p definition it is clear that M and M † are submodules of OX,x , M ⊆ M and M † ⊆ M.

Example 1.2.2. ([41] Example 1.4) Suppose X = C2 and that M ⊆ O22 is generated by {(x, 0), (0, y), (y, x)}. Then, M = m2 O22 . Indeed, let ϕ : (C, 0) → (C2 , 0). Then, (ϕ * (M))O1 is generated by {(t n , 0), (0,t n )}, where n = min {o(ϕ1 ), o(ϕ2 )} and o(ϕi ) denotes the order of vanishing of ϕi , for i ∈ {1, 2}. But these are the generators of (ϕ * (m2 )O22 )O1 as well.

33

1.2. The integral closure of modules

p If (X, x) has several components, then it is clear that M induces a submodule MV in OV,x , V any component of (X, x). It is also clear from the definition that h ∈ M if and only if hV ∈ MV p for all components V of (X, x), where hV is the element of OV,x induced by h. It is clear from the definition that M = M.

The following generalization of Nakayama’s Lemma is often useful. p Proposition 1.2.3. ([41] Proposition 1.5) Suppose N, M are submodules of OX,x , and m = mX,x .

a) If (mM + N) = M then N = M; b) If M ⊆ mM + N ⊆ M then N = M. The link between the integral closure of ideals and modules is very strong. If M is a p submodule of OX,x , and [M] is a matrix of generators of M, let Jk (M) denote the ideal generated by O

p

k × k minors of [M]. This is the same as the (p − k)-th Fitting ideal of MX,x , hence is independent p of the choice of generators of M (see [36] and [68]). If h ∈ OX,x , let (h, M) denote the submodule generated by h and M. The following lemma, which is a generalization of Cramer’s rule, is helpful in establishing the connection between M and Jk (M). p p , M is an OX,x -submodule of OX,x , Jk+1 (h, M) = Lemma 1.2.4. ([41] Lemma 1.6) Suppose h ∈ OX,x 0 and no element of Jk (M) is a zero divisor on OX,x . Then

Jk (M) · h ⊆ Jk (h, M) · M. The link between M and Jk (M) is established by p , X irreducible, Proposition 1.2.5. ([41] Proposition 1.7) Suppose M is a submodule of OX,x p h ∈ OX,x . Then h ∈ M if and only if Jk (h, M) ⊆ Jk (M), where k is the largest integer such that Jk (h, M) ̸= 0.

The proof of the above proposition in [41] shows that h ∈ M implies Jk (h, M) ⊆ Jk (M) for all k with no assumptions on X or the generic rank of M. If X is not irreducible, one obtains: Corollary 1.2.6. ([41] Corollary 1.8) Suppose (X, x) is a complex analytic germ with irreducible p components (Vi ), and M is a submodule of OX,x . Then h ∈ M if and only if Jki (h, Mi ) ⊆ Jki (Mi ), p for all i, where Mi is the submodule of OVi ,x induced from M and ki is the generic rank of (h, Mi ) on Vi . In [87] Rees defined the notion of the integral closure of M in K ⊗ M, where R is a R

noetherian domain, K its field of fractions and M is a finitely generated torsion-free R-module.

34


His definition is based on the theory of discrete valuations. However, the previous proposition p and the Theorem 1.2 of [87] show that M in our language is exactly the set of elements of OX,x which are integral over M in Rees’s language. Using Proposition 1.2.5 and Corollary 1.2.6, Gaffney proved the next result. p p Proposition 1.2.7. ([41] Proposition 1.9) Suppose M is a submodule of OX,x and h ∈ OX,x . Then h ∈ M if and only if on each component V of (X, x), there exists an ideal I of OV,x , I ̸= 0 such p that I · h ⊆ I · M in OV,x .

In the particular case when the generic rank of M is 0 or maximal, Gaffney obtained a description of the integral closure of M stronger than the Proposition 1.2.7. p of generic rank p Proposition 1.2.8. ([41] Proposition 1.10) Suppose M is a submodule of OX,x p or 0 on each component of X, and let h ∈ OX,x . Then h ∈ M if and only if there exists a faithful submodule I of OX,x such that I · h ⊆ I · M.

The next proposition contains the growth condition for integral closure in the module case. The version we work here was suggested by Looijenga. In what follows, we let Γ(E) denote sections of a vector bundle. p p and that M is a submodule of OX,x . Proposition 1.2.9. ([41] Proposition 1.11) Suppose h ∈ OX,x Then h ∈ M if and only if for each choice of generators {si } of M there exists a neighborhood U of x such that for all ϕ ∈ Γ(Hom(C p , C)),

‖ϕ(z) · h(z)‖ ≤ C sup ‖ϕ(z) · si (z)‖ i

for all z ∈ U. We consider the "sheafification"of the construction of the integral closure of modules. The connection between M and Jk (M) allows us to show that the integral closure gives rise to a coherent sheaf. The proof uses a description of M in terms of blowing up. Proposition 1.2.10. ([41] Proposition 1.12) Suppose M is a coherent sheaf of submodules of OXp . Then, there exists a unique coherent sheaf M on X such that for each x ∈ X, (M )x = Mx . There is another way to make a connection between the integral closure of modules and ideals. Let us define the ideal sheaf ρ(M) on X × P p−1 associated to a submodule sheaf M of OXp (see [48]): Given h = (h1 , ..., h p ) ∈ OXp and (x, [t1 , ...,t p ]) ∈ X × P p−1 , with ti ̸= 0, we define p

T

ρ(h) as the germ of the analytic map given by ∑ h j (z) Tij which is well-defined on a Zariski open j=1

× P p−1

subset of X that contains the point (x, [t1 , ...,t p ]). We define ρ(M) as the ideal generated by {ρ(h) | h ∈ M}.

35

1.2. The integral closure of modules

The next result gives a relation between the integral closure of modules and ideals. p Proposition 1.2.11. ([48] Proposition 3.4) Let h ∈ OX,x . Then h ∈ M at x if and only if ρ(h) ∈ ρ(M) at all point (x, [t1 , ...,t p ]) ∈ V (ρ(M)).

The Multiplicity of a Pair of Modules We end this section defining the multiplicity of a pair of modules following the lines of [42]. The multiplicity of an ideal or module or pair of modules is one of the most important invariants we can associate to an m-primary module. It is intimately connected with integral closure. Let (X, x) be a germ of an analytic space and assume that (X, x) has dimension d. Let M ⊆ N be sheaves of OX -submodules of OXp , where M has finite colength in N. Denote M the associated ideal sheaf ρ(M). Let R(M) and R(N) the Rees algebra of M and N, respectively. Then, we have canonical maps which give us the following diagram:

BM (Pro jan(R(N)))

πN

Proj(R(N))

πM

πXN

Pro jan(R(M))

X

πXM

On the blow up BM (Pro jan(R(N))) we have two tautological bundles. One is the pullback of the bundle on Pro jan(R(N)). The other comes from Pro jan(R(M)). Denote the corresponding Chern classes by cM and cN , and denote the exceptional divisor by DM,N . Suppose the generic rank of N (and hence M) is g. Then, the multiplicity of a pair of modules (M, N) at x is: d+g−2 Z

e(M, N) :=

∑

d+g−2− j

DM,N · cM

j

· cN .

j=0

Kleiman and Thorup show that this multiplicity is well defined at x ∈ X as long as M = N on a deleted neighborhood of x. This condition implies that DM,N lies in the fiber over x, hence p p is compact. It also possible to conclude that, when N = OX,x , and M has finite colength in OX,x p then e(M, OX,x ) is the Buchsbaum-Rim multiplicity e(M) defined in [19]. There is a fundamental result due to Kleiman and Thorup, the principle of additivity [62], which states that given a sequence of OX,x -modules M ⊆ N ⊆ P such that the multiplicity of the pairs is well defined, then e(M, P) = e(M, N) + e(N, P).

36


Also if M = N at x then e(M, N) = 0, and the converse also holds if X is equidimensional. These results will be used in this work.

1.3

Whitney equisingularity and the integral closure of modules

In this section, we follow the Gaffney’s approach about Whitney equisingularity and integral closure of modules presented in [42]. The way how Gaffney got algebraic conditions that characterizes the Whitney equisingularity using the integral closure of modules have inspired us to develop some algebraic tools in this work that will be useful in order to describe the bi-Lipchitz Equisingularity. We start with some notation to describe a family of analytic sets. Setup 1.3.1. Let (X, 0) ⊆ (Cn+k , 0) be the germ of the analytic space defined by an analytic map F : Cn × Ck → C p , where X is a sufficient small representative such that Y = Ck = 0 × Ck ⊆ X is the singular set of X. Let F1 , . . . , Fp : Cn × Ck → C be the coordinates functions of F, for each y ∈ Y let fy : Cn → C p given by fy (z) := F(z, y) and let Xy := fy−1 (0). Assume that Xy has an isolated singularity in 0, for all y ∈ Y and n ≥ p. Let z1 , . . . , zn , y1 , . . . , yk be the coordinates on Cn+k , let mY be the ideal of OX generated by {z1 , . . . , zn }, let JM(X) be the jacobian module of X, let JM(X)Y be the module generated by { ∂∂yF , . . . , ∂∂ yF } and let JMz (X) be the module 1 k generated by { ∂∂ zF , . . . , ∂∂ zFn }. In this setup we assume that F defines X with reduced structure 1 and X is equidimensional. Usually Y is called the parameter space of the family X. Given a family as above, we say the family is holomorphically trivial if there exists a holomorphic family of origin preserving bi-holomorphic germs ry such that ry (X0 ) = Xy . If the map-germs are only homeomorphisms, we say the family is C0 trivial. Example 1.3.2. ([42] Example 1.1) Let X ⊆ C3 be the family of four moving lines with equation F(x, y, z) = xz(z + x)(z − (1 + y)x) = 0. Here y is the parameter, the x and z axis are fixed as is the line z + x = 0, while the line z = (1 + y)x moves with y. For each y ∈ C let fy : (C2 , 0) → C given by fy (z) = F(x, y, z). It is easy to see that if θ : C2 → C2 is a C-linear map which preserves the lines x = 0, z = 0 and z = −x on C2 then θ is a multiple of the identity on C2 . Suppose by contradiction that X is a holomorphically trivial family. So we have a family ry y ∈ C, of bi-holomorphic germs such that ry (X0 ) = Xy , for all y ∈ C. Take y ∈ C, y ̸= 0. Since ry (X0 ) = Xy then the differential Dry (0) : C2 → C2 takes the tangent lines of X0 into the tangent lines of Xy , i.e: : (C2 , 0) → (C2 , 0),

{z = −x} ⊆ X0 maps to {z = −x} ⊆ Xy through Dry (0)

1.3. Whitney equisingularity and the integral closure of modules

37

Figure 1 – Four moving lines

{z = 0} ⊆ X0 maps to {z = 0} ⊆ Xy through Dry (0) {x = 0} ⊆ X0 maps to {x = 0} ⊆ Xy through Dry (0) {z = x} ⊆ X0 maps to {z = x} ⊆ Xy through Dry (0). In particular Dry (0) is a C-linear map which preserves the lines x = 0, z = 0 and z = −x on Thus, Dry (0) is a multiple of the identity, i.e, there exists a ∈ C such that Dry (0) = a.IdC2 . Taking x ̸= 0 sufficiently nearby 0, once we apply Dry (0) at the point (x, x), we conclude that a = 1 and a = 1 + y, which implies that y = 0, contradiction. C2 .

Therefore, the family X is not holomorphically trivial. However it should be equisingular for any reasonable definition of equisingularity. This example shows that we need a notion of equisingularity that is less restrictive than holomorphic equivalence. The Whitney conditions imply C0 -triviality but also imply the family is well-behaved at the infinitesimal level. Here we denote K as the field R or C. Definition 1.3.3. (Whitney’s conditions) Suppose X is a subset of Kn , X0 is the set of smooth points on X and that Y is a smooth subset of X. ∙ The pair (X0 ,Y ) satisfies Whitney’s condition A at y ∈ Y if for all sequences (xi ) of points of X0 such that xi → y Txi X → T implies TyY ⊆ T ;

38


∙ The pair (X0 ,Y ) satisfies Whitney’s condition B at y ∈ Y if for all sequences (xi ) of points of X0 and (yi ) of points of Y such that xi → y yi → y Txi X → T sec(xi , yi ) → L implies L ⊆ T . John Mather first pointed out that Whitney’s condition B implies Whitney’s condition A in the notes of his lectures at Harvard in 1970 [69]. Example 1.3.4. Let us go back to the Example 1.3.2, and to prove that the family of four moving lines satisfies the Whitney’s conditions. As pointed out by Mather, we only have to check the condition B. Let Y be a smooth subspace of X, u ∈ Y , (xi )i∈N a sequence of points of X0 and (yi )i∈N a sequence of points of Y such that xi → u yi → u Txi X → T sec(xi , yi ) → L. Let M, N and P the planes x = 0, z = 0 and z = −x, respectively, and let R the surface z = (1+y)x. So M ∩ N ∩ P ∩ R is the y-axis and X = M ∪ N ∪ P ∪ R. Consider M˜ := {i ∈ N | xi ∈ M} N˜ := {i ∈ N | xi ∈ N} P˜ := {i ∈ N | xi ∈ P} R˜ := {i ∈ N | xi ∈ R}. Clearly N = M˜ ∪ N˜ ∪ P˜ ∪ R˜ and one of these sets has to be not bounded. (i) Suppose that M˜ is not bounded. So we have a subsequence (Txi X)i∈M˜ → T , hence T = M. We also have that u = lim xi = lim xi ∈ M = M which implies that Tu M = M. Since i→∞

i∈M˜

Y is a smooth subset of X and u ∈ M there exists an open subset U of Y such that u ∈ U and U is a submanifold of M (this occurs because M, N, P and R intersects transversely pairwise). Since (xi ), (yi ) → u then xi , yi ∈ U ⊆ M, ∀i ∈ N sufficiently large. Therefore, the set Mˆ := {i ∈ N | xi , yi ∈ M} is not bounded. Since M is a plane then sec(xi , yi ) ⊆ M, ∀i ∈ Mˆ and since this subsequence also converges to L then L ⊆ M = T .

1.3. Whitney equisingularity and the integral closure of modules

39

The same argument can be applied when N˜ or P˜ once these subsets of X are planes as well. (ii) Suppose that R˜ is not bounded. Analogously to the previous case, we have that u ∈ R, there exists an open subset U of Y such that u ∈ U such that U is a submanifold of R and the set Rˆ := {i ∈ N | xi , yi ∈ R}. Since R and U is a submanifold of R then the pair (R,U) satisfies the Whitney’s conditions. Since (xi )i∈Rˆ → u (yi )i∈Rˆ → u (Txi R)i∈Rˆ → T (sec(xi , yi ))i∈Rˆ → L. then L ⊆ T . Therefore, the family of four moving lines satisfies the Whitney’s conditions. Example 1.3.5. ([42] Example 1.4) Consider X ⊆ R3 the family defined by F(x, y, z) = z3 + x2 − y2 z2 = 0. The members of the family Xy consist of node singularities where the loop is pulled smaller and smaller as y tends to zero becoming a cusp at y = 0 (see the picture below). The singular locus is the y-axis. Whitney A holds because every limiting tangent plane contains the y-axis. But Whitney B fails. Notice that the parabola z = y2 is in the surface, and letting xi = (0,ti ,ti2 ) and yi = (0,ti , 0), (ti ) any sequence tending to 0, we see that the limiting secant line is the z-axis, while the limiting tangent plane along this curve is the xy-plane. We see that the dimension of the limiting tangent planes at the origin is 1, while it is zero elsewhere on the y-axis. This kind of drastic change at the infinitesimal level is prevented by the Whitney conditions. Reading: There are many places that we can read about the Whitney conditions. An important reference is the Chapter III of [98], which is more in the spirit of the way that Gaffney develops the subject. Surely one can see the works of Whitney [109, 110] as well.

Verdier’s Condition W (or (w)-regularity) The next condition, while is equivalent to the Whitney conditions in the complex analytic case (proved by Teissier in [98]), is easier to work using algebra. The (w)-regularity says that the distance between the tangent space to X at a point xi of X0 and the tangent space to Y at y goes to zero as fast as the distance between xi and Y . We first need to define what we mean by the distance between two linear spaces. Suppose A, B are linear subspaces of KN . We define the distance from A to B as

‖⟨u, v⟩‖ ⊥ dist(A, B) = sup | u ∈ B − {0} and v ∈ A − {0} . ‖u‖ ‖v‖

40


Figure 2 – X: y2 z2 = z3 + x2

In the applications, B is the "big space"and A is the "small" space. The inner product is Hermitian inner product when we work over C. Example 1.3.6. ([42] Example 1.5) Work in R3 . Let A = x-axis, B a plane with unit normal u0 , then the distance from A to B is cos θ , where θ is the small angle between u0 and the x-axis, in the plane they determine. So when the distance is 0, B contains the x-axis. We recall Verdier’s condition W , also called (w)-regularity. Definition 1.3.7. Suppose Y ⊆ X, where X,Y are strata in a stratification of an analytic space. Then the pair (X,Y ) satisfies the Verdier’s condition W at 0 ∈ Y if there exists a positive real number C such that dist(TY0 , T Xx ) ≤ Cdist(x,Y ) for all x close to Y . The next theorem was proved first by Teissier in the case of a family of analytic hypersurfaces, and generalized by Gaffney, for any codimension. Theorem 1.3.8. ([41] Theorem 2.5) With the setup 1.3.1, the Verdier’s condition W holds for (X −Y,Y ) at (0, 0) if and only if (X −Y,Y ) satisfies the Whitney’s conditions at (0, 0) if and only if JM(X)Y ⊆ mY JMz (X).

1.4

The Lipschitz saturation and the double of an ideal

Following the approach of Pham-Teissier in [85], let A be a commutative local ring over C, and A its normalization (we can assume that A is the local ring of an analytic space X at the

41

1.4. The Lipschitz saturation and the double of an ideal

origin in Cn ). Let I be the kernel of the inclusion A⊗A → A⊗A C

A

In this construction, the tensor product is the analytic tensor product which has the right universal property for the category of analytic algebras, and which gives the analytic algebra for the analytic fiber product. ˜ to consist of all Pham and Teissier then defined the Lipschitz saturation of A, denoted A, elements h ∈ A such that h ⊗ 1 − 1 ⊗ h ∈ A ⊗ A is in the integral closure of I (for related results C

see [66]). The connection between this notion and that of Lipschitz functions is as follows. If we pick generators (z1 , ..., zn ) of the maximal ideal of the local ring A, then zi ⊗ 1 − 1 ⊗ zi ∈ A ⊗ A C

give a set of generators of I. Choosing zi so that they are the restriction of coordinates on the ambient space, using the supremum criterion given by Lejeune-Jalabert and Teissier in [64], we see the integral closure condition is equivalent to |h(z1 , ..., zn ) − h(z′1 , ..., z′n )| ≤ C sup{|zi − z′i |} i

holding on some neighborhood U of (0, 0) on X × X. This last inequality is what is meant by the meromorphic function h being Lipschitz at the origin on X. Note that the integral closure condition is equivalent to the inequality holding on a neighborhood U for some C for any set of generators of the maximal ideal of the local ring. The constant C and the neighborhood U will depend on the choice. If (X, x) is normal then passing to the Lipschitz saturation does not add any functions. Definition 1.4.1. Let I be an ideal of OX,x , SBI (X) the Lipschitz saturation of the blow-up, which is the space whose analytic sheaf of rings OSBI (X) is the Lipschitz saturation of the analytic sheaf of rings OBI (X) , i.e, OSB (X) = OeB (X) . I

I

Let πS : SBI (X) → X be the projection map. The Lipschitz saturation of the ideal I is denoted IS , and is the ideal IS := {h ∈ OX,x | πS* (h) ∈ πS* (I)}. Since the normalization of a local ring A contains the Lipschitz Saturation of A then I ⊆ IS ⊆ I. In particular, if I is integrally closed then IS = I. Here is a viewpoint on the Lipschitz saturation of an ideal I, which will be useful later. Given an ideal I, and an element h that we want to check for inclusion in IS , we can consider (BI (X), π), π * (I) and h ∘ π. Since π * (I) is locally principal, working at a point z on the exceptional divisor E, we have a local generator f ∘ π of π * (I). Consider the quotient (h/ f ) ∘ π. Then h ∈ IS if and only if at the generic point of any component of E, (h/ f ) ∘ π is Lipschitz with respect to a system of local coordinates. If this holds we say h ∘ π ∈ (π * (I))S .

42


We can also work on the normalized blow-up (NBI (X), πN ). Then we say h ∘ πN ∈ if (h/ f ) ∘ πN satisfies a Lipschitz condition at the generic point of each component of the exceptional divisor of (NBI (X), πN ) with respect to the pullback to (NBI (X), πN ) of a system of local coordinates BI (X) at the corresponding points of BI (X). As usual, the inequalities at the level of NBI (X) can be pushed down and are equivalent to inequalities on a suitable collection of open sets on X.

(πN* (I))S

This definition can be given in an equivalent statement using the theory of integral closure of modules. Since Lipschitz conditions depend on controlling functions at two different points as the points come together, we should look for a sheaf defined on X × X. We describe a way of moving from a sheaf of ideals on X to a sheaf on X × X. Let π1 , π2 : X × X → X be the projections to the i-th factor, i ∈ {1, 2}, and let h ∈ OX,x . 2 Define hD ∈ OX×X,(x,x) as (h ∘ π1 , h ∘ π2 ), called the double of h. We define the double of the 2 ideal I, denoted ID , as the submodule of OX×X,(x,x) generated by hD , where h is an element of I. If I is an ideal sheaf on a space X then intuitively, h ∈ I if h tends to zero as fast as the elements of I do as you approach a zero of I. If hD is in ID then the element defined by (1, −1) · (h ∘ π1 , h ∘ π2 ) = h ∘ π1 − h ∘ π2 should be in the integral closure of the ideal generated by applying (1, −1) to the generators of ID , namely the ideal generated by g ∘ π1 − g ∘ π2 , g any element of I. This implies the difference of h at two points goes to zero as fast as the difference of elements of I at the two points go to zero as the points approach each other. It is reasonable that elements in IS should have this property. In fact, as we can see in [40], the following result gives a nice link between Lipschitz saturation and integral closure of modules. Theorem 1.4.2. ([40] Theorem 2.3) Suppose (X, x) is a complex analytic set germ, I ⊆ OX,x is an ideal and h ∈ OX,x . Then h ∈ IS if and only if hD ∈ ID . Corollary 1.4.3. ([39] Corollaries 3.4 and 3.5) Let I ⊆ J ⊆ I be ideals of OX,x , with X equidimensional. Suppose that ID has finite colenght in JD and JD has finite colenght in (I)D . Then e(ID , (I)D ) = e(JD , (I)D ) if and only if ID = JD if and only if IS = JS . Here is an example showing the difference between the integral closure of the jacobian ideal and its saturation, in particular, the difference between the integral closure and saturation.

1.4. The Lipschitz saturation and the double of an ideal

43

Example 1.4.4. ([39]) Consider F(x, y) = x2 + y p , p ≥ 3 odd. Denote by X the plane curve defined by F. Then X has a normalization given by φ (t) = (t p ,t 2 ). The elements in the integral closure of the jacobian ideal are just those ring elements h such that h ∘ φ ∈ φ * (J(F)) = (t p ). Now, yq ∘ φ = t 2q , so yq ∈ J(F), for q > 2p . Denote a matrix of generators for J(F)D by [J(F)D ]. Consider the curve mapping into X × X given by Φ(t) = (t p ,t 2 ,t p , ct 2 ), where c is a pth root of unity different from 1. Now consider the ideal generated by the entries of the vector (1, −1)[J(F)D ] ∘ Φ(t). This ideal is generated by (y p−1 − y′p−1 , (y − y′ )(x, x′ , y p−1 , y′p−1 )) ∘ Φ(t) = (t p+2 ). Meanwhile q ′q the order in t of (1, −1)(yq , y′q ) ∘ Φ(t) is 2q. If p < 2q < p + 2, i.e, q = p+1 2 , then (y , y ) cannot be in J(F)D , hence yq ∈ / J(F)S , but yq ∈ J(F). Using the Lipschitz saturation of ideals (and doubles), in [39] we have set the infinitesimal Lipschitz conditions for hypersurfaces. Setup: Let X n+k , 0 ⊆ Cn+1+k , 0 be a hypersurface, containing a smooth subset Y embedded in Cn+1+k as 0 × Ck , with pY the projection to Y . Assume Y = S(X), the singular set of X. Suppose F is the defining equation of X, (z, y) coordinates on Cn+1+k . Denote by fy (z) = F(z, y) the family of functions defined by F and Xy := fy−1 (0). Assume that fy has an isolated singularity at the origin. Let mY denote the ideal defining Y , and J(F)Y , the ideal generated by the partial derivatives with respect to the y coordinates, Jz (F), those with respect to the z coordinates. Here we work with the double relative to Y , which means that we work with the projections π1 and π2 defined on the fibered product X × X. Y

Definition 1.4.5. We say the pair (X,Y ) satisfies the infinitesimal Lipschitz condition mY (iLmY ) at the origin if either of the two equivalent conditions hold: 1. J(F)Y ⊆ (mY Jz (F))S ; 2. (J(F)Y )D ⊆ (mY Jz (F))D . An analogous condition for iLmY is J(F)Y ⊆ mY Jz (F). This is the equivalent to the Verdier’s condition W or the Whitney conditions. Next we give the definition of iLA . Definition 1.4.6. We say the pair (X,Y ) satisfies the infinitesimal Lipschiz condition A (iLA ) at the origin if either of the two equivalent conditions hold: 1. J(F)Y ⊆ (Jz (F))S ; 2. (J(F)Y )D ⊆ (Jz (F))D . The analogous condition is J(F)Y ⊆ Jz (F). If one works on the ambient space, then this is equivalent to the AF condition.

44


Since there are different ways in which the total space X n+k can be made into a family of spaces, it is natural to ask if the conditions we have defined depend on the projection to Y which defines the family. The following result says the iLmY does not depend on the projection to Y . Proposition 1.4.7. ([39] Proposition 3.8) In the above setup the following conditions are equivalent: (a) (J(F)Y )D ⊆ (mY Jz (F))D ; (b) (J(F)Y )D ⊆ (mY J(F))D . While a similar result for iLA does not make sense, if we ask that (J(F)Y )D ⊆ (Jz (F))†D then an analogous result holds. In Proposition 4.1 of [39] is proved that the cosupport of (mY Jz (F))D and (Jz (F))D on X × X are equal, and consist of Y

∆(X) ∪ (X × 0) ∪ (0 × X). Y

Y

This set is where makes sense to ask about the infinitesimal Lipschitz conditions. The next result says us that both iL conditions hold on the above set off (0, 0) ×Y . Proposition 1.4.8. ([39] Proposition 4.2) iLA and iLmY hold at all points of ∆(X) − ((0, 0) ×Y ), and both conditions hold at all point of (0 × X) ∪ (X × 0) − ((0, 0) ×Y ) if W condition holds at Y

Y

all point (0, y), y ∈ Y . In the chapter 3 we generalize this results for an arbitrary codimension of X. Finally, in (0, 0) ×Y , Gaffney proved that the iLA condition holds generically. Theorem 1.4.9. ([39] 4.3) In the setup of this section, there exists a Zariski open subset U of Y such that iLA holds for the pair (X −Y,U ∩Y ) along Y .

1.5

Determinantal Varieties

We first recall the definition of determinantal varieties. Let Σt ⊆ Hom(Cn , C p ) be the subset consisting of the maps that have rank less than t, with 1 ≤ t ≤ min(n, p). It is possible to show that Σt is an irreducible singular algebraic variety of codimension (n −t + 1)(p −t + 1) (see [18]). Moreover the singular set of Σt is exactly Σt−1 . The set Σt is called a generic determinantal variety of size (n, p) obtained from t × t minors. The representation of the variety Σt as the union Σi ∖ Σi−1 , i = 1, . . . ,t is a stratification of Σt , which is locally holomorphically trivial, and it is called the rank stratification of Σt .

1.5. Determinantal Varieties

45

Definition 1.5.1. Let U ⊆ Cr be an open domain, F = (mi j (x)) be an n × p matrix whose entries are complex analytic functions on U, 0 ∈ U and f the function defined by the t × t minors of M. We say that X = V ( f ) is a determinantal variety if it has codimension (n − t + 1)(p − t + 1). Currently, determinantal varieties have been an important object of study in Singularity Theory. For example, we can refer to the works of Damon [29], Frühbis-Krüger [37, 38], Gaffney [39, 47], Grulha [47], Nuño-Ballesteros [2, 76], Oréfice-Okamoto [2, 76], Pereira [83, 89], Pike [29], Ruas [47, 89], Tomazella [2, 76], Zhang [114] and others. In the case where X is a codimension two determinantal variety, we can use the HilbertBurch theorem to obtain a good description of X and its deformations in terms of its presentation matrix. In fact, if X is a codimension two Cohen- Macaulay variety, then X can be defined by the maximal minors of a n × (n + 1) matrix. Moreover, any perturbation of a n × (n + 1) matrix gives rise to a deformation of X and any deformation of X can be obtained through a perturbation of the presentation matrix (see [93]). We can use this correspondence to study properties of codimension two Cohen-Macaulay varieties through their presentation matrix. This is the approach of Frühbis-Krüger, Pereira and Ruas. In order to introduce the notion of EIDS and relate it with the classical approach of singularity theory, let us recall some concepts in this field. Let R be the group of coordinate changes (on the source) in (Cr , 0). We denote GLi (Or ) the group of invertible matrices of size i × i with entries in the local ring Or . Consider the group H := GL p (Or ) × GLn (Or ). Given two matrices, we are interested in studying these germs according to the following equivalence relation. Definition 1.5.2. Let G := R n H be the semi-direct product of R and H . We say that two germs F1 , F2 ∈ Mat(n,p) (Or ) are G -equivalent if there exist (φ , R, L) ∈ G such that F1 = L−1 (φ * F2 )R. It is not difficult to see that G is one of Damon’s geometric subgroups of K (see [83]), hence as a consequence of Damon’s result ([27]) we can use the techniques of singularity theory, for instance, those concerning finite determinacy. The notions of G -equivalence and K∆ -equivalence, where ∆ consists of the subvariety of matrices of rank less than the maximal rank [27], coincide for finitely determined germs (see [17]). The next result is a Geometric Criterion of Finite Determinacy for families of n × p matrices and was proved by Pereira in her Ph. D. thesis. Theorem 1.5.3. (Geometric Criterion of Finite Determinacy, [83] Theorem 2.4.1) A representative of a germ F : Cr , 0 −→ Mat(n,p) (C) is G -finitely determined if and only if F is transverse to the strata of the rank stratification of Mat(n,p) (C) outside the origin.

46


It follows that if F is a n × p matrix with entries in the maximal ideal of Or , defining an isolated singularity, then F is G -finitely determined. Moreover if F is G -finitely determined, then the germ of X at a singular point is holomorphic to either the product of Σt with an affine space or a transverse slice of Σt . This motivates the following definition ([30]): Definition 1.5.4. A point x ∈ X = F −1 (Σt ) is called essentially non-singular if, at the point x, the map F is transversal to the corresponding stratum of the variety Σt . A germ (X, 0) ⊆ (Cr , 0) of a determinantal variety has an essentially isolated singular point at the origin (or is an essentially isolated determinantal singularity: EIDS) if it has only essentially non-singular points in a punctured neighborhood of the origin in X. If X = F −1 (Σt ) then a perturbation of X is obtained by perturbing the entries of F. This yields an unfolding of F, and if X is an EIDS then happens to also give a deformation of X which is transverse to the strata of Hom(Cn , C p ). In the particular case where X is Cohen-Macaulay of codimension 2, it is a consequence of the Auslander-Buchsbaum formula and the Hilbert-Burch Theorem that any deformation of X can be given as a perturbation of the presentation matrix (see [38]). Therefore we can study these varieties and their deformations using their representation matrices. We can express the normal module N(X), and the space of the first order deformations TX1 , in terms of matrices, hence we can treat the base of the semi-universal deformation using matrix representation and we can express the normal module N(X) in terms of matrices.

1.6

Bi-Lipschitz geometry and Lipschitz stratifications

In this section we follow the lines of [70] for an understanding of the bi-Lipschitz geometry. Let M and N be metric spaces. Definition 1.6.1. A map f : M → N is called Lipschitz if there exists a constant C > 0 such that d( f (x), f (y)) ≤ Cd(x, y), for all x, y ∈ M. A homeomorphism f : M → N is called bi-Lipschitz if f is Lipschitz and the inverse f −1 : N → M is Lipschitz as well. In this case we say that f is a bi-Lipschitz equivalence and that M and N are bi-Lipschitz equivalent. In order to understand the bi-Lipschitz equisingularity, we recall here some important notions developed by Mostowski and Parusiński.

1.6. Bi-Lipschitz geometry and Lipschitz stratifications

47

We denote K the field R or C. Definition 1.6.2. A filtration of a subset X ⊆ Kn is a sequence of subsets of X X = X d1 ⊃ X d2 ⊃ ... ⊃ X dl ̸= 0/ di with d1 > d2 > ... > dl , and Xsing ⊆ X di+1 , ∀i ∈ {1, ..., l}. The superscripts di denote dimensions, the X di are called skeletons, the manifolds X˚ di := X di − X di+1 strata and the collection {X˚ di } is

called a stratification of X. The stratification {X˚ di } is called weakly Lipschitz if for every i, for all point p ∈ X˚ di and vector v ∈ Tp X˚ di , there is a Lipschitz vector field v˜ defined in a neighborhood of p in Kn such that v(p) ˜ = v and the local flow vt generated by v satisfies vt (X) ⊆ X. The weakly Lipschitz stratifications have the following property: if v is a Lipschitz vector field defined in a neighborhood of a point p ∈ X˚ di in X˚ di , which is tangent to X˚ di , then v extends to a Lipschitz vector field defined in a neighborhood of p in Kn , whose local flow preserves X. Definition 1.6.3. Under the above notation, we say that X is locally Lipschitz equisingular ˚ di if for every p ∈ X˚ di there is a neighborhood U0 of p in X˚ di , a tubular along a stratum X neighborhood U of U0 in Kn with a projection π : U → U0 , and a bi-Lipschitz homeomorphism h : U → U0 × H0 , where H is the complement of TpU0 in Tp Kn = Kn , i.e, Tp Kn = TpU0 ⊕ H, H0 is a neighborhood of p in H, such that 1. pr ∘ h = π; 2. h(U ∩ X) = U0 × (H0 ∩ X), where pr : U0 × H0 → U0 is the projection. It is clear that X is locally Lipschitz equisingular along every stratum of a weakly Lipschitz stratification. The existence of weakly Lipschitz stratifications is proved in [72]. By dist(?, X) we mean the function of distance to X. If X = 0/ we set dist(?, 0) / ≡ 1. For n any linear operator A defined on K , we define the norm of A as ‖A‖ = sup{‖A(v)‖ | v ∈ Kn , ‖v‖ = 1} For q ∈ X˚ d j , let Pq : Kn → Tq X˚ d j be the orthogonal projection onto the tangent space of X˚ d j at q, and let Pq⊥ = IdKn − Pq : Kn → Tq⊥ X˚ d j be the orthogonal projection onto the normal space.

48


For technical reasons we replace the distance functions dist(q, X d j ) by semianalytic functions (i.e, continuous functions with semianalytic graphs) such that 1 dist(q, X d j ) ≤ ρ j (q) ≤ 2ndist(q, X d j ). 2n For a proof of the existence of these functions ρ j ’s see [67]. If X d j ’s are algebraic then the distance functions themselves can be used [28]. Definition 1.6.4. (Chain) Let c0 be a fixed constant, c0 ≥ 2n. A c0 -chain (or chain) for a point q ∈ X˚ d j is a strictly increasing sequence of indices j = j1 , j2 , ..., jr = l and a sequence of points q js ∈ X˚ d js such that j1 = j, q j1 = q and each js (s ≥ 2) is the smallest integer greater than js−1

for which ρ js −1 (q) ≥ 2c20 ρ js (q), and q − q js ≤ c0 ρ js (q). Now we are able to recall the following definition (see [80]). Definition 1.6.5. (Mostowski’s conditions for Lipschitz stratifications) We call a stratification {X˚ di } a Lipschitz stratification if for some constant C > 0, every c0 -chain q = q j1 , q j2 , ..., q jr and every k ∈ {1, ..., r} we have:

‖q−q j2 ‖

⊥

1. Pq ∘ Pq j2 ∘ ... ∘ Pq jk ≤ C . dj dist(q,X

k−1 )

d j−1

) then If q′ ∈ X˚ d j such that ‖q − q′ ‖ ≤ dist(q,X 2c0

′

2. (Pq − Pq′ ) ∘ Pq j2 ∘ ... ∘ Pq jk ≤ C ‖q−qd j‖ ; dist(q,X

3. Pq − Pq′ ≤ C

k−1 )

‖q−q′ ‖ . d dist(q,X j−1 )

Example 1.6.6. Consider the cone X given by the equation z2 = x2 + y2 . If we take X 2 := X and X 0 := (0, 0, 0) then {X 0 , X 2 − X 0 } is a stratification of the cone X which satisfies the three above Mostowski conditions.

Figure 3 – Cone

1.6. Bi-Lipschitz geometry and Lipschitz stratifications

49

An alternate definition of Lipschitz stratification is given in [70]. Definition 1.6.7. We call a stratification {X˚ di } a Lipschitz stratification if for some constant C > 0, for every compact set F such that X di+1 ⊆ F ⊆ X di and every Lipschitz vector field v, defined on F and tangent to strata (i.e v(x) ∈ Tx X˚ d j for all x ∈ X˚ d j , j > i, or x ∈ X˚ di ∩ F), there exists a Lipschitz vector field v, ˜ defined on Kn , tangent to strata and extending v, such that the best Lipschitz constant for v˜ is less than or equal to C. We state the main result of this section, due to Mostowski. Theorem 1.6.8. ([72] Proposition 1.2) Every complex analytic set admits a complex analytic Lipschitz stratification. Moreover such Lipschitz stratification are locally bi-Lipschitz trivial along strata In [78], Parusiński obtained the above theorem for analytic real varieties. After, he generalized this result for subanalytic sets. Theorem 1.6.9. ([82] Theorem 1.4) Every subanalytic set admits a subanalytic Lipschitz stratification. Moreover such Lipschitz stratification are locally bi-Lipschitz trivial along strata There is a construction of a Lipschitz stratification of an analytic hypersurface in [72], but it was not made canonically. Hence, one important thing to investigate is when it is possible to construct a canonical vector field satisfying the previous Lipschitz conditions. Example 1.6.10. ([101] Example 5.1) Consider R3 with coordinates x, y and z. For each z consider Xz = {y = 0} ∪ {(x, xz , z)}. Then the bi-Lipschitz type of Xz and Xz′ are different if z ̸= z′ , both greater than 1. Let us to verify this for natural numbers n ̸= m. For each n consider the map-germ fn : (R, 0) → (R, 0) given by fn (x) = xn . Thus fn is finitely C0 -K -determined map germs. Since deg fn ̸= deg fm whenever n ̸= m then by Theorem 3.1 of [21] fn and fm are not topologically K -equivalent if n ̸= m. In particular, fn and fm are not bi-Lipschitz K -equivalent for n ̸= m, and therefore the fibers Xn and Xm have not the same bi-Lipschitz type. Hence, there is no bi-Lipschitz trivial stratification in this case. For more results toward the contact equivalence, see [50, 96]. Example 1.6.11. ([59] Example 1.5) This is a semialgebraic example due to Satoshi Koike, showing that (w)- regularity does not imply local bi-Lipschitz triviality. Let X = {y2 = z2 x2 + x3 , x ≥ 0} in R3 . Let Y be the z-axis. Then the pair (X −Y,Y ) is (w)-regular, but the bi-Lipschitz type of the germ of X at point (0, 0, 0) ∈ Y is different from the bi-Lipschitz type of the germ of X at points (0, 0, z) ∈ Y if z ̸= 0.

50


Figure 4 – {y = 0} ∪ {y = xz }, z ≥ 0

In fact, suppose that X0 = {y2 = x3 } and Xz = {y2 = z2 x2 +x3 } have the same bi-Lipschitz type, with z ̸= 0. In particular X0 and Xz are homeomorphic, hence X0 − {0} and Xz − {0} as well, which is a contradiction because these sets have 2 and 3 connected components, respectively.

Figure 5 – Cusp and Node

51

CHAPTER

2 THE DOUBLE OF A MODULE

In order to deal with bi-Lipschitz equisingularity in a family of hypersurfaces and to get algebraic infinitesimal Lipschitz conditions, Gaffney defined in [39, 40] the double of an ideal in a quite natural way. In this chapter we extend this notion for modules and we generalize several results of [39].

2.1

The double of a module and basic properties Let X ⊆ Cn be an analytic space. Consider the projection maps X ×X π1 X

π2 X

where πi is the projection onto the ith -factor, for every i ∈ {1, 2}. p Definition 2.1.1. Let h ∈ OX,x . The double of h is defined as the element 2p hD := (h ∘ π1 , h ∘ π2 ) ∈ OX×X,(x,x) . p The double of a submodule M ⊆ OX,x is denoted by MD , and is defined as the OX×X,(x,x) 2p submodule of OX×X,(x,x) generated by {hD | h ∈ M}.

Let M be a sheaf of OX -submodules of OXp . We can assume that M is finitely generated by global sections. If h ∈ OXp (X) is a global section, given an arbitrary point (x, x′ ) ∈ X × X, we 2p 2p can consider hD ∈ OX×X,(x,x ′ ) , the double of h, as the germ of (h ∘ π1 , h ∘ π2 ) ∈ OX×X (X × X) at the point (x, x′ ). Thus, we define the double of the sheaf M , denoted MD , as the OX×X 2p submodule of OX×X generated by {hD | h ∈ M (X) is a global section}. Unless we say the

52

Chapter 2. The Double of a Module

opposite, we denote by M the stalk of M at an arbitrary point of X and MD the stalk of MD in an arbitrary point of X × X. If we take the germ of h at a point, we use the same notation, if there is no confusion. For any sheaf of OX -modules F , the notation h ∈ F means that h is a global section of F . We want to recover some results which are true in the ideal case, i.e, when p = 1 and M = I is an ideal. We start by obtain a set of generators for MD from a set of generators of M. Consider z1 , ..., zn the coordinates on Cn . Lemma 2.1.2. Under the above notation, we have the following properties: a) (αh)D = −(0O p , (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) + (α ∘ π1 )hD , for all α ∈ OX and h ∈ OXp ; X×X

b) (0O p , (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) ∈ MD , for all h ∈ M and α ∈ OX ; X×X

c) α ∘ π1 − α ∘ π2 ∈ I(∆(X)) = (z1 ∘ π1 − z1 ∘ π2 , . . . , zn ∘ π1 − zn ∘ π2 ), for all α ∈ OX ; d) (g + h)D = gD + hD , for all g, h ∈ OXp ; e) (0O p , (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) ∈ MD at (x, x′ ), for all h ∈ Mx′ and α ∈ OX . The X×X analogous property holds if we look at the first p coordinates instead the last. Proof. (a) We have: (αh)D = ((αh) ∘ π1 , (αh) ∘ π2 ) = ((α ∘ π1 )(h ∘ π1 ), (α ∘ π2 )(h ∘ π2 )) = −(0O p , (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) + ((α ∘ π1 )(h ∘ π1 ), (α ∘ π1 )(h ∘ π2 )) X×X = −(0O p , (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) + (α ∘ π1 )hD . X×X

(0O p

X×X

(b) Since h ∈ M then αh ∈ M, so hD ∈ MD and (αh)D ∈ MD . Thus, by (a) we have that , (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) = (α ∘ π1 )hD − (αh)D ∈ MD . (c) Obviously α ∘ π1 − α ∘ π2 vanishes on the diagonal of X, which finishes the proof of

(c). (d) Notice that: (g + h)D = ((g + h) ∘ π1 , (g + h) ∘ π2 ) = (g ∘ π1 + h ∘ π1 , g ∘ π2 + h ∘ π2 ) = (g ∘ π1 , g ∘ π2 ) + (h ∘ π1 , h ∘ π2 ) = gD + hD . (e) Let φ : (C, 0) → (X × X, (x, x′ )) be an arbitrary analytic curve, φ = (φ1 , φ2 ). Since h ∈ Mx′ then we can write h ∘ φ2 = ∑ αl (vl ∘ φ2 ) l

with αl ∈ OC,0 and vl ∈ Mx′ , for all l. So we have (0, (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) ∘ φ = (0, (α ∘ φ1 − α ∘ φ2 )(h ∘ φ2 )) = ∑ αl (0, (α ∘φ1 −α ∘φ2 )(vl ∘φ2 )) = ∑ αl ((0, (α ∘ π1 − α ∘ π2 )(vl ∘ π2 )) ∘φ ) ∈ φ * (MD ). Hence, | {z } l l ∈MD

(0, (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) ∈ MD .

53

2.1. The double of a module and basic properties

The next proposition gives a set of generators of MD , from a known set of generators of M, which will be useful in the proof of many results in this section and in the section 2.2. Proposition 2.1.3. Suppose that M is generated by {h1 , . . . , hr }. Then, the following sets are generators of MD : a) B = {(h1 )D , . . . , (hr )D } ∪ {(0O p , (zi ∘ π1 − zi ∘ π2 )(h j ∘ π2 )) | i ∈ {1, . . . , n} and X×X j ∈ {1, . . . , r}}; b) B ′ = {(h1 )D , . . . , (hr )D } ∪ {((zi ∘ π1 − zi ∘ π2 )(h j ∘ π1 ), 0O p ) | i ∈ {1, . . . , n} and X×X j ∈ {1, . . . , r}}; c) B ′′ = {(h1 )D , . . . , (hr )D } ∪ {(zi h j )D | i ∈ {1, . . . , n} and j ∈ {1, . . . , r}}. 2p Proof. (a) Let N be the submodule of OX×X generated by B. By Lemma 2.1.2 (b) we have that N ⊆ MD . Now, in order to verify that MD ⊆ N, it is enough to check that hD ∈ N, ∀h ∈ M. Indeed, r

if h ∈ M we can write h = ∑ α j h j , for some α j ∈ OX . By Lemma 2.1.2 (a) and (d) we have that j=1 ! r

r

hD =

∑ α jh j j=1 r

=

= ∑ (α j h j )D j=1 D !

∑ (α j ∘ π1 )(h j )D − j=1

!

r

p , (α j ∘ π1 − α j ∘ π2 )(h j ∘ π2 )) . Clearly the first sum is ∑ (0OX×X

j=1

in N. By Lemma 2.1.2 (c) we have that each α j ∘ π1 − α j ∘ π2 belongs to the ideal I(∆(X)), so the second sum is in N, which finishes the proof. (b) This is completely analogous to the item (a). 2p (c) We use (a). Let N be the submodule of OX×X generated by B ′′ . For all j ∈ {1, ..., r} and i ∈ {1, ..., n} we have

(zi h j )D = (zi ∘ π1 )(h j )D − (0O p , (zi ∘ π1 − zi ∘ π2 )(h j ∘ π2 )) ∈ MD , X×X

by the previous lemma. Hence, N ⊆ MD . Now, in order to check that MD ⊆ N, it suffices to verify that all the generators of MD given in (a) are in N. We already have (h j )D ∈ N, for all j ∈ {1, ..., r}. Also, for all j and i we have (0O p , (zi ∘ π1 − zi ∘ π2 )(h j ∘ π2 )) = (zi ∘ π1 )(h j )D − (zi h j )D ∈ N X×X

which finishes the proof. We can develop the same notion for the double in the family case. Suppose that X ⊆ Cn+k is an analytic space and let Y = 0×Ck ⊆ X. Identifying Y = 0×Ck = Ck we have that X ⊆ Cn ×Y . Let p : X ⊆ Cn ×Y → Y be the projection, X × X the fibered product relative to the projection Y

onto Y , with the projections maps π1 , π2 : X × X → X. Y

54


X ×X

X ×X

Y

Y

π1

π2 X p Y

Let h ∈ OXp . The double of h relative to Y is defined by 2p hD,Y := hD := (h ∘ π1 , h ∘ π2 ) ∈ OX× X. Y

In the same way, the double of a sheaf of OX -submodules M of OXp relative to Y is defined as the 2p OX×X -submodule of OX× X generated by {hD,Y | h ∈ M (X) is a global section}, and is denoted Y

by MD (or MD,Y ).

Y

Let z1 , . . . , zn , y1 , . . . , yk be the coordinates on Cn+k . It is easy to see that the Lemma 2.1.2 still holds when we are working with the projections restricted to the fibered product X × X and, Y

since each yl ∘ π1 − yl ∘ π2 vanishes on the fibered product, then we get the following proposition, whose proof is completely analogous to the proof of the Proposition 2.1.3. Proposition 2.1.4. Suppose that M is generated by {h1 , . . . , hr }. Then, the following sets are generators of MD relative to Y : a) B = {(h1 )D , . . . , (hr )D } ∪ {(0O p , (zi ∘ π1 − zi ∘ π2 )(h j ∘ π2 )) | i ∈ {1, . . . , n} and X×X j ∈ {1, . . . , r}}; b) B ′ = {(h1 )D , . . . , (hr )D } ∪ {((zi ∘ π1 − zi ∘ π2 )(h j ∘ π1 ), 0O p ) | i ∈ {1, . . . , n} and X×X j ∈ {1, . . . , r}}; c) B ′′ = {(h1 )D , . . . , (hr )D } ∪ {(zi h j )D | i ∈ {1, . . . , n} and j ∈ {1, . . . , r}}. Example 2.1.5. Consider the ideal I = (x2 , y2 ) of OC2 ,0 . Notice that ID has generators further the doubles of x2 and y2 . In fact, by Proposition 2.1.3 the double ID is generated by B := {(x2 )D , (y2 )D , (0, (x − x′ )x′ ), (0, (x − x′ )y′ ), (0, (y − y′ )x′ ), (0, (y − y′ )y′ )}. It is easy to see that no element of B can be generated by the another elements of B, i.e, B is a minimal set of generators of ID . Hence, we need further elements than the doubles of x2 and y2 to generate the double of I. Remember that for any sheaf F of OX -modules, the support of F is defined as the subset of X given by supp(F ) := {x ∈ X | Fx ̸= 0}.

55


If G is a sheaf of OX -submodules of F , the cosupport of G in F is defined as F cosupp(G ) := supp G Let us to fix some notations. Let us assume that M is generated by global sections {h1 , . . . , hr } of OXp , let [M ] be the p × r matrix whose columns are the generators of M , and for each x ∈ X let [M (x)] be the p × r matrix whose columns are h1 (x), . . . , hr (x). Let Σ(M ) := {x ∈ X | rank[M (x)] < p}. This set is called the singular set of M . 2p In the next theorem we compute the cosupport of the double MD in OX×X .

Theorem 2.1.6. Let M be a sheaf of OX -submodules of OXp . Then cosupp(MD ) = ∆(X) ∪ (X × Σ(M )) ∪ (Σ(M ) × X). Proof. Let (x, x′ ) ∈ X × X and suppose that (x, x′ ) is not in the cosupp(MD ). (1) In a neighborhood of this point, by Proposition 2.1.3 we can write p p (1, 0, . . . , 0, 0O ) = ∑ αi (hi ∘ π1 , hi ∘ π2 ) + ∑ βi j ((zi ∘ π1 − zi ∘ π2 )(h j ∘ π1 ), 0O ) X×X

X×X

for some αi , βi j ∈ OX×X . If (x, x′ ) ∈ ∆(X), applying the above equation in this point, the second sum vanishes and we get (1, 0, . . . , 0, 0C p ) = ∑ αi (x, x)(hi (x), hi (x)). Notice that the right hand side of the above equation is in the diagonal of C p and the left hand side is not, which is a contradiction. So, (x, x′ ) ∈ / ∆(X). (2) Suppose by contradiction that (x, x′ ) ∈ Σ(M ) × X. Then rank[M (x)] < p, so there exists v ∈ C p such that v is not a linear combination of h1 (x), . . . , hr (x). Consider the element 2p (v, 0O p ) ∈ OX×X (thinking v as a constant function). Since (x, x′ ) is not in the cosupport of X×X MD then in a neighborhood of this point, by Proposition 2.1.3 we can write (v, 0O p ) = ∑ αi (hi ∘ π1 , hi ∘ π2 ) + ∑ βi j (0O p , (zi ∘ π1 − zi ∘ π2 )(h j ∘ π2 )) X×X

X×X

for some αi , βi j ∈ OX×X . Now, comparing the first p coordinates in above equation and applying in (x, x′ ) we have that v = ∑ αi (x, x′ )hi (x), so v is a linear combination of h1 (x), . . . , hr (x), which is a contradiction. Hence, (x, x′ ) ∈ / Σ(M ) × X. (3) Analogously we can conclude that (x, x′ ) ∈ / X × Σ(M ) by using again the Proposition 2.1.3 with the natural changes. The three above steps prove the inclusion cosupp(MD ) ⊇ ∆(X) ∪ (X × Σ(M )) ∪ (Σ(M ) × X).

56


Now, let (x, x′ ) ∈ X × X such that x ̸= x′ , [M (x)] and [M (x′ )] have rank ≥ p. For each z ∈ X denote by [M (z)] p the p × p submatrix given by the first p columns of [M (z)]. Since [M (x)] and [M (x′ )] have rank ≥ p, without loss of generality we can assume that [M (x)] p and [M (x′ )] p are invertibles. Thus, there exist U and U ′ open subsets of X containing x and x′ , respectively, such that [M (z)] p and [M (z′ )] p are invertibles, ∀(z, z′ ) ∈ U ×U ′ . Since x ̸= x′ then there exists k ∈ {1, . . . , n} such that (zk ∘ π1 − zk ∘ π2 )(x, x′ ) ̸= 0, so zk ∘ π1 − zk ∘ π2 is invertible in OX×X,(x,x′ ) . Let us to prove that (ei , 0O p ) ∈ MD at (x, x′ ), ∀i ∈ {1, . . . , p}. In fact, given a such X×X i, consider the analytic map α : U × U ′ → C p given by α(z, z′ ) := [M (z)]−1 p · ei , with α j its p

coordinates functions. So, it is easy to see that ∑ α j (z, z′ )h j (z) = ei , ∀(z, z′ ) ∈ U ×U ′ . Thus, in j=1 p

the open subset U × U ′ (which contains the point (x, x′ )) we have ei = ∑ α j (h j ∘ π1 ), which j=1

implies that p

(ei , 0O p ) = X×X

−1 p α (z ∘ π − z ∘ π ) (z ∘ π − z ∘ π )(h ∘ π ), 0 j 1 2 1 k k ∑ j k 1 k 2 OX×X {z } | j=1 ∈MD

⇒ (ei , 0O p ) ∈ MD . X×X

2p Analogously, (0O p , ei ) ∈ MD at (x, x′ ), ∀i ∈ {1, . . . , p}, hence MD = OX×X,(x,x ′ ) and X×X ′ (x, x ) ∈ / cosupp(MD ).

It is easy to see this theorem still holds in the family case, replacing the corresponding sets by X × Σ(M ) and Σ(M ) × X. Y

Y

Remark 2.1.7. Notice that if the singular set of M is empty then cosupp(MD ) = ∆(X) the diagonal of X and if the singular set of M is the whole X then cosupp(MD ) = X × X. Example 2.1.8. Let F : Cn → C p be an analytic map with n < p and X = F −1 (0). Let JM(X) be the jacobian module of X. In this case, the rank of JM(X) at any point of X is less than or equal to min{n, p} = n < p, then Σ(JM(X)) = X and cosupp((JM(X))D ) = X × X. Example 2.1.9. Let F : Cn → C p be an analytic map and suppose that F is a submersion. Thus X = F −1 (0) is a smooth submanifold of Cn of dimension n − p and the singular set of the jacobian module JM(X) is empty. Therefore, cosupp((JM(X))D ) = ∆(X). Example 2.1.10. Let us consider the cusp deformation X = F −1 (0) ⊆ C3 given by the equation F(x, y, z) = x3 + z2 x2 − y2 = 0. Thus the gradient of F at a point (x, y, z) is (3x2 + 2z2 x, −2y, 2zx2 ) and it is easy to see that the singular set of the jacobian module of X is the z-axis. Therefore, cosupp((JM(X))D ) = ∆(X) ∪ (X × (z-axis)) ∪ ((z-axis) × X). The next proposition generalizes the Corollary 3.4 of [39] for modules.

57


Proposition 2.1.11. Let M ⊆ N ⊆ M be OX -submodules of OXp , with X equidimensional. Suppose that MD has finite colenght in ND and ND has finite colenght in (M)D . Then e(MD , (M)D ) = e(ND , (M)D ) if and only if MD = ND . Proof. By the principle of additivity (see Theorem 6.7 of [62]), we have that e(MD , (M)D ) = e(MD , ND ) + e(ND , (M)D ). Notice that all these multiplicities are well-defined by hypothesis. So, e(MD , (M)D ) = e(ND , (M)D ) if and only if e(MD , ND ) = 0, which is equivalent to the equality MD = ND , since X is equidimensional. The following proposition and corollary will be useful later to make a relation between the saturation and the double of a module, and to work with the infinitesimal Lipschitz conditions. Proposition 2.1.12. Let h ∈ OXp . a) If hD ∈ MD at (x, x′ ) then h ∈ M at x and x′ ; b) If hD ∈ (MD )† at (x, x′ ) then h ∈ M † at x and x′ . The same result still holds in the family case. Proof. (a) Let us to prove that h ∈ M at x (the case at x′ is completely analogous). Let φ : (C, 0) → (X, x) be an arbitrary analytic curve. Define γ : (C, 0) → (X × X, (x, x′ )) given by γ(t) = (φ (t), x′ ). Since hD ∈ MD then hD ∘ γ ∈ MD ∘ γ, so we can write hD ∘ γ = ∑ α j ((g j )D ∘ γ) j

with g j ∈ M and α j ∈ OC,0 . Since π1 ∘ γ = φ , comparing the first p coordinates of the above equation, we get h ∘ φ = ∑ α j (g j ∘ φ ) ∈ M ∘ φ . Therefore, h ∈ M at x. j

(b) Let t be the coordinate on C. Let us to prove that h ∈ M † at x (the case at x′ is completely analogous). Let φ : (C, 0) → (X, x) be an arbitrary analytic curve. Define γ : (C, 0) → (X × X, (x, x′ )) given by γ(t) = (φ (t), x′ ). Since hD ∈ (MD )† then hD ∘ γ ∈ m1 (MD ∘ γ), so we can write hD ∘ γ = ∑ tα j ((g j )D ∘ γ) j

with g j ∈ M and α j ∈ OC,0 . Since π1 ∘ γ = φ , comparing the first p coordinates of the above equation, we get h ∘ φ = ∑ tα j (g j ∘ φ ) ∈ m1 (M ∘ φ ). Therefore, h ∈ M † at x. j

The proof in the family case is quite analogous, working on the fibered product X × X. Y

58


Example 2.1.13. As in Example 1.4.4, consider F(x, y) = x2 + y p , p ≥ 3 odd. Denote by X the q plane curve defined by F. We have seen that if we take q = p+1 2 then y ∈ J( f ) and (yq )D ∈ / (J( f ))D . Hence, the converse of the Proposition 2.1.12 (a) is not true in general. Despite the converse of the Proposition 2.1.12 (a) is not true in general, the following proposition gives a condition such that the converse holds. p Proposition 2.1.14. Let M be an Ox,x -submodule of OX,x and suppose that I∆⊕p ⊕ I∆⊕p ⊆ MD .

If h ∈ M then: a) (h ∘ π1 , h ∘ π1 ) ∈ MD ; b) (h ∘ π2 , h ∘ π2 ) ∈ MD ; c) hD = (h ∘ π1 , h ∘ π2 ) ∈ MD ; d) (h ∘ π2 , h ∘ π1 ) ∈ MD . The same result still holds in the family case. Proof. (a) Assume first that h ∈ M. We can write (h ∘ π1 , h ∘ π1 ) = hD + (0, h ∘ π1 − h ∘ π2 ). In this case we have that hD ∈ MD ⊆ MD and by hypothesis (0, h ∘ π1 − h ∘ π2 ) ∈ 0 ⊕ I∆⊕p ⊆ MD . Hence, (h ∘ π1 , h ∘ π1 ) ∈ MD . Now, let us go back to the general case. Suppose that h ∈ M and take an arbitrary analytic curve φ : (C, 0) → (X × X, (x, x)) written as φ = (φ1 , φ2 ). Since h ∈ M then we can write h ∘ φ1 = ∑ αi (gi ∘ φ1 ) with gi ∈ M and αi ∈ OC,0 , for all i. By what we have proved before, we i

know that (gi ∘ π1 , gi ∘ π1 ) ∈ MD for alli. Thus, (h ∘ π1 , h ∘ π 1 ) ∘ φ = (h ∘ φ1 , h ∘ φ1 )   = (∑ αi (gi ∘ φ1 ), ∑ αi (gi ∘ φ1 )) = ∑ αi (gi ∘ π1 , gi ∘ π1 ) ∘φ  ∈ φ * (MD ). Hence, | {z } i i i ∈MD

(h ∘ π1 , h ∘ π1 ) ∈ MD = MD . (b) The proof is analogous to the item (a) using the fact that I∆⊕p ⊕ 0 ⊆ MD . (c) We can write hD = (h ∘ π1 , h ∘ π1 ) + (0, h ∘ π2 − h ∘ π1 ). By the item (a) we already know that (h ∘ π1 , h ∘ π1 ) ∈ MD and since (0, h ∘ π2 − h ∘ π1 ) ∈ 0 ⊕ I∆⊕p ⊆ MD then hD ∈ MD . (d) Notice that (h ∘ π2 , h ∘ π1 ) = (h ∘ π2 − h ∘ π1 , 0) + (h ∘ π1 , h ∘ π1 ). By the item (a) we already know that (h ∘ π1 , h ∘ π1 ) ∈ MD and since (h ∘ π2 − h ∘ π1 , 0) ∈ I∆⊕p ⊕ 0 ⊆ MD then (h ∘ π2 , h ∘ π1 ) ∈ MD .

59


Corollary 2.1.15. Let M and N be sheaves of OX -submodules of OXp . a) If MD ⊆ ND at (x, x′ ) then M ⊆ N at x and x′ ; b) If MD ⊆ (ND )† at (x, x′ ) then M ⊆ N

†

at x and x′ .

The same result still holds in the family case. Proof. (a) Suppose that h ∈ M at x and x′ . So, hD ∈ MD at (x, x′ ), and since MD ⊆ ND at (x, x′ ) then hD ∈ ND at (x, x′ ). By Proposition 2.1.12 (a) we conclude that h ∈ N at x and x′ . (b) Suppose that h ∈ M at x and x′ . So, hD ∈ MD at (x, x′ ), and since MD ⊆ (ND )† at (x, x′ ) then hD ∈ (ND )† at (x, x′ ). By Proposition 2.1.12 (b) we conclude that h ∈ N † at x and x′ . Looking to the Example 2.1.13, we conclude that the converse of the Corollary 2.1.15 (a) also is not true in general. In the next corollary, we prove that the double of a module is contained in the integral closure of the double of another module, provided the doubles of the generators are. p . Suppose that M is generated Corollary 2.1.16. Let M and N be OX,x -submodules of OX,x 2p ′ by {g1 , ..., gr } and let MD be the submodule of OX×X,(x,x) generated by {(g1 )D , ..., (gr )D }. If MD′ ⊆ ND then MD ⊆ ND . The same result still holds in the family case.

Proof. Let us prove that (0, (zi ∘π1 −zi ∘π2 )(g j ∘π2 )) ∈ ND , for all i ∈ {1, ..., n} and j ∈ {1, ..., r}. In fact, let i and j be arbitrary. Since (g j )D ∈ MD′ ⊆ ND then by Proposition 2.1.12 (a) we have that g j ∈ N. By Lemma 2.1.2 (e) we conclude that (0, (zi ∘ π1 − zi ∘ π2 )(g j ∘ π2 )) ∈ ND . Therefore, all the generators of MD belong to ND and so MD ⊆ ND . In the next theorem we compute the generic rank of the double of a module. Theorem 2.1.17. Let (X, x) be an irreducible analytic complex germ of dimension d ≥ 1 and p M ⊆ OX,x a submodule of generic rank k. Then MD has generic rank 2k at (x, x). The same result still holds in the family case. 

 | |   Proof. Let {g1 , ..., gr } be a set of generators of M. We can write [M] = g1 . . . gr  = | |   −− v1 −−   ..  , where v1 , ..., v p ∈ O r are the rows of [M]. Since M has generic rank k then . X,x   −− v p −−

60


we can choose k rows of [M] which are OX,x -linearly independent. Without loss of generality, we can suppose that v1 , ..., vk are linearly independent. For each l ∈ {1, ..., p}, let n.r ) wl := (vl ∘ π1 , 0OX,x

and w¯ l := (vl ∘ π2 , (z1 ∘ π1 − z1 ∘ π2 )(vl ∘ π2 ), ..., (zn ∘ π1 − zn ∘ π2 )(vl ∘ π2 )). Thus, w1 , ..., w p , w¯ 1 , ..., w¯ p are the rows of the matrix [MD ] given by the generators of MD induced by the generators of M. ¯ 1 , ..., w ¯ k } is OX×X,(x,x) -linearly independent. Claim 1: {w1 , ..., wk , w In fact, let α1 , ..., αk , β1 , ..., βk ∈ OX×X,(x,x) be arbitrary such that k

k

∑ αl wl + ∑ βl w¯ l = 0O r+n.r l=1

l=1

.

X×X,(x,x)

Then, we get two equations: k

k

r ∑ αl (vl ∘ π1) + ∑ βl (vl ∘ π2) = 0OX×X,(x,x)

(1)

n.r ∑ βl ((z1 ∘ π1 − z1 ∘ π2)(vl ∘ π2), ..., (zn ∘ π1 − zn ∘ π2)(vl ∘ π2)) = 0OX×X,(x,x)

(2)

l=1

l=1

k l=1

The equality (2) implies that k r (zi ∘ π1 − zi ∘ π2 ) ∑ βl (vl ∘ π2 ) = 0OX×X,(x,x)

(3)

l=1

for all i ∈ {1, ..., n}. Since d ≥ 1 then X is not a point. If zi ∘ π1 − zi ∘ π2 = 0OX×X,(x,x) for all i then X × X ⊆ V (I(∆)) = ∆ which implies X × X = ∆, so X is a point, contradiction. So zi ∘ π1 − zi ∘ π2 ̸= 0OX×X,(x,x) , for some i. Since (X, x) is irreducible then (X × X, (x, x)) is irreducible, so OX×X,(x,x) is a domain, and by the equation (3) we have k r . ∑ βl (vl ∘ π2) = 0OX×X,(x,x)

(4)

l=1

We can find an open subset U of X, with x ∈ U, such that all v1 , ..., vk are defined on U, and all β1 , ..., βk are defined on U ×U. For each z ∈ U define βlz : U −→ C w ↦−→ βl (z, w) Then, the germ βlz ∈ OX,x , ∀z ∈ U and l ∈ {1, ..., k}.

61


Let us prove that βlz = 0OX,x , ∀z ∈ U and l ∈ {1, ..., k}. In fact, given z and l, by the equation (4), for all w ∈ U we have k

k

0Cr = ( ∑ βl (vl ∘ π2 ))(z, w) = l=1

k

∑ βl (z, w)vl (w) = ( ∑ βlzvl )(w) l=1

l=1

k

z r . Since v1 , ..., vk are OX,x -linearly independent then β = 0O . which implies ∑ βlz vl = 0OX,x X,x l l=1

Hence, βl = 0OX×X,(x,x) , for all l ∈ {1, ..., k}. So, the equation (1) becomes k r ∑ αl (vl ∘ π1) = 0OX×X,(x,x)

l=1

which is similar to the equation (4). Using the same argument, we conclude that αl = 0OX×X,(x,x) , for all l ∈ {1, ..., k}. Thus, the Claim 1 is proved. Hence, the generic rank of MD is ≥ 2k. Claim 2: It does not exist more than k rows OX×X,(x,x) -linearly independent on the first p rows of [MD ]. In fact, suppose that there exist. So, we have at least k + 1 rows linearly independent on the first p rows of [MD ]. Without loss of generality, we can assume that the first k + 1 rows satisfy it. Let us prove that {v1 , ..., vk+1 } is OX,x -linearly independent. Indeed, let α1 , ..., αk+1 ∈ OX,x k+1

r . Take λl := αl ∘ π1 , ∀l ∈ {1, ..., k + 1}. For all (z, w) in a small enough such that ∑ αl vl = 0OX,x

l=1

neighborhood of (x, x) we have k+1

k+1

( ∑ λl wl )(z, w) = (( ∑ αl vl )(z), 0Cn.r ) = 0Cr+n.r l=1

l=1

k+1

which implies that ∑ λl wl = 0O r+n.r

X×X,(x,x)

l=1

. Since {wl }k+1 l=1 is linearly independent then λl =

0OX×X,(x,x) , for all l ∈ {1, ..., k + 1}. Thus, for all z in a small neighborhood of x we have 0 = λl (z, z) = αl ∘ π1 (z, z) = αl (z), hence αl = 0OX,x , ∀l ∈ {1, ..., k + 1}. Then, v1 , ..., vk+1 is linearly independent and the generic rank of M is ≥ k + 1, contradiction. Therefore, the Claim 2 is proved. Claim 3: It does not exist more than k rows OX×X,(x,x) -linearly independent on the last p rows of [MD ]. In fact, suppose that there exist. So, we have at least k + 1 rows linearly independent on the last p rows of [MD ]. Without loss of generality, we can assume that w¯ 1 , ..., w¯ k+1 are linearly independent. Let us prove that {v1 , ..., vk+1 } is OX,x -linearly independent. Indeed, let k+1

r . Take λl := αl ∘ π2 , ∀l ∈ {1, ..., k + 1}. For all (z, w) α1 , ..., αk+1 ∈ OX,x such that ∑ αl vl = 0OX,x

l=1

in a small enough neighborhood of (x, x) we have k+1

k+1

k+1

k+1

( ∑ λl w¯ l )(z, w) = (( ∑ αl vl )(w), (z1 −w1 )(( ∑ αl vl )(w)), ..., (zn −wn )(( ∑ αl vl )(w))) = 0Cr+n.r . l=1

l=1

l=1

l=1

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Since {w¯ l }k+1 l=1 is linearly independent then λl = 0OX×X,(x,x) , for all l ∈ {1, ..., k +1}, and αl = 0OX,x , ∀l ∈ {1, ..., k + 1}. Then, v1 , ..., vk+1 is linearly independent and the generic rank of M is ≥ k + 1, contradiction. Therefore, the Claim 3 is proved. Finally, suppose by contradiction that the generic rank of MD is ≥ 2k + 1. Then we have 2k + 1 rows of [MD ] which are OX×X,(x,x) -linearly independent. By the Claim 2, we have at most k of these rows on the first p rows of [MD ]. Then, we have at least k + 1 of these rows on the last p rows, which contradicts the Claim 3. Therefore, the generic rank of MD is 2k. The proof for the family case is completely analogous working on the fibered product X × X. Y

Corollary 2.1.18. Let {Vi } be the irreducible components of (X, x). For each i, if M has generic rank ki on Vi then MD has generic rank 2ki on Vi ×Vi . p of generic rank k, and suppose that Remark 2.1.19. Let M be an OX,x -submodule of OX,x dim X = 0. Then X is a point and OX,x ∼ = C. In this case, M is a k-dimensional C-vector p subspace of C . Notice that, since X × X is also a point, then π1 = π2 . Hence, the map

M ⊆ C p −→ MD ⊆ C2p h ↦−→ (h, h) is an isomorphism of C-vector spaces which implies that dimC MD = k, i.e, MD has generic rank k. In next proposition we prove that the integral closure of modules commutes with finite direct sum of modules. q

p Proposition 2.1.20. Let M ⊆ OX,x and N ⊆ OX,x be OX,x -submodules. Then

M ⊕ N = M ⊕ N. Proof. First we prove that M ⊕ 0 ⊆ M ⊕ N. Let v ∈ M ⊕ 0. Then we can write v = (g, 0) with g ∈ M. Let φ : (C, 0) → (X, x) be an arbitrary analytic curve. Since g ∈ M then we can write g ∘ φ = ∑ αi (gi ∘ φ ), with gi ∈ M and αi ∈ OC,0 , for all i. Thus, v ∘ φ = (g ∘ φ , 0) i

= (∑ αi (gi ∘φ ), 0) = ∑ αi (gi ∘φ , 0) = ∑ αi φ * (gi , 0) ∈ φ * (M ⊕N). Hence, v ∈ M ⊕ N and M ⊕0 ⊆ i

i

i

M ⊕ N. Analogously, 0 ⊕ N ⊆ M ⊕ N. Therefore, M ⊕ N ⊆ M ⊕ N. q

p Conversely, let v ∈ M ⊕ N. We can write v = (g, h) with g ∈ OX,x and h ∈ OX,x . Let φ : (C, 0) → (X, x) be an arbitrary analytic curve. Then we can write φ * (v) = ∑ αi φ * (gi , hi ) i

where gi ∈ M, hi ∈ N and αi ∈ OC,0 , for all i. Comparing the coordinates of the last equation we conclude that φ * (g) = ∑ αi φ * (gi ) and φ * (h) = ∑ αi φ * (hi ). Thus, g ∈ M and h ∈ N. Therefore, i

v ∈ M ⊕ N.

i

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2.1. The double of a module and basic properties pi Corollary 2.1.21. Let Mi ⊆ OX,x be OX,x -submodules, i ∈ {1, ..., r}. Then r M

Mi =

i=1

r M

Mi .

i=1

Proof. Induction on r and apply the Proposition 2.1.20. Let us recall some basic results in commutative algebra. Remark 2.1.22. Let (R, m) be a local ring and consider the residue field k := mR . a) If M is a free R-module of rank p then

M mM

is a k-vector space of dimension p;

b) If M is a free R-submodule of R p of rank p then M = R p . Proof. (a) By hypothesis there exists an R-basis {v1 , ...v p } of M. Clearly the images of these M M elements in mM form a generator set of mM . So, it remains to prove that these elements are R-linearly independent. Suppose we have an equation p

∑ (αi + m)(vi + mM) = 0 + mM

i=1

p

p

with αi ∈ R, for all i ∈ {1, ..., p}. Then ∑ αi vi ∈ mM which implies that we can write ∑ αi vi = p

i=1

i=1

∑ βi vi , for some β1 , ..., β p ∈ m. Since {v1 , ..., v p } is R-linearly independent then αi = βi ∈ m, i=1

for all i ∈ {1, ..., p}. Hence, αi + m = 0 + m, for all i ∈ {1, ..., p}. (b) By hypothesis the submodule M has an R-basis {v1 , ..., v p }. Thus, {v1 , ..., v p } is an R-linearly independent subset of R p . By the proof of the item (a) we have that Rp {v1 + mR p , ..., v p + mR p } is a k-linearly independent subset of mR p . By the item (a) we have Rp Rp p p dimk mR p = p, hence {v1 + mR , ..., v p + mR } is a k-basis of mR p . By Nakayama’s Lemma {v1 , ..., v p } is a generator set of R p and we conclude that M = R p . p p p Remark 2.1.23. Let M be an OX,x -submodule of OX,x . If M = OX,x then M = OX,x .

In fact, let s be the generic rank of M. Take an analytic curve φ : (C, 0) → (X, x) such p p that the rank of φ * (M) is generically s. Since M = OX,x then φ * (M) = φ * (OX,x ) which has rank p, hence s = p. Thus, M has generic rank p and there exists a free OX,x -submodule M p of M of p p rank p. By Remark 2.1.22 (b) we have that M p = OX,x . Since M p ⊆ M then M = OX,x . p p Therefore, in order to verify the equation M = OX,x it suffices to check that M = OX,x , which may be easier sometimes.

In the next proposition we calculate the stalk of the double sheaf MD of a sheaf of modules M in a pair of points of X such that one of them is not a singular point of M , i.e, the matrix [M ] in one of these points has maximal rank p.

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Proposition 2.1.24. Let M ⊆ OXp be a sheaf of submodules. Let (x, x′ ) ∈ X × X with x ̸= x′ . p p ′ then MD = OX×X,(x,x a) If Mx = OX,x ′ ) ⊕ (Mx′ ∘ π2 ) at (x, x ); p p ′ b) If Mx′ = OX,x ′ then MD = (Mx ∘ π1 ) ⊕ OX×X,(x,x′ ) at (x, x ).

The same result still holds in the family case, working on the fibered product X × X. Y

Proof. We may assume that M is generated by global sections {g1 , ..., gr }. Since x ̸= x′ then (zl ∘ π1 − zl ∘ π2 )(x, x′ ) ̸= 0, for some l ∈ {1, ..., n}, thus zl ∘ π1 − zl ∘ π2 is an invertible element of OX×X,(x,x′ ) . (a) Let us prove that (0, h ∘ π2 ) ∈ MD at (x, x′ ), ∀h ∈ Mx′ . In fact, let φ : (C, 0) → (X × X, (x, x′ )) be an arbitrary analytic curve. Write φ = (φ1 , φ2 ). Since h ∈ Mx′ then we can write h ∘ φ2 = ∑ αi (gi ∘ φ2 ) with αi ∈ OC,0 , for all i. Since (0, (zl ∘ π1 − zl ∘ π2 )(gi ∘ π2 )) ∈ MD i

at (x, x′ ), for all i, and zl ∘ π1 − zl ∘ π2 is invertible then (0, gi ∘ π2 ) ∈ MD at (x, x′ ), for all i. Hence, (0, h∘π2 )∘φ = (0, h∘φ2 ) = ∑ αi (0, gi ∘φ2 ) = ∑ αi ((0, gi ∘π2 )∘φ ) ∈ φ * (MD ). Therefore, i

i

(0, h ∘ π2 ) ∈ MD at (x, x′ ) and we conclude that 0 ⊕ (Mx′ ∘ π2 ) ⊆ MD at (x, x′ ). Let e1 , ..., e p be the canonical basis elements of OXp . Since all these elements belong to M at x then ((zl ∘π1 −zl ∘π2 )(e j ∘π1 ), 0) ∈ MD at (x, x′ ), ∀ j ∈ {1, ..., p}. The element zl ∘π1 −zl ∘π2 p is invertible, hence (e j ∘π1 , 0) ∈ MD at (x, x′ ), ∀ j ∈ {1, ..., p}. Thus, OX×X,(x,x ′ ) ⊕0 ⊆ MD ⊆ MD p ′ ′ at (x, x ), and OX×X,(x,x′ ) ⊕ (Mx′ ∘ π2 ) ⊆ MD at (x, x ). p p ′ Clearly MD ⊆ OX×X,(x,x ′ ) ⊕ (Mx′ ∘ π2 ) ⊆ OX×X,(x,x′ ) ⊕ (Mx′ ∘ π2 ) at (x, x ). Therefore, p MD ⊆ OX×X,(x,x ′ ) ⊕ (Mx′ ∘ π2 ) ⊆ MD p at (x, x′ ) and using the Proposition 2.1.20 we conclude that MD = OX×X,(x,x ′ ) ⊕ (Mx′ ∘ π2 ) = p p ′ OX×X,(x,x ′ ) ⊕ (Mx′ ∘ π2 ) = OX×X,(x,x′ ) ⊕ (Mx′ ∘ π2 ) at (x, x ).

(b) It is quite analogous to the item (a). Corollary 2.1.25. Let M ⊆ OXp be a sheaf of submodules. Let (x, x′ ) ∈ X × X with x ̸= x′ . If p p Mx = OX,x or Mx′ = OX,x ′ then (M )D = MD at (x, x′ ). The same result still holds in the family case, working on the fibered product X × X. Y p Proof. Assume that Mx = OX,x (the another case is analogous). By Proposition 2.1.24 (a) we have that p MD = OX×X,(x,x ′ ) ⊕ (Mx′ ∘ π2 )

65

2.1. The double of a module and basic properties p at (x, x′ ). Since Mx = OX,x then applying again the Proposition 2.1.24 (a) we get p (M )D = OX×X,(x,x ′ ) ⊕ (Mx′ ∘ π2 )

at (x, x′ ). We already know that Mx′ = Mx′ . Therefore (M )D = MD at (x, x′ ). Now we calculate the stalk of the double sheaf MD at a point of the diagonal of X whose corresponding point in X is a non-singular point of M . p Proposition 2.1.26. Let M ⊆ OXp be a sheaf of submodules, x ∈ X such that Mx = OX,x . Then:

a) MD = I∆⊕p ⊕ I∆⊕p + E at (x, x), where E is the submodule generated by {(ei )D | i ∈ {1, ..., p}}, where e1 , ..., e p are the canonical basis elements of OXp ; b) (M )D = MD at (x, x). The same result still holds in the family case, working on the fibered product X × X. Y

Proof. For each i ∈ {1, ..., p} we have that ei ∈ Mx , which implies that ((z j ∘ π1 − z j ∘ π2 )(ei ∘ π1 ), 0), (0, (z j ∘ π1 − z j ∘ π2 )(ei ∘ π2 )) ∈ MD at (x, x), for all j ∈ {1, ..., n} and i ∈ {1, ..., p}. Hence, I∆⊕p ⊕I∆⊕p ⊆ MD at (x, x). Since E ⊆ MD at (x, x) then I∆⊕p ⊕I∆⊕p +E ⊆ MD at (x, x). Notice that (0, (z j ∘ π1 − z j ∘ π2 )(ei ∘ π2 )) ∈ I∆⊕p ⊕ I∆⊕p , for all j ∈ {1, ..., n} and i ∈ {1, ..., p}. Thus, all the generators of MD at (x, x) belong to I∆⊕p ⊕ I∆⊕p + E, hence MD ⊆ I∆⊕p ⊕ I∆⊕p + E at (x, x). Therefore, MD = I∆⊕p ⊕ I∆⊕p + E at (x, x). p (b) Since Mx = OX,x then Mx = Mx at x, which implies that (M )D = MD at (x, x), and

taking the integral closure we conclude that (M )D = MD at (x, x). We use the previous results to obtain an algebraic invariant associated to an analytic variety with isolated singularity. Let F : (Cn , 0) → (C p , 0) be an analytic map, n ≥ p such that X = F −1 (0) has an isolated singularity at the origin, d = dim X and JM(F) the jacobian module defined on X. In 2p general the submodule (JM(F))D does not have finite colength in OX×X , even if X is an isolated hypersurface singularity. Nevertheless, the multiplicity of the pair offers a way around this. The module (JM(F))D has a simple description, as we will see, off the origin in its cosupport, and any integral closure condition we wish to use is easily checked because of this structure. In the notation of [43], let H2d−1 ((JM(F))D ) be the largest sheaf of modules whose integral closure

66


agrees with (JM(F))D off the origin. This is the integral hull of (JM(F))D of codimension 2d − 1, which means the integral closure of (JM(F))D and H2d−1 ((JM(F))D ) agree off a set of codimension 2d, i.e, off (0, 0) in X d × X d . The next result identifies H2d−1 ((JM(F))D ) and is a generalization of the Lemma 3.3 of [39]. Corollary 2.1.27. Let F : (Cn , 0) → (C p , 0) be an analytic map with n ≥ p and suppose that X = F −1 (0) has isolated singularity at the origin. Let JM(F) be the jacobian module defined in X. Then the multiplicity of the pair of modules e((JM(F))D , (JM(F))D ) is well defined at the origin and if X is an ICIS then H2d−1 ((JM(F))D ) = JM(F) . D

Proof. By Corollary 2.1.25 and Proposition 2.1.26 (b) we have that (JM(F))D = (JM(F))D off the origin. As a corollary of the proof of the last result, we have that H2d−1 (MD ) = (M )D for any sheaf of submodules M ⊆ OXp of finite colength in OXp .

2.2

The infinitesimal Lipschitz conditions iLA and iLmY

Now we use some of the results presented in last section to recover some properties about the infinitesimal Lipschitz conditions for the following more general setup. Setup 2.2.1. Let (X, 0) ⊆ (Cn+k , 0) be the germ of the analytic space defined by an analytic map F : Cn × Ck → C p , where X is a sufficient small representative such that Y = Ck = 0 × Ck ⊆ X is the singular set of X. Let F1 , . . . , Fp : Cn × Ck → C be the coordinates functions of F, for each y ∈ Y let fy : Cn → C p given by fy (z) := F(z, y) and let Xy := fy−1 (0). Assume that Xy has an isolated singularity in 0, for all y ∈ Y and n ≥ p. Let z1 , . . . , zn , y1 , . . . , yk be the coordinates on Cn+k , let mY be the ideal of OX generated by {z1 , . . . , zn }, let JM(X) be the jacobian module of X, let JM(X)Y be the module generated by { ∂∂yF , . . . , ∂∂ yF } and let JMz (X) be the module 1 k generated by { ∂∂ zF , . . . , ∂∂ zFn }. 1

In this section we work with the doubles relative to Y and with the projections X ×X Y

π1 X

π2 X

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2.2. The infinitesimal Lipschitz conditions iLA and iLmY

Since Xy has an isolated singularity in 0, for all y ∈ Y , then Σ(JMz (X)) = {(0, y) | y ∈ Y } = Y , so by Theorem 2.1.6 we have that cosupp((JMz (X))D ) = ∆(X) ∪ (X × 0) ∪ (0 × X). Y

Y

Definition 2.2.2. Under the above setup, we introduce the infinitesimal Lipschitz conditions. ∙ The pair (X,Y ) satisfies the infinitesimal Lipschitz condition mY (iLmY ) at (x, x′ ) ∈ X × X Y

if (JM(X)Y )D ⊆ (mY JMz (X))D at

(x, x′ );

∙ The pair (X,Y ) satisfies the infinitesimal Lipschitz condition A (iLA ) at (x, x′ ) ∈ X × X Y

if (JM(X)Y )D ⊆ (JMz (X))D at (x, x′ ). Notice that iLmY implies iLA . By Theorem 2.5 of [41], the pair (X,Y ) satisfies the Verdier (w)-regularity at x ∈ X if and only if JM(X)Y ⊆ mY JMz (X) at x. Thus, the iLmY condition is the Lipschitz version of the W condition. The iLA condition is the Lipschitz version of the AF condition, which is JM(X)Y ⊆ JMz (X), if this holds on the ambient space. Example 2.2.3. (Family of cusps) Consider X ⊆ C3 given by the equation F(x, y, z) = z2 −x3 = 0. Since the gradient of F at a point (x, y, z) is (−3x2 , 2z, 0) then the singular locus of X is the y-axis. In this case, it is easy to see that JM(X)Y = {0} and therefore both infinitesimal Lipschitz conditions are trivially satisfied at any point of X × X. Y

Figure 6 – Family of Cusps

In general, if X is defined by an analytic map which does not depend of the parameter space then JM(X)Y = {0} and both infinitesimal Lipschitz conditions are trivially satisfied. Example 2.2.4. Consider the analytic map f (x, y) = x2 + y3 . Taking q := 3+1 2 = 2 we have seen in Example 1.4.4 that y2 ∈ J( f ) but (y2 )D ∈ / (J( f ))D . In this case, J( f )Y is generated by y2 . Then, (J( f )Y )D * (J( f ))D . Hence, neither of the infinitesimal Lipschitz conditions are satisfied.

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We want to see if the condition iLmY at a point of (0, 0) ×Y depends of the projection p : X → Y . The next lemma will be useful to answer this question. Lemma 2.2.5. Consider the setup 2.2.1. For any parameter y ∈ Y we have: a) If (JM(X)Y )D ⊆ (mY JM(X))D at (0, y, 0, y) ∈ (0, 0) ×Y then JM(X)Y ⊆ mY JMz (X) at (0, y), i.e, the W condition holds at (0, y); b) If (JM(X)Y )D ⊆ ((mY JM(X))D )† at (z, y, z, y) ∈ X × X then Y

JM(X)Y ⊆ (mY JMz (X))† at (z, y). Proof. (a) Let φ : (C, 0) → (X, (0, y)) be an arbitrary analytic curve. By the hypothesis and Corollary 2.1.15 (a) we have that JM(X)Y ⊆ mY JM(X), hence φ * (JM(X)Y ) ⊆ φ * (mY JM(X)). Now, since φ * (mY ) ⊆ m1 , JM(X) = JM(X)Y + JMz (X) and φ * preserves the module operations then: φ * (JM(X)Y ) ⊆ φ * (mY JM(X))) = φ * (mY JM(X)Y + mY JMz (X)) ⊆ m1 φ * (JM(X)Y ) + φ * (mY JMz (X)). By Nakayama’s Lemma we conclude that φ * (JM(X)Y ) ⊆ φ * (mY JMz (X))). Therefore, (a) is proved by the curve criterion. (b) Let φ : (C, 0) → (X, (z, y)) be an arbitrary analytic curve. By the hypothesis and Corollary 2.1.15 (b) we have that JM(X)Y ⊆ (mY JM(X))† , hence φ * (JM(X)Y ) ⊆ m1 φ * (mY JM(X)), and furthermore, φ * (JM(X)Y ) ⊆ m1 φ * (mY JM(X)) = m1 φ * (mY JM(X)Y ) + m1 φ * (mY JMz (X)) ⊆ m1 φ * (JM(X)Y ) + (m1 φ * (mY JMz (X))). By Nakayama’s Lemma we have that φ * (JM(X)Y ) ⊆ m1 φ * (mY JMz (X)). Therefore, (b) is proved. The next result gives us a kind of “independence of the projection onto Y ” of the iLmY condition at the origin . Proposition 2.2.6. Consider the setup 2.2.1. At any point of (0, 0)×Y , we have that (JM(X)Y )D ⊆ (mY JM(X))D if and only if (JM(X)Y )D ⊆ (mY JMz (X))D . Proof. The implication (⇐=) is obvious. Let us to prove (=⇒). Let φ : (C, 0) → (X × X, (0, y, 0, y)) be an arbitrary analytic curve. Write φ = (φ1 , φ2 ), Y

with φ1 , φ2 : (C, 0) → (X, (0, y)). Claim 1: ((zi ∘ φ1 − zi ∘ φ2 )( ∂∂ yF ∘ φ1 ), 0) ∈ φ * ((mY JMz (X))D ), for all i ∈ {1, ..., n} and l l ∈ {1, ..., k}.

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In fact, by Lemma 2.2.5 (a) we have (JM(X))Y ⊆ (mY JMz (X)), so ∈ (mY JMz (X)) =⇒ ∂∂ yF ∘ φ1 ∈ φ1* (mY JMz (X)) and we can write l ∂F ∂F ∘ φ = β ((z ) ∘ φ ), with βr j ∈ OC,0 . Then, ∑ r j r 1 1 ∂y ∂zj ∂F ∂ yl

l

r, j

((zi ∘ φ1 − zi ∘ φ2 )( ∂∂ yF ∘ φ1 ), 0) = ∑ βr j ((zi ∘ φ1 − zi ∘ φ2 )((zr ∂∂ zFj ) ∘ φ1 ), 0) l

r, j

∂F = ∑ βr j φ * (((zi ∘ π1 − zi ∘ π2 )((zr ) ∘ π1 ), 0)) ∈ φ * ((mY JMz (X))D ). So the Claim 1 is proved. ∂zj r, j | {z } ∈φ * ((mY JMz (X))D )

Claim 2: φ * ((mY JM(X)Y )D ) ⊆ m1 φ * ((mY JM(X))D ) + φ * ((mY JMz (X))D ). In fact, it is enough to look to the images of the generators of (mY JM(X)Y )D . For all i, j ∈ {1, ..., n} and l ∈ {1, ..., k} we have φ * ((zi

∂F ∂F ∂F ∂F )D ) = (zi ∘ φ2 )( ∘ φ1 , ∘ φ2 ) + ((zi ∘ φ1 − zi ∘ φ2 )( ∘ φ1 ), 0). ∂ yl ∂ yl ∂ yl ∂ yl

We have that zi ∘ φ2 ∈ m1 . By hypothesis we have φ * ((JM(X)Y )D ) ⊆ φ * ((mY JM(X))D ), so φ * (( ∂∂ yF )D ) ∈ φ * ((mY JM(X))D ). Thus, the first term on the right hand side of the above sum l is in m1 φ * ((mY JM(X))D ). The second term on the right hand side of the above sum is in φ * ((mY JMz (X))D ) by Claim 1. Hence, φ * ((zi ∂∂ yF )D ) ∈ m1 φ * ((mY JM(X))D ) + φ * ((mY JMz (X))D ). Also, we have l ∂F * ∘ φ1 ), 0), so the Claim 2 φ ((zi ∘ π1 − zi ∘ π2 )((z j ∂∂ yF ) ∘ π1 ), 0) = (z j ∘ φ1 ) ((zi ∘ φ1 − zi ∘ φ2 )( l ∂ yl | {z } ∈φ * ((mY JMz (X))D )

is proved. Thus, φ * ((mY JM(X))D ) = φ * ((mY JM(X)Y )D ) + φ * ((mY JMz (X))D ) ⊆ (m1 φ * ((mY JM(X))D ) + φ * ((mY JMz (X))D )) + φ * ((mY JMz (X))D ) = m1 φ * ((mY JM(X))D ) + φ * ((mY JMz (X))D ). By Nakayama’s Lemma we conclude that φ * ((mY JM(X))D ) ⊆ φ * ((mY JMz (X))D ). Since φ * ((JM(X)Y )D ) ⊆ φ * ((mY JM(X))D ) then φ * ((JM(X)Y )D ) ⊆ φ * ((mY JMz (X))D ). By the curve criterion we conclude that (JM(X)Y )D ⊆ (mY JMz (X))D . While a similar result for iLA does not make sense, if we work on the strict iLA , then we get an analogous result, as we can see in the following proposition. Proposition 2.2.7. Consider the setup 2.2.1. At any point (x, x′ ) ∈ X × X we have: Y

(JM(X)Y )D ⊆ (JM(X)D )† if and only if (JM(X)Y )D ⊆ (JMz (X)D )† . Proof. The implication (⇐=) is obvious. Now, suppose that (JM(X)Y )D ⊆ (JM(X)D )† , and let φ : (C, 0) → (X × X, (x, x′ )) be an arbitrary curve. Then, Y

φ * ((JM(X)Y )D ) ⊆ m1 φ * (JM(X)D ) = m1 φ * ((JM(X)Y )D )+m1 φ * ((JMz (X))D ). By Nakayama’s Lemma we conclude that φ * ((JM(X)Y )D ) ⊆ m1 φ * (JMz (X)D ), which finishes the proof.

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Lemma 2.2.8. Consider the setup 2.2.1. If x = (z, y) ∈ X with z ̸= 0 then mY JMz (X) = JMz (X) = OXp at x. Proof. Since z ̸= 0 then the jacobian matrix [J( fy )(z)] = [JMz (X)(x)] has maximal rank p. Without loss of generality we can assume that the p × p submatrix formed by the first p columns of [JMz (X)(x)] is invertible. Denote this matrix by [JMz (X)(x)] p . Then, there exists an open subset U of X containing x such that [JMz (X)(w)] p is invertible, ∀w ∈ U. Let e1 , ..., e p be the canonical generators of OXp . Let us to prove that e j ∈ JMz (X) at x, ∀ j ∈ {1, ..., p}. In fact, given such j, consider the analytic map α : U → C p given by α(w) := [JMz (X)(w)]−1 p · e j , with coordinates function p

α1 , ..., α p . So, we have e j = ∑ αi ∂∂ Fzi on U, hence e j ∈ JMz (X) at x. Therefore, JMz (X) = OXp at i=1 x. Now, since z ̸= 0 then zk (x) ̸= 0 for some k, so zk is invertible in OX,x . Since ∂F z−1 k (zk ∂ zi ), for all i ∈ {1, ..., n} then JMz (X) = mY JMz (X) at x.

∂F ∂ zi

=

The next result generalizes the Proposition 4.1 of [39]. Proposition 2.2.9. Consider the setup 2.2.1. Then cosupp((mY JMz (X))D ) = cosupp((JMz (X))D ) = ∆(X) ∪ (X × 0) ∪ (0 × X). Y

Y

Proof. Since (mY JMz (X))D ⊆ (JMz (X))D at any point, then cosupp(JMz (X)D ) ⊆ cosupp(mY JMz (X))D . We already know that cosupp((JMz (X))D ) = ∆(X) ∪ (X × 0) ∪ (0 × X). Y

Y

Let us to prove that X × X − (cosupp((JMz (X))D )) ⊆ X × X − (cosupp((mY JMz (X))D )). Y

Y

2p ′ In fact, if (x, x′ ) is not in the cosupport of (JMz (X))D then (JMz (X))D = OX× X at (x, x ), and we Y

can write x = (z, y) and x′ = (z′ , y), with z ̸= 0, z′ ̸= 0 and z ̸= z′ . So there exist t, k, r ∈ {1, ..., n} such that zt ∘ π1 , zk ∘ π2 and zr ∘ π1 − zr ∘ π2 are invertibles in OX×X,(x,x′ ) . Y

Notice that: for all j ∈ {1, ..., n} we have: ∂F ∂F ∂F −1 ( ∂ z j )D = (zt ∘ π1 ) (zt ) ∘ π1 , (zt ) ∘ π2 ∂zj ∂zj | {z } ∈(mY JMz (X))D ∂F −1 −1 − (zt ∘ π1 ) (zr ∘ π1 − zr ∘ π2 ) 0, (zr ∘ π1 − zr ∘ π2 )((zt ) ∘ π2 ) ∂zj | {z } ∈(mY JMz (X))D ∂F −1 −1 + (zk ∘ π2 ) (zr ∘ π1 − zr ∘ π2 ) 0, (zr ∘ π1 − zr ∘ π2 )((zk ) ∘ π2 ) ∈ (mY JMz (X))D . ∂zj | {z } ∈(mY JMz (X))D

71


Furthermore, for all i, j ∈ {1, ..., n} we have (0, (zi ∘ π1 − zi ∘ π2 )( ∂∂ zFj ∘ π2 )) = (zk ∘ π2 )−1 (0, (zi ∘ π1 − zi ∘ π2 )((zk ∂∂ zFj ) ∘ π2 )) ∈ (mY JMz (X))D . 2p ′ Thus, we conclude that (mY JMz (X))D = (JMz (X))D = OX× X at (x, x ) Y

⇒ (x, x′ ) ∈ / cosupp((mY JMz (X))D ).

Hence, cosupp(mY JMz (X))D ⊆ cosupp(JMz (X))D , which finishes the proof. The next result generalizes the Proposition 4.2 of [39], and states that both infinitesimal Lipschitz conditions hold in any point of X × X off (0, 0) ×Y . Y

Proposition 2.2.10. Consider the setup 2.2.1. a) If (x, x) ∈ ∆(X) − ((0, 0) ×Y ) then (JM(X)Y )D ⊆ (JMz (X))D and (JM(X)Y )D ⊆ (mY JMz (X))D at (x, x). In particular, iLA and iLmY hold at (x, x); b) If (x, x′ ) ∈ (X × 0) ∪ (0 × X) − ((0, 0) ×Y ) and the W condition holds in any point (0, y), Y

Y

y ∈ Y , then iLA and iLmY conditions hold at (x, x′ ). Proof. (a) We can write x = (z, y) with z ̸= 0. By Lemma 2.2.8 we have that JM(X)Y ⊆ OXp = JMz (X) = mY JMz (X) at x, which implies that (JM(X)Y )D ⊆ (JMz (X))D and (JM(X)Y )D ⊆ (mY JMz (X))D at (x, x). (b) We can assume that (x, x′ ) ∈ (X × 0) − ((0, 0) × Y ) (the other case is completely Y

analougous). So, we can write x = (z, y) and x′ = (0, y), with z ̸= 0. So zk (x) ̸= 0 for some k, and follows that (zk ∘ π1 − zk ∘ π2 )(x, x′ ) = zk (x) − 0 = zk (x) ̸= 0 ⇒ zk ∘ π1 and zk ∘ π1 − zk ∘ π2 are p ′ ′ invertibles in OX×X(x,x′ ) . Let e1 , ..., e p be the canonical generators of OX× X and e1 , ..., e p be the Y

Y

canonical generators of OXp . Thus e j = e′j ∘ π1 , ∀ j ∈ {1, ..., p}. Since z ̸= 0 then by Lemma 2.2.8 we have JMz (X) = OXp at x. Let us to prove that (e j , 0) ∈ (mY JMz (X))D , ∀ j ∈ {1, ..., p} at (x, x′ ). In fact, given a such n

n

j, we have e′j ∈ OXp = JMz (X) at x, so we can write e′j = ∑ αi ∂∂ Fzi ⇒ (e j , 0) = ∑ (αi ∘ π1 )(zk ∘ i=1 i=1 ∂ F ) ∘ π1 , 0 , hence (e j , 0) ∈ (mY JMz (X))D . π1 − zk ∘ π2 )−1 (zk ∘ π1 )−1 (zk ∘ π1 − zk ∘ π2 )(zk ∂ zi | {z } ∈(mY JMz (X))D

Let us to prove that (0, ∂∂ yF ∘ π2 ) ∈ (mY JMz (X))D ) at (x, x′ ), ∀l ∈ {1, ..., k}. In fact, given l a such l, we will use the curve criterion. Let φ : (C, 0) → (X × X, (x, x′ )) be an arbitrary analytic Y

curve. Write φ = (ϕ, γ). Since γ : (C, 0) → (X, x′ ) is an analytic curve and at x′ = (0, y) then

∂F ∂ yl

∈ mY JMz (X) n

∘ γ = g ∘ γ, for some g ∈ mY JMz (X) at x′ . So, we can write g = ∑ βr j zr ∂∂ zFj . r, j=1 ∂F −1 Thus, we have that (0, g∘π2 ) = ∑(βr j ∘π2 )(zk ∘π1 −zk ∘π2 ) 0, (zk ∘ π1 − zk ∘ π2 )(zr ) ∘ π2 ∂zj | {z } ∂F ∂ yl

∈(mY JMz (X))D

72


⇒ (0, g ∘ π2 ) ∈ (mY JMz (X))D . Then, (0, ∂∂ yF ∘ π2 ) ∘ φ l = (0, g ∘ π2 ) ∘ φ ∈ (mY JMz (X))D ∘ φ . By curve criterion we conclude that (0, ∂∂ yF ∘ π2 ) ∈ (mY JMz (X))D at (x, x′ ). l

Finally, let us to prove that (JM(X)Y )D ⊆ (mY JMz (X))D at (x, x′ ). In fact, for all i ∈ {1, ..., n} and l ∈ {1, ...k} we have that (0, (zi ∘ π1 − zi ∘ π2 )( ∂∂ yF ∘ π2 )) = l ∂F (zi ∘ π1 − zi ∘ π2 ) (0, ∘ π2 ) ∈ (mY JMz (X))D and ∂ yl | {z } ∈(mY JMz (X))D



 p

(

∂ Fj ∂F  )D =  ∑ ( ∘ π1 ) ∂ yl j=1 ∂ yl

(e j , 0) | {z }

∈(mY JMz (X))D

∂F  ∘ π2 ) ∈ (mY JMz (X))D .  + (0, ∂ yl | {z } ∈(mY JMz (X))D

Therefore, (JM(X)Y )D ⊆ (mY JMz (X))D at (x, x′ ) and the iLmY and iLA condition are satisfied in (x, x′ ). The next result generalizes the Theorem 4.3 of [39] and states that the infinitesimal Lipschitz condition A holds generically along the parameter space Y . Theorem 2.2.11. Consider the setup 2.2.1. Then there exists a dense Zariski open subset U of Y such that the infinitesimal Lipschitz condition A holds for the pair (X −Y,U ∩Y ) along Y . Proof. We can write a matrix of generators of (JMz (X))D as " # JMz (x) ∘ π1 0 [(JMz (X))D ] = JMz (X) ∘ π2 (0, (zi ∘ π1 − zi ∘ π2 )( ∂∂ zFs ∘ π2 ))ni,s=1 2p whose entries are in OX×X . Since (JMz (X))D is a sheaf of submodules of OX× X then, choosing Y

Y

S1 , ..., S2p as the homogeneous coordinates on P2p−1 , we can consider the sheaf of ideals of OX×X×P2p−1 induced by (JMz (X))D , namely ρ((JMz (X))D ), which is generated by the entries Y

of the vector [1

S2 S1

S2p ] · [(JMz (X))D ] S1

...

on the chart U1 := {[S1 , ..., S2p ] ∈ P2p−1 | S1 ̸= 0} which is a dense Zariski open subset of P2p−1 . Denote by N := NBρ((JMz (X))D ) (X × X × P2p−1 ) the normalized blow-up of X × X × Y

Y

P2p−1 with respect to the sheaf of ideals ρ((JMz (X))D ) of OX×X×P2p−1 . Consider the projection Y

map π : N → X × X × P2p−1 and let E ⊆ N be the normalized exceptional divisor. To prove this Y

theorem we use the Corollary 2.1.16 and the module criterion (see Proposition 3.5 in [48]), i.e, in order to verify the condition (JM(X)Y )D ⊆ (JMz (X))D in a dense Zariski open subset U of Y , it suffices to check that on each component of the exceptional divisor, the pullback of the element

73


ρ(( ∂∂Fy )D ) to the normalized blow-up is in the pullback of ρ((JMz (X))D ), for every coordinate y in the parameter space. Let p : X ⊆ Cn × Y → Y be the projection onto Y . For each ` ∈ {1, 2} consider the projection map p` : X × X × P2p−1 → X on the `th factor and π¯` : N → X given π¯` := p` ∘ π. Y

N π¯1 p Y

p1

X

π X × X × P2p−1

π¯2 p2

p X

Y

Y

π1

p¯

π2

X ×X Y

By Proposition 2.2.10 we need only consider those components of the exceptional divisor which projects to Y under the map to X × X. Since we are working over a dense Zariski open Y

subset of Y we may assume that every such components maps surjectively onto Y . Since N is a normal space and E has codimension 1 in N then we can work at a point q of the normalized exceptional divisor E such that E is smooth at q, N is smooth at q and the projection to Y is a submersion at q. Thus we can choose coordinates (y′ , u′ , x′ ) such that y′ = y ∘ p, u′ defines E ′ locally with reduced structure and ∂∂ uy′ = 0, i.e, u′ and y′ are independent coordinates. Working on the subset U1 ⊂ P2p−1 , since X is defined by F then the germ of 

[1

S2 S1

 F1 ∘ p1  .   ..      Fp ∘ p1  S2p   =0 ... S1 ] ·   F ∘ p  1 2  .   ..    Fp ∘ p2

is identically zero on X × X × P2p−1 . Pull this back to N by π and take the partial derivative with Y

respect to y′ at q. We get by the chain rule:

74


n



∂ F1  ∂y

[1

S2 S1

∂ F1 ∂ zi

∂ (zi ∘π¯1 ) ∂ y′



∘ π¯1 + ∑ ∘ π¯1  i=1   ..   .     n ∂F  ∂ Fp  ¯ ∂ (z ∘ π ) i 1  ∂ y ∘ π¯1 + ∑ ∂ zip ∘ π¯1  ′ ∂ y  S2p  i=1 ... S1 ] ·  =0 n ∂ (zi ∘π¯2 )  ∂ F1  ∂ F1 ∘ π¯ + ∑ ¯ ∘ π ′ 2 2  ∂y  ∂ zi ∂y   i=1   . .   .   n ∂F  ∂ Fp ∂ (zi ∘π¯2 )  p ¯ ¯ ∘ π + ∘ π ∑ 2 2 ∂y ∂ zi ∂ y′

(?)

i=1

Since Fj ∘ π¯1 = Fj ∘ π¯2 = 0 for all j ∈ {1, ..., p} then there is no term involving the derivatives of the homogeneous coordinates with respect to y′ . Notice that all zi vanish along Y and zi ∘ π¯1 and zi ∘ π¯2 vanish along E at q then we can assume that the order of vanishing of z1 ∘ π¯` is minimal among {zi ∘ π¯` } and that the strict transform of z1 ∘ π¯` do not pass through q, ∀` ∈ {1, 2}.   ∂F 1 ¯ ∘ π 1  ∂y .   ..      ∂ Fp ¯ S2p  ∂ y ∘ π1  S2 ∂F  By the equation (?) we have that ρ(( ∂ y )D ) ∘ π = [1 S ... S ] ·  ∂ F1  = −v, 1 1  ∂ y ∘ π¯2   .   ..    ∂ Fp ¯ ∂ y ∘ π2 where



v := [1

S2 S1

n

∂ F1 ∂ zi

∂ (zi ∘π¯1 ) ∂ y′



∘ π¯1 ∑ i=1    .   ..   n   ∂F ∂ (zi ∘π¯1 )   ∑ ∂ zip ∘ π¯1  ∂ y′   S ... S2p1 ] · i=1 . n ∂ (zi ∘π¯2 )  ∂ F1 ∑ ¯ ∘ π 2   ∂ y′ i=1 ∂ zi    ..   .    n ∂ Fp ∂ (zi ∘π¯2 )  ∑ ∂ z ∘ π¯2 ∂ y′ i=1

i

In order to simplify the notation, for each i ∈ {1, ...n} define

75




wi := [1

S2 S1

∂ F1 ∂ zi

∂ (zi ∘π¯1 ) ∂ y′



∘ π¯1     ..   .    ∂ Fp ∂ (zi ∘π¯1 )  ¯ ∂ zi ∘ π1 ∂ y′  S2p    ... S1 ] ·  ∂ F ∂ (zi ∘π¯1 )  1 ¯  ∂ zi ∘ π2  ∂ y′   ..     .   ∂ F ¯ ∂ (zi ∘π1 ) p ¯ ∂ z ∘ π2 ∂ y′ i

and 

w˜ i := −[1



0 .. .

          S2p  0  ... S1 ] ·  ∂ F1 . ¯ ∂ (zi ∘π¯1 ) − ∂ (z∂i ∘yπ′ 2 )   ∂ zi ∘ π¯2 ′ ∂ y     ..   .   ∂ F ∂ (zi ∘π¯1 ) ∂ (zi ∘π¯2 ) p ¯ ∂ zi ∘ π 2 ∂ y′ − ∂ y′

S2 S1

n

Clearly v = ∑ (wi + w˜ i ). For every i ∈ {1, ..., n} we have that wi =

i=1 ∂ (zi ∘π¯1 ) * ∂F * π (ρ(( ∂ y′ ∂ zi )D )) ∈ π (ρ((JMz (X))D )).

Now it suffices to check that w˜ i ∈ π * (ρ((JMz (X))D )), ∀i ∈ {1, ..., n}. Since the pullback of the ideal ρ((JMz (X))D ) is locally principal then we can work at a point q such that π * (ρ((JMz (X))D )) is generated by u′r , a power of u′ . Since ON,q is a normal ring then the Lemma 1.12 of [64] implies that the ideal π * (ρ((JMz (X))D )) is integrally closed, i.e, π * (ρ((JMz (X))D )) = π * (ρ((JMz (X))D )). So, it is enough to prove that w˜ i ∈ π * (ρ((JMz (X))D )), for all i ∈ {1, ..., n}. Let i ∈ {1, ..., n} be arbitrary. We use the curve criterion. Let φ˜ : (C, 0) → (N, q) be an analytic curve. We can choose φ˜ such that φ : (C, 0) → (X × X × P2p−1 , π(q)) given Y h i ψ2p 2 by φ := π ∘ φ˜ meets the dense Zariski open subset U1 , φ = (φ1 , φ2 , ψ) and ψ = 1, ψ , ..., ψ1 ψ1 . Further, φ˜ can be chosen such that φ˜ is transverse to the component so that u′ ∘ φ˜ = t, where t is the generator of the maximal ideal of OC,0 . Hence, the pullback of the ideal π * (ρ((JMz (X))D )) is generated by t r . Consider the element 

wˆ i := −[1

S2 S1

0 .. .



          0   S ... S2p1 ] ·  ∂ F1 .  ∂ zi ∘ π¯2 (zi ∘ π¯1 − zi ∘ π¯2 )     ..   .  ∂ F  p ¯ ¯ ¯ ∘ π (z ∘ π − z ∘ π ) i i 2 1 2 ∂ zi

76


∂F ∘ π2 )))) ∈ π * (ρ((JMz (X))D )). Since y′ and u′ Notice that wˆ i = −π * (ρ((0, (zi ∘ π1 − zi ∘ π2 )( ∂ zi | {z } ∈(JMz (X)D )

are independent coordinates then the order of ∂ (z∂i ∘yπ′ 1 ) − ∂ (z∂i ∘yπ′ 2 ) in u′ is the same as the order of zi ∘ π¯1 − zi ∘ π¯2 in u′ . Then the pullback of both have the same order in t, so there exists an invertible element αi ∈ OC,0 such that ¯

˜*

φ

p

Hence, φ˜ * (w˜ i ) = − ∑

∂ (zi ∘ π¯1 ) ∂ (zi ∘ π¯2 ) − ∂ y′ ∂ y′

∂ Fj ∂ zi

∘ φ2

¯

= αi (φ˜ * (zi ∘ π¯1 − zi ∘ π¯2 )).

ψ p+ j ¯ ¯ φ˜ * ∂ (zi ∘π′ 1 ) − ∂ (zi ∘π′ 2 )

j=1 = αi φ˜ * (wˆ i ) ∈ φ˜ * (π * (ρ((JMz (X))D ))).

∂y

∂y

ψ1

Therefore, w˜ i ∈ π * (ρ((JMz (X))D )), for all i ∈ {1, ..., n} which finishes the proof.

2.3

The genericity theorem applied in a family of hyperplane sections

Given X an analytic variety with isolated singularity at the origin, we can consider the sections of X by hyperplanes. One natural question is if there exists a generic set of hyperplanes for which the family of hyperplanes sections satisfies the infinitesimal Lipschitz condition A. We will show this is true. First, we recall some important notions in order to make precise statements. Fore more details see [44]. Let us work on the Grassmanian modification of X = f −1 (0), defined by an analytic map f : (Cn , 0) → (C p , 0), X with isolated singularity at the origin, n ≥ p. For each y = [y1 , ..., yn ] ∈ Pn−1 , consider the hyperplane on Cn given by n

Hy := {z = (z1 , ..., zn ) ∈ Cn | z · y := ∑ zi yi = 0}. i=1

Let En−1 be the canonical bundle over Pn−1 , i.e, En−1 := {(z, y) ∈ Cn × Pn−1 | z ∈ Hy }. Consider the projection map β : En−1 → Cn . We call X˜ := β −1 (X) the (n − 1)-Grassmanian modification of X. Here we will simply refer to the (n − 1)-modification as the Grassmanian modification of X. We can see Pn−1 embedded into En−1 as the zero section of the bundle En−1 , ˜ Note that the projection to 0 × Pn−1 which allows us to think of 0 × Pn−1 as a stratum of X. makes X˜ a family of analytic sets with 0 × Pn−1 as the parameter space, which we denote by Y . The members of this family are just {Hy ∩ X} as y varies in Pn−1 .

77

2.3. The genericity theorem applied in a family of hyperplane sections

X˜

,→

En−1

β |X˜

β

X

Y

Cn

Y

Consider the chart Un := {[y1 , ..., yn ] ∈ Pn−1 | yn ̸= 0} = {[y1 , ..., yn−1 , −1] | (y1 , ..., yn−1 ) ∈ Cn−1 } ≡ Cn−1 which is a dense Zariski open subset of Pn−1 . Working on the dense Zariski open subset En−1 ∩ (Cn ×Un ) of En−1 , we have local coordinates given by (z1 , ..., zn , y1 , ..., yn−1 ). In these coordinates, the projection map β satisfies the equation n−1

β (z1 , ..., zn , y1 , ..., yn−1 ) = (z1 , ..., zn−1 , ∑ yi zi ). i=1

Consider the analytic map F : En−1 ∩ (Cn ×Un ) → Cp (z, y) ↦→ f ∘ β (z, y) ˜ hence X˜ is defined by F. For each y = (y1 , ..., yn−1 ) Thus, F −1 (0) = β −1 ( f −1 (0)) = β −1 (X) = X, ≡ [y1 , ..., yn−1 , −1] ∈ Un , let Fy : Cn → C p given by Fy (z) := F(z, y) and let X˜y := F −1 (0). In these coordinates, clearly X˜y = ( f −1 (0)) ∩ Hy = X ∩ Hy . Therefore, F defines the family of sections of X by the hyperplanes Hy , as y varies on the dense Zariski open subset Un of Pn−1 . The next result generalizes the Theorem 4.4 of [39]. Theorem 2.3.1. Suppose (X, 0) is a germ of an isolated singularity analytic variety defined by an analytic map-germ f : (Cn , 0) → (C p , 0), n ≥ p, and consider the Grassmanian modification X˜ of X. Then, there exists a non-empty Zariski open subset U of Pn−1 , such that the iLA condition holds for the pair (X˜ −U,U) along U. Proof. As we have seen, X˜ is a family defined by the above analytic map F. Let us prove that X˜y has isolated singularity at (0, y) for all y varying in a non-empty Zariski open subset U ′ of Un . In fact, we already know that the set of limiting tangent hyperplanes of X at the origin is a Zariski proper closed subset of Pn−1 . Call this set W . Let U ′ := Un − (W ∩Un ). Since Pn−1 is irreducible then Un is irreducible. Since Un is a dense subset of Pn−1 then W ∩Un also is a proper Zariski closed subset of Un , hence U ′ is a dense Zariski open subset of Un . Let y ∈ U ′ . We want to show that (0, y) is an isolated singularity of X˜y . By hypothesis, Hy is not a limiting tangent hyperplane of X at the origin, and by Lemma 4.1 (a) of [48] we have that JM(X)Hy = JM(X) at the origin, where JM(X)Hy := { ∂∂ vf | v ∈ Hy }. Thus, in a neighborhood of the origin, the generic rank of JM(X) and JM(X)Hy is the same. Thus, if we take

78


z in this neighborhood, such that z ∈ Hy , z ̸= 0 then the generic rank of JM(X˜y ) = JM(X ∩ Hy ) at z is the generic rank of JM(X)Hy at z, which is the generic rank of JM(X) at z. Since z ̸= 0 and X has isolated singularity at the origin then we can choose this neighborhood so that z is a non-singular point of X, which implies that z is not a singular point of X˜y . Therefore, X˜y has isolated singularity at the origin, for all y ∈ U ′ . Now, the existence of U follows from the Theorem 2.2.11. Let us go back to the discussion before the last theorem. We have seen that X˜ is defined by the map F : En−1 ∩ (Cn ×Un ) → C p given by F(z, y) = f ∘ β (z, y). From the chain rule we have that   1 0 ... 0 0 0 ... 0    0 1 ... 0 0 0 ... 0     0 0 ... 0 0 0 ... 0    [dF(z,y) ] = [d fβ (z,y) ] ·  . .  . . . . .. .. .. ..   .. ..      0 0 ... 1 0 0 ... 0  y1 y2 ... yn−1 0 z1 ... zn−1 n−1 which implies that ∂∂ yFi = zi ∂∂zfn ∘ β and ∂∂ Fzi = ∂∂ zfi ∘ β + ∑ y j ∂∂zfn ∘ β , for all i ∈ {1, ..., n − 1}, j=1

∂F ∂ zn

and = 0. Thus, we have immediately the next result, which is a generalization of the Corollary 4.5 of [39]. n Corollary 2.3.2. The point(0, P) ∈ En−1 ∩ (C ×Un ) belongs to the Zariski open subset of the ˜ D at (0, P), for all i ∈ {1, ..., n − 1}. last theorem if and only if zi ( ∂∂zfn ∘ β ) ∈ (JMz (X)) D

The corollary tell us that in order to check if a hyperplane is generic, it is enough to ˜ ˜ verify that for all curves φ1 , φ2 , tangent to Pat the origin with lifts φ1 , φ2 , respectively, with φ := (φ1 , φ2 ) and φ˜ := (φ˜1 , φ˜2 ), we have that zi ( ∂∂zfn ∘ β ) ∘ φ is an OC,0 -linear combination of ! n on−1 n−1 ∂f ∂f ˜ ∘φ , for all i ∈ {1, ..., n − 1}. ∑ y j ∂ zn ∘ β ∂ zs ∘ φ + j=1

s=1

In [39] Gaffney gave a description of these generic hyperplanes using analytic invariants in the jacobian ideal case. Now we generalize this description for the jacobian module case. For the rest of this section we assume that the hyperplanes Hy are not limiting tangent hyperplanes of (X, 0). As we have seen, this implies that X˜y = X ∩ Hy has isolated singularity at the origin and JM(X)Hy = JM(X) at the origin. The invariants that we use here appeared earlier in section 2.1. Since X˜y has isolated singularity at the origin then by Corollary 2.1.27 the multiplicity of the pair e((JM(X˜y ))D , (JM(X˜y ))D ) is well defined. These invariants have been used in this context before. As pointed out by Gaffney in [39], for ICIS singularities, and more generally in [43], to check for whether or not a hyperplane is in

2.3. The genericity theorem applied in a family of hyperplane sections

79

the generic set of hyperplanes for which the hyperplane sections form a Whitney equisingular family, you can use the multiplicity of the pair (JM(X˜y ), OXp˜ ), which is the Buchsbaum-Rim y multiplicity e(JM(X˜y )). The hyperplane is generic if this multiplicity is minimal, and the minimal number is the sum of the Milnor number of X ∩ Hy and X ∩ Hy ∩ Hy′ , where Hy and Hy′ are generic hyperplanes. The proof that the minimal value of e((JM(X˜y ))D , (JM(X˜y ))D ) identifies generic hyperplanes will be done using the Multiplicity Polar Theorem (see Corollary 1.4 [45]). Now we identify the modules we will use. We will work on the fibered product X˜ × X˜ ⊆ X × Pn−1 × X. Let N := (β * (JM(X)))D Pn−1

˜ D , considering X˜ defined by the analytic map F : En−1 ∩ (Cn ×Un ) → C p , and M := (JMz (X)) given by F(z, y) = f ∘β (z, y). Clearly M restricted to the fiber of the family X˜ over the hyperplane Hy is just (JM(X ∩ Hy ))D and N restricted to Hy is (JM(X) |Hy )D . Further, since we are assuming that Hy is not a limiting tangent hyperplane of (X, 0) then JM(X) |Hy = JM(X)Hy , hence N restricted to Hy is (JM(X˜y ))D . Therefore, the multiplicity of the pair (M |Hy , N |Hy ) is the same as e((JM(X˜y ))D , (JM(X˜y ))D ). The next result is a generalization of the Theorem 4.6 of [39]. Theorem 2.3.3. Under the above notations, let U be the set of hyperplanes which are not limiting tangent hyperplanes of (X, 0). Suppose that N has no polar variety with the same codimension of U. Then: a) The map U → Z Hy ↦→ e((JM(X˜y ))D , (JM(X˜y ))D ) is upper semicontinuous on U; b) The iLA condition holds along U at a hyperplane Hy for which the value of e((JM(X˜y ))D , (JM(X˜y ))D ) is minimal. Proof. (a) By the definition of U, (JM(X˜y ))D is the restriction of N to the fiber X˜y = X ∩ Hy . Since N has no polar variety with the same codimension of U then the multiplicity polar theorem implies that e((JM(X˜y ))D , (JM(X˜y ))D ) is upper semicontinuous. (b) Suppose Hy ∈ U gives the minimal value of the multiplicity. Since this value already is minimal then it cannot go down, hence it must be constant. This implies that the polar variety of M of the same codimension as U is empty, which puts restrictions on the size of the fiber of Pro j(R(M)). We already know that { ∂∂ yFi }n−1 i=1 are in M generically. Since the dimension of the fiber of Pro j(R(M)) is bounded then by Theorem A1 of [62] we have that { ∂∂ yFi }n−1 i=1 are in M at Hy , which finishes the proof.

80


In order to check the hypothesis over N, we have to check if the fiber dimension of Pro j(R(N)) is small enough so that N has no polar variety of the same codimension of U.

81

CHAPTER

3 THE LIPSCHITZ SATURATION OF A MODULE

Motivated by some properties that the Lipschitz saturation has on the ideal case, we define some kinds of Lipschitz saturation for modules.

3.1

The Lipschitz saturation of a module and basic properties

Let X ⊆ Cn be an analytic set and let M be a sheaf of OX -submodules of OXp generated by global sections {g1 , ..., gr }. Consider the ideal sheaf ρ(M ) on X × P p−1 induced by M . Then ρ(M ) is generated by {ρ(g1 ), ..., ρ(gr )}, so we can consider the blowup Bρ(M ) (X × P p−1 ) ⊆ (X × P p−1 ) × Pr−1 and the blowup map π : Bρ(M ) (X × P p−1 ) → X × P p−1 . Let T1 , ..., Tp and S1 , ..., Sr be the homogeneous coordinates of P p−1 and Pr−1 , respectively. Definition 3.1.1. Let x ∈ X. The 1-Lipschitz saturation of M at x is denoted by (MS1 )x and is defined by p (MS1 )x := {h ∈ OX,x | ρ(h) ∈ (ρ(M ))S at all (x, [t1 , ...,t p ]) ∈ V (ρ(M ))}.

In the family case, the 1-Lipschitz saturation of M at x ∈ X relative to Y is defined as above taking the Lipschitz saturation of ρ(M ) relative to Y . Unless we say the opposite, we denote by MS1 the stalk of the sheaf MS1 at an arbitrary point of X.

82

Chapter 3. The Lipschitz Saturation of a Module

Write h = (h1 , ..., h p ) and each generator g j = (g1 j , ..., g p j ). Suppose t1 ̸= 0 and take (x, [t1 , ...,t p ], [s1 , ..., sr ]) ∈ π −1 ((x, [t1 , ...,t p ])), with s1 ̸= 0. Then, π * (ρ(M )) is locally generated ρ(h)∘π by ρ(g1 ) ∘ π, and ρ(h) ∈ (ρ(M ))S at (x, [t1 , ...,t p ]) if and only if ρ(g )∘π is Lipschitz with respect to the system of coordinates

T z1 , ..., zn , TT11 , ..., T1p , SS11 , ..., SS1r .

1

Remark 3.1.2. MS1 is the Lipschitz saturation of an ideal if p = 1. In fact, in this case ρ(h) = h, p ∀h ∈ OX,x and ρ(M ) = M , and the condition ρ(h) ∈ (ρ(M ))S at all (x, [1]) ∈ V (ρ(M )) is equivalent to h ∈ MS at x. Hence, MS1 = MS at x if M is an ideal sheaf of OX . Proposition 3.1.3. Let M be a sheaf of OX -submodules of OXp . Then: a) MS1 is an OX -submodule of OXp ; b) M ⊆ MS1 ⊆ M. In particular, M is a reduction of MS1 and, if M has finite colenght in MS1 then e(M, MS1 ) = 0. Proof. Let x ∈ X be an arbitrary point. (a) Let h, h′ ∈ MS1 at x and α ∈ OX,x . So, in every point (x, [t1 , ...,t p ]) ∈ V (ρ(M )) we have ρ(αh + h′ ) = αρ(h) + ρ(h′ ) ∈ (ρ(M ))S . So, αh + h′ ∈ MS1 at x, which finishes the proof of (a). (b) It is pretty obvious that M ⊆ MS1 . Let us to prove that MS1 ⊆ M. Let h ∈ MS1 at x. So ρ(h) ∈ (ρ(M))S in every point (x, [t1 , ...,t p ]) ∈ V (ρ(M)). By Proposition 2.1 of [40] we have (ρ(M))S ⊆ ρ(M), hence ρ(h) ∈ ρ(M) in every point (x, [t1 , ...,t p ]) ∈ V (ρ(M)). By Proposition 3.4 of [48] we conclude that h ∈ M at x. Therefore, MS1 ⊆ M. Definition 3.1.4. Consider the Setup 2.2.1. We define the infinitesimal Lipschitz conditions with respect to the 1-Lipschitz saturation of a module. ∙ The pair (X,Y ) satisfies the 1-infinitesimal Lipschitz condition mY (1-iLmY ) at x ∈ X if JM(X)Y ⊆ (mY JMz (X))S1 at x; ∙ The pair (X,Y ) satisfies the 1-infinitesimal Lipschitz condition A (1-iLA ) at x ∈ X if JM(X)Y ⊆ (JMz (X))S1 at x. Here we consider the 1-Lipschitz saturation relative to the parameter space Y . Notice that 1-iLmY implies 1-iLA . Corollary 3.1.5. Consider the Setup 2.2.1. If x = (z, y) ∈ X with z ̸= 0 then 1-iLmY and 1-iLA hold at x. Proof. By Lemma 2.2.8 we have that JM(X)Y ⊆ mY JMz (X) at x. By Proposition 3.1.3 the inclusion mY JMz (X) ⊆ (mY JMz (X))S1 holds and therefore the condition 1-iLmY holds at x. In particular, 1-iLA holds at x.

83

3.1. The Lipschitz saturation of a module and basic properties

Thus, it remains to investigate the 1-infinitesimal Lipschitz conditions along the parameter space Y ≡ 0 ×Y . p is denoted by MS2 , and Definition 3.1.6. The 2-Lipschitz saturation of the submodule M ⊆ OX,x is defined by p MS2 := {h ∈ OX,x | hD ∈ MD at (x, x)}.

In the family case, the 2-Lipschitz saturation of M at x ∈ X relative to Y is defined as above taking the double relative to Y . If M is an OX -submodule of OXp then we can consider the sheaf of OX -modules MS2 where each stalk(MS2 )x is (Mx )S2 . Unless we say the opposite, we denote by MS2 the stalk of the sheaf MS2 at an arbitrary point of X. Remark 3.1.7. MS2 is the Lipschitz saturation of an ideal if p = 1. In fact, by Theorem 1.4.2 we have that h ∈ MS2 if and only if hD ∈ MD if and only if h ∈ MS . Therefore, MS2 = MS if M is an p ideal of OX,x . Proposition 3.1.8. Let M be a sheaf of OX -submodules of OXp . Then: a) MS2 is an OX -submodule of OXp ; b) M ⊆ MS2 ⊆ M. In particular, M is a reduction of MS2 and, if M has finite colenght in MS2 then e(M, MS2 ) = 0. The same result holds in the family case. Proof. Let x ∈ X be an arbitrary point. (a) Let α ∈ OX,x and h, h′ ∈ MS2 at x. Claim: (0, (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) ∈ MD at (x, x). In fact, let φ : (C, 0) → (X ×X, (x, x)) be an arbitrary curve. Since hD ∈ MD then by Proposition 2.1.12 (a) we have that h ∈ M at x. If we write φ = (φ1 , φ2 ), we have that h ∘ φ2 ∈ M ∘ φ2 , so we can write h ∘ φ2 = ∑ α j (h j ∘φ2 ), with α j ∈ OC,0 and h j ∈ M. By the Lemma 2.1.2 (b) we have that (0, (α ∘π1 − α ∘π2 )(h j ∘π2 )) ∈ MD , for all j. Thus, we get: (0, (α ∘π1 −α ∘π2 )(h∘π2 ))∘φ = (0, (α ∘φ1 −α ∘ φ2 )(h ∘ φ2 )) = (0, (α ∘ φ1 − α ∘ φ2 )(∑ α j (h j ∘ φ2 )) = ∑ α j ((0, (α ∘ π1 − α ∘ π2 )(h j ∘ π2 )) ∘ φ ) ∈ {z } | MD ∘ φ , which proves the claim.

∈MD

Finally, we have that hD , h′D ∈ MD at (x, x), hence: (αh + h′ )D = (α ∘ π1 )hD + (0, (α ∘ π1 − α ∘ π2 )(h ∘ π2 )) + h′D , and each factor on the right hand side belongs to MD at (x, x), so the proof of (a) is done.

84


(b) If h ∈ M at x then hD ∈ MD ⊆ MD at (x, x), so h ∈ MS2 at x. Therefore, M ⊆ MS2 . Furthermore, if h ∈ MS2 at x then hD ∈ MD at (x, x), and by Proposition 2.1.12 (a) we have that h ∈ M at x. Therefore, MS2 ⊆ M. Definition 3.1.9. Consider the Setup 2.2.1. We define the infinitesimal Lipschitz conditions with respect to the 2-Lipschitz saturation of a module. ∙ The pair (X,Y ) satisfies the 2-infinitesimal Lipschitz condition mY (2-iLmY ) at x ∈ X if JM(X)Y ⊆ (mY JMz (X))S2 at x; ∙ The pair (X,Y ) satisfies the 2-infinitesimal Lipschitz condition A (2-iLA ) at x ∈ X if JM(X)Y ⊆ (JMz (X))S2 at x. Here we consider the 2-Lipschitz saturation relative to the parameter space Y . Notice that 2-iLmY implies 2-iLA . Clearly the above definition is equivalent to the Definition 2.2.2. Corollary 3.1.10. Consider the Setup 2.2.1. If x = (z, y) ∈ X with z ̸= 0 then 2-iLmY and 2-iLA hold at x. Furthermore, there exists a dense Zariski open subset U of Y such the 2-iLA holds in (0, y), for all y ∈ U. Proof. By Lemma 2.2.8 we have that JM(X)Y ⊆ mY JMz (X) at x. By Proposition 3.1.8 the inclusion mY JMz (X) ⊆ (mY JMz (X))S2 holds and therefore the condition 2-iLmY holds at x. In particular, 2-iLA holds at x. The second statement is a consequence of the Theorem 2.2.11. Before to define the third Lipschitz saturation, Let us fix some notations. For each ψ : X → Hom(C p , C) analytic map, ψ = (ψ1 , ..., ψ p ) and h = (h1 , ..., h p ) ∈ OXp , p

we define ψ · h ∈ OX given by (ψ · h)(z) := ∑ ψi (z)hi (z). We define ψ · M as the ideal of OX i=1

generated {ψ · h | h ∈ M}. Lemma 3.1.11. Under the above notation, we have the following properties: a) ψ · (αg + h) = α(ψ · g) + (ψ · h), ∀g, h ∈ OXp and α ∈ OX . b) If M is generated by {h1 , ..., hr } then ψ · M is generated by {ψ · h1 , ..., ψ · hr }. Proof. It is easy to see that (b) is a straightforward consequence of (a). Now, write g = (g1 , ..., g p ) p

and h = (h1 , ..., h p ). Then, for every z we have (ψ · (αg + h))(z) = ∑ ψi (z)(α(z)gi (z) + hi (z)) = p

p

i=1

α(z) ∑ ψi (z)gi (z) + ∑ ψi (z)hi (z) = (α(ψ · g) + (ψ · h))(z), so (a) is proved. i=1

i=1

85


p Lemma 3.1.12. Suppose the submodule M ⊆ OX,x has generic rank k on each component of X. If I = (i1 , ..., ik ) and J = ( j1 , ..., jk ) are indexes with j1 = 1 then there exists ψ : X → Hom(C p , C) such that:

p a) ψ · h = JIJ (h, M), ∀h ∈ OX,x ;

b) ψ · M ⊆ Jk (M). 

 | |   Proof. Let us fix a matrix of generators of M, [M] = g1 . . . gr . We have JIJ (h, M) = | |   hi gi1 , j2 −1 . . . gi1 , jk −1  .1 .. ..  .. det[h, M]IJ , where [h, M]IJ =  . .   , for all h = (h1 , ..., h p ). Let Gil , js −1 hik gik , j2 −1 . . . gik , jk −1 be the (l, s)-cofactor of [h, M]IJ , for all l, s ∈ {1, ..., k}. Notice that the (l, 1)-cofactors Gil ,0 k

does not depend of h. Then, JIJ (h, M) = det[h, M]IJ = ∑ Gil ,0 · hil , for all h (taking the cofactor l=1

expansion at the first column, since j1 − 1 = 0). Take ψ : X → Hom(C p , C) given by (ψ1 , ..., ψ p ) p where ψil = Gil ,0 , for all l ∈ {1, ...k}, and ψ j = 0, for every index j off I. Thus, for all h ∈ OX,x we get JIJ (h, M) = ψI · hI = ψ · h. So, (a) is proved. Now, let g ∈ M arbitrary. Then (g, M) = M. By (a) we have ψ ·g = JI,J (g, M) ∈ Jk ((g, M)) = Jk (M), hence ψ · M ⊆ Jk (M).

The Propositions 1.2.5 and 1.2.11 are characterizations for the integral closure of modules using integral closure of ideals. Notice the next proposition is another characterization of the same type. p p Proposition 3.1.13. Let h ∈ OX,x and suppose that the submodule M ⊆ OX,x has generic rank k on each component of X. Then, h ∈ M at x if and only if ψ · h ∈ ψ · M at x, for every ψ : X → Hom(C p , C) analytic map.

Proof. (=⇒) Let φ : (C, 0) → (X, x) be an arbitrary curve. Since h ∈ M then h ∘ φ ∈ φ * (M), so (ψ ∘ φ )(h ∘ φ ) ∈ (ψ ∘ φ )φ * (M) ⇒ (ψ · h) ∘ φ ∈ φ * (ψ · M). Hence, ψ · h ∈ ψ · M. (⇐=) The proof use 1.2.5 and 1.2.6. By these results it is enough to check that JIJ (h, M) ∈ Jk (M), for all indexes I and J.   | |   Write I = (i1 , ..., ik ) and J = ( j1 , ..., jk ). Let [M] = g1 . . . gr  be a matrix of generators of | |

86




 | | |   M. Write h = (h1 , ..., h p ). Then, [h, M] = h g1 . . . gr . If j1 > 1 then JIJ (h, M) is a k × k | | | minor taken only among the generators of M, hence JIJ (h, M) ∈ Jk (M) ⊆ Jk (M). Now suppose that j1 = 1. By the Lemma 3.1.12 there exists ψ : X → Hom(C p , C) such that ψ · h = JI,J (h, M) and ψ · M ⊆ Jk (M). By the hypothesis we have ψ · h ∈ ψ · M, hence JI,J (h, M) = ψ · h ∈ ψ · M ⊆ Jk (M), which finishes the proof. The last proposition will be useful to prove that the 3-Lipschitz Saturation satisfies the same statement of the Propositions 3.1.3 and 3.1.8. p Definition 3.1.14. The 3-Lipschitz saturation of the submodule M ⊆ OX,x is denoted by MS3 , and is defined by p MS3 := {h ∈ OX,x | ψ · h ∈ (ψ · M)S , ∀ψ : X → Hom(C p , C) analytic map}.

In the family case, the 3-Lipschitz saturation of M at x ∈ X relative to Y is defined as above taking the Lipschitz saturation of ψ · M relative to Y . If M is an OX -submodule of OXp then we can consider the sheaf of OX -modules MS3 where each stalk(MS3 )x is (Mx )S3 . Unless we say the opposite, we denote by MS3 the stalk of the sheaf MS3 at an arbitrary point of X. Remark 3.1.15. MS3 is the Lipschitz saturation of an ideal if p = 1. First, let us prove that α · MS ⊆ (α · M)S , ∀α ∈ OX,x . In fact consider the canonical projection π : SBM (X) → X. We have a canonical projection Bα·M (X) = B(α) (BM (X)) → BM (X) which gives rise a canonical projection πα : SBα·M (X) → SBM (X) and π ∘ πα : SBα·M (X) → X is the canonical projection for the Lipschitz saturation of the blowup of X with respect to the ideal α · M. We already know that MS = {h ∈ OX,x | π * (h) ∈ π * (M)} and (α · M)S = {h ∈ OX,x | (π ∘ πα )* (h) ∈ (π ∘ πα )* (α · M)}. Thus, given h ∈ MS we have that π * (h) ∈ π * (M) which implies that (π ∘ πα )* (αh) = πα* ∘ π * (αh) = πα* (π * (α)) · πα* (π * (h)) ∈ (π ∘ πα )* (α) · (π ∘ πα )* (M) ⊆ (π ∘ πα )* (α · M). Hence αh ∈ (αM)S and therefore α · MS ⊆ (α · M)S . Finally, let us prove that MS3 = MS . Indeed, if h ∈ MS3 , taking ψ0 : X → Hom(C, C) = C, ψ0 ≡ 1 we have that h = ψ0 · h ∈ (ψ0 · M)S = MS . Conversely, let h ∈ MS . For every ψ : X → Hom(C, C) = C we can look to the germ ψ ∈ OX,x and ψ · h ∈ ψ · MS ⊆ (ψ · M)S , i.e, ψ · h ∈ (ψ · M)S . Hence, h ∈ MS3 .


87

Proposition 3.1.16. Let M be a sheaf of OX -submodules of OXp . Then: a) MS3 is an OX -submodule of OXp ; b) M ⊆ MS3 ⊆ M. In particular, M is a reduction of MS3 and, if M has finite colenght in MS3 then e(M, MS3 ) = 0. The same result holds in the family case. Proof. Let x ∈ X be an arbitrary point. (a) Let h, h′ ∈ MS3 and α ∈ OX at x, and let ψ : X → Hom(C p , C)} arbitrary. Then, ψ · h, ψ · h′ ∈ (ψ · M)S at x, and by Lemma 3.1.11 (a) we have ψ · (αh + h′ ) = α(ψ · h) + ψ · h′ ∈ (ψ · M)S at x. Therefore, αh + h′ ∈ MS3 at x. (b) If h ∈ M then ψ · h ∈ ψ · M ⊆ (ψ · M)S , so h ∈ MS3 and M ⊆ MS3 . Now, let h ∈ MS3 . Then, ψ · h ∈ (ψ · M)S ⊆ ψ · M, ∀ψ : X → Hom(C p , C). By Proposition 3.1.13 we conclude that h ∈ M. Definition 3.1.17. Consider the Setup 2.2.1. We define the infinitesimal Lipschitz conditions with respect to the 3-Lipschitz saturation of a module. ∙ The pair (X,Y ) satisfies the 3-infinitesimal Lipschitz condition mY (3-iLmY ) at x ∈ X if JM(X)Y ⊆ (mY JMz (X))S3 at x; ∙ The pair (X,Y ) satisfies the 3-infinitesimal Lipschitz condition A (3-iLA ) at x ∈ X if JM(X)Y ⊆ (JMz (X))S3 at x. Here we consider the 3-Lipschitz saturation relative to the parameter space Y . Notice that 3-iLmY implies 3-iLA . Corollary 3.1.18. Consider the Setup 2.2.1. If x = (z, y) ∈ X with z ̸= 0 then 3-iLmY and 3-iLA hold at x. Furthermore, there exists a dense Zariski open subset U of Y such the 3-iLA holds in (0, y), for all y ∈ U. Proof. By Lemma 2.2.8 we have that JM(X)Y ⊆ mY JMz (X) at x. By Proposition 3.1.16 the inclusion mY JMz (X) ⊆ (mY JMz (X))S3 holds and therefore the condition 3-iLmY holds at x. In particular, 3-iLA holds at x. The second statement is a consequence of the Theorem 2.2.11 and the Proposition 3.1.22.

88


p Proposition 3.1.19. Suppose the submodule M ⊆ OX,x has generic rank k on every component of X. Let πS : SBJk (M) (X) → BJk (M) (X), p : BJk (M) (X) → X and π = p ∘ πS be the projections maps.

πS

SBJk (M) (X)

BJk (M) (X) p

π X

p Let h ∈ OX,x . Then:

Jk (h, M) ⊆ (Jk (M))S if and only if π * (h) ∈ π * (M). Proof. Fix a set of generators {g1 , ..., gr } of M. We work at x′ ∈ E, E = πS−1 (EB ), EB the exceptional divisor of BJk (M) (X). Suppose first that Jk (h, M) ⊆ Jk (M)S . Let π * (JI,J (M)) be a local generator of the principal ideal π * (Jk (M)). Then, by Cramer’s rule we can write (1)

(JI,J (M) ∘ π)(hI ∘ π) =

∑ (JI, j (hI , MI ) ∘ π)(mI, j ∘ π) j∈J

with m j ∈ M. We claim in fact that (JI,J (M) ∘ π)(h ∘ π) =

∑ (JI, j (hI , MI ) ∘ π)(m j ∘ π). j∈J

To see this pick a curve φ : (C, 0) → (BJk (M) (X)S , x′ ) and choose φ so that the rank of π * (M)|φ is generically k. Since by hypothesis h ∈ M then h ∘ π ∘ φ ∈ φ * (π * (M)). So, the element ? = h ∘ π ∘ φ − ∑( j∈J

JI, j (hI , MI ) ∘ π ∘ φ )(m j ∘ π ∘ φ ) ∈ φ * (π * (M)) JI,J (M) ∘ π

because the above quotients by hypothesis are regular functions on SBJk (M) (X). By equation (1), the above element has 0 for the entries indexed by I. So, if ? is not zero then φ * (π * (M)) has rank at least k + 1, which is a contradiction. Since the images of φ fill up a Z-open set it follows that (see Lemma 3.3 of [48]) ? is locally zero. Therefore, h∘π =

∑( j∈J

JI, j (hI , MI ) ∘ π )(m j ∘ π) ∈ π * (M). JI,J (M) ∘ π

Conversely, suppose that π * (h) ∈ π * (M). Then we can write r

(2)

h∘π =

∑ α j (g j ∘ π)

j=1

89


where α j ∈ OeBJ (M) (X),x′ , ∀ j ∈ {1, ..., r}. k

Let c be an arbitrary generator of Jk (h, M). Then we can write c = det[h, M]IJ for some k-indexes I and J. If the k-index J does not pick the first column of [h, M], then c is a k × k minor of the matrix [M], hence c ∈ Jk (M) ⊂ (Jk (M))S and we are done. Thus we may assume that J pick the first column of [h, M]. Then, we can write h i c = det hI (g j1 )I ... (g jk−1 )I . Applying the pullback of the projection map we have h i π * (c) = det hI ∘ π (g j1 )I ∘ π ... (g jk−1 )I ∘ π . By the equation (2), if we look only to the entries of the k-index I, we get r

hI ∘ π =

∑ α j ((g j )I ∘ π).

j=1

r

Using the linearly of the determinant in the first column, we conclude that π * (c) = ∑ α j (v j ∘ π), j=1

where h i v j := det (g j )I (g j1 )I ... (g jk−1 )I , for all j ∈ {1, .., r}. Clearly, v j ∈ Jk (M), ∀ j ∈ {1, ..., r}. Hence, π * (c) ∈ π * (Jk (M)) and by the definition of the Lipschitz saturation of an ideal, we conclude that c ∈ (Jk (M))S . Therefore, Jk (h, M) ⊆ (Jk (M))S . p Corollary 3.1.20. Suppose the submodule M ⊆ OX,x has generic rank k on every component of X. If Jk (h, M) ⊆ (Jk (M))S then there exists an open covering of X such that h can be written on each U of the covering as an element of M by using Lipschitz functions (Lipschitz with respect to J ′ ′ (M) (zi ) and ( JII,J,J (M) ), where JI,J (M) is the minor which gives the local generator on the preimage of U in SBJk (M) (X)).

Proof. It is a straightforward consequence of the proof of the last proposition, more precisely, the equation (2) above. p Proposition 3.1.21. Suppose the submodule M ⊆ OX,x has generic rank k on each component of X. If h ∈ MS3 then Jk (h, M) ⊆ (Jk (M))S .

Proof. By Theorem 2.3 of [40] it is enough to prove that (Jk (h, M))D ⊆ (Jk (M))D at (x, x). So, we have to prove that all the generators (JIJ (h, M))D and (0, (zi ∘ π1 − zi ∘ π2 )(JIJ (h, M)) ∘ π2 ) are in (Jk (M))D , for all i ∈ {1, ...n} and for all indexes I and J. Write I = (i1 , ..., ik ) and J = ( j1 , ..., jk ). We have two cases. Suppose j1 > 1. Then, as we have seen, JIJ (h, M) ∈ Jk (M), so (JIJ (h, M))D ∈ (Jk (M))D , in particular, (JIJ (h, M))D ∈ (Jk (M))D . Furthermore, since JIJ (h, M) ∈ Jk (M), by Lemma 2.1.2

90


(b) we have that (0, (zi ∘ π1 − zi ∘ π2 )(JIJ (h, M)) ∘ π2 ) ∈ (Jk (M))D ⊆ (Jk (M))D . Suppose j1 = 1. By the Lemma 3.1.12 there exists ψ : X → Hom(C p , C) such that ψ · h = JIJ (h, M) and ψ · M ⊆ Jk (M). Since h is in the 3-Lipschitz Saturation of M at x then ψ · h ∈ (ψ · M)S , which is equivalent to (ψ · h)D ∈ (ψ · M)D . Thus, (JI,J (h, M))D = (ψ · h)D ∈ (ψ · M)D ⊆ (Jk (M))D . Let us prove that (0, (zi ∘ π1 − zi ∘ π2 )(JIJ (h, M) ∘ π2 )) ∈ (Jk (M))D by using the curve criterion. Let φ : (C, 0) → (X × X, (x, x)) be an analytic curve. Write φ = (φ1 , φ2 ). Since ψ · h ∈ (ψ · M)S ⊆ ψ · M then we can write (ψ · h) ∘ φ2 = ∑ β j ((ψ · g j ) ∘ φ2 ), with β j ∈ OC,0 and g j ∈ M. Thus, we get (0, (zi ∘ j

π1 − zi ∘ π2 )(JIJ (h, M) ∘ π2 )) ∘ φ = ∑ β j ((0, (zi ∘ π1 − zi ∘ π2 )(ψ · g j ) ∘ π2 ) ∘ φ ) ∈ (ψ · M)D ∘ φ ⊆ j

(Jk (M))D ∘ φ , which finishes the proof. Proposition 3.1.22. Let M be a sheaf of OX -submodules of OXp . Then, MS2 ⊆ MS3 in every point x ∈ X. Proof. Let h = (h1 , ..., h p ) ∈ MS2 at x. Then hD ∈ MD at (x, x). It is enough to check that (ψ · h)D ∈ (ψ · M)D , for all ψ : X → Hom(C p , C). Let φ = (φ1 , φ2 ) : (C, 0) → (X × X, (x, x)) be an arbitrary analytic curve. Since hD ∈ MD then we can write hD ∘ φ = ∑ α j φ * ((g j )D ) j

with g j = (g1 j , ..., g p j ) ∈ M and α j ∈ OC,0 for all j. Looking to the above equation and comparing the 2p coordinates we conclude that hi ∘ φ1 = ∑ α j (gi j ∘ φ1 ) and hi ∘ φ2 = ∑ α j (gi j ∘ φ2 ), for all j

j

i ∈ {1, ..., p}. Let ψ1 , ..., ψ p be the coordinates functions of ψ. Thus: (ψ · h)D ∘ φ = ((ψ · h) ∘ φ1 , (ψ · h) ∘ φ2 ) = (∑(ψi ∘ φ1 ) · (hi ∘ φ1 ), ∑(ψi ∘ φ2 ) · (hi ∘ φ2 )) i

i

= (∑(ψi ∘ φ1 )α j (gi j ∘ φ1 ), ∑(ψi ∘ φ2 )α j (gi j ∘ φ2 )) i, j

i, j

= ∑ α j (∑(ψi ∘ φ1 ) · (gi j ∘ φ1 ), ∑(ψi ∘ φ2 ) · (gi j ∘ φ2 )) j

i

i

= ∑ α j ((ψ · g j ) ∘ φ1 , (ψ · g j ) ∘ φ2 ) j

= ∑ α j ((ψ · g j )D ∘ φ ) ∈ (ψ · M)D ∘ φ , and the proof is done. | {z } j ∈(ψ·M)D ∘φ

p Definition 3.1.23. Suppose that M ⊆ OX,x is an OX,x -submodule of generic rank k on each component of X. The 4-Lipschitz saturation of M is denoted by MS4 , and is defined by p MS4 := {h ∈ OX,x | Jk (h, M) ⊆ (Jk (M))S }.

In the family case, the 4-Lipschitz saturation of M at x ∈ X relative to Y is defined as above taking the Lipschitz saturation of Jk (M) relative to Y .


91

If M is an OX -submodule of OXp then we can consider the sheaf of OX -modules MS4 where each stalk(MS4 )x is (Mx )S4 . Unless we say the opposite, we denote by MS4 the stalk of the sheaf MS4 at an arbitrary point of X. Remark 3.1.24. MS4 is the Lipschitz saturation of an ideal if p = 1. In fact, since p = 1 then the generic rank of M is 0 or 1. If the generic rank is 0 then M = MS4 = MS = (0). Suppose the generic rank is 1. Then: h ∈ MS4 ⇔ J1 (h, M) ⊆ (J1 (M))S ⇔ (h, M) ⊆ MS ⇔ h ∈ MS . The next proposition allows us to see MS4 as a sheaf of OX -submodules of OXp . Proposition 3.1.25. Let M be a sheaf of OX -submodules of OXp of generic rank k on each component of X. Then: a) MS4 is an OX -submodule of OXp ; b) M ⊆ MS4 ⊆ M. In particular, M is a reduction of MS4 and, if M has finite colenght in MS4 then e(M, MS4 ) = 0. The same result holds in the family case. Proof. (a) Let g, h ∈ MS4 at x ∈ X and α ∈ OX,x . By the basic properties of determinants we have that Jk (αg + h, M) ⊆ αJk (g, M) + Jk (h, M) ⊆ (Jk (M))S . Hence, αg + h ∈ MS4 at x. (b) Since Jk (M) ⊆ (Jk (M))S then M ⊆ MS4 . Now, let h ∈ MS4 . Thus, Jk (h, M) ⊆ (Jk (M))S ⊆ Jk (M) which implies that h ∈ M. Therefore, the inclusion MS4 ⊆ M is proved. Definition 3.1.26. Consider the Setup 2.2.1. We define the infinitesimal Lipschitz conditions with respect to the 4-Lipschitz saturation of a module. ∙ The pair (X,Y ) satisfies the 4-infinitesimal Lipschitz condition mY (4-iLmY ) at x ∈ X if JM(X)Y ⊆ (mY JMz (X))S4 at x; ∙ The pair (X,Y ) satisfies the 4-infinitesimal Lipschitz condition A (4-iLA ) at x ∈ X if JM(X)Y ⊆ (JMz (X))S4 at x. Here we consider the 4-Lipschitz saturation relative to the parameter space Y . Notice that 4-iLmY implies 4-iLA . Corollary 3.1.27. Consider the Setup 2.2.1. If x = (z, y) ∈ X with z ̸= 0 then 4-iLmY and 4-iLA hold at x. Furthermore, there exists a dense Zariski open subset U of Y such the 4-iLA holds in (0, y), for all y ∈ U.

92


Proof. By Lemma 2.2.8 we have that JM(X)Y ⊆ mY JMz (X) at x. By Proposition 3.1.25 the inclusion mY JMz (X) ⊆ (mY JMz (X))S4 holds and therefore the condition 4-iLmY holds at x. In particular, 4-iLA holds at x. The second statement is a consequence of the Theorem 2.2.11, Propositions 3.1.21 and 3.1.22. Notice that we can rephrase the Proposition 3.1.21 by saying that MS3 ⊆ MS4 at every point x ∈ X. Example 3.1.28. Consider X = C with coordinate t and p = 2. Consider the OC,0 -submodule 2 generated by {(t, 0)}. Let h := (0,t) ∈ O 2 . Notice that the generic rank of M M ⊆ OC,0 C,0 " # " # t 0 t is 1, [M] = and [h, M] = . Thus, J1 (h, M) = (t) and J1 (M) = (t). In particular, 0 t 0 J1 (h, M) ⊆ J1 (M) ⊆ (J1 (M))S which implies that h ∈ MS4 . Now consider ψ : C → Hom(C2 , C) given by ψ(t) := (t, 1). Then ψ · M = (t 2 ) and ψ · h = t. Since t ∈ / (t 2 ) then ψ · h ∈ / ψ · M, in particular, ψ · h ∈ / (ψ · M)S . Hence, h ∈ / MS3 and we conclude that MS3 ( MS4 . In [39] Gaffney gave a geometric interpretation of the infinitesimal Lipschitz condition A, presenting vector fields defined in a open covering of BJz (G) (X), where X is a hypersurface defined by G. Here we extend this interpretation for the jacobian module case. Consider X a family of analytic varieties as on the setup 2.2.1. Let s be the generic rank of M := JMz (X). Let g be the number of generators of Js (M), i.e, g = #{(I, J) | I and J are s-indexes}. Let TIJ be the homogeneous coordinates of Pg−1 and MIJ be the generators of Js (M), (I, J) varying on the set of all pairs of s-indexes. Consider the diagram SBJs (M) (X)

πS

BJs (M) (X)

⊆ X × Pg−1

p

π X

as in the Proposition 3.1.19. Let VIJ be the dense Zariski open subset of Pg−1 defined by TIJ ̸= 0 and let UIJ := BJs (M) (X) ∩ (X ×VIJ ). Clearly the collection {UIJ }(I,J) is an open covering of BJs (M) (X). At each point of UIJ , MIJ ∘ π is a local generator of the principal ideal sheaf π * (Js (M)). Suppose that JM(X)Y ⊆ (JMz (X))St for some t ∈ {2, 3, 4}. By Propositions 3.1.21 and 3.1.22 we have JM(X)Y ⊆ (JMz (X))S4 . By the proof of the Proposition 3.1.19, for each ` ∈ {1, ..., k} we can write MIJ, j,` ∘ π ∂ F ∂F ∘π = ∑ ∘π ∂ y` ∂zj j∈J MIJ ∘ π

93

3.2. The generic equivalence among the Lipschitz saturations

where MIJ, j,` is the determinant of the matrix obtained by replacing the jth column of MIJ by ( ∂∂ yF )I , for all j ∈ J. `

Thus, for each pair of s-indexes (I, J), we have s.k vector fields tangent to X → v IJ, j,` M

:=

MIJ, j,` ∂ ∂ − ∂ y` MIJ ∂ z j →

∘π

defined on π(UIJ ). Since MIJ,IJj,`∘π is Lipschitz, for all j ∈ J and ` ∈ {1, ..., k} then v IJ, j,` is a Lipschitz vector field relative to Y .

3.2

The generic equivalence among the Lipschitz saturations

Now we look for conditions ensuring the above notions of Lipschitz saturations are generically equivalent. The next theorem gives us a such condition. p Theorem 3.2.1. Let M be an OX,x -submodule of OX,x with generic rank k on each component of p 2p X and h ∈ OX,x . Suppose that there exists an ideal I of OX×X,(x,x) such that

I.J2 ((Jk (M))D ) ⊆ J2k (MD ) J2k (hD , MD ) ⊆ I.J2 ((Jk (h, M))D ). Then, the following condition are equivalent: (a) h ∈ MS2 ; (b) h ∈ MS3 ; (c) h ∈ MS4 . Proof. (a) =⇒ (b) Follows from the Proposition 3.1.22. (b) =⇒ (c) Follows from the Proposition 3.1.21. (c) =⇒ (a) Suppose that Jk (h, M) ⊆ (Jk (M))S . Then, (Jk (h, M))D ⊆ (Jk (M))D and so (Jk (h, M))D = (Jk (M))D . Let us prove that J2k (hD , MD ) ⊆ J2k (MD ). In fact, let φ : (C, 0) → (X × X, (x, x)) be an arbitrary analytic curve. Since (Jk (h, M))D = (Jk (M))D then φ * ((Jk (h, M))D ) = φ * ((Jk (M))D ). Using the inclusions of the hypothesis and the curve criterion for the integral closure of modules we have φ * (J2k (hD , MD )) ⊆ φ * (I.J2 ((Jk (h, M))D )) = φ * (I).φ * (J2 ((Jk (h, M))D ))

94


= φ * (I).J2 (φ * ((Jk (h, M))D )) = φ * (I).J2 (φ * ((Jk (M))D )) = φ * (I).φ * (J2 ((Jk (M))D )) = φ * (I.J2 ((Jk (M))D )) ⊆ φ * (J2k (MD )). Hence, the desired inclusion is proved. By Theorem 2.1.17 we have that MD has generic rank 2k on each component of (X × X, (x, x)). Therefore, by Corollary 1.2.6 we conclude that hD ∈ MD . p Corollary 3.2.2. Let M ⊆ OX,x be an OX,x -submodule of generic rank k on each component of 2p X. Suppose that there exists an ideal I of OX×X,(x,x) such that

I.J2 ((Jk (M))D ) ⊆ J2k (MD ) J2k (hD , MD ) ⊆ I.J2 ((Jk (h, M))D ), ∀h ∈ M. Then MS2 = MS3 = MS4 In order to use the above criterion, we prove some useful lemmas that allow us to get the desired equivalence. Notation: For each object A, we denote A = A ∘ π1 and A′ = A ∘ π2 . For k-indexes I, J, we denote by MIJ the submatrix of [M] defined by these k-indexes. p Lemma 3.2.3. Let M ⊆ OX,x be a submodule and k ∈ N. Then ′ (zt1 − zt′1 )...(ztk − zt′k ) det(MIJ ) det(MKL ) ∈ J2k (MD ) at (x, x)

for every t1 , ...,tk ∈ {1, ..., n} and k-indexes I, J, K, L. Proof. Write MKL = (grs ). The 2k × 2k matrix  MIJ | 0k×k  − − − − − − −− − − − − − − − − − − −−   (zt1 − zt′1 )g′11 ... (ztk − zt′k )g′1k   .. .. ′  MIJ | . .  (zt1 − zt′1 )g′k1 ... (ztk − zt′k )g′kk

        

′ ), is a submatrix of [MD ]. So, its determinant, which is (zt1 − zt′1 )...(ztk − zt′k ) det(MIJ ) det(MKL belogns to J2k (MD ).

The next result gives us the first condition of the Theorem 3.2.1 in terms of the ideal coming from the diagonal. p Lemma 3.2.4. Let M ⊆ OX,x be a submodule and k ∈ N.

a) I∆k J2 ((Jk (M))D ) ⊆ J2k (MD ) at (x, x); b) If Jk (M) is principal then I∆k−1 J2 ((Jk (M))D ) ⊆ J2k (MD ) at (x, x).


95

Proof. (a) We have that # " det(MIJ ) ... 0 [(Jk (M))D ] = ′ ) ′ ′ det(MIJ ) ... (zi − zi ) det(MKL varying the k-indexes I, J, K, L and i ∈ {1, ..., n}. Thus, the desired inclusion is a straightforward consequence of the previous lemma. (b) Since Jk (M) is principal then there exist k-indexes I, J such that g = det(MIJ ) and Jk (M) is generated by {g}.Thus we can write " # g 0 [(Jk (M))D ] = ′ g (zi − z′i )g′ varying i ∈ {1, ...n}. Thus, in this case J2 ((Jk (M))D ) is generated by {g.g′ (zi − z′i ) | i ∈ {1, ..., n}}. So, by previous lemma we have that [(zt1 − zt′1 )...(ztk−1 − zt′k−1 )].[g.g′ (zi − z′i )] ∈ J2k (MD ), for all t1 , ...,tk−1 , i ∈ {1, ..., n}. Therefore, I∆k−1 J2 ((Jk (M))D ) ⊆ J2k (MD ) at (x, x). Now, we start to get conditions in order to obtain the second condition of the Theorem 3.2.1. p ˜ the matrix such Lemma 3.2.5. Let M be an OX,x -submodule of OX,x and k ∈ N. Denote by [M] that " # [M] 0 [MD ] = . ˜ [M]′ [M]

Let J2k (MD ) be the subideal of J2k (MD ) generated by {det(MIJ ) det(M˜ KL ) | I, J, K, L are k-indexes}. Then, J2k (MD ) ⊆ I∆k−1 J2 ((Jk (M))D ) at (x, x). Proof. It suffices to prove that each generator of J2k (MD ) belongs to I∆k−1 J2 ((Jk (M))D ). The columns of M˜ KL are columns of [M]′ possibly in a different order with terms of type zi − z′i multiplied on each column. If the k-indexes K, L pick repeated columns then det(M˜ KL ) = 0 and we are done. If this does not occur then det(M˜ KL ) = (zi1 − z′i1 )...(zik − z′ik )(± det(MK′ ′ L′ ))

96


for some reorganization k-indexes K ′ , L′ , where i1 , ..., ik ∈ {1, ..., n} are the indexes which zi1 − z′i1 , ..., zik − z′ik appear on each of the k columns of M˜ KL . ′ ) ∈ J ((J (M)) ) Since ±(zi1 −z′i1 )...(zik−1 −z′ik−1 ) ∈ I∆k−1 and det(MIJ )(zik −z′ik ) det(MKL D 2 k k−1 ˜ then det(MIJ ) det(MKL ) ∈ I∆ J2 ((Jk (M))D ), which finishes the proof.

The next result ensures that any free submodule satisfies the second condition of the Theorem 3.2.1 for a suitable power of the ideal coming from the diagonal. p Lemma 3.2.6. Let M be a free OX,x -submodule of OX,x of rank k. Then

J2k (MD ) ⊆ I∆k−1 J2 ((Jk (M))D ) at (x, x). Proof. We have that 

 [M] p×k | 0   [MD ] = − − − − −− | − − − − − − − [M]′p×k | (zi − z′i )[M]′ varying i ∈ {1, ..., n}. Let d ∈ J2k (MD ) at (x, x) be an arbitrary generator. Then d = det N where N is a 2k × 2k submatrix of [MD ]. (i) Suppose first that there are k + t columns of N taken on the part " # 0 (zi − z′i )[M]′ of [MD ], with 1 ≤ t ≤ k. Then we can write 

 (MIJ )k×(k−t) | 0k×(k+t)   N = − − − − − − −− | − − − − − − −− ˜ KL (MIJ )′k×(k−t) | (M) 

 MIJ 0k×t | 0k×k   = − − − − − − | − − − − −− ′ MIJ * | ** for some matrices * hand **, where i ** is a square matrix of size k × k. So in this case we have that d = det N = det MIJ 0k×t . det(**) = 0 ∈ I∆k−1 J2 ((Jk (M))D ). (ii) Now suppose that we have exactly k columns of N taken on the part " # 0 (zi − z′i )[M]′


97

of [MD ]. Then we can write 

 MIJ | 0k×k   N = − − − − −− | − − − − −− . ′ MIJ | M˜ KL Thus, d = det(MIJ ) det(M˜ KL ) ∈ J2k (MD ) and by previous lemma we conclude that d ∈ I∆k−1 J2 ((Jk (M))D ). As a consequence of the previous lemma, we get another lemma which states that the second condition of the Theorem 3.2.1 holds in a dense Zariski open subset of X in a sheaf of submodules of OXp . Lemma 3.2.7. Let M be a sheaf of OX -submodules of OXp of generic rank k on each component of X. Then there exists a dense Zariski open subset U of X such that U ∩V is a dense Zariski open subset of V , ∀V component of X, Jk (M ) is principal at x and J2k (MD ) ⊆ I∆k−1 J2 ((Jk (M ))D ) at (x, x), for all x ∈ U. Proof. Consider [M ] a matrix of generators of M . Using the Cramer’s rule, we can choose k OX -linear independents columns of [M ] such that these columns generates M in a such dense Zariski open subset U of X. Let Mk be the OX -submodule of OXp generated by the columns chosen above. Thus, given x ∈ U, we have that Mx = (Mk )x is a free OX,x -submodule of p OX,x of rank k and the desired inclusion is a consequence of the previous lemma. The ideal Jk (M ) = Jk (Mk ) is principal at x, because [Mk ] is a square k × k matrix. p Lemma 3.2.8. Let M be an OX,x -submodule of OX,x of generic rank k on each component of X, where OX,x is a reduced ring. If h ∈ M then (h, M) also has generic rank k on each component of X.

Proof. Let V be an arbitrary component of (X, x) and let t be the generic rank of (h, M) at V . O Since V is irreducible then there exists a prime ideal Ix of OX,x such that OV,x = IX,x . Since Ix is x a prime ideal (hence radical ideal) then OV,x is a domain (hence reduced ring). Since M ⊆ (h, M) then t ≥ k. Suppose by contradiction that t > k. Then Jt (M) = (0), p once M has generic rank k. Since h ∈ M then (see [41]) Jt (h, M) ⊆ Jt (M) = (0) ⊆ (0) = (0) at the component V . Hence, Jt (h, M) = (0) at V , contradiction. Therefore, t = k. The next theorem gives us the equivalence desired.

98


Theorem 3.2.9. Let M be a sheaf of OX -submodules of OXp of generic rank k on each component of X, and suppose that X has reduced structure. Then there exists a dense Zariski open subset U of X such that U ∩V is a dense Zariski open subset of V , for all V component of X and MS2 = MS3 = MS4 along U. In the case when k = p is the maximal rank, the same conclusion holds even if X does not have reduced structure. Proof. Since X ⊆ Cn is an analytic complex variety then OX is a noetherian sheaf of rings, hence OXp is a noetherian sheaf of OX -modules. Since MS4 is a sheaf of OX -modules then MS4 is finitely generated by global sections h1 , ..., hr . Since hi ∈ M , ∀i ∈ {1, ..., r} then by previous lemma (hi , M ) also has generic rank k on each component of X, ∀i ∈ {1, ..., r}. By Lemma 3.2.7, for each i ∈ {1, ..., r} there exists a dense Zariski open subset Ui of X such that Ui ∩V is a dense Zariski open subset of V , for all V component of X, and J2k ((hi )D , MD ) ⊆ I∆k−1 J2 ((Jk (hi , M ))D ) at (x, x), ∀x ∈ Ui . Also by Lemma 3.2.7 there exists a dense Zariski open subset U0 of X such that U0 ∩V is a dense Zariski open subset of V , for all component V of X, and Jk (M) is principal at x, ∀x ∈ U0 . From the Lemma 3.2.4(b), we conclude that I∆k−1 J2 ((Jk (M ))D ) ⊆ J2k (MD ) at (x, x), ∀x ∈ U0 . Take U :=

r T

U j . Then U is a dense Zariski open subset of X and U ∩ V is a dense

j=0

Zariski open subset of V , for all V component of X. We already know that MS2 ⊆ MS3 ⊆ MS4 at any point of X, in particular, along U. Let us prove that MS4 ⊆ MS2 along U. In fact, let x ∈ U. Given an arbitrary i ∈ {1, ..., r}, since x ∈ U0 and x ∈ Ui then I∆k−1 J2 ((Jk (M ))D ) ⊆ J2k (MD ) at (x, x) and J2k ((hi )D , MD ) ⊆ I∆k−1 J2 ((Jk (hi , M ))D ) at (x, x). Since hi ∈ MS4 at x then by Theorem 3.2.1 we have that hi ∈ MS2 at x, ∀i ∈ {1, ..., r}. Since MS4 is generated by h1 , ..., hr at x, then MS4 ⊆ MS2 at x, and the theorem is proved.

99

3.3. Geometric applications

3.3

Geometric applications

The next result gives us conditions envolving the double and the integral closure of modules so that we can construct Lipschitz vector fields. Proposition 3.3.1. Let X ⊆ Cn be an analytic curve. Let M be an OX -submodule of OXp of generic rank k, with k + 1 generators. Let Mk be a reduction of M generated by the first k columns of [M ], whose rank is k. Let h ∈ OXp . Let c(h) ∈ OXp be defined by the solution of the equation [Mk ] · c(h) = h in a dense Zariski open subset U of X (using the Cramer’s Rule). Let x ∈ U, M = Mx , Mk = (Mk )x and m = mX,x . If h ∈ mM and hD ∈ (Mk )D at (x, x) then the vector field c(h) is Lipschitz, p i.e, c(h) − c(h)′ ∈ I∆ OX×X at (x, x). Proof. Let us use the curve criterion. Let φ : C, 0 → X × X, (x, x) be a curve, with coordinates φ1 , φ2 . First suppose φ1 (t) ≡ 0. In this case we have " # " # " # h ∘ φ1 0 0 = = ′ h ∘ φ2 h ∘ φ2 [Mk ] · c(h)

p mod φ2* (mMk′ )OC,0

because h ∘ φ2 ∈ φ2* (mMk′ ). Let m1 , ..., mk be the columns of [Mk ]. Suppose that zi ∘ φ2 is a generator of φ2* (m). For each j ∈ {1, ..., k} look to h i h i ′k−1 ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ det zi m1 ... zi m j−1 h zi m j+1 ... zi mk = zi det m1 ... m j−1 h m j+1 ... mk . Then,

h i h i ′ ′ ′ ′k−1 det m′ ... m′ det z′i m′1 ... z′i m′j−1 h′ z′i m′j+1 ... z′i m′k z 1 j−1 h m j+1 ... mk h i = i ′k det[Mk ]′ zi det z′i m′1 ... z′i m′k = z1′ c(h)′j which is analytic along φ2 , ∀ j ∈ {1, ..k}. Thus, c(h)′ ∈ z′i OXp along φ2 and since i p c(h) = 0 along φ1 then φ * (c(h) − c(h)′ ) ∈ φ * (I∆ OX×X ). The case where φ2 ≡ 0 is analogous. Now assume that φ1 , φ2 ̸= 0. Since (hD ) ∈ (Mk )D then (0, [Mk ]′ .(c(h)′ − c(h)) ∈ (Mk )D . p Since Mk ∘ φ1 has rank k generically, then Mk′ (c(h)′ − c(h)) ∈ Mk′ I∆ OX×X at (x, x) along p ′ φ , so c(h) − c(h) ∈ I∆ OX×X along φ at (x, x), which finishes the proof.

100


Bi-Lipschitz equisingularity in an ICIS family of irreducible curves Let X ⊆ C × Cn be an ICIS family of irreducible curves. Assume that we have a normalization F : C × C → X ⊆ C × Cn which is a homeomorphism, and suppose that the family X is Whitney equisingular. Let p be the multiplicity of X and assume we can write F(t, s) = (t, F1 (t, s), ..., Fn−1 (t, s), s p ) where for each parameter t the order of s ↦→ Fi (t, s) is greater than p, ∀i ∈ {1, ..., n − 1}. We already know that if equisingular.

∂ F1 ∂ Fn ∂t , ..., ∂t

are Lipschitz functions on X then X is bi-Lipschitz

Let G : C × Cn → Cq be an analytic map that defines X, i.e, X = G−1 (0). Consider the jacobian module JM(X) = JM(G). Here we consider the double and the 2-Lipschitz saturation relative to the parameter space Y = C × 0 ≡ C. The following proposition characterizes the above condition in terms of the 2-Lipschitz saturation (or the double), which gives us an infinitesimal condition for the bi-Lipschitz equisingularity. Proposition 3.3.2. With the above notations, the functions ∂∂tF1 , ..., ∂∂tFn are Lispchitz if and only if ∂G )D ∈ (DGn−1 )D ( ∂t where DGn−1 denotes the submodule of JMz (X) generated by the first n − 1 partial derivatives of G with respect to the z coordinates. In particular, if JM(X)Y ⊆ (JMz (X))S2 then X is bi-Lipschitz equisingular. Proof. Since G ∘ F ≡ 0 then  0=

∂ (G ∘ F) = ∂t

h

∂G ∂t

∘F

∂G ∂ z1

∘ F ...

∂G ∂ zn−1

∘F

which implies that 

∂ F1 ∂t



 .  ∂G .  ∘ F = −[DGn−1 ∘ F] ·   . . ∂t ∂F n−1

∂t

1



 ∂ F1    i  ∂t   ..  ∂G  ∂ zn ∘ F ·   .   ∂ Fn−1   ∂t  0

101

3.3. Geometric applications

We also have that 

∂ F1 ∂s



 .   ..  ∂ (G ∘ F)  0= = [JM(G) ∘ F] ·   ∂ Fn−1  . ∂s   ∂s

ps p−1

Hence,  x˙1 (s) − ps p−1   ∂G .. .  ∘ F = [DGn−1 ∘ F] ·  .  ∂ zn x˙n−1 (s) − ps p−1 

Thus, for each parameter t we have that JMz (X)|Xt ⊆ DGn−1 |Xt . Since X is a Whitney equisingular family of curves then the multiplicity of the members of the family does not depend on the parameter and we can apply the Principle of Specialization (Theorem 1.8 of [48]) to conclude that JMz (X) ⊆ DGn−1 , i.e, DGn−1 is a reduction of JMz (X). Let c( ∂∂tG ) be the vector field associated to the Cramer’s rule in order to solve the equation [

∂G ] = [DGn−1 ] · ξ . ∂t



   − ∂∂tF1 − ∂∂tF1  .   .  ∂G  .  .  Since ∂∂tG ∘ F = [DGn−1 ∘ F] ·   .  then c( ∂t ) ∘ F =  . . − ∂ F∂tn−1 − ∂ F∂tn−1 Therefore, ∂∂tF1 , ..., ∂∂tFn are Lipschitz functions if and only if c( ∂∂tG ) ∘ F is Lipschitz which is equivalent to ( ∂∂tG )D ∈ (DGn−1 )D .

103

CHAPTER

4 BI-LIPSCHITZ EQUISINGULARITY OF DETERMINANTAL SURFACES

In [70] Mostowski shows that every analytic variety has a stratified vector field which is Lipschitz, however, this vector field is not canonical from the variety. In [40], Gaffney got conditions so that a family of irreducible curves has a canonical vector field which is Lipschitz, namely ∂ ∂y

+ ∑ ∂∂ yf · ∂∂zi e

i

e where Fe : C × C −→ C × Cn given by F(y,t) = (y, f˜(y,t)) defines the family of curves. In this ˜ to be independent of the parameter y. case, the main condition is the Segre number s2 (ID (F)) ˜ I∆ ) to Gaffney showed that the last condition is equivalent to the multiplicity of the pair e(ID (F), be independent of the parameter y, where I∆ is the ideal which defines the diagonal on C × C and ˜ is the ideal generated by the doubles of the components of F. ˜ For more about the Segre ID (F) numbers see [46]. In this chapter we get conditions so that the above vector field is Lipschitz on the context of determinantal varieties. It following the approach of Pereira and Ruas [89], we see that for the special case of determinantal surfaces, there are deformations f˜ : Cq −→ Hom(Cn , Cm ) such that the above vector field always is Lipschitz.

104

4.1

Chapter 4. Bi-Lipschitz Equisingularity of Determinantal Surfaces

Simple isolated Cohen-Macaulay of codimension 2 singularities in C4 and C5

In [37], Frübis-Krüger and Neumer determine a complete classification of simple CohenMacaulay codimension 2 singularities. This classification was obtained up to isomorphism of germs. A singularity is called simple if it can only deform into finitely many different isomorphism classes. The Cohen-Macaulay singularities in codimension 2 are particularly important. One special reason is that not all of them are complete intersection, however, the theorem of HibertBruch provides a powerful tool in order to describe these singularities and their deformations. First they classified the possible candidates of 1-jets with 4 or more variables. Theorem 4.1.1. ([37] Lemma 3.2) Let M be a 3 × 2 matrix with entries in the maximal ideal of C{x1 , ..., xm }. Then, j1 M is contact-equivalent to one of the jets in the following tables or is contact-equivalent to a 1-jet containing only 3 or less variables.

J (6,1)

6 variables ! x y v J (5,1) z w u J (5,2)

5 variables ! z y v J (4,1) z w x! x y v J (4,2) z w 0 J (4,3) J (4,4) J (4,5) J (4,6)

4 variables ! w y x z w y! w y x z w 0! 0 y x z w 0! x y z z w 0! x y 0 z w 0! x y z w 0 0

Table 1 – Possible candidates of 1-jets with 4 or more variables

In [37] the authors have seen that 1-jets containing only 2 or fewer variables cannot be simple in dimension 4, so we have to consider only the 1-jets with 3 and 4 variables. We denote τ as the Tjurina number.

4.1. Simple isolated Cohen-Macaulay of codimension 2 singularities in C4 and C5

105

Theorem 4.1.2. ([37] Theorem 3.3) The list of simple isolated Cohen-Macaulay in codimension 2 singularities in (C4 , 0) are the following ones:

Jet-type

Type

J (4,1)

Λ1,1

J (4,2)

Λk,1

J (4,3)

Λk,l

J (4,4)

J (3,1) J (3,2)

J (3,3)

Presentation Matrix ! w y x z w y! w y x z w yk ! wl y x z w yk ! z y x 2 x w y + zk ! z y x x w yz + yk w ! z y x x w yz + yk ! z y x 2 x w z + yw! z y x 2 x w z + y3 ! z y + wl wm wk y x! z y xl + w2 wk x y ! z y + wl xw wk x y ! z y xw + wl wk x y ! z y + w2 x2 wk x y! z y x2 + w3 wk x y ! z y xw + wk y x z ! z y xw y x z + wk ! z y xw y + wk x z

τ

Name of triple point in [100]

2

A0,0,0

k≥2

k+1

A0,0,k−1

k≥l≥2

k+l

A0,l−1,k−1

k≥2

k+3

Ck+1,0

k≥1

2k + 4

B2k+2,0

k≥3

2k + 1

B2k−1,0

7

D0

8

F0

k, l, m ≥ 2

k+l +m−1

Ak−1,l−1,m−1

k, l ≥ 2

k+l +2

Cl+1,k−1

k, l ≥ 2

k + 2l + 1

B2l,k−1

k ≥ 2, l ≥ 3

k + 2l

B2l+1,k−1

k≥2

k+6

Dk−1

k≥2

k+7

Fk−1

3k + 1

H3k

3k + 2

H3k+1

3k + 3

H3k+2

106


z y z y z y

! y w2 x z + x2 ! y x3 + w2 x z ! y x2 x z + w2

8 9 9

Table 2 – Simple isolated Cohen-Macaulay in codimension 2 singularities in (C4 , 0)

In the case in (C5 , 0), in [37] the authors realized that we only need to consider matrices whose 1-jet involves at least 4 variables. The methods are basically the same as in the previous case, with one exception: For the case J (5,2) , the problem of classification and of finding adjacencies can be reduced to the corresponding problem for plane curve singularities and deformations with sections thereof. Theorem 4.1.3. ([37] Theorem 3.5) The simple isolated Cohen-Macaulay in codimension 2 singularities in (C5 , 0) are the following ones: Jet-type

Type

J (5,1)

A]0

J (5,2)

A]k D]k E6] E7] E8]

J (4,1) J (4,2)

Πk

Presentation Matrix ! x y z w v x ! x y z w v xk+1 + y2 ! x y z 2 w v xy + xk−1 ! x y z w v x3 + y4 ! x y z 3 w v x + xy3 ! x y z w v x3 + y5! w y x z w y + vk ! w y x k z w y + v2 ! w y x z w yv + vk ! w + vk y x z w yv ! w + v2 y x z w y2 + vk

τ 1 k≥1

k+2

k≥4

k+2 8 9 10

k≥2

2k − 1

k≥2

k+2 2k 2k + 1 k+3

4.1. Simple isolated Cohen-Macaulay of codimension 2 singularities in C4 and C5

J (4,3)

J (4,4)

! w y x z w y2 + v3 ! v2 + wk y x z w v2 ! + yl v2 + wk y x z w yv ! v2 + wk y x z w y2 + vl ! wv + vk y x z w yv + vk ! wv + vk y x z w yv ! wv + v3 y x + v3 z w y2 ! wv y x 2 z w y + v3 ! w2 + v3 y x z w y2 + v!3 z y x 2 x w v + y2 + zk ! z y x 2 x w v + yz + yk w ! z y x x w v2 + yz + yk+1 ! z y x x w v2 + yw + z2! z y x 2 x w v + y3! + z2 z y x + v2 x w vy + z2 ! z y x + v2 x w vz + y2! z y x + v2 x w y2 + z2

107

7 l≥k≥2

k+l +1

k≥2

k+4

k ≥ 2, l ≥ 3

k+l +2

k≥3

2k + 1

k≥3

2k + 2 8 9 9

k≥2

k+4

k≥1

2k + 5

k≥2

2k + 4 8 9 7 8 9

Table 3 – Simple isolated Cohen-Macaulay in codimension 2 singularities in (C5 , 0)

108

4.2


The bi-Lipschitz equisingularity on determinantal varieties Let us fix some notations.

We work with one parameter deformations and unfoldings. The parameter space is denoted by Y = C ≡ C × 0. Let h ∈ OCN . The (1,-1)-double of h is the element denoted by ID (h) ∈ OC2N defined by the equation ID (h)(z, z′ ) := h(z) − h(z′ ). If h = (h1 , ..., hr ) is a map, with hi ∈ OCN , ∀i ∈ {1, ..., r}, then we define ID (h) as the the ideal of OC2N generated by {ID (h1 ), ..., ID (hr )}. The above notions was defined by Gaffney in [39]. Now we get a relation between the integral closure of the double and the canonical vector field induced by a one parameter unfolding to be Lipschitz. Let F˜ : C × Cq −→ C × Cn be an analytic map, which is a homeomorphism onto its ˜ x) = (y, f˜(y, x)), with f˜(y, x) = ( f˜1 (y, x), ..., fñ (y, x)). Let image, and such that we can write F(y, us denote by n ∂ ∂ fe ∂ +∑ · ∂ y j=1 ∂ y ∂ z j

˜ × Cq ) −→ C × Cn given by the vector field v : F(C v(y, z) = (1, Proposition 4.2.1. The vector field

∂ f˜1 ˜ −1 ∂ fñ ˜ −1 (F (y, z)), ..., (F (y, z))). ∂y ∂y ∂ ∂y

n

+ ∑ j=1

ID (

∂ fe ∂ ∂y · ∂zj

is Lipschitz if and only if

∂ F˜ ˜ ) ( ID (F). ∂y

Proof. Since we are working in a finite dimensional C-vector space then all the norms are equivalent. To simplify the argument, we use the notation ‖.‖ for the maximum norm on C × Cq and C × Cn , i.e, ‖(x1 , ..., xn+1 )‖ = maxn+1 i=1 {‖xi ‖}. Suppose that the canonical vector field is Lipschitz. By hypothesis there exists a constant c > 0 such that ‖ v(y, z) − v(y′ , z′ ) ‖≤ c ‖ (y, z) − (y′ , z′ ) ‖ ˜ × Cq ). ∀(y, z), (y′ , z′ ) ∈ U, where U is a non-empty open subset of F(C

109

4.2. The bi-Lipschitz equisingularity on determinantal varieties

Thus, given (y, x), (y′ , x′ ) ∈ F˜ −1 (U), applying the above inequality on these points, we get ‖(

∂ f˜j ˜ x) − F(y ˜ ′ , x′ ) ‖ )D (y, x, y′ , x′ ) ‖≤ c ‖ F(y, ∂y

˜ ˜ for all j ∈ {1, ...n}. By Theorem 1.1.23 each generator of ID ( ∂∂Fy ) belongs to ID (F). ˜ ˜ Using the hypothesis and again the Lejeune-Teissier Now suppose that ID ( ∂∂Fy ) ( ID (F). theorem, for each j ∈ {1, ...n} there exists a constant c j > 0 and an open subset U j ( C × Cq such that ∂ f˜j ˜ x) − F(y ˜ ′ , x′ ) ‖ )D (y, x, y′ , x′ ) ‖≤ c j ‖ F(y, ‖( ∂y

∀(y, x), (y′ , x′ ) ∈ U j . Take U :=

n T j=1

˜ U j , c := max{c j }nj=1 and V := F(U), which is an

˜ × Cq ), since F˜ is a homeomorphism onto its image. Hence, it is easy to see open subset of F(C that ‖ v(y, z) − v(y′ , z′ ) ‖≤ c ‖ (y, z) − (y′ , z′ ) ‖ ∀(y, z), (y′ , z′ ) ∈ V . Therefore, the vector field

∂ ∂y

n ∂ f˜ j ∂y j=1

+ ∑

· ∂∂z j is Lipschitz .

Now, we have an application for a special case of determinantal surfaces. Proposition 4.2.2. Suppose that F˜ : C × Cq −→ C × Hom(Cm , Cn ) is an analytic map and a homeomorphism onto its image, and suppose we can write ˜ x) = (y, F(x) + yθ (x)). F(y, a) The vector field

∂ ∂y

+ ∑ ∂∂ yf · ∂∂z j is Lipschitz if and only if e

j

˜ ID (θ ) ( ID (F). b) If θ is constant then the vector field

∂ ∂y

+ ∑ ∂∂ yf · ∂∂z j is Lipschitz. e

j

Proof. (a) It is a straightforward consequence of the last proposition because

∂ f˜ ∂y

= θ.

(b) Since θ is constant then the doubles of the components of θ are all zero, so ID (θ ) is ˜ the zero ideal, which ensures the inclusion ID (θ ) ( ID (F). Remark 4.2.3. In [2] and [76], the authors consider a one parameter deformation with a constant θ . As showed above, for all these deformations the canonical vector field is Lispchitz. In Example 4.2.4 we see a case where the deformation does not come from a constant θ , and the canonical vector field remains Lipschitz. In Example 4.2.5 we have another deformation that does not come from a constant θ where the canonical vector field is not Lipschitz.

110


As we have seen before, the canonical vector field is naturally associated to the 1unfolding of the variety. However, its behaviour for the Lipschitz equisingularity is not the same. This behaviour depends on the type of the normal form, as we will see later. Mat

(O )

r The space of the first order deformations TX1 can be identified with T(n,p) G F , where T G F is the extended G -tangent space of the matrix F (Lemma 2.3, [37]). Hence we can treat the base of the semi-universal deformation using matrix representation and F is G -finitely determined if and only if TX1 is a finite dimensional module. From now on, the element θ is taken Mat (Or ) as an element of the space of the first order deformations TX1 ∼ = T(n,p) GF .

Example 4.2.4. Consider wl y x z w yk

F=

!

with l, k ≥ 2, which is one of the normal forms obtained in Table 2. Consider the matrix of deformation l−1  i 0  ∑w 0 i=0   θ = k−1  j 0 0 ∑ y j=0

˜ x, y, z, w) = (u, F(x, y, z, w) + uθ (x, y, z, w). Notice that θ ∈ and F(u, l−1

k−1

i=0

j=0

Mat(2,3) (O4 ) TG F

and ID (θ ) is

generated by { ∑ (wi − w′i ), ∑ (y j − y′ j )}. So, the generators are multiples of w − w′ and ˜ Therefore, ID (θ ) ( ID (F). ˜ By y − y′ , respectively, and these linear differences belong to ID (F). Proposition 4.2.2 we conclude that the canonical vector field is Lipschitz. Example 4.2.5. z y + w2 x2 wk x y

F=

!

with k ≥ 2, which is one of the normal forms obtained in Table 2. Consider the matrix of deformation 

0

θ = k−1 i ∑ w

 1 xw + w  w w

i=0

˜ x, y, z, w) = (u, F(x, y, z, w) + uθ (x, y, z, w). Notice that θ ∈ and F(u, k−1

Mat(2,3) (O4 ) , ID (θ ) TG F

is gener-

˜ is generated by {z − z′ , (y − ated by {w − w′ , w(x + 1) − w′ (x′ + 1), ∑ (wi − w′i )} and ID (F) i=0 k−1

k−1

i=0

i=0

y′ ) + w2 − w′2 , x2 − x′2 + u(xw + w) − u′ (x′ w′ + w′ ), wk − w′k + u ∑ wi − u′ ∑ w′i , x + uw − x′ − u′ w′ , y + uw − y′ − u′ w′ }.

111

4.2. The bi-Lipschitz equisingularity on determinantal varieties

Consider the curve φ : C −→ (C×C4 )×(C×C4 ) given by φ (t) = (0, 0, 0, 0, 2t, 0, 0, 0, 0,t). ˜ ∘ φ is the ideal generated by t 2 . Then, ID (θ ) ∘ φ is the ideal of O1 generated by t, and ID (F) ˜ ∘ φ and by the curve criterion for the integral closure of ideals we Hence, ID (θ ) ∘ φ * ID (F) ˜ Therefore, Proposition 4.2.2 ensures that the canonical vector conclude that ID (θ ) * ID (F). field is not Lipschitz. Theorem 4.2.6. Consider X a variety given by some F : Cq −→ Hom(C3 , C2 ) and F˜ : C × Mat(2,3) (Oq ) . Suppose that X is a Cq −→ C × Hom(C3 , C2 ), q ∈ {4, 5} as in 4.2.2, where θ ∈ TG F simple isolated Cohen-Macaulay variety of codimension 2. If F is of 1-jet-type Jq,k from Table 1 then the canonical vector field is Lipschitz, otherwise it is not. Mat

(Oq )

(2,3) then Proof. Suppose that F is of 1-jet-type from Lemma 3.2 of [37]. Since θ ∈ TG F the q order 1 entries of the matrix F stay unperturbed, thus the differences of the monomial ˜ In particular the ideal I∆ from the diagonal satisfies generators of the maximal ideal are in ID (F). ˜ Let θi , i ∈ {1, ..., 6} be the components of θ . Notice that every (θi )D the inclusion I∆ ⊆ ID (F). vanishes on the diagonal ∆ which implies that all the generators of ID (θ ) belong to I∆ . Therefore, ˜ and the Proposition 4.2.2 ensures that the canonical vector field is Lipschitz. ID (θ ) ⊆ I∆ ⊆ ID (F)

Suppose the opposite. In this case, one of the generators of the maximal ideal is not an entry of the 1-jet of the matrix F. Without loss of generality, we may assume this is the first coordinate x. Since F˜ is a semiuniversal unfolding then x − x′ certainly appears as a part of a generator set of ID (θ ). Take the curve φ : (C, 0) → (C × Cq ) × (C × Cq ) given by ˜ ∘ φ is generated by t m , for some m > 1. Since (x − x′ ) ∘ φ (t) = (0, 2t, 0, ..., 0,t, 0, ...). Then ID (F) ˜ ∘ φ , and by the curve criterion we conclude that x − x′ ∈ ˜ φ = t then (x − x′ ) ∘ φ ∈ / ID (F) / ID (F). ˜ and the Proposition 4.2.2 ensures that the canonical vector field is not Therefore, ID (θ ) * ID (F) Lipschitz. Remark 4.2.7. (Frühbis-Krüger) We can rephrase the condition on the jet-type by stating: a) The canonical vector field is Lipschitz if the ideal of 1-minors of the matrix of X defines a reduced point. b) The canonical vector field is not Lipschitz if the ideal of 1-minors of the matrix of X defines a fat point. We can re-state the Theorem 4.2.6 in a more explicit way using the tables of classification of the Theorems 4.1.2 and 4.1.3. Theorem 4.2.8. Consider X a variety given by some F : Cq −→ Hom(C3 , C2 ) and F˜ : C × Mat(2,3) (Oq ) Cq −→ C × Hom(C3 , C2 ), q ∈ {4, 5} as in 4.2.2, where θ ∈ . Suppose that X is a TG F simple isolated Cohen-Macaulay variety of codimension 2.

112


a) Suppose q = 4. If the normal form of F is one of those first eight normal forms on 4.1.2, then the canonical vector field is Lipschitz. For the other normal forms, the canonical vector field is not Lipschitz. b) Suppose q = 5. If the normal form of F is one of those first six normal forms on 4.1.3, then the canonical vector field is Lipschitz. For the other normal forms, the canonical vector field is not Lipschitz. Looking to tables 2 and 3 one can realize that the canonical vector field is Lipschitz exactly when the normal form has all the generators of the maximal ideal as an entry of the matrix.

113

CHAPTER

5 CATEGORICAL ASPECTS OF THE DOUBLE STRUCTURE

In this section, our main goal is look to the categorical properties of the double structure, under an algebraic viewpoint.

5.1

The double homomorphism and basic properties Let R be a ring.

Definition 5.1.1. Let T (R) be the category of the R-modules M which are R-submodules of R p , for some natural number p. Let X ⊆ Cn be an analytic space and let OX be the analytic sheaf of local rings over X, and let x ∈ X. Here we work on the categories T (OX,x ) and T (OX×X,(x,x) ). Consider the projection maps π1 , π2 : X × X → X. The first result is a quite useful tool many times when we work with the double. p p Proposition 5.1.2. Let M, N ⊆ OX,x submodules and h, g ∈ OX,x . Then:

a) h=g if and only if hD = gD ; b) h ∈ M if and only if hD ∈ MD ; c) M ⊆ N if and only if MD ⊆ ND ; d) M = N if and only if MD = ND .

114

Chapter 5. Categorical Aspects of the Double Structure

Proof. (a) The implication (=⇒) is obvious. Suppose that hD = gD . In particular, h ∘ π1 = g ∘ π1 , and for all z in a neighborhood of x we have h(z) = h ∘ π1 (z, x) = g ∘ π1 (z, x) = g(z), hence h = g. (b) The implication (=⇒) is obvious. Suppose now that hD ∈ MD . Then, we can write hD = ∑ αi (gi )D with gi ∈ M and αi ∈ OX×X,(x,x) . In particular, h ∘ π1 = ∑ αi (gi ∘ π1 ). Taking αix ∈ OX,x given by αix (z) := αi (z, x), ∀i, we get h = ∑ αix gi which belongs to M. (c) The implication (=⇒) is obvious. Suppose that MD ⊆ ND . Let h ∈ M arbitrary. Then hD ∈ MD ⊆ ND , so by the item (b) we conclude that h ∈ N. Therefore, M ⊆ N. (d) It is a straightforward consequence of the item (c). p , the natural map Corollary 5.1.3. For each OX,x -submodule M of OX,x

DM : M −→ MD h ↦−→ hD is an injective group homomorphism. In particular, we can see M as an additive subgroup of MD . Proof. It is a straightforward consequence of the definition of the double that DM is a group homomorphism. The Proposition 5.1.2 (a) gives the injectivity. Our main goal is to give a categorical sense for the double structure. The next theorem is the key for it. q

p Theorem 5.1.4. Let M ⊆ OX,x and N ⊆ OX,x be OX,x -submodules. If φ : M → N is an OX,x module homomorphism then there exists a unique OX×X,(x,x) -module homomorphism φD : MD → ND such that φD (hD ) = (φ (h))D , ∀h ∈ M, i.e, the following diagram commutes:

M

φ

DM MD

N DN

φD

ND

The map φD is called the double of φ . Proof. Since MD is generated by {hD | h ∈ M} then we can define φD : MD → ND in a quite natural way: for each u = ∑ αi (hi )D with αi ∈ OX×X,(x,x) and hi ∈ M we define i

φD (u) := ∑ αi (φ (hi ))D i

115

5.1. The double homomorphism and basic properties

which belongs to ND . Claim : φD is well defined. In fact, suppose that ∑ αi (hi )D = ∑ β j (g j )D , with αi , β j ∈ i

OX×X,(x,x) and hi , g j ∈ M. So, we get two equations:

j

∑ αi(hi ∘ π1) = ∑ β j (g j ∘ π1)

(1)

∑ αi(hi ∘ π2) = ∑ β j (g j ∘ π2).

(2)

i

j

i

j

Take U an open neighborhood of x in X where αi , β j are defined on U ×U, and hi , g j are defined on U. For each w ∈ U define αiw , β jw ∈ OX,x given by the germs of the maps αiw : U −→

β jw : U −→

C

↦−→ αi (z, w)

z

z

C

↦−→ β j (z, w)

The equation (1) implies that ∑ αiw hi = ∑ β jw g j , ∀w ∈ U. Applying φ (which is an OX,x i

j

homomorphism) in both sides of the last equation we get ∑ αiw φ (hi ) = ∑ β jw φ (g j ), ∀w ∈ U. This i

j

implies that (∑ αi (φ (hi ) ∘ π1 ))(z, w) = (∑ β j (φ (g j ) ∘ π1 ))(z, w) i

j

∀(z, w) ∈ U ×U, hence

∑ αi(φ (hi) ∘ π1) = ∑ β j (φ (g j ) ∘ π1). i

(3)

j

Analogously, using the equation (2), we get

∑ αi(φ (hi) ∘ π2) = ∑ β j (φ (g j ) ∘ π2). i

(4)

j

The equations (3) and (4) implies that

∑ αi(φ (hi))D = ∑ β j (φ ((g j ))D i

j

and the Claim is proved. Now, by the definition of φD , it is clear that φD is an OX×X,(x,x) -module homomorphism and is the unique satisfying the property φD (hD ) = (φ (h))D , ∀h ∈ M. From now on, all the modules are objects in T (OX,x ) and their doubles are objects in T (OX×X,(x,x) ). Notice that if idM : M → M and idMD : MD → MD are the identity homomorphisms of M and MD , then (idM )D = idMD . The next proposition gives us a relation between images and kernels of a module homomorphism.

116


Proposition 5.1.5. Let φ : M → N be an OX,x -module homomorphism and φD : MD → ND its double. Then: a) Im(φD ) = (Im(φ ))D ; b) (ker(φ ))D ⊆ ker(φD ). Proof. (a) By definition of φD , it is clear that Im(φD ) ⊆ (Im(φ ))D . Now, if g ∈ Im(φ ) then we can write g = φ (h), for some h ∈ M. So, gD = (φ (h))D = φD (hD ) ∈ Im(φD ). Thus, gD ∈ Im(φD ), ∀g ∈ Im(φ ), hence (Im(φ ))D ⊆ Im(φD ). (b) For every h ∈ ker(φ ), we have that φD (hD ) = (φ (h))D = (0N )D = 0ND , so hD ∈ ker(φD ). The next proposition shows that the double homomorphism has a good behavior with respect to sum and composition. Proposition 5.1.6. Let φ , φ ′ : M → N and γ : N → P be OX,x -module homomorphisms. a) φ = φ ′ if and only if φD = φD′ ; b) (γ ∘ φ )D = γD ∘ φD ; c) (φ + φ ′ )D = φD + φD′ . Proof. (a) (=⇒) Suppose φ = φ ′ . For all h ∈ M, φD (hD ) = (φ (h))D = (φ ′ (h))D = φD′ (hD ). Since the module MD is generated by {hD | h ∈ M} then φD = φD′ . (⇐=) Suppose φD = φD′ . Given h ∈ M arbitrary, we have (φ (h))D = φD (hD ) = φD′ (hD ) = (φ ′ (h))D . By Proposition 5.1.2 (a), φ (h) = φ ′ (h). Hence, φ = φ ′ . (b) For every h ∈ M we have (γ ∘ φ )D (hD ) = (γ ∘ φ (h))D = γD ((φ (h))D ) = γD ∘ φD (hD ), which proves (b). (c) For every h ∈ M we have (φ + φ ′ )D (hD ) = ((φ + φ ′ )(h))D = (φ (h) + φ ′ (h))D = (φ (h))D + (φ ′ (h))D = (φD + φD′ )(hD ), which proves (c). In the next corollary we compare algebraic properties of a homomorphism and its double.

117


Corollary 5.1.7. Let φ : M → N be an OX,x -module homomorphism. Then: a) φ : M → N is surjective if and only if φD : MD → ND is surjective; b) If φD : MD → ND is injective then φ : M → N is injective; c) φ : M → N is an OX,x -isomorphism if and only if φD : MD → ND is an OX×X,(x,x) isomorphism; d) φ : M → N is the zero homomorphism if and only if φD : MD → ND is the zero homomorphism. Proof. (a) By Propositions 5.1.2 (d) and 5.1.5 (a) we have: φ is surjective ⇐⇒ Im(φ ) = N ⇐⇒ (Im(φ ))D = ND ⇐⇒ Im(φD ) = ND ⇐⇒ φD is surjective. (b) If φD is injective then ker(φD ) = 0MD , and the Proposition 5.1.5 (b) implies that (ker(φ ))D ⊆ ker(φD ) = 0MD = (0M )D . By Proposition 5.1.2 (c) we conclude that ker(φ ) = 0M , hence φ is injective. (c) (=⇒) Since φ : M → N is an isomorphism then there exists an OX,x -homomorphism γ : N → M such that γ ∘ φ = idM and φ ∘ γ = idN , then, by Proposition 5.1.6 (b) we have (γ)D ∘ (φ )D = idMD and (φ )D ∘ (γ)D = idND . Hence, φD is an isomorphism. (⇐=)It follows immediately from (a) and (b). (d) By Propositions 5.1.2 (d) and 5.1.5 (a) we have: φ is the zero homomorphism ⇐⇒ Im(φ ) = 0N ⇐⇒ (Im(φ ))D = (0N )D ⇐⇒ Im(φD ) = 0ND ⇐⇒ φD is the zero homomorphism.

q

p Definition 5.1.8. We say that an OX,x -homomorphism φ : M ⊆ OX,x → N ⊆ OX,x is induced by a q × p matrix if there exists A ∈ Matq×p (OX,x ) such that φ (h) = A · h, ∀h ∈ M. q

p Lemma 5.1.9. An OX,x -homomorphism φ : M ⊆ OX,x → N ⊆ OX,x is induced by a q × p matrix q p if and only if there exists an OX,x -homomorphism φ˜ : OX,x → OX,x such that φ˜ (M) ⊆ N and φ˜ |M = φ .

Proof. (=⇒) By hypothesis there exists a q × p matrix A with entries in OX,x such that φ (h) = q p A · h, ∀h ∈ M. From this matrix A, we can define φ˜ : OX,x → OX,x given by φ˜ (g) := A · g, which is an OX,x -homomorphism. Clearly, φ˜ |M = φ , and for all h ∈ M we have φ˜ (h) = φ (h) ∈ N, so φ˜ (M) ⊆ N. p (⇐=) Let e1 , ..., e p be the canonical elements in OX,x . Let A be the q × p matrix whose p columns are φ (e1 ), ..., φ (e p ). Then φ˜ (g) = A · g, ∀g ∈ OX,x . Since φ˜ |M = φ then φ (h) = φ˜ (h) = A · h, ∀h ∈ M. Therefore, φ is induced by a q × p matrix.

118

Chapter 5. Categorical Aspects of the Double Structure q

p Proposition 5.1.10. If φ : M ⊆ OX,x → N ⊆ OX,x is an OX,x -homomorphism induced by a q × p matrix then 2q

2p φD : MD ⊆ OX×X,(x,x) → ND ⊆ OX×X,(x,x)

is an OX×X,(x,x) -homomorphism induced by a 2q × 2p matrix. Proof. By hypothesis there exists a q× p matrix " A with#entries " in OX,x such # that " φ (h) = A·h, ∀h#∈ φ (h) ∘ π1 (A · h) ∘ π1 (A ∘ π1 ) · (h ∘ π1 ) M. Then, for all h ∈ M we have φD (hD ) = = = . φ (h) ∘ π2 (A · h) ∘ π2 (A ∘ π2 ) · (h ∘ π2 ) So, taking the 2q × 2p matrix " # A ∘ π1 0q×p B := 0q×p A ∘ π2 we conclude that φD (hD ) = B · hD , and the proposition is proved, once MD is generated by hD , h ∈ M. As an application of the double homomorphism, we prove in the next theorem that the double structure is compatible with finite direct sum of modules. q

p Theorem 5.1.11. Let M ⊆ OX,x and N ⊆ OX,x be OX,x -submodules. Then

(M ⊕ N)D ∼ = MD ⊕ ND 2(p+q)

as OX×X,(x,x) -submodules of OX×X,(x,x) . Furthermore, there exists an isomorphism η : (M ⊕ N)D −→ MD ⊕ ND such that η((h, g)D ) = (hD , gD ), for all h ∈ M and g ∈ N. Proof. Consider the canonical projections and inclusions: ψ1 : M ⊕ N −→ (h, g) ↦−→ δ1 :

M h

M h

−→ M ⊕ N ↦−→ (h, 0N )

ψ2 : M ⊕ N −→ (h, g) ↦−→ δ2 :

N g

N g

−→ M ⊕ N ↦−→ (0M , g)

Thus, we get the double homomorphism of each one above:

(ψ1 )D : (M ⊕ N)D −→ MD

(ψ2 )D : (M ⊕ N)D −→ ND

(δ1 )D : MD −→ (M ⊕ N)D

(δ2 )D : ND −→ (M ⊕ N)D


119

Define: η : (M ⊕ N)D −→ MD ⊕ ND w ↦−→ ((ψ1 )D (w), (ψ2 )D (w))

δ : MD ⊕ ND −→ (M ⊕ N)D (u, v) ↦−→ (δ1 )D (u) + (δ2 )D (v) which are OX×X,(x,x) -module homomorphisms. Claim 1: η((h, g)D ) = (hD , gD ), for all h ∈ M and g ∈ N. In fact, η((h, g)D ) = ((ψ1 )D ((h, g)D ), (ψ2 )D ((h, g)D )) = ((ψ1 (h, g))D , (ψ2 (h, g))D ) = (hD , gD ). Claim 2: δ (hD , gD ) = (h, g)D , for all h ∈ M and g ∈ N. In fact, δ (hD , gD ) = (δ1 )D (hD )+(δ2 )D (gD ) = (δ1 (h))D +(δ2 (g))D = (h, 0N )D +(0M , g)D = ((h, 0N ) + (0M , g))D = (h, g)D . By the Claims 1 and 2 we have that δ ∘ η((h, g)D ) = (h, g)D , η ∘ δ (hD , 0ND ) = (hD , 0ND ) and η ∘ δ (0MD , gD ) = (0MD , gD ), for all h ∈ M and g ∈ N. Since {(h, g)D | h ∈ M and g ∈ N} is a generator set of (M ⊕ N)D and {(hD , 0ND ), (0MD , gD ) | h ∈ M and g ∈ N} is a generator set of MD ⊕ ND then we conclude that δ ∘ η = id(M⊕N)D and η ∘ δ = idMD ⊕ND , which finishes the proof of the theorem. pi be OX,x -submodules, for each i ∈ {1, ..., r}. Then Corollary 5.1.12. Let Mi ⊆ OX,x

(M1 ⊕ ... ⊕ Mr )D ∼ = (M1 )D ⊕ ... ⊕ (Mr )D 2(p +...+pr )

1 as OX×X,(x,x) -submodules of OX×X,(x,x)

through an isomorphism such that

(h1 , ..., hr )D ↦−→ ((h1 )D , ..., (hr )D ) for all hi ∈ Mi . Proof. Induction on r and use the previous theorem. The next proposition compares the length of a module and its double. p Proposition 5.1.13. Let M ⊆ N be OX,x -submodules of OX,x .

a) If MD has finite length then M has finite length and `(M) ≤ `(MD ); b) If MD has finite colength in ND then M has finite colength in N.

120


Proof. (a) If r ∈ N and (Mi )ri=0 is an ascending series of M, then ((Mi )D )ri=0 is an ascending series of MD , which has finite length. Thus, r ≤ `(MD ). Therefore `(M) is finite and `(M) ≤ `(MD ). N (b) Let r ∈ N and consider an arbitrary ascending series of M of length r. This series can be given on the form N0 N1 Nr−1 Nr N ( ( ... ( ( = M M M M M where N0 ( N1 ( ... ( Nr−1 ( Nr = N are OX,x -submodules of N which contain M. Then,

(Nr−1 )D (Nr )D ND (N0 )D (N1 )D ( ( ... ( ( = (MD ) MD MD MD MD is an ascending series of is finite.

ND MD ,

ND N ) and `( M ) which has finite length by hypothesis. Hence, r ≤ `( M D

We want to use the double homomorphism in order to get an equivalence between the second and third Lipschitz saturations. The next proposition gives the "persistence" of the integral closure of modules. q

p → N ⊆ OX,x be a homomorphism of OX,x -modules which Proposition 5.1.14. Let ϕ : M ⊆ OX,x q p ˜ → cOX,x , given by ϕ(h) = A · h, where A is a q × p can be extended to a homomorphism ϕ˜ : OX,x p matrix with entries in OX,x . Let h ∈ OX,x .

˜ a) If h ∈ M then ϕ(h) ∈ ϕ(M); b) Suppose q = p. If A is an invertible matrix and ϕ is an isomorphism of OX,x -modules then: ˜ h ∈ M if and only if ϕ(h) ∈ ϕ(M); ˜ c) Suppose q = p and ϕ is injective. If ϕ(h) ∈ ϕ(M) then h ∈ M. Proof. (a) Let φ : (C, 0) → (X, x) be an arbitrary analytic curve. By hypothesis φ * (h) ∈ φ * (M) ˜ and we can write φ * (h) = ∑ αi φ * (gi ), for some gi ∈ M and αi ∈ OC,0 . Thus: ϕ(h) ∘ φ = (A · h) ∘ φ = [A ∘ φ ] · [h ∘ φ ] = [A ∘ φ ] · ∑ αi φ * (gi ) = ∑ αi ([A · gi ] ∘ φ ) ∈ φ * (ϕ(M)). ˜ Hence, ϕ(h) ∈ ϕ(M). (b) It suffices apply (a) in ϕ˜ −1 . ˆ (c) It suffices consider the isomorphism ϕˆ : M → ϕ(M) given by ϕ(h) = ϕ(h) and apply the item (b). For each i ∈ {1, ..., p} consider the i-th canonical global section of the vector bundle Hom(C p , C), ξi : X → Hom(C p , C) given by ξi (x) = (0, .., 1, ...0), where 1 is on the i-th place. p Notice that if M is an OX,x -submodule of OX,x then M ⊆ ξ1 · M ⊕ ... ⊕ ξ p · M.


121

p Theorem 5.1.15. Let M ⊆ OX,x be a submodule. Suppose that MD is a reduction of (ξ1 · M ⊕ ... ⊕ ξ p · M)D . Then, MS2 = MS3 .

Proof. Consider the inclusion i : M → ξ1 · M ⊕ ... ⊕ ξ p · M. Then we can consider the inclusion iD : MD → (ξ1 · M ⊕ ... ⊕ ξ p · M)D which is induced by an invertible 2p × 2p matrix. By Corollary 5.1.12 there is an isomorphism γ : (ξ1 · M ⊕ ... ⊕ ξ p · M)D → (ξ1 · M)D ⊕ ... ⊕ (ξ p · M)D and by the proof of this corollary, this isomorphism is induced by an invertible 2p × 2p matrix. Taking the composition of iD with γ, we get an injective homomorphism η : MD → (ξ1 · M)D ⊕ ... ⊕ (ξ p · M)D induced by an invertible 2p × 2p matrix B which extends to the isomorphism 2p 2p η˜ : OX×X,(x,x) → OX×X,(x,x)

given by the multiplication by B, which satisfies the property (g1 , ..., g p )D ↦→ ((g1 )D , ..., (g p )D ). By Proposition 3.1.22 we have the inclusion MS2 ⊆ MS3 . So, it suffices to check the another inclusion. Let h ∈ MS3 . In particular, (ξi · h)D ∈ (ξi · M)D , ∀i ∈ {1, ..., p}. Let φ : (C, 0) → (X × X, (x, x)) be an arbitrary analytic curve. Then φ * ((ξi · h)D ) ∈ φ * ((ξi · M)D ), ∀i ∈ {1, ..., p} and ˜ D )) = φ * ((ξ1 · h)D , ..., (ξ p · h)D ) = (φ * ((ξ1 · h)D ), ..., φ * ((ξ p · h)D )) which belongs to φ * (η(h ˜ D ) ∈ (ξ1 · M)D ⊕ ... ⊕ (ξ p · M)D . φ * ((ξ1 · M)D ⊕ ... ⊕ (ξ p · M)D ). Hence, η(h Since MD is a reduction of (ξ1 · M ⊕ ... ⊕ ξ p · M)D then by the previous proposition we ˜ D ) ∈ η(MD ), and by the have that η(MD ) is a reduction of (ξ1 · M)D ⊕ ... ⊕ (ξ p · M)D . Thus, η(h previous proposition we conclude that hD ∈ MD , therefore h ∈ MS2 . p Example 5.1.16. Notice that any p-symmetric submodule M ⊆ OX,x satisfies the hypothesis of the previous theorem, so MS2 = MS3 . Here, p-symmetric means that given (h1 , ..., h p ) ∈ M arbitrary, then (hσ (1) , ..., hσ (p) ) ∈ M for every p-permutation σ . Observe that is equivalent to M = ξ1 · M ⊕ ... ⊕ ξ p ·M.

122

5.2


Homological aspects of the double structure

In this section we define the double chain of a given chain complex, and we compare homological properties between these complexes. First, we need the following result. Proposition 5.2.1. Let φ

M

N

γ

P

be a sequence of OX,x -module homomorphism and consider the double sequence

MD

φD

ND

γD

PD

If Im(φ ) ⊆ ker(γ) then Im(φD ) ⊆ ker(γD ). Proof. Since Im(φ ) ⊆ ker(γ) then (Im(φ ))D ⊆ (ker(γ))D . Hence, Im(φD ) = (Im(φ ))D ⊆ (ker(γ))D ⊆ ker(γD ). We realize that the double homomorphism gives a natural way to study the homology of the double structure. Definition 5.2.2 (The double chain complex). Let C = (M∙ , φ∙ ) be a chain complex in T (OX,x ). We define CD := ((M∙ )D , (φ∙ )D ) and by Proposition 5.2.1 we have that CD is a chain complex in T (OX×X,(x,x) ). The chain complex CD is called the double of C . Proposition 5.2.3. Let C = (M∙ , φ∙ ) be a chain complex. If CD is an exact sequence then C is an exact sequence. In other words, if CD has trivial homology then C has trivial homology. Proof. Let i ∈ Z be arbitrary. We have the commutative diagram: Mi+1

(Mi+1 )D

φi+1

(φi+1 )D

Mi

(Mi )D

φi

(φi )D

Mi−1

(Mi−1 )D

123

5.2. Homological aspects of the double structure

where the vertical arrows are the canonical morphisms. We already know that Im(φi+1 ) ⊆ ker(φi ). Since CD is an exact sequence then Im((φi+1 )D ) = ker((φi )D ). By Proposition 5.1.5 we have (ker(φi ))D ⊆ ker((φi )D ) = Im((φi+1 )D ) = (Im(φi+1 ))D . By Proposition 5.1.2 (c), we conclude that ker(φi ) ⊆ Im(φi+1 ). Therefore, Im(φi+1 ) = ker(φi ). Proposition 5.2.4. Let C = (M∙ , φ∙ ) and C ′ = (M∙′ , φ∙′ ) be chain complexes. If α : C −→ C ′ is a chain complex morphism then αD : CD −→ CD′ given by {(αi )D : (Mi )D → (Mi′ )D | i ∈ Z} is a chain complex morphism, called the the double morphism of α. Proof. Let i ∈ Z. We have the following commutative diagram Mi

φi

αi Mi′

Mi−1 αi−1

φi′

′ Mi−1

By Proposition 5.1.6 (b) follows that (φi′ )D ∘(αi )D = (φi′ ∘αi )D = (αi−1 ∘φi )D = (αi−1 )D ∘ (φi )D , and the following diagram is also commutative: (Mi )D

(φi )D

(αi )D

(Mi−1 )D (αi−1 )D

(Mi′ )D

(φi′ )D

′ ) (Mi−1 D

Corollary 5.2.5. If α : C −→ C ′ and β : C ′ −→ C ′′ are chain morphisms then (β ∘ α)D = βD ∘ αD . Proof. It is a straightforward consequence of the Proposition 5.1.6 (b). Now, we get some results related to chain homotopy. Let C = (M∙ , φ∙ ) and C ′ = (M∙′ , φ∙′ ) be chain complexes. Let µ : C → C ′ be a homomorphism of degree 1, i.e, µ is a collection of OX,x -module homomorphisms {µi : Mi → ′ Mi+1 | i ∈ Z}. We know this homomorphism induces a chain morphism µ˜ : C → C ′ given by ′ ∘µ +µ {µ˜i : Mi → Mi′ | i ∈ Z}, where µ˜i := φi+1 i i−1 ∘ φi , ∀i ∈ Z. If α, β : C → C ′ are chain morphisms, remember that µ : C → C ′ is defined as a homotopy between α and β when µ˜ = α − β , and we denote α ≃ β . µ

124


Lemma 5.2.6. Consider µD : CD → CD′ the homomorphism of degree 1 given by the double fD = (µ) ˜ D. homomorphisms of µ : C → C ′ . Then µ ′ ) ∘ (µ ) + (µ fD )i = (φi+1 Proof. For all i ∈ Z we have (µ D i D i−1 )D ∘ (φi )D ′ ∘µ +µ = (φi+1 i i−1 ∘ φi )D = ( µ˜i )D , and the lemma is proved.

Proposition 5.2.7. Let α, β : C → C ′ be chain morphisms and µ : C → C ′ a homomorphism of degree 1. Then: µ is a homotopy between α and β if and only if µD is a homotopy between αD and βD . Proof. We have that µ is a homotopy between α and β if and only if µ˜ = α − β . By Proposition ˜ D = (α − β )D ⇐⇒ 5.1.6 (a) and (c) and the previous lemma we have: µ˜ = α − β ⇐⇒ (µ) fD = αD − βD ⇐⇒ µD is a homotopy between αD and βD . µ Corollary 5.2.8. If α : C → C ′ is a chain homotopy equivalence then αD : CD → CD′ is a chain homotopy equivalence. Proof. By hypothesis there exists a chain morphism β : C ′ → C such that β ∘ α ≃ idC and α ∘ β ≃ idC ′ . By the previous proposition, we have (β ∘ α)D ≃ (idC )D and (α ∘ β )D ≃ (idC ′ )D , and therefore (β )D ∘ (α)D ≃ idCD and αD ∘ βD ≃ idCD′ . Corollary 5.2.9. If C is a contractible chain complex then CD is also contractible. Proof. Since C is contractible then idC ≃ 0C , and the Proposition 5.2.7 implies that (idC )D ≃ (0C )D , thus idCD ≃ 0CD . Hence, CD is contractible. Notice the nice relation between the Proposition 5.2.3 and Corollary 5.2.9. We already know every contractible chain complex is an exact sequence. The Proposition 5.2.3 says that the exactness on the double level implies the exactness on the "single" level. The Corollary 5.2.9, which treats about contractible (stronger than exactness), says the opposite. It is clear that all the results obtained in this section can be naturally translated to the cohomology language.

125

5.3. The Double category

5.3

The Double category Let us define the category D(OX,x ).

The objects of D(OX,x ) consist of the double of modules in T (OX,x ). Given MD , ND objects in D(OX,x ), we define Mor(MD , ND ) := {φD : MD → ND | φ : M → N is an OX,x -module homomorphism}. Working with the standard composition of maps, we have the category D(OX,x ), called the Double category of (X, x). The morphisms in D(OX,x ) are called OX,x -double morphisms. Observe that the OX,x doubles morphisms are OX×X,(x,x) -homomorphisms with an addictional property: they preserve the double structure. Notice that D(OX,x ) is a subcategory of T (OX×X,(x,x) ). Theorem 5.3.1. The covariant functor D:

T (OX,x ) −→ D(OX,x ) M ↦−→ MD φ : M → N ↦−→ φD : MD → ND

is an isomorphism of categories. Proof. The Proposition 5.1.2 (d) proves that the map between the objects is a bijection, and the Proposition 5.1.6 (a) proves that the map between the morphisms is a bijection. Hence, D is an isomorphism of categories. Corollary 5.3.2. T (OX,x ) can be seen as a subcategory of T (OX×X,(x,x) ). The Theorem 5.3.1 implies that T (OX,x ) and D(OX,x ) are essentially the same category, so they have the same behavior in all of the categorical statements. But, one of them is reasonable to emphasize, in the next corollary, which is interesting to compare with the result obtained in the Corollary 5.1.7. Corollary 5.3.3. Let φ : M → N be an OX,x -module homomorphism. Then: a) φ : M → N is an OX,x -monomorphism of modules if and only if φD : MD → ND is an OX,x -double monomorphism; b) φ : M → N is an OX,x -epimorphism of modules if and only if φD : MD → ND is an OX,x double epimorphism; c) φ : M → N is an OX,x -isomorphism of modules if and only if φD : MD → ND is an OX,x double isomorphism.

126


Here it is reasonable to emphasize the difference between the notions of injective homomorphism and monomorphism of modules, which are not the samething in the category of modules, since we are understanding the term monomorphism in the categorical sense, i.e, there is a left-inverse morphism. The same remark has to be done between surjective homomorphism and epimorphism. Remark 5.3.4. The covariant functor D:

T (OX,x ) −→ T (OX×X,(x,x) ) M ↦−→ MD φ : M → N ↦−→ φD : MD → ND

is not an isomorphism of categories anymore. In fact, suppose (X, x) irreducible. It is proved in 2.1.17 that the generic rank of the double of every module M in T (OX,x ) has generic rank even. Thus, the map between the objects cannot be surjective.

5.4

The double in a quotient of a free OX -module of finite rank p Let W be an OX,x -submodule of OX,x and consider the quotient map

π

p : OX,x

−→

p OX,x

.

W

2p Then, WD is an OX×X,(x,x) -submodule of OX×X,(x,x) . p . We define the double of h +W ∈ Let h ∈ OX,x

(h +W )D := hD +WD ∈

p

OX,x W

as

2p OX×X,(x,x)

WD

.

Notice the definition of (h +W )D does not depend of the choice of the representative h. In fact, if h +W = g +W then h − g ∈ W =⇒ (h − g)D ∈ WD =⇒ hD − gD ∈ WD =⇒ hD +WD = gD +WD . O

p

Now, we want to define the double of a submodule M of WX,x . We have that π −1 (M) is p 2p a submodule of OX,x and π −1 (M) ⊃ W , hence (π −1 (M))D is a submodule of OX×X,(x,x) which contains WD . Then, we define the double of M as MD :=

(π −1 (M))D WD

127

5.5. The double homomorphism relative to an analytic map germ 2p

which is an OX×X,(x,x) -submodule of

OX×X,(x,x) . WD

If we call M˜ := π −1 (M) then M=

M˜ M˜ D and MD = . W WD

p Rewriting with standard notation, we conclude that, if M is a submodule of OX,x and M ⊃ W then M MD = W D WD

and is generated by {(h +W )D | h ∈ M}.

5.5

The double homomorphism relative to an analytic map germ

Let (Y, y) and (X, x) be germs of analytic spaces, and let ϕ : (Y, y) → (X, x) be an analytic map germ. The pullback map ϕ * : OX,x → OY,y is a ring homomorphism, which induces an OX,x -algebra structure in OY,y . Thus, every OY,y -module is also an OX,x -module through this ring homomorphism. We see that there is a natural OX×X,(x,x) -algebra structure in OY ×Y,(y,y) induced by the pullback of ϕ. In fact, let µX,x : OX,x ⊗ OX,x −→ OX×X,(x,x) C

be the C-algebra homomorphism such that µX,x ( f ⊗ g) is the germ of the map C

U ×U → C (u, v) ↦→ f (u).g(v) and let µY,y : OY,y ⊗ OY,y −→ OY ×Y,(y,y) C

the same for (Y, y). Since ϕ * : OX,x → OY,y is a ring homomorphism then we have a natural C-algebra homomorphism ϕ ⊗ : OX,x ⊗ OX,x −→ OY,y ⊗ OY,y C

C

such that ϕ ⊗ ( f ⊗ g) = ϕ * ( f ) ⊗ ϕ * (g), ∀ f , g ∈ OX,x . In fact, the map C

C

OX,x × OX,x −→

OY,y ⊗ OY,y C

( f , g)

↦−→ (ϕ * ( f )) ⊗ (ϕ * (g)) C

128


is C-bilinear. So, the existence and uniqueness of ϕ ⊗ is provided by the universal property of the tensor product. It is known that µX,x and µY,y are C-algebra isomorphisms, so we can consider the C-algebra homomorphism εϕ : OX×X,(x,x) → OY ×Y,(y,y) such that the following diagram is commutative: OX,x ⊗ OX,x

µX,x

OX×X,(x,x)

C

εϕ

ϕ⊗ OY,y ⊗ OY,y C

µY,y

OY ×Y,(y,y)

Since µX,x and µY,y are C-algebra isomorphisms then we can identify εϕ ∼ = ϕ ⊗ , and ϕ ⊗ : OX×X,(x,x) → OY ×Y,(y,y) induces in OY ×Y,(y,y) an OX×X,(x,x) -algebra structure. Lemma 5.5.1. Let α ∈ OX×X,(x,x) . Suppose that U is an open subset of X containing x where a representative of α is defined on U ×U. For each w ∈ U let α w ∈ OX,x be the germ of the map αw : U → C z ↦→ α(z, w) For each y2 ∈ ϕ −1 (U) let (ϕ ⊗ (α))y2 ∈ OY,y be the germ of the map (ϕ ⊗ (α))y2 : ϕ −1 (U) → C y1 ↦→ (ϕ ⊗ (α))(y1 , y2 ) Then ϕ * (α ϕ(y2 ) ) = (ϕ ⊗ (α))y2 , ∀y2 ∈ ϕ −1 (U). Proof. We can write α = ∑( fi ⊗ gi ), with fi , gi ∈ OX,x . For all y1 ∈ ϕ −1 (U) we have: i

C

ϕ * (α ϕ(y2 ) )(y1 ) = α ϕ(y2 ) (ϕ(y1 )) = α(ϕ(y1 ), ϕ(y2 )) ∑( fi (ϕ(y1 )) ⊗ gi (ϕ(y2 ))) C i * * = ∑(ϕ ( fi )) ⊗ ((ϕ (gi )) (y1 , y2 ) = (ϕ ⊗ (α))(y1 , y2 ) = (ϕ ⊗ (α))y2 (y1 ), and the lemma i

C

is proved. Clearly we get the analogous result if we fix the first coordinate instead the second one. Consider the projections π1X , π2X : X × X → X and π1Y , π2Y : Y ×Y → Y .

129

5.5. The double homomorphism relative to an analytic map germ q

p Theorem 5.5.2. Let M ⊆ OX,x and N ⊆ OY,y be submodules. If φ : M → N is an OX,x -module homomorphism then there exists a unique OX×X,(x,x) -module homomorphism φD,ϕ = φD : MD → ND such that the following diagram is commutative:

M

N φ

DM

DN

MD

ND

φD,ϕ

The map φD,ϕ = φD is called the double of φ relative to ϕ : (Y, y) → (X, x). Proof. Since MD is generated by {hD | h ∈ M} then we can define φD : MD → ND in a natural way: for each u = ∑ αi (hi )D with αi ∈ OX×X,(x,x) and hi ∈ M we define i

φD (u) := ∑ αi (φ (hi ))D = ∑ ϕ ⊗ (αi )(φ (hi ))D i

i

which belongs to ND . Claim : φD is well defined. In fact, suppose that ∑ αi (hi )D = ∑ β j (g j )D , with αi , β j ∈ i

OX×X,(x,x) and hi , g j ∈ M. So, we obtain two equations:

j

∑ αi(hi ∘ π1X ) = ∑ β j (g j ∘ π1X )

(1)

∑ αi(hi ∘ π2X ) = ∑ β j (g j ∘ π2X ).

(2)

i

j

i

j

Take U an open neighborhood of x in X where representatives of αi , β j are defined on U × U, and representatives of hi , g j are defined on U. For each w ∈ U define αiw , β jw ∈ OX,x given by the germs of the maps αiw : U −→ z

β jw : U −→

C

↦−→ αi (z, w)

z

C

−→ β j (z, w)

The equation (1) implies that ∑ αiw hi = ∑ β jw g j , ∀w ∈ U. Applying φ (which is a OX,x i

j

homomorphism) in both sides of the last equation we get

∑ αiwφ (hi) = ∑ β jwφ (g j ), ∀w ∈ U. i

j

By the OX,x -module structure on N induced by ϕ * , the last equation boils down to

∑ ϕ *(αiw)φ (hi) = ∑ ϕ *(β jw)φ (g j ), ∀w ∈ U. i

j

130


By Lemma 5.5.1 we conclude that

∑(ϕ ⊗(αi))y2 φ (hi) = ∑(ϕ ⊗(β j ))y2 φ (g j ), ∀y2 ∈ ϕ −1(U). i

j

Hence,

∑ ϕ ⊗(αi)(φ (hi) ∘ π1Y ) = ∑ ϕ ⊗(β j )(φ (g j ) ∘ π1Y ). i

j

Working with the analogous result of the Lemma 5.5.1, the equation (2) implies that

∑ ϕ ⊗(αi)(φ (hi) ∘ π2Y ) = ∑ ϕ ⊗(β j )(φ (g j ) ∘ π2Y ). i

j

Therefore,

∑ ϕ ⊗(αi)(φ (hi))D = ∑ ϕ ⊗(β j )(φ (g j ))D i

j

and φD is well-defined. Now, by the definition of φD , it is clear that φD is an OX×X,(x,x) -module homomorphism and is the unique satisfying the property φD (hD ) = (φ (h))D , ∀h ∈ M, i.e, φD (h ∘ π1X , h ∘ π2X ) = (φ (h) ∘ π1Y , φ (h) ∘ π2Y ).

Notice that this approach generalizes what we have defined in Section 5.1, taking ϕ : (X, x) → (X, x) as the identity map. The main motivation of this approach is the fact that when we work with integral closure of modules, the analytic curves ϕ : (C, 0) → (X, x) has a key role. Clearly the Propositions 5.1.5, 5.1.6 (a,c) and the Corollary 5.1.7 (a,b,d) still hold for the double homomorphism relative to an analytic map. We can write the Proposition 5.1.6 (b) on this new language as follows: Proposition 5.5.3. Let ϕ : (Y, y) → (X, x) and ϕ ′ : (Z, z) → (Y, y) be analytic map germs, M ⊆ q p r submodules. Let φ : M → N be an O -module homomorphism OX,x , N ⊆ OY,y and P ⊆ OZ,z X,x ′ ′ and φ : N → P be an OY,y -module homomorphism. Then, φ ∘ φ : M → P is an OX,x -module homomorphism, considering P with the OX,x -module structure induced by the pullback of ϕ ∘ ϕ ′ : (Z, z) → (X, x) and ′ (φ ′ ∘ φ )D,ϕ∘ϕ ′ = φD,ϕ ′ ∘ φD,ϕ .

Proof. For all α ∈ OX,x and h ∈ M, working with the module structures induced by the pullbacks of the analytic map germs, we have:

5.5. The double homomorphism relative to an analytic map germ

131

φ ′ ∘φ (αh) = φ ′ (αφ (h)) = φ ′ (ϕ * (α)φ (h)) = ϕ * (α)φ ′ (φ (h)) = ϕ ′* (ϕ * (α))(φ ′ ∘φ (h)) = (ϕ ∘ ϕ ′ )* (α)(φ ′ ∘ φ (h)) = α(φ ′ ∘ φ (h)). So φ ′ ∘ φ : M → P is an OX,x -module homomorphism ′ and (φ ′ ∘ φ )D,ϕ∘ϕ ′ is well defined and clearly is equal to φD,ϕ ′ ∘ φD,ϕ .

133

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Determinantal

Varieties,

143

INDEX

(w)-regularity, 40 Bi-Lipschitz equisingularity on determinantal varieties, 108 Chain homotopy, 123 Chain of points, 48 Determinantal Varieties, 44 Double chain complex, 122 Double homomorphism, 114, 129 Double of a module, 51 Double of an ideal, 42 EIDS, 46 Genericity of the iLA condition, 72 Infinitesimal Lipschitz conditions, 43, 67 Integral closure of ideals, 27 Integral closure of modules, 32 Lipschitz equisingularity, 47 Lipschitz saturation of a module, 81, 83, 86, 90 Lipschitz saturation of an ideal, 41 Lipschitz stratifications, 48 Multiplicity of a pair of modules, 35 Reduction of ideals, 30 Simple isolated Cohen-Macaulay singularities, 104 Stratifications, 47 The Double Category, 125 The generic rank of the double of a module, 59 Whitney equisingularity, 37

Instituto de Ciências Matemáticas e de Computação

UNIVERSIDADE DE SÃO PAULO