based on lectures of j. sancho - unex.esmatematicas.unex.es/~navarro/degree.pdfand constructions to...

Notes for a Licenciatura

Based on Lectures of J. Sancho

February 19, 2020

Preface

During the seventies, the Licenciatura –the university studies– in Mathematics at the Universityof Salamanca had a highly coherent structure devised by late Prof. Juan B. Sancho Guimerá.In his lectures he taught us how the main ideas of Grothendieck pervade mathematics:

How separable extensions of a field k may be understood as coverings of a one point spaceSpec k, the localization process being the change of the base field and the globalization processencoding Galois theory.

How any morphism X → T may be understood as a family of spaces parameterized by T ,so that absolute statements about a space X really refer to the projection X → p onto the onepoint space, and how absolute statements always have relative versions about families X → T ,which greatly simplify the theory because of the freedom they provide.

How any morphism T → X may be understood as a point of X (parameterized by T ) andthat obviously such functor of points of X fully determines the whole structure of X, so reducingdefinitions of morphisms to the silly case of sets –provided that they are natural definitions–and constructions to the problem of representing a functor.

How sheaf cohomology is well-suited for Algebraic Geometry, Analysis and Topology.

How Grothendieck’s representability theorem is very simple in the case of categories ofabelian sheaves, and that it directly provides the existence of dualizing sheaves and dualiz-ing complexes in the case of smooth curves (representing the functor H1(C,−)∗, Riemann-Roch’s theorem), smooth varieties (Serre’s duality), topological manifolds (Poincaré’s duality)and proper morphisms (Grothendieck’s duality).

How toposes may unify Algebraic Geometry, Arithmetic and Topology.

And above all, how deeply rooted we have the wish of simple and natural definitions, state-ments and proofs, and that such yearning may be always overwhelmingly accomplished.

The aim of these student notes based on the lectures of Sancho and his collaborators, updat-ing them and redacting any topic as best as I know today, is to achieve a unified presentation of asignificant part of the university teaching in Mathematics. And also to develop the courses withcomplete proofs, in a concise and coordinated way; mainly focusing on the logical and meaninginterdependencies of the involved topics, devoid of anything that would put the central pointsin the shade. Hence the pages will be filled with definitions, statements and proofs, while withscarce examples, applications and motivations. And the aim of this preface is to bring thesedry pages into life, to invite the reader to place them into the perspective of the much broaderGrothendieck’s oeuvre and life.

Fortunately Grothendieck has given us a very inspired and appealing description of its ownmathematical works in the Promenade à travers une oeuvre – ou l’enfant et la Mère1 at the verybeginning of Récoltes et Semailles2, an unpublished work where (following the path of Descartes,Pascal, Leibnitz,...) he has contributed to embed Mathematics as a significant part of a muchbroader spiritual adventure: man’s self-conscience unfolding. In this Promenade he presents therelationship of any man with the spiritual goods under a double aspect. On the one hand, theluminous aspect (the passion for knowledge) represented by the figure of a child. On the otherhand, the obscure aspect of la peur et de ses antidotes vaniteux, et les insidieux blocages de la

1Promenade through an oeuvre – or the child and the Mother.2Reapings and Sowings.

ii

creativité qui en derivent3. A deep and valuable lesson of the main body of Récoltes et Semaillesis summarized at the end of the Introduction:

Si quelque chose pourtant est saccagé et mutilé, et desamorcé de sa force originelle,c’est en ceux qui oublient la force qui repose en eux-mêmes et qui s’imaginent saccagerune chose à leur merci, alors qu’ils se coupent seulement de la vertue créatrice de cequi est à leur disposition comme elle est à la disposition de tous, mais nullement àleur merci ni au pouvoir de personne.4

This Promenade touched me deeply. First by the innocence of the child making mathematics,of our children untiringly asking why? and what is it? instead of the weary and clumsy what isit for? of adults. And above all by the silently, quite, attentive, feminine attitude of listeningour whispering interior voice and readiness to accept it, resounding me Mary’s fiat: be it untome according to your word. But the exigence of solitude in any creative work shocked me, andwhen I wrote Grothendieck in 1987 that I missed it, he said me in a letter:

Vous soulignez, à juste titre, la difficulté psychique de la création solitaire... C’estlà la situation qui a été la mienne plus ou moins pendant toute ma vie, depuis monenfance, tant sur le plan de la création mathématique, que dans mon itinéraire spi-rituel... Et sans cela, rien de grand ne s’accomplit, ni dans l’aventure individuelle,ni dans l’aventure collective – que ce soit au plan intellectuel, ou au plan spirituel...Au niveau spirituel, la plus grande ouevre (à mes yeux) qu’un homme ait accomplie,était la Passion du Christ et sa mort sur la croix... Cette oeuvre était et ne pouvaitêtre que solitaire. Et même c’était la solitude suprême, car Dieu Lui-même s’estretiré, pour que l’Oeuvre s’accomplisse sans le secours d’une consolation.5

In fact any great creative labour usually has a harsh period of solitude, and Grothendieck andhis parents had a very tough life, which is a paradigmatic and astonishing incarnation of thewhole XXth century history. A life that we may foresee in the autobiographical chapters III andVI of La Clef des Songes6, another unpublished work where he embeds mathematics into man’sreligious adventure. He says:

Les lois mathématiques peuvent être découvertes par l’homme, mais elles ne sontcréés ni par l’homme ni même par Dieu. Que deux plus deux égale quatre n’est pasun décret de Dieu, qu’Il aurait été libre de changer en deux plus deux égale trois,ou cinq. Je sens les lois mathématiques comme faisant partie de la nature même deDieu – une partie infime, certes, la plus superficielle en quelque sorte, et la seule quisoit accesible à la seule raison.7

3the fear and the antidotes of vanity, and the insidious ways they block creativity (Avant-propos, page A3).4If something is despoiled and mutilated, defused of its initial force, it is in those unaware of the force resting

inside them, who think they are despoiling something at their mercy, while they are cutting themselves off fromthe creative virtue of that which is at their disposal as this virtue is at the disposal of everyone, but not at theirmercy nor under the power of anyone (Introduction II.10, page xxii).

5You remark, deservedly, the psychic difficulty of any lonely creation... Such has been my situation all my life,since my childhood, in the mathematical creation and in my spiritual itinerary... And without that, nothing greatis accomplished, in the individual adventure, nor in the collective adventure – at the intellectual level or at thespiritual level... At the spiritual level, the biggest oeuvre (in my eyes) accomplished by a man, was the Passionof the Christ and his death on the cross... This oeuvre was and could only be solitary. It was even the supremesolitude, since God Itself moved away, so that the Oeuvre is accomplished without the help of any consolation.

6The Key of Dreams.7Mathematical laws may be discovered by man, but they are not created by man, nor even by God. That two

plus two equals four is not a decree of God that He is free to change into two plus two equals three, or five. Isense the mathematical laws as being part of the very nature of God – a tiny part, certainly, the most superficialin some sense, and the only part accessible to reason alone (III 31, Les retrouvailles perdues..., page 100).

iii

And without this skin, without that flesh and blood, and without that fire resting inside us,the following hundreds of tight pages will only be a nude and dead skeleton.

Juan A. Navarro González

Contents

Introduction 1

First Year 13

1 Analysis I 13

1.1 Integer and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Cardinal and Ordinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Real and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Metric Spaces and Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.1 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.2 Completion of a Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6 Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.7 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.7.1 Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Linear Algebra 41

2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.1 The Quotient Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4 The Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5 Euclidean Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.6 Diagonalization of Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.7 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.7.1 Alternate Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Algebra I 69

3.1 The Quotient Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3 Roots and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.1 Quadratic Irrationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.2 Simple Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3.3 Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.4 Root Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.3.5 Multiple Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4 Rings of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 The Resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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vi CONTENTS

Second Year 91

4 Analysis II 91

4.1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


4.3 The Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4 Integral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.4.1 Change of Variables Formula . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5 Convergence in Cm(U) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5 Algebra II 117

5.1 Actions of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2.1 Injective and Projective Modules . . . . . . . . . . . . . . . . . . . . . . . 122

5.2.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.3 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Categories and Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.3.1 Grothendieck Representability Theorem . . . . . . . . . . . . . . . . . . . 132

5.4 The Spectrum of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134


5.6 Finite Algebras over a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.6.1 Trivial and Separable Algebras . . . . . . . . . . . . . . . . . . . . . . . . 143

5.6.2 Galois Theory of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.6.3 The Frobënius Automorphism . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.6.4 Radical Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6.5 Inseparable Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6 Projective Geometry 155

6.1 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.1.1 Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.2 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.2.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.2.2 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.3 Modules over a Principal Ideal Domain . . . . . . . . . . . . . . . . . . . . . . . . 169

6.3.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.3.2 Grothendieck K-Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.4 Classification of Pairs of Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.5 Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.5.1 The Brauer Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Third Year 188

7 Commutative Algebra 189

7.1 The Spectrum of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.2 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.3 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

7.4 Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

7.5 Finite Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.6 Valuations and Dedekind Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

CONTENTS vii

7.7 Birrational Finite Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

7.8 Faithfully Flat Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.9 Galois Theory of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

7.9.1 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8 Topology 215

8.1 Lattice Semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.2 Compact Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8.2.1 Proper Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

8.3 Separation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.4 Noetherian and Finite Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8.5 Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8.6 Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

8.7 Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

8.8 Galois Theory of Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8.8.1 The Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

8.8.2 Triangulated Compact Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 244

9 Analysis III 249

9.1 Rings of Smooth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

9.2 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

9.2.1 Uniparametric Groups and Lie Derivative . . . . . . . . . . . . . . . . . . 254

9.2.2 Pfaff Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

9.3 Integration of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

9.3.1 Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

9.3.2 De Rham Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

9.4 Functions of a Complex Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

9.4.1 Meromorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

9.4.2 Convergence in O(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

10 Differential Geometry I 275

10.1 Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.1.1 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

10.1.2 Smooth Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

10.2 Linear Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

10.2.1 Torsion and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

10.3 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

10.3.1 Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

10.3.2 Surfaces of Constant Curvature . . . . . . . . . . . . . . . . . . . . . . . . 292

10.4 Riemannian Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

10.5 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Fourth Year 300

11 Algebraic Geometry I 301

11.1 Sheaves and Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

11.2 Sheaf Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

11.3 Schemes and Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

11.4 Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

viii CONTENTS

11.4.1 Calculation of the Canonical Sheaf . . . . . . . . . . . . . . . . . . . . . . 316

11.5 Projective Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

11.5.1 Projective Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

11.6 Complete Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

12 Algebraic Topology I 327

12.1 Cohomology with Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

12.2 Homological Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

12.2.1 The Functors TorAn and ExtnA . . . . . . . . . . . . . . . . . . . . . . . . . 333

12.2.2 Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

12.3 Inverse Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

12.4 Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

12.4.1 Universal Coefficients and Künneth’s Theorem . . . . . . . . . . . . . . . 342

12.5 Locally Trivial Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

12.5.1 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

12.6 Local Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

12.6.1 Topological Intersection Theory . . . . . . . . . . . . . . . . . . . . . . . . 351

12.7 Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

12.7.1 Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

12.8 Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

12.9 Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

13 Analysis IV 369

13.1 Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

13.1.1 Uniformization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

13.2 Fréchet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

13.2.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

13.2.2 The Transpose Linear Mapping . . . . . . . . . . . . . . . . . . . . . . . . 381

13.3 Compact Riemann Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

13.3.1 Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

14 Differential Geometry II 395

14.1 Valued Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

14.1.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

14.2 Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

14.2.1 Problems in Dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

14.3 Natural Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

14.4 Chern Classes and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Fifth Year 411

15 Algebraic Geometry II 411

15.1 Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

15.2 Local Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

15.2.1 Regular Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

15.2.2 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

15.2.3 Local Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

15.3 Quasi-Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

15.4 K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

CONTENTS ix

15.4.1 Graded K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42715.4.2 Cohomology theories and Chern Classes . . . . . . . . . . . . . . . . . . . 43115.4.3 Grothendieck’s Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . 435

15.5 Duality Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44015.5.1 Calculation of the Dualizing Complex . . . . . . . . . . . . . . . . . . . . 44215.5.2 Local Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44515.5.3 Biduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

15.6 Formal Functions Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44915.7 Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

15.7.1 The Faithfully Flat Topology . . . . . . . . . . . . . . . . . . . . . . . . . 45515.7.2 Sheafification of a Presheaf . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Exercises 4631. Analysis I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4632. Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4663. Algebra I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4724. Analysis II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4755. Algebra II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4786. Projective Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4897. Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5008. Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5059. Analysis III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

10. Differential Geometry I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51511. Algebraic Geometry I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52012. Algebraic Topology I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52613. Analysis IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53114. Differential Geometry II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53315. Algebraic Geometry II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

Index of Terms 543

Bibliography 559

x CONTENTS

Introduction

On the occasion of the Symposium in memory of late Prof. J. Sancho Guimerá held atSalamanca in 2014, I decided to write some notes based on Sancho’s lectures, with the sameinitial aim as Bourbaki at the 40’s: to undertake the university mathematical teaching from theoutset, with precise definitions, clear statements and complete proofs.

But, contrary to Bourbaki’s book, without avoiding sheaves and categories.

Hence, all the theorems included in these notes have concise and complete proofs, but thisis not a textbook for self-study of students in a standard sense:

1. It tries to reflect the terse style of the study notes of a student, always filled in with theexplanations of a lecturer. And always assuming the atmosphere of each course, with someimplicit assumptions and conventions, that surely will be clear to anyone who carefully readsany section from the beginning. However, the defined concepts are always written in boldfaceand they figure in the index of terms.

2. It focuses on the core of each course, the simple concepts and ideas that directly lead tothe main results, leaving aside details, examples and applications that surely are required toassimilate the theory, but would put the central points in the shadow.

3. All the courses of a year must be studied simultaneously, as students actually do, and nological order may be imposed upon them, each one massively using definitions, results andideas from the companion chapters, sometimes placed at subsequent pages, due to the linearcharacter of any book. It is non sense to read a chapter alone, and all the courses of a yearmust be studied simultaneously as a whole. So the course Linear Algebra uses some propertiesof polynomials, proved in the companion but posterior chapter Algebra I ; the course AnalysisIII massively uses from the start smooth manifolds, vector fields and differential forms (andby the end metrics of constant curvature), studied in the companion but posterior chapterDifferential Geometry I ; and so on.

I do not pretend to develop each course in a self-contained way, without references to thesimultaneous courses. On the contrary, I pretend to enlighten the mutual connections, to showthat it would be unnatural to parcel out mathematics in several unrelated fields (algebra, anal-ysis, differential geometry, topology,...), that the main ideas are simple and everywhere fruitful,so reflecting the essential unity of Mathematics. Even if this choice force us to use concepts,results and ideas eventually placed in a posterior chapter, when all the courses of a year arewritten in a single book.

I have tried to be brief, since my aim is to exhibit many university courses, underliningthe logical interdependencies and the central concepts and methods. To show that, when eachcourse is developed with a regard to the resources and needs of the other parts of Mathematics,the main results have more natural and simple proofs. To show that, again and again, when agentle hand provides the correct definitions, the adequate point of view, surprisingly the knotsuntie, the difficulties dissolve and the backbone of each theory may be exposed in a handful ofpages.

By the moment, I have been able to put black on white fifteen annual courses, with thecourses of each year assumed to be studied at the same time:

2 INTRODUCTION

First Year

• Analysis I: Cardinal and ordinal numbers. Real and complex numbers. Metric spaces andtopological spaces. Differential and integral calculus. Power series.

• Linear Algebra: Vector spaces and linear maps. The dual space. Euclidean vector spaces.Tensors and p-forms.

• Algebra I: The quotient ring. Principal ideal domains. Field extensions and roots. Rings offractions. Elimination.

Second Year

• Analysis II: σ-compact spaces. The Lebesgue measure. Differential and integral calculuswith several real variables. Convergence in rings of differentiable functions.

• Algebra II: Actions of a group. Modules. Categories. Spectrum of a ring. Module ofdifferentials. Finite algebras over a field. Galois theory of fields.

• Projective Geometry: Projective and affine spaces. Classification of metrics, modules overa principal ideal domain, and pairs of metrics. Semisimple rings and the Brauer group.

Third Year

• Commutative Algebra: Noetherian rings. Primary decomposition. Completion. Dimen-sion theory. Integral dependence. Dedekind domains. Galois theory of rings.

• Topology: Lattice semirings. Separation properties. Noetherian spaces. Compactifications.Dimension theory. Uniform spaces. Galois theory of coverings. The fundamental group.

• Analysis III: Rings of smooth functions. Ordinary differential equations. Pfaff systems.Integration of differential forms. De Rham cohomology. Riemann surfaces. Meromorphicfunctions. The modular λ function. Riemann mapping theorem.

• Differential Geometry I: Smooth manifolds. Tensor fields and the exterior differentialcalculus. Linear connections. Riemannian metrics. Lie groups.

Fourth Year

• Algebraic Geometry I: Sheaf cohomology. Schemes and coherent sheaves. Algebraic curvesand the Riemann-Roch theorem. The projective spectrum.

• Algebraic Topology I: The cohomology ring. Homological algebra. Local cohomology.Duality theorem. Characteristic classes. Spectral sequences.

• Analysis IV: Uniformization theorem. Compact Riemann surfaces.

• Differential Geometry II: Valued differential calculus. Calculus of variations. Naturalbundles. Chern classes and curvature.

Fifth Year

• Algebraic Geometry II: Local algebra. Quasi-coherent sheaves. K-theory. Duality theory.Grothendieck topologies and descent theory.

3

so that many significant parts of the university teaching are still lacking: functional analysis andpartial differential equations, homology and homotopy theory, number theory, measure theoryand statistics, mathematical physics, etc.

The first three years are mainly devoted to the concepts of structure and morphism, thecentral results being “Galois type” theorems stating the equivalence of two different structures(i.e. of two categories), while the last two years focus on the concepts of sheaf and cohomology,the central results being duality theorems. Moreover, representable functors and the conceptsof point and space pervade the notes from start to finish, as well as the clarification of theadjectives natural, intrinsic, canonical and universal that we frequently use in a naive sense.

Here it is a brief description of the sense and content of each annual course:

Analysis I: We begin with the construction of integer, rational, real and complex numbers,the genetic method of reducing all mathematical concepts to the process of counting (sets andnatural numbers); so initiating the accomplishment of the fantastic pythagorean vision8: all isa music of numbers. This genetic method is the backbone of Analysis and culminates in thetheory of functions of several complex variables and the theory of partial differential equations.

We study cardinal numbers, and we put |X| ≤ |Y | when there is an injective map X → Y .The main result is that any set of cardinal numbers has a first element. So, there is the firstinfinite cardinal ℵ0 = |N|, the first cardinal ℵ1 bigger than ℵ0,... and so we may parameterizethe infinite cardinals by the ordinal numbers.

We construct the real numbers using Cauchy sequences of rational numbers, and generalizethis method to complete any metric space. We also prove that the compact sets in Rn are justthe bounded closed sets, a central result that readily proves that the image of any continuousfunction f : [a, b] → R is a closed interval, f([a, b]) = [m,M ], a statement encoding the mainresults on continuous functions.

After an exposition of the Infinitesimal Calculus, using the Riemann integral, we show thata bounded function f : [a, b] → R is Riemann-integrable if and only if it is almost everywherecontinuous. We end with an introduction to the power series expansions, and we use them tostudy the exponential and trigonometric functions.

Linear Algebra: In this chapter we take the path of the axiomatic method. Geometry burstsluminously in Greece when the Greek genius realized that properties of such subtle and funda-mental concepts as point, line, right angle, etc., should not be reduced to properties of previousconcepts, but that we should better put on show their basic mutual relations, stating them asaxioms or postulates of the theory, and then derive the remaining properties. In modern terms,any question is submerged in an implicit structure, and a breaking point is to put it into light.Besides, in the 19th century the German world discovered that it is crucial to clarify what struc-tures are considered to be equal (the isomorphisms) and it placed as the central problem of anytheory the classification of all possible structures up to isomorphisms9. As a first example of the

8Aristotle (Metaphysics, Book I, Part 5, 986a): τα των αριθµων στωιχ�ια των oντων στωιχ�ια παντωνυπ�λαβoν �ιναι, και τoν oλoν oυρανoν αρµoνιαν �ιναι και αριθµoν. ([The pythagoreans supposed] thenumbers to be the elements of all things, and the whole heaven to be a musical scale and a number).

9One of the first to glimpse such classification problems was Goethe, as he wrote in his diary entries (Naples,May 17, 1787): Die Urpflanze wird das wunderlichste Geschöpf von der Welt, um welches mich die Natur selbstbeneiden soll. Mit diesem Modell und dem Schlüssel dazu kann man alsdann noch Pflanzen ins Unendlicheerfinden, die konsequent sein müssen, das heißt, die, wenn sie auch nicht existieren, doch existieren könntenund nicht etwa malerische oder dichterische Schatten und Scheine sind, sondern eine innerliche Wahrheit undNotwendigkeit haben. (The Urpflanze is going to be the most wonderful creation of the world, which Nature herselfshall envy me. With this model and the key to it, one can then go on inventing plants ad infinitum, which mustbe consistent; that is, even if they do not exist, they could exist and not as picturesque or poetic shadows andillusions, but with inner truth and necessity). In his quiet walks along the Italian botanical gardens, he dreamed

4 INTRODUCTION

great insight of the German point of view, just compare the genetic definition of the additionof integer numbers given in p. 13 with the definition steaming from the classification of cyclicgroups (p. 44): it is the unique possible operation defining a structure of infinite cyclic group(so recovering our infantile vision: what is added does not matter, only how to do the addition).

After a very concise introduction to the structures of group and ring, the course focuseson the structure of vector space (so that we may exhibit for the first time the structure of theEuclidean Geometry10, at least with a fixed point and a fixed length unit) and we prove thatvector spaces are classified by the dimension, whenever it is finite.

A big emphasis is put on universal properties (our first contact with representable functors)which simplify the theory in many points. For example in the intrinsic definition of the con-traction of indices, otherwise cumbersome and in need of some checking, and in the proof of thenatural isomorphism ΛpE = (ΛpE)

∗. The systematic use of linear maps and exact sequencesfrom the very beginning of the course also simplifies the theory.

Algebra I: This chapter is devoted to the study of the structure of ring. We understand itas a generalized arithmetic: numbers are ideals and the divisibility relation is the inclusionrelation, so that prime numbers are maximal ideals, the quotient ring encoding the gaussiantheory of congruences. The main result is Kronecker’s formula for a root of any polynomialequation p(x) = 0, discovered when he realized that the extraction of a root involves two quitedifferent steps. The first one, subtle, humble and usually of long gestation, is the highlight of astructure of field (radical expressions, real or complex numbers, etc.); while the second one is toexpress a root inside such structure. But men were looking for a root without questioning theconvenience of the structure where they limited the search, enclosed in an invisible iron circle11

that Kronecker innocently crossed when he discovered that, if p(x) is irreducible, an obviousroot is just the class [x] in the field of residue classes modulo p(x), so reducing the extraction ofroots to the problem of decomposing polynomials into irreducible factors.

about the plant structure and its possible realizations, so that he could know all the plants that Linnaeus hadcatalogued, the unknown plants of Africa and America that brave and courageous explorers were looking for, theextinct plants, the future ones and those that will remain in God’s mind forever. Even if he could not achievethis wonderful dream, it is not a mere coincidence that the periodic table of elements and the first classificationsof mathematical structures and elementary particles were so close to Goethe, in time, space and spirit.

10Stricto senso, the higher dimensional real Euclidean geometries studied in this course were the first “NonEuclidean” geometries, glimpsed in 1747 by Immanuel Kant (Gedanken von der wahren Schätzung der lebendi-gen Kräfte §10–11): Diesem zu folge halte ich dafür: daß die Substanzen in der existirenden Welt, wovon wirein Theil sind, wesentliche Kräfte von der Art haben, daß sie in Vereinigung miteinander nach dem doppeltenumgekehrten Verhältniß der Weiten ihre Wirkungen von sich ausbreiten; zweitens, daß das Ganze, was daherentspringt, vermöge dieses Gesetzes die Eigenschaft der dreifachen Dimension habe; drittens, daß dieses Gesetzwillkürlich sei, und da Gott dafür ein anderes, zum Exempel des umgekehrten dreifachen Verhältnisses, hättewählen können; daß endlich viertens aus einem andern Gesetze auch eine Ausdehnung von andern Eigenschaftenund Abmessungen geflossen wäre. Eine Wissenschaft von allen diesen möglichen Raumesarten wäre unfehlbar diehöchste Geometrie, die ein endlicher Verstand unternehmen könnte... Wenn es möglich ist, daß es Ausdehnungenvon andern Abmessungen gebe, so ist es auch sehr wahrscheinlich, daß sie Gott wirklich irgendwo angebracht hat.Denn seine Werke haben alle die Größe und Mannigfaltigkeit, ist, daß es die sie nur fassen können. (Thoughtson the True Estimation of Living Forces §10–11: Accordingly, I am of the opinion that substances in the existingworld, of which we are a part, have essential forces of such a kind that they propagate their effects in union witheach other according to the inverse square of the distances; secondly, that the whole to which this gives rise has,by virtue of this law, the property of being three dimensional; thirdly, that this law is arbitrary, and that Godcould have chosen another, e.g., the inverse-cube, relation; fourthly, and finally, that an extension with differentproperties and dimensions would also have resulted from a different law. A science of all these possible kinds ofspace would undoubtedly be the highest geometry that a finite understanding could undertake... If it is possiblethat there are extensions of different dimensions, then it is also very probable that God has really produced themsomewhere. For his works have all the greatness and diversity that they can possibly contain).

11Our questions always have an implicit horizon embracing all the possible answers. To make it clear, and toextend it if necessary, are the strong moments of man’s self-conscience unfolding.

5

At first glance Kronecker’s astonishing answer may seem a mere tautology or a sterile for-mality. Certainly it is a tautology presented with all the formal rigor we are able, as any othertheorem; but, after applying it to prove D’Alembert’s and Hamilton-Cayley theorems (p. 76and p. 59), and to solve the millennial questions on ruler-and-compass constructions (p. 78),we leave to the reader the consideration that the adjectives mere and sterile deserve.

Analysis II: After a brief study of topological spaces, we study the differential calculus withfunctions of several real variables. The main tool is a key lemma in the calculus with infinitesi-mals: If f is a function of class Cm on an open ball U around the origin 0 ∈ Rn and f(0) = 0,then f =

∑i fixi for some functions fi of class Cm−1. This key lemma is proved using Barrow’s

rule and the differentiation rule under the integral sign, and it readily gives Schwarz’s theorem,∂i∂jf = ∂j∂if , and Taylor’s expansions; and it also gives the inverse mapping theorem, usingthe relative version along the diagonal: any function f ∈ Cm(U ×U), vanishing on the diagonal,is f =

∑i fi(yi − xi) for some functions fi ∈ Cm−1(U × U).

Then we study the Lebesgue measure and the corresponding integration of functions, themain results being the change of variables formula and Sard’s theorem. We prove both usingthe key lemma along the diagonal, that shows that in a neighborhood of a point p, the image ofany cube is very close to the image by the tangent linear map at p.

Finally, we introduce topological vector spaces and we prove that Cm(U) is complete withthe topology of the compact convergence of the functions and the iterated partial derivatives.

Algebra II: This is a crucial course with an explosion of many central ideas and constructionsthat will pervade the remaining chapters.

1. A big step is given in the clarification of the concepts of point and function. Any commutativering A defines a topological space SpecA whose points correspond to the prime ideals of A,the elements of A are viewed as functions on SpecA, and the ideals of A as equations ofgeometric loci. Integers, gaussian integers, etc., may be understood as functions, and we mayapply them our geometric intuitions and resources, so starting the unification of Arithmeticand Geometry dreamed by Kronecker.

2. Localization of modules, proving that most properties of modules are local12, in the sensethat they only have to be checked on the local rings of the points of SpecA.

3. Tensor products. Hence the base change of algebras and modules, so that we may introducethe concepts of geometric property (stable under changes k → K of the base field) and localproperty (valid whenever it holds after some base change k → K).

4. Categories and functors, aiming to give a first rigorous approximation to the fundamentalconcepts of structure, natural and canonical.

5. Points of SpecA with coordinates in a field K are viewed as morphisms SpecK → SpecA and,in general, Grothendieck’s representability theorem gives necessary and sufficient conditionsfor a functor to be a functor of points.

6. The module of differentials, giving an algebraic treatment of the differential calculus.

7. Injective and projective modules, a basic tool of the forthcoming Homological Algebra.

12A basic fact that we constantly use throughout all the notes, even without further mention. Surprisingly, anyother property is easily reduced to the critical property of being 0. Here, as in any other case, nothingness, assoon as it is senseful, becomes a driving force.

6 INTRODUCTION

The central result is Galois theorem, stated as an antiequivalence of the category13 of k-algebras trivial on a given Galois extension k → L of Galois group G with the category of finiteG-sets. The proof14 directly follows from the fundamental exact sequence k → A⇒ A⊗kA andthe stability of the algebra of invariants AG under base changes, (AG)⊗k L = (A⊗k L)G.

We also introduce the Frobënius automorphism of a polynomial q(x) ∈ Z[x] at a prime p, sorelating the global properties of q(x) to the behavior of its reductions q̄(x) ∈ Fp[x].

Finally, we prove that the maximal separable subalgebra is stable under base changes k → K,and the elementary theory of inseparable k-algebras readily follows.

Projective Geometry: We introduce the projective space P(E) attached to a vector space E,but only as a mere set, without elucidating the underlying structure. To figure out the implicitstructure grounding projective geometry has been a major topic in the history of Geometry,and so it will be in these notes. Nevertheless, in this course at least we know the group ofautomorphisms of such elusive structure: an isomorphism of a k-vector space E into a k′-vectorspace E′ clearly should be defined as a group isomorphism f : E → E′ such that f(λe) =σ(λ)f(e) for some ring isomorphism σ : k → k′; hence the group of automorphisms of P(E) isthe group PSl(E) of all bijections P(E) → P(E) represented by a semilinear automorphismE → E.

Klein, in his foundational Erlangen Programme, states that the structure of the projectivespace is given by the action of the group PGl(E) of all projectivities15 on P(E), and that ingeneral any action of a group G on a set defines a geometry: concepts of the geometry beingjust G-invariant concepts, statements being relations between concepts, and theorems beingtrue relations. Considering different subgroups of PGl(E), in this course we also study affinegeometry, Euclidean geometry and Non-Euclidean geometries. In this course we show thatthe structure of the projective spaces of dimension ≥ 2 is captured by the lattice of linearsubvarieties16; but to figure out the structure of the real projective line as a ringed space (sogrounding projective geometry on non-algebraically closed fields) was a first success of the theoryof schemes.

On the other hand, any concept of a geometry gives rise to a classification problem, con-sidered as the determination of the set of orbits of the induced G-action. In this course westudy the projective and affine classifications of quadrics (reduced to the linear classification ofsymmetric metrics), the projective classification of projectivities (reduced to the linear classi-fication of endomorphims) and the projective classification of pencils of quadrics (reduced tothe classification of metrics on modules over the finite k-algebras k[x]/(p(x)) considered in thecompanion course Algebra II ). In this sense, this is a second course in Linear Algebra.

We classify endomorphisms, and finitely generated modules over a principal ideal domain A,using the crucial fact that B = A/pnA is an injective B-module when p is irreducible, a resultthat follows directly from the Ideal Criterion studied in Algebra II.

It is worthy of attention that, in the former course Linear Algebra, the characteristic polyno-mial of an endomorphism was introduced in a non-intrinsic way, defining it in a base and checkingthat it does not depend on the base. Now we introduce the K-group to give Grothendieck’s

13In fact the category of finite direct sums of intermediate fields, so that the theorem not only determines theintermediate fields but also their k-morphisms.

14Two alternative proofs are sketched in exercises 106 and 107, p. 485. The first one reduces the proof to thesimple case k = L, once the exactness properties of the involved functors are checked; while the second one usesGrothendieck’s characterization of the forgetful functor on the category of finite G-sets.

15so that the group of automorphisms of P(E) is just the normalizer of PGl(E) in the group of all bijectionsP(E)→ P(E), which may be shown (ex. 8, p. 489) to be PSl(E).

16The so called fundamental theorem of projective geometry (p. 159); but today such fundamental theorem,illuminating the structure of projective spaces, is the universal property of P(E) given in p. 322.

7

astonishing definition of the characteristic polynomial: it is the universal additive function onthe finitely generated torsion k[x]-modules.

Finally, we study semisimple rings and the Brauer group, obtaining that the quaternions arethe unique non-commutative finite extension of the real numbers.

Commutative Algebra: First we introduce the “structural sheaf” of the spectrum of a ring,so that we may present all the topics with its full geometric comprehension and flavor:

1. Primary decomposition of ideals and completion.

2. Krull dimension theory and regular rings.

3. Finite morphisms and birrational finite morphisms, obtaining the desingularization of curves,and the calculation of intersection numbers, by quadratic transformations.

Finally we show that the theory of unramified finite flat ring morphisms A → B is totallyanalogous to the Galois theory of separable finite k-algebras and that, when considering theinduced morphism SpecB → SpecA, it is totally analogous to the theory of coverings of atopological space, so obtaining the definition of the fundamental group π1(SpecA, p) of SpecAat a geometric point p : Spec k̄ → SpecA.

In these notes the Galois theory of fields, of noetherian rings and of coverings of a topologicalspace, have a unified presentation, with equal statements and proofs.

Topology: When studying continuous functions, the particular value of a function f at a pointx usually does not matter, only wether f(x) = 0 or f(x) 6= 0. So it is natural to identify all thenon-zero real numbers, and so we obtain K = {0, g}, with a closed point 0 and a dense point g(the generic real number). We have a natural structure of ”field” with unity g = 1 on K,

0 + 0 = 0 0 + g = g + 0 = g g + g = g0 · 0 = 0 0 · g = g · 0 = 0 g · g = g

and it is clear that any closed set Y in a topological space X is the zero-set of a unique continuousfunction fY : X → K, so that the lattice A(X) of all closed sets in a topological space X maybe viewed as a ”ring” of continuous functions, A(X) = C(X,K), the addition corresponding tothe intersection, and the product to the union17,

fY + fZ = fY ∩Z , fY · fZ = fY ∪Z .

Many operations introduced in the elementary theory of rings (quotient ring, localization,tensor product, spectrum, etc.) remain meaningful in the framework of lattice semirings, andthey preserve their geometric meanings. Moreover, most statements and proofs remain valid,and the purpose of this chapter is to show that this comprehension of the lattice of closed setsA(X) as a ”ring” of functions provides a handy tool in some topics:

1. The functor Spec clarifies the theory of compactifications.

2. Finite topological spaces are dual to finite lattice semirings, and essentially encode the theoryof polyhedra.

17In fact A(X) is not a true ring, since the existence of opposite element fails, but it comes with the extraproperties 1 + a = 1 and a2 = a. We name them lattice semirings. In a suggestive way, Connes uses to say thatsemirings where 1 + 1 = 1 have characteristic 1, so that K is a semifield of characteristic 1.

8 INTRODUCTION

3. Typically the Krull dimension of A(X) is infinite, but Sancho realized that the minimal Krulldimension of a base18 B ⊆ A(X) provides a good definition for the dimension of a topologicalspace X, since it coincides with Grothendieck’s dimension (the maximal length of a chain ofirreducible closed sets) when X is a noetherian space, and with the inductive dimension andLebesgue cover dimension when X is a separable metric space.

Finally we study uniform spaces, and the theory of coverings X → S is developed parallel tothe Galois theory of fields and rings, taking advantage of the existence of a universal coveringwith Galois group π1(S, p).

Analysis III: First we study partitions of unity and we show how a smooth manifold X maybe reconstructed from the ring of smooth functions C∞(X), the basic facts being the naturalhomeomorphism X = SpecR C∞(X) and the coincidence of C∞(U) with the ring of fractions ofC∞(X) with respect to the multiplicative system of all smooth functions without zeros in theopen subset U .

In the theory of ordinary differential equations, we show that the argument of the contrac-tive map, classically used to prove the existence and uniqueness of a solution, also proves thecontinuous and differentiable dependence on the initial conditions when considering adequatefunctional spaces.

Then we study Pfaff systems, the central result being a geometrically obvious proof of aprojection theorem given necessary and sufficient condition for a Pfaff system P on a smoothmanifold X to be projectable by a regular projection π : X → Y , in the sense that P = π∗Q forsome Pfaff system Q on Y . As a direct consequence we obtain Frobënius theorem (involutivedistributions are integrable) and Darboux’s classification of germs of 1-forms.

We also study the integration of differential forms on smooth manifolds, obtaining the fun-damental Stoke’s theorem. We apply it to harmonic functions, De Rham cohomology and thetheory of complex analytic functions.

Finally, we introduce Riemann surfaces as ringed spaces. It is easy to prove that locally anynon constant analytic morphism is u = zn for some exponent n ≥ 1 and, as a direct consequenceof this local classification of morphisms, most elementary properties of Riemann surfaces follow.We prove the Riemann mapping theorem with a systematic use of the hyperbolic plane geometry.

Differential Geometry I: We introduce smooth manifolds as ringed spaces X locally isomor-phic to open sets in Euclidean spaces with the sheaf of smooth functions (so eliminating thecumbersome atlases) and submanifolds as topological subspaces Y ↪→ X such that (Y, C∞Y ) isa smooth manifold, where C∞Y is the sheaf of restrictions of smooth functions on X. Then wedevelop the tensor calculus and the exterior differential calculus on smooth manifolds.

We study linear connections, the main result being that any torsion free linear connection ofnull curvature is locally Euclidean, and riemannian manifolds, proving that any simply connectedand complete riemannian surface of constant curvature is isometric to the Euclidean plane, thehyperbolic plane or the sphere. Then we present the theory of riemannian embeddings.

Finally we present a bit of the theory of Lie groups, up to the point of obtaining the classi-fication of abelian Lie groups.

Algebraic Geometry I: This chapter may be considered the hearth of these notes. We in-troduce sheaves and we define the cohomology groups Hp(X,F) using Godement’s resolution.Then we prove a cohomological bound theorem

Hp(X,F) = 0, p > dimX,18a subring whose complements form a base of open sets of X in the usual sense.

9

when X is a noetherian space, and the acyclicity theorem for arbitrary rings,

Hp(SpecA, M̃ ) = 0, p > 0.

Now, after a massive use of the spectrum of a ring in the courses Algebra II, Topology andCommutative Algebra, and the comprehension of smooth manifolds and Riemann surfaces asringed spaces in Differential Geometry I and Analysis III, we may naturally introduce schemesas ringed spaces locally isomorphic to (SpecA, Ã ). The major example of scheme in this coursewill be the Riemann variety of a finite extension of k(t), the Riemann variety of k(t) itself beingthe projective line P1 over the field k.

First we calculate the cohomology groups of the structural sheaf of P1, projecting it onto afinite space with two closed points and a dense point, and it readily follows the determinationof the cohomology groups of the line sheaves OP1(n). Then we prove the fundamental finitenesstheorem for coherent sheaves, dimHp(X,M) < ∞, projecting the curve X onto P1, since thedirect image preserves cohomology.

Now the weak Riemann-Roch theorem for line sheaves LD is immediate,

dimH0(X,LD)− dimH1(X,LD) = 1− g + degD.

Since, H0(X,LK−D) = HomOX (LD, LK), to prove the strong Riemann-Roch theorem weonly have to see that

dimH1(X,LD) = dim HomOX (LD, LK),

where K stands for a canonical divisor of the curve X. Since H2(X,M) = 0, Grothendieck’srepresentability theorem directly gives the existence of a dualizing sheaf:

H1(X,M)∗ = HomOX (M,DX)

for some quasi-coherent sheaf DX , and the problem is to determine DX .It is easy to show that DX is a line sheaf, but it is hard to prove that it is the sheaf of

differentials. We present two proofs of this fundamental result, the first one based on a laboriouslocal calculation of the conductor of a projection X → P1, while the more natural second proofuses the diagonal embedding X → X ×k X and the stability of the cohomology of coherentsheaves under base changes (so pointing that the theory of curves is naturally entangled withthe theory of higher dimensional varieties).

Finally we introduce the projective spectrum of a graded ring, and determine the cohomologygroups of the line sheaves OPd(n) over the projective space Pd = Proj k[x0, . . . , xd].

Algebraic Topology I: This chapter is devoted to the cohomology of sheaves over σ-compactspaces. After introducing the fundamental exact sequences (Mayer-Vietoris, closed subspaceand local cohomology exact sequences) we determine the cohomology groups Hp([0, 1]n,Z) ofcubes, hence of spheres and many classical spaces.

Then we introduce the inverse image and the cup product, and we prove some fundamentalresults: calculation of the cohomology of the fibres, theorem of base change, finiteness theorem,universal coefficients formula, Künneth theorem, and the classification of line sheaves. We alsouse local cohomology groups to define the cohomology class of a closed submanifold and todevelop a topological intersection theory.

Now we get down to the duality theorem. So as to include non locally Euclidean spaces, wedo not consider the dual of the last cohomology group Hnc (X,F), but of a complex determiningall the cohomology groups with compact supports. Again the representability theorem gives theexistence of a dualizing complex D•X such that

RHom(RΓc(F),Z) = RHom(F ,D•X)

10 INTRODUCTION

and in this case it is not hard to determine it when X is a topological manifold of dimension n:the dualizing complex D•X has a unique non-zero cohomology sheaf19 TX , locally constant andplaced at degree −n, so that HomZ(Hnc (X,F),Z) = Hom(F ,TX).

Then we determine the cohomology groups of a projective bundle, a result grounding thetheory of characteristic classes for vector bundles.

Finally we introduce some useful spectral sequences.

Analysis IV: After using Perron’s method to solve the Dirichlet problem, we use it and sheafcohomology to prove the uniformization theorem: any simply connected Riemann surface is thecomplex plane, the complex projective line or the unit disk.

Then we present a brief introduction to Fréchet spaces and the duality theory of locallyconvex spaces, up to the point of obtaining Schwartz theorem, the crucial technical tool to prove(using Cech cohomology) that locally free sheaves on Riemann compact surfaces always havefinite dimensional cohomology groups.

This finiteness theorem readily gives the existence of non constant meromorphic functions,and the equivalence of the category of compact Riemann surfaces to the category of completeand non singular algebraic curves over the field C, studied in the companion course AlgebraicGeometry I. Hence, most of the results there obtained for complete non singular curves overarbitrary fields may be extended to compact Riemann surfaces, in particular the fundamentalRiemann-Roch theorem. However, we show that now the residue theorem provides a very simpleand natural proof of the coincidence of the dualizing sheaf with the sheaf of analytic differentials,the fundamental result that was so hard to prove for algebraic curves.

Differential Geometry II: Many mathematical and physical concepts may be viewed as dif-ferential forms with values in a vector bundle (or a locally free sheaf, the point of view adoptedin this chapter) and the differential calculus with such forms requires a linear connection onthe vector bundle. We develop this fruitful differential calculus up to the point of obtainingBianchi’s identities and Cartan’s structure equations. We also determine the Chern classes of acomplex vector bundle in terms of the curvature 2-form of any linear connection.

We present the variational calculus, obtaining the fundamental Poincaré-Cartan form andNoether’s theorem on the infinitesimal symmetries of a variational problem.

Finally, since most bundles used in Geometry and Physics are natural bundles, we includethe Galois theory of natural bundles (of a given finite order). The statements are parallel tothose of the Galois theory of fields, rings and coverings, but unfortunately proofs differ.

Algebraic Geometry II: We begin with a study of Local Algebra, obtaining Serre’s theoremon regular rings and some characterizations of Cohen-Macaulay rings. Then, after a brief studyof quasi-coherent sheaves (including Deligne’s formula and the base change theorem) we getdown to the central topics of this course: the Riemann-Roch and duality theorems.

1. First we study the K-theory of coherent sheaves up to the point of obtaining Chern classes ofvector bundles with values in the graded K-theory GK•(X). Then we prove the fundamentalresult: the K-theory is the universal multiplicative20 cohomology theory.

Now, if a cohomology theory A(X) follows the additive law c1(L⊗L′) = c1(L) + c1(L′), thenwe may modify the direct image in A(X) ⊗ Q with an exponential, so that it follows themultiplicative law and we have a functorial ring morphism K(X) → A(X) ⊗ Q preservingthe new direct image. This is just the Riemann-Roch-Grothendieck theorem.

19that would be obtained in case of dualizing the last cohomology group, as in Algebraic Geometry I.20in the sense that Chern classes of line sheaves follow the law c1(L⊗L′) = c1(L) + c1(L′)− c1(L)c1(L′) of the

multiplicative group. Remark that (1− x)(1− y) = 1− (x+ y − xy).

11

A direct consequence, in the complex case, is the existence of a natural ring morphismGK•(X) ⊗ Q → H2•(Xan,Q) preserving inverse and direct images, hence Chern classes.Since GK0(SpecC) ⊗ Q = H0(pt,Q) = Q, the algebraic and topological definitions of anynumerical cohomological invariant coincide. For example, if X is a projective smooth variety,the topological and algebraic self-intersection numbers of the diagonal coincide,∑

i(−1)idimQH i(Xan,Q) =∑

p,q(−1)p+qdimCHp(X,ΩqX).

2. Given a projective morphism f : X → S, the representability theorem gives the existence ofa dualizing complex DX/S such that for any quasi-coherent OX -module M,

RHomOS (Rf∗(M),OS) = RHomOX (M,DX/S)

and again the crux is to calculate it in the case of a smooth morphism; to prove that it is thehighest exterior power of the sheaf of differentials, DX/S ' ΩdX/S .Using the Koszul complex, we may see that the dualizing sheaf of a closed regular embeddingY → X of codimension d is just ΛdNY/X , where NY/X stands for the normal bundle. Nowthe calculation of the dualizing sheaf of a smooth projective morphism X → S is a simplequestion. Just consider the composition

X∆−−−→ X ×S X

π1−−−−→ X

of the diagonal embedding with the first projection (obviously it is the identity), and thenormal bundle N to the diagonal embedding (which is dual to the sheaf of differentials ΩX/Sby the very definition). We have

OX = DX/X = DX/X×X ⊗∆∗DX×X/X = ΛdN ⊗∆∗(π∗1DX/S) = ΛdN ⊗DX/S

and we are done: DX/S = (ΛdN)∗ = ΩdX/S . Once the question is placed in the relative andgeneral setting (morphisms and arbitrary dimension instead of the absolute case of curves)the obvious compatibility properties of the theory dissolve the question.

Then we prove the local duality theorem. Again, dualizing the local cohomology groups ata point x, the representability theorem directly gives the existence of a local dualizing complex,and the central point is to show that, in the case of a projective variety, it is just the stalk at x ofthe global dualizing complex. In the crucial case of a smooth variety X, we prove it determiningthe local cohomology groups Hpx(X,OX).

Finally, we introduce Grothendieck topologies, proving that quasi-coherent OS-modules andS-schemes define sheaves on the category of S-schemes with the fpqc topology, and that locallyquasi-coherent sheaves are quasi-coherent (descent of quasi-coherent sheaves).

Requirements: From a logical point of view, only the elementary properties of natural numbersand the intuitive set theory (including Zorn’s lemma) are required, and further foundationalquestions are avoided. In fact, I feel that Zermelo-Fraenkel’s axioms do not ground properly settheory: they include unions of arbitrary sets and assume that the elements of a set are alwayssets, while in mathematics, only unions of subsets of a given set are sensible, and it is non-senseto consider the elements of the elements of any given set21.

21So, even if the spectrum SpecA of a ring A is defined to be the set of all prime ideals of A, the statements0 ∈ x ∈ SpecA and 1 /∈ x ∈ SpecA are non-sense: points of SpecA are not prime ideals of A, but naturallycorrespond to prime ideals. And the elements of the quotient set are not equivalence classes, but naturallycorrespond to equivalence classes.

12 INTRODUCTION

However, along the five years, an increasing maturity level is assumed.

Notations: Exceptionally we use some non-standard notations. The covariant derivative of atensor field T in the direction of a vector field D is denoted D∇T , instead of the usual ∇DT ;and the Lie derivative DLT , instead of the usual LDT .

When a fixed map f : X → Y is unambiguously understood, and Z ⊆ Y , we put

X ∩ Z := f−1(Z) = {x ∈ X : f(x) ∈ Z},

so that X ∩ Z is a subset of X but not a subset of Z.

Acknowledgements: My debt, gratitude and admiration to my teachers at the University ofSalamanca in the seventies, mainly D. Juan Bautista Sancho Guimerá and his collaborators D.Antonio Pérez-Rendón Collantes, D. Pedro Luis Garćıa Pérez, D. Cristóbal Garćıa-Loygorri yUrzaiz, D. Jesús Muñoz Dı́az and D. Jaime Muñoz Masqué.

Also my grateful acknowledgement to many companions who have written notes on thesecourses, with many improvements of their own, since I have made good use of them: DanielHernández, José M. Muñoz, Juan Sancho Jr. and his brothers Teresa, Carlos, Pedro and Fer-nando, my friend Ricardo Faro and my sons José and Alberto.

Chapter 1

Analysis I

1.1 Integer and Rational Numbers

Definitions: A relation ≡ on a set X is an equivalence relation if it is

1. Reflexive: x ≡ x, ∀x ∈ X.

2. Symmetric: x, y ∈ X, x ≡ y ⇒ y ≡ x.

3. Transitive: x, y, z ∈ X, x ≡ y, y ≡ z ⇒ x ≡ z.

The equivalence class of x ∈ X is x̄ = [x] = {y ∈ X : x ≡ y}.A subset C ⊆ X is an equivalence class when C = [x] for some x ∈ X.The quotient set X/≡ is the set of all equivalence classes (or even better, it is a set whose

elements correspond to the equivalence classes).The surjective map π : X → X/≡ , π(x) = [x], is the canonical projection.

Theorem: In X/≡ , only equivalent elements are identified; [x] = [y]⇔ x ≡ y .

Proof: If [x] = [y], then y ∈ [y] = [x], and x ≡ y.Conversely, if x ≡ y, since ≡ is reflexive, it is enough to show that [y] ⊆ [x].Now, if z ∈ [y], then y ≡ z; hence x ≡ z and z ∈ [x].

Corollary: Each element of X is in a unique equivalence class.

Proof: If x ∈ [y], then y ≡ x; hence [y] = [x].

Construction of Z: Let N = {0, 1, 2, 3, . . .} be the set of natural numbers. The set of integernumbers Z is defined to be the quotient set of N× N by the equivalence relation

(m,n) ≡ (m′, n′) when m+ n′ = m′ + n,

and m− n denotes the class of (m,n). Any natural number n defines an integer number n− 0,and so we may identify N with a subset of Z = {. . . ,−2,−1, 0, 1, 2, . . .}.

The sum and the product of a = m− n, b = r − s, are defined to be

a+ b = (m+ r)− (n+ s) , a · b = (mr + ns)− (nr +ms)

and both are well-defined. In fact, if a = m′ − n′, then m+ n′ = m′ + n, and

m+ n′ + r + s = m′ + n+ r + s

(m+ n′)r + (m′ + n)s = (m′ + n)r + (m+ n′)s

13

14 CHAPTER 1. ANALYSIS I

and we also have a+ b = m′ + r − (n′ + s), ab = m′r + n′s− (n′r +m′s).So we reduce statements on Z to statements on N. The theory of integer numbers is just a

part of the theory of natural numbers.If the natural numbers are free from contradiction, so are the integer numbers.

Example: Let us fix a natural number n ≥ 2. The congruence relation, a ≡ b (mod. n) whenb− a ∈ nZ, is an equivalence relation on Z, and [a] = a+ nZ = {a+ cn : c ∈ Z}.

The quotient set Z/nZ = {[1], [2], . . . , [n] = [0]} has n elements.

Construction of Q: The set of rational numbers Q is defined to be the quotient set of thedirect product Z× (Z− {0}) by the equivalence relation

(a, s) ≡ (b, t) when at = bs,

and as denotes the class of (a, s). Each integer a defines a rational numbera1 and we may identify

Z with a subset of Q. The sum and the product of q = as and r =bt are defined to be

q + r =at+ bs

st, q · r = ab

st

and both are well-defined. In fact, if q = a′

s′ , then as′ = a′s, so that

(at+ bs)s′t = a′tst+ bss′t = (a′t+ bs′)st,

abs′t = a′bst,

and we also have q + r = a′t+bs′

s′t , qr =a′bs′t .

We say that a rational number is ≥ 0 when it may be represented as a quotient nm of twonatural numbers. If q, r ∈ Q, we put q ≤ r when r − q ≥ 0.

So we reduce statements on Q to statements on Z. The theory of rational numbers is just apart of the theory of integer numbers.

If the natural numbers are free from contradiction, so are the rational numbers.

1.2 Cardinal and Ordinal Numbers

Definitions: Let X, Y be two sets. A map f : X → Y assigns to any element x ∈ X a uniqueelement f(x) ∈ Y , and the composition with another map g : Y → Z is

g ◦ f : X → Z , (g ◦ f)(x) = g(f(x)

).

If A ⊆ X, we put f(A) = {f(a) : a ∈ A} ⊆ Y .If B ⊆ Y , we put B ∩X := f−1(B) = {x ∈ X : f(x) ∈ B} ⊆ X.A map f : X → Y is injective if f(x) = f(x′) ⇒ x = x′, surjective when f(X) = Y , and

bijective if it is injective and surjective: any element y ∈ Y is the image of a unique elementf−1(y) ∈ X, so that f−1 : Y → X also is a bijection and f−1 ◦ f = IdX , f ◦ f−1 = IdY .

Compositions of injective (surjective, bijective) maps are injective (surjective, bijective).

Definition: Let X, Y be two sets. If there exists a bijection f : X → Y , then we say that Xand Y have the same cardinal and we put |X| = |Y |. If there exists an injection f : X → Y ,then we say that |X| ≤ |Y |, and we put |X| < |Y | if moreover |X| 6= |Y |.

Sets of cardinal ≤ ℵ0 := |N| are said to be countable.

Schröder-Bernstein Theorem: If |X| ≤ |Y | and |Y | ≤ |X|, then |X| = |Y |.

1.2. CARDINAL AND ORDINAL NUMBERS 15

Proof: Let f : X → Y and g : Y → X be injective maps, so that h = gf : X0 := X → X2 := h(X)is a bijection, and we have X2 = (gf)(X) ⊆ X1 := g(Y ), where |Y | = |X1|.

Put Xn+2 := h(Xn), X∞ :=⋂nXn, and Zn := Xn −Xn+1.

The map h defines bijections Z2n → Z2n+2, n ≥ 0, and the identity defines bijectionsZ2n+1 → Z2n+1, n ≥ 0. We see that |X| = |X1| = |Y |, since we have a bijection

X = Z0 ∪ Z1 ∪ Z2 ∪ . . . ∪X∞ → Z1 ∪ Z2 ∪ Z3 ∪ . . . ∪X∞ = X1 = g(Y ).

Definitions: Given two cardinals a = |X|, b = |Y |, the sum a+ b is the cardinal of the disjointunion X q Y , the product ab is the cardinal of the cartesian product X × Y , and the powerab is the cardinal of the set XY of all maps1 Y → X.

Any subset Y of a set X may be viewed as a map IY : X → {0, 1}, where IY (Y ) = 1 andIY (X − Y ) = 0; hence the cardinal of the set P(X) of all subsets of X is just 2|X|.

Cantor’s Theorem: |X| < 2|X|.

First proof : Obviously |X| ≤ |P(X)|.If there exists a bijection f : X → P(X), we consider the subset Y := {x ∈ X : x /∈ f(x)}.Since f is surjective, we have Y = f(y) for some y ∈ X.If y ∈ Y , then y /∈ f(y) = Y . If y /∈ Y , then y ∈ f(y) = Y . Absurd.Hence such bijection does not exist, and |X| 6= |P(X)| = 2|X|.

Second proof : Let us give an alternative proof, using Cantor’s Diagonalverfahren. Given a mapφ : X → {0, 1}X , φ 7→ φx, we may consider the map ϕ : X → {0, 1} such that ϕ(x) = 1 whenφx(x) = 0, and ϕ(x) = 0 when φx(x) = 1, so that ϕ 6= φx, ∀x ∈ X, and we see that φ is notsurjective. Hence |X| 6= |{0, 1}X | = 2|X|.

Definitions: A relation ≤ on a set X is an order relation if it is

1. Reflexive: x ≤ x, ∀x ∈ X.

2. Antisymmetric: x, y ∈ X, x ≤ y, y ≤ x ⇒ x = y.

3. Transitive: x, y, z ∈ X, x ≤ y, y ≤ z ⇒ x ≤ z.

and we put x < y when x ≤ y and x 6= y.Any order on X clearly induces an order on any subset Y ⊆ X.Let X, X ′ be two ordered sets. A map f : X → X ′ is a morhism of ordered sets when

x1 ≤ x2 ⇒ f(x1) ≤ f(x2).

It is an isomorphism of ordered sets when it is bijective and f−1 also is a morphism (it isbijective and x1 ≤ x2 ⇔ f(x1) ≤ f(x2)), and it is an anti-isomorphism when it is bijectiveand x1 ≤ x2 ⇔ f(x1) ≥ f(x2).

Definitions: Let X be an order. We say that x ∈ X is the first element when x ≤ y, ∀y ∈ X(if it exists, it is unique) and that it is the last element when y ≤ x, ∀y ∈ X. The first (resp.last) element of a subset Y ⊆ X is the maximum (resp. minimum) of Y .

We say that x ∈ X is maximal when x ≤ y ⇒ x = y, and minimal when y ≤ x⇒ y = x.We say that x ∈ X is an upper (resp. lower) bound of a subset Y ⊆ X when y ≤ x (resp.

x ≤ y) for all y ∈ Y . The supremum of Y is the first element of the subset of all upper bounds1We assume that any set X admits a unique map ∅ → X (in particular there is a unique map ∅ → ∅, the

identity), while there is no map X → ∅ when X 6= ∅.


of Y (if it exists, it is unique) and the infimum of Y is the last element of the subset of alllower bounds of Y (if it exists, it is unique). An order is a lattice if has first and last element,and any pair x1, x2 ∈ X has supremum and infimum.

An order X is total if any pair is comparable (x ≤ y or y ≤ x for any x, y ∈ X) and it is awell-order when every non empty subset has a first element (hence it is a total order).

The chains in an order X are the totally ordered subsets of X.

In the founding principles of Mathematics we admit the so named Zorn’s lemma:

Zorn’s Lemma: Let X be a non empty order. If any chain of X has an upper bound in X,then X has some maximal element.

Axiom of Choice: Any surjective map p : X → Y admits a section s : Y → X (a map suchthat p ◦ s = IdY ).

Proof: Let S be the set of pairs (A, s) where A ⊆ Y and s : A→ X is a map such that p◦s = IdA.The set S is not empty (just take a point y ∈ Y , a point x ∈ p−1(y), and put s(y) = x) and

we order S as follows:

(A, s) ≤ (B, t) when A ⊆ B and s = t|A.

Any chain {(Ai, si)}i∈I admits the upper bound (⋃iAi, s), where s(a) = si(a) when a ∈ Ai.

Hence, by Zorn’s lemma, S has a maximal element (Z, s).

If Z 6= Y , and y ∈ Y −Z, we may extend s to Z ∪{y} (just put s(y) = x, where x ∈ p−1(y))against the maximal character of (Z, s). Hence Z = Y and s is a section of p.

Corollary: If p : X → Y is a surjective map, then |Y | ≤ |X|.

Proof: Let s be a section of p. Since s is injective, we have |Y | = |s(Y )| ≤ |X|. q.e.d.

1. N× N is a well-order set with the lexicographical order 00 < 01 < . . . < 10 < 11 < . . .

2. Any set of cardinals is an ordered set.

3. The subsets of a set, subgroups of a group, ideals of a ring, vector subspaces of a vector space,etc., ordered by inclusion, form a lattice.

4. Any ordered set (X,≤) defines a dual order X∗ = (X,≤∗), where we put x ≤∗ y ⇔ y ≤ x.It is clear that X∗∗ = X, and that a map f : X → Y is an anti-isomorphism if and only iff : X∗ → Y is an isomorphism. Moreover, X is a lattice if and only if so is X∗.

5. Since f : Z→ N, f(n) = 2n when n ≥ 0 and f(−n) = 2n− 1 when n ≥ 1, is bijective, we seethat |Z| = ℵ0.

6. Since f : N × N → N, f(n,m) = 2n3m, is injective, we see that ℵ20 = ℵ0; hence ℵn0 = ℵ0,n ≥ 1.

7. Since Q is a quotient of Z× (N− 0), we see that |Q| = ℵ0; hence |Qn| = ℵ0, n ≥ 1.

8. The set of infinite decimal expressions 0.c1c2c3 . . . has cardinal 10ℵ0 : it is uncountable.

9. Cantor’s theorem shows that “the set X of all sets” is non sense: since P(X) ⊆ X, we wouldhave 2|X| = |P(X)| ≤ |X|. Analogously there is no “set of all groups”, etc.

1.2. CARDINAL AND ORDINAL NUMBERS 17

Ordinal Numbers

Definition: Let X be a well-order. A set Y ⊆ X is an initial ray of X when y < x for ally ∈ Y , x ∈ X − Y . When Y is incomplete, Y 6= X, then Y = (← x) := {y ∈ X : y < x}, wherex is the supremum of the ray, so that X is isomorphic to the set of incomplete initial rays withthe inclusion order.

Theorem: Any set X admits a well-order.

Proof: Consider the set W of all pairs (Y,≤) where Y ⊆ X and ≤ is a well-order on Y .The set W is not empty (finite subsets admit a well-order) and we order W as follows:

(Y ′,≤′) ≤ (Y,≤) when (Y ′,≤′) is an initial ray of (Y,≤).

In W any chain {(Yi,≤i)}i∈I admits an upper bound:In fact Y =

⋃i Yi has an obvious order ≤ inducing ≤i on Yi, and Yi is an initial ray of Y :

if y ∈ Y, y /∈ Yi, then y is in some Yj containing Yi so that y bounds above Yi. Finally, Y is awell-ordered set: if ∅ 6= Z ⊆ Y , we have ∅ 6= Yi ∩ Z for some index i, and the first element ofYi ∩ Z is the first element of Z, because Yi is an initial ray of Y .

According to Zorn’s lemma, W has a maximal element (Y,≤).If Y 6= X, and x ∈ X − Y , we may extend the well-order of Y to Y ∪ {x} so that y ≤ x,

∀y ∈ Y , against the maximal character of (Y,≤). Hence Y = X and ≤ is a well-order on X.

Zermelo’s Theorem: Let X be a well ordered set. If f : X → X is a strictly increasing map(x < y ⇒ f(x) < f(y)) then x ≤ f(x), ∀x ∈ X.

Proof: Otherwise, let x be the first element such that f(x) < x, so that f(x) ≤ f(f(x)).Since f is increasing, we have f(f(x)) < f(x). Absurd. q.e.d.

1. Any well-order X has a unique automorphism: the identity.

If τ : X → X is an automorphism and x < τ(x), then τ−1(x) < x. Impossible.

2. If two well-orders are isomorphic, then the isomorphism is unique. (So, we may identifyisomorphic well-orders and name them ordinal numbers. The ordinal of N is ω.)

If f, g : X → X ′ are isomorphisms, then g−1f is an automorphism; hence g−1f = IdX .

3. Any two initial rays R1 ⊂ R2 of a well-order never are isomorphic.If f : R2 → R1 ⊂ R2 is an isomorphism and x ∈ R2 −R1, then f(x) < x. Impossible.

Definition: Let α, β be two ordinal numbers. We put β < α when β is isomorphic to anincomplete initial ray of α. By the above results, ≤ defines an order on any set of ordinalnumbers, and the set (← α) of ordinal numbers β < α is a well-order, since it is just α.

Lemma: Let X,Y be two well-orders. If any incomplete initial ray of X is isomorphic to anincomplete initial ray of Y , then X ≤ Y .

Proof: Any incomplete initial ray (← x) of X is isomorphic to a unique incomplete initial ray(← f(x)

)of Y . If x < x′, by the above results it is clear that f(x) < f(x′), and that f is an

isomorphism of X onto an initial ray of Y (eventually complete).

Lemma: If α and β are two ordinal numbers, then α ≤ β or β ≤ α.


Proof: Otherwise, by the lemma we may consider the first incomplete initial ray (← x) of α(resp. (←, y) of β) not isomorphic to an incomplete initial ray of β (resp. of α).

Now (← x) and (←, y) satisfy the hypothesis of the above lemma, so that (← x) ≤ (← y)and (← y) ≤ (← x). Hence (← x) and (← y) are isomorphic. Absurd.

Theorem: If S is a set of ordinal numbers, then (S,≤) is a well-order.

Proof: If R ⊆ S is not empty, we consider an ordinal α ∈ R.The initial rays of α give all the ordinals β < α, so that {β ∈ R : β ≤ α} has a first element

β0 and, by the lemma, β0 ≤ α < γ for any other γ ∈ R.

Cardinal Numbers: Let ω be the ordered set of all finite ordinal numbers, so that |ω| = ℵ0.Let ω1 be the ordered set of all ordinal numbers of cardinal ≤ ℵ0, so that ω1 is the first

ordinal number of cardinal > ℵ0. We put ℵ1 := |ω1|, so that ℵ0 < ℵ1 and there is no intermediatecardinal: if |σ| < ℵ1, then σ < ω1, so that |σ| ≤ ℵ0.

Let ω2 be the ordered set of all ordinal numbers of cardinal ≤ ℵ1, so that ω2 is the firstordinal number of cardinal > ℵ1, and we put ℵ2 := |ω2|. So we obtain an increasing sequenceℵ0 < ℵ1 < ℵ2 < . . . of cardinals with no intermediate cardinal. Now we put ℵω := |

⋃n ωn|, and

so on:

ℵ0 < ℵ1 < ℵ2 < . . . < ℵω < ℵω+1 < . . .

In general, if α = β + 1, we consider the ordered set ωα of all ordinal numbers of cardinal≤ ℵβ, and we put ℵα := |ωα|. Otherwise, we consider ωα =

⋃β

1.3. REAL AND COMPLEX NUMBERS 19

For the product, consider a constant c ∈ N bounding (qn) and (q′n). Given ε ∈ Q+, there isan index nε such that |qn − qm|, |q′n − q′m| < ε, ∀m,n ≥ nε, so that

|qnq′n − qmq′m| = |qn(q′n − q′m) + (qn − qm)q′m| ≤ |qn| · |q′n − q′m|+ |qn − qm| · |q′m| < 2cε.

Finally the null sequences clearly form an additive subgroup and, when (q′n) is a null sequence,we consider a constant c bounding (qn) and an index nε such that |q′n| < εc , ∀n ≥ nε, so that|qnq′n| ≤ c|q′n| < ε, and we see that (qnq′n) also is a null sequence.

Definition: The ring of real numbers R is the quotient of the ring of Cauchy sequences ofrational numbers by the ideal of null sequences.

A real number is ≥ 0 when it may be represented by some Cauchy sequence (qn) with allthe terms qn ≥ 0. If a, b ∈ R, we put a ≤ b when b− a ≥ 0.

We have a canonical injective ring morphism Q→ R, q 7→ [(q)], (preserving inequalities).

A Cauchy sequence (qn) and any subsequence (qin) represent the same real number.

Lemma: Given a real number x 6= 0, there exists k ∈ N such that x may be represented by aCauchy sequence (qn) with all the terms qn ≥ 1k , or with all the terms qn ≤ −

1k .

Proof: By definition, any non-null Cauchy sequence admits a subsequence (qn) with all the terms|qn| ≥ 1k for some constant k ∈ N.

Now (qn) admits a subsequence of positive terms or a subsequence of negative terms.

Theorem: The ring R is a field and ≤ is total order compatible with the additive and multi-plicative structure:

1. If a ≤ b, then a+ c ≤ b+ c.

2. If 0 ≤ a and 0 ≤ b, then 0 ≤ ab.

Proof: Clearly ≤ is reflexive and transitive. It is antisymmetric: If a ≤ b and b ≤ a, then b− aand a− b are represented by Cauchy sequences (qn) and (q′n) with all the terms qn, q′n ≥ 0. Now(qn + q

′n) is a null sequence and so is (qn), since |qn| ≤ |qn + q′n|, so that b− a = 0.

It is a total order: If a 6= b, by the above lemma b− a > 0 or a− b > 0; hence a < b or b < a.

Finally we prove that R is a field: By the lemma, any non-zero real number is representedby a Cauchy sequence (qn) with all the terms |qn| ≥ 1k for some constant k ∈ N. If (q

−1n ) is a

Cauchy sequence, then 1 = [(qn)] · [(q−1n )], and we conclude. Now,∣∣∣∣ 1qn − 1qm∣∣∣∣ = ∣∣∣∣qm − qnqnqm

∣∣∣∣ ≤ |qm − qn|k2 ·Definition: The absolute value of x ∈ R is defined to be |x| = max{x,−x}.

A sequence of real numbers (xn) is a Cauchy sequence when for every positive real numberε ∈ R+ there is an index nε such that |xn − xm| < ε, ∀m,n ≥ nε; and it converges to a limitx ∈ R (and we put x = limxn) when there is an index nε such that |xn − x| < ε, ∀n ≥ nε.

Any convergent sequence is a Cauchy sequence.

Theorem: For all x ∈ R and ε ∈ Q+ there exists q ∈ Q with |q − x| < ε.

Proof: If (qn) represents x, then there exists n0 ∈ N such that |qn − qn0 | < ε2 , ∀n ≥ n0.Hence |x− qn0 | ≤ ε2 < ε.


Theorem: Any Cauchy sequence (xn) of real numbers converges to a unique real limit.

Proof: Let us see the existence of the limit:Pick qn ∈ Q such that |qn − xn| < 1n . Then (qn) is a Cauchy sequence of rational numbers:Given ε ∈ Q+, there exists an index n0 such that n0 > 3ε and |xn − xm| <

ε3 , ∀n ≥ n0, so

that we have |qn − qm| ≤ |qn − xn|+ |xn − xm|+ |xm − qm| < ε.In particular, if we put x = [(qn)], we have |qn − x| ≤ ε, ∀n ≥ n0.Now, given ε > 0, there exists an index nε such that nε >

2ε and |qn − x| <

ε2 , ∀n ≥ nε, so

that we have |xn − x| ≤ |xn − qn|+ |qn − x| < 1n +ε2 ≤ ε, ∀n ≥ nε. Hence x = limxn.

If y ∈ R is a different limit, there is an index n0 such that |xn−x| < |y−x|2 and |xn−y| <|y−x|

2 ,

∀n ≥ n0, and we get a contradiction: |y

based on lectures of j. sancho - unex.esmatematicas.unex.es/~navarro/degree.pdfand constructions to...

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