Notes for a Licenciatura

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Notes for a Licenciatura

Based on Lectures of J. Sancho

November 16, 2021

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Preface

During the seventies, theLicenciatura –the university studies– in Mathematics at the University of Salamanca had a highly coherent structure devised by late Prof. Juan B. Sancho Guimer´a.

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 space Speck, the localization process being the change of the base field and the globalization process encoding Galois theory.

How any morphism X → T may be understood as a family of spaces parameterized by T, so that absolute statements about a spaceX really refer to the projectionX →ponto the one point 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 ofX (parameterized byT) and that obviously such functor of points ofXfully determines the whole structure ofX, so reducing definitions 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 of abelian sheaves, and that it directly provides the existence of dualizing sheaves and dualizing complexes in the case of smooth curves (representing the functor H¹(C,−)^∗, Riemann- Roch’s theorem), smooth varieties (Serre’s duality), topological manifolds (Poincar´e’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, statements 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 a significant part of the university teaching in Mathematics. And also to develop the courses with complete proofs, in a concise and coordinated way; mainly focusing on the logical and meaning interdependencies of the involved topics, devoid of anything that would put the central points in the shade. Hence the pages will be filled with definitions, statements and proofs, while with scarce examples, applications and motivations. And the aim of this preface is to bring these dry pages into life, to invite the reader to place them into the perspective of the much broader Grothendieck’s oeuvre and life.

Fortunately Grothendieck has given us a very inspired and appealing description of its own mathematical works in thePromenade à travers une oeuvre – ou l’enfant et la Mère¹at the very beginning ofRécoltes 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 much broader spiritual adventure: man’s self-conscience unfolding. In thisPromenade he presents the relationship of any man with the spiritual goods under a double aspect. On the one hand, the luminous aspect (the passion for knowledge) represented by the figure of a child. On the other hand, 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.

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creativit´e qui en derivent³. A deep and valuable lesson of the main body ofR´ecoltes et Semailles is 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 saccager une chose à leur merci, alors qu’ils se coupent seulement de la vertue créatrice de ce qui est à leur disposition comme elle est à la disposition de tous, mais nullement à leur 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? andwhat is it? instead of the weary and clumsywhat is it for? of adults. And above all by the silently, quite, attentive, feminine attitude of listening our whispering interior voice and readiness to accept it, resounding me Mary’s fiat: be it unto me according to your word. But the exigence of solitude in any creative work shocked me, and when 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’est là la situation qui a été la mienne plus ou moins pendant toute ma vie, depuis mon enfance, tant sur le plan de la création mathématique, que dans mon itinéraire spirituel... 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 (`a mes yeux) qu’un homme ait accomplie,

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etait la Passion du Christ et sa mort sur la croix... Cette oeuvre ´etait et ne pouvait ˆ

etre que solitaire. Et même c’était la solitude suprême, car Dieu Lui-même s’est retiré, 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 and his parents had a very tough life, which is a paradigmatic and astonishing incarnation of the whole XXth century history. A life that we may foresee in the autobiographical chapters III and VI of La Clef des Songes⁶, another unpublished work where he embeds mathematics into man’s religious adventure. He says:

Les lois mathématiques peuvent être découvertes par l’homme, mais elles ne sont créés ni par l’homme ni même par Dieu. Que deux plus deux égale quatre n’est pas un 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 de Dieu – une partie infime, certes, la plus superficielle en quelque sorte, et la seule qui soit 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 from the creative virtue of that which is at their disposal as this virtue is at the disposal of everyone, but not at their mercy 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 great is accomplished, in the individual adventure, nor in the collective adventure – at the intellectual level or at the spiritual level... At the spiritual level, the biggest oeuvre (in my eyes) accomplished by a man, was the Passion of the Christ and his death on the cross... This oeuvre was and could only be solitary. It was even the supreme solitude, 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. I sense the mathematical laws as being part of the very nature of God – a tiny part, certainly, the most superficial in some sense, and the only part accessible to reason alone (III 31,Les retrouvailles perdues..., page 100).

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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 Gonz´alez

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Introduction

On the occasion of the Symposium in memory of late Prof. J. Sancho Guimer´a held at Salamanca in 2014, I decided to write some notes based on Sancho’s lectures, with the same initial aim as Bourbaki at the 40’s: to undertake the university mathematical teaching from the outset, 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 this is 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 the explanations of a lecturer. And always assuming the atmosphere of each course, with some implicit assumptions and conventions, that surely will be clear to anyone who carefully reads any section from the beginning. However, the defined concepts are always written in boldface and they figure in the index of terms.

2. It focuses on the core of each course, the simple concepts and ideas that directly lead to the main results, leaving aside details, examples and applications that surely are required to assimilate 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 no logical order may be imposed upon them, each one massively using definitions, results and ideas from the companion chapters, sometimes placed at subsequent pages, due to the linear character of any book. It is non sense to read a chapter alone, and all the courses of a year must be studied simultaneously as a whole. So the courseLinear Algebrauses some properties of polynomials, proved in the companion but posterior chapterAlgebra I; the courseAnalysis III massively uses from the start smooth manifolds, vector fields and differential forms (and by the end metrics of constant curvature), studied in the companion but posterior chapter Differential Geometry I; and so on.

I do not pretend to develop each course in a self-contained way, without references to the previous or simultaneous courses. On the contrary, I pretend to enlighten the mutual connections, to show that it would be unnatural to parcel out mathematics in several unrelated fields (algebra, analysis, 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 are written in a single book.

I have tried to be brief, since my aim is to exhibit many university courses, underlining the logical interdependencies and the central concepts and methods. To show that, when each course 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 a gentle hand provides the correct definitions, the adequate point of view, surprisingly the knots untie, the difficulties dissolve and the backbone of each theory may be exposed in a handful of pages.

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

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2 INTRODUCTION

First Year

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

• Linear Algebra: Vector spaces and linear maps. The dual space. Euclidean vector spaces.

Tensors andp-forms.

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

Second Year

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

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

• Projective Geometry: Projective and affine spaces. Classification of metrics, modules over a 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. Meromorphic functions. Riemann mapping theorem.

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

Fourth Year

• Algebraic Geometry I:Sheaf cohomology. Schemes and coherent sheaves. Algebraic curves and 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. Fr´echet spaces. Compact Riemann surfaces.

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

Fifth Year

• Algebraic Geometry II:Local algebra. Quasi-coherent sheaves. K-theory. Duality theory.

Grothendieck topologies and descent theory.

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Many significant parts of the university teaching lack (functional analysis and partial differential equations, homology and homotopy theory, number theory, measure theory and statis- tics, mathematical physics, etc.), but I think that these courses suffice to show that many Grothendieck ideas may be introduced at the undergraduate level when the courses are developed in a coordinate way.

The first three years are mainly devoted to the concepts of structure and morphism, the central 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 ofsheaf and cohomology, the central results being duality theorems. Moreover, representable functors and the concepts of point and space pervade the notes from start to finish, as well as the clarification of the adjectivesnatural,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, thegenetic method of reducing all mathematical concepts to the process of counting (sets and natural numbers); so initiating the accomplishment of the fantastic pythagorean vision⁸: all is a music of numbers. This genetic method is the backbone of Analysis and culminates in the theory 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 first infinite cardinal ℵ₀ =|N|, the first cardinal ℵ₁ bigger than ℵ₀,... and so we may parameterize the infinite cardinals by the ordinal numbers.

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

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

Linear Algebra: In this chapter we take the path of theaxiomatic method. Geometry bursts luminously in Greece when the Greek genius realized that properties of such subtle and fundamental concepts as point, line, right angle, etc., should not be reduced to properties of previous concepts, but that we should better put on show their basic mutual relations, stating them as axioms 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 structures are considered to be equal (the isomorphisms) and it placed as the central problem of any theory the classification of all possible structures up to isomorphisms⁹. As a first example of the

8Aristotle (Metaphysics, Book I, Part 5, 986a): τ α τ ων αριθµων στ ωιχια τ ων oντ ων στ ωιχια παντ ων υπλαβoν ιναι, και τ oν oλoν oυρανoν αρµoνιαν ιναι και αριθµoν. ([The pythagoreans supposed] the numbers 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 selbst beneiden soll. Mit diesem Modell und dem Schlüssel dazu kann man alsdann noch Pflanzen ins Unendliche erfinden, die konsequent sein müssen, das heißt, die, wenn sie auch nicht existieren, doch existieren könnten und nicht etwa malerische oder dichterische Schatten und Scheine sind, sondern eine innerliche Wahrheit und Notwendigkeit haben. (TheUrpflanzeis going to be the most wonderful creation of the world, which Nature herself shall envy me. With this model and the key to it, one can then go on inventing plants ad infinitum, which must

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4 INTRODUCTION

great insight of the German point of view, just compare the genetic definition of the addition of integer numbers given in p. 13 with the definition steaming from the classification of cyclic groups (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, onlyhow to do the addition).

After a very concise introduction to the structures of group and ring, the course focuses on the structure of vector space (so that we may exhibit for the first time the structure of the Euclidean Geometry¹⁰, at least with a fixed point and a fixed length unit) and we prove that vector 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 the natural isomorphism (Λ^pE)^∗ = Λ^pE^∗. The systematic use of linear maps and exact sequences from 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 it as a generalized arithmetic: numbers are ideals and the divisibility relation is the inclusion relation, so that prime numbers are maximal ideals, the quotient ring encoding the gaussian theory of congruences. The main result is Kronecker’s formula for a root of any polynomial equation p(x) = 0, discovered when he realized that the extraction of a root involves two quite different steps. The first one, subtle, humble and usually of long gestation, is the highlight of a structure of field (radical expressions, real or complex numbers, etc.); while the second one is to express a root inside such structure. But men were looking for a root without questioning the convenience 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 obvious

be consistent; that is, even if they do not exist, they could exist and not as picturesque or poetic shadows and illusions, but with inner truth and necessity). In his quiet walks along the Italian botanical gardens, he dreamed about the plant structure and its possible realizations, so that he could know all the plants that Linnaeus had catalogued, the unknown plants of Africa and America that brave and courageous explorers were looking for, the extinct plants, the future ones and those that will remain in God’s mind forever. Even if he could not achieve this wonderful dream, it is not a mere coincidence that the periodic table of elements and the first classifications of 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 “Non Euclidean” 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 wir ein Theil sind, wesentliche Kräfte von der Art haben, daß sie in Vereinigung miteinander nach dem doppelten umgekehrten Verhältniß der Weiten ihre Wirkungen von sich ausbreiten; zweitens, daß das Ganze, was daher entspringt, vermöge dieses Gesetzes die Eigenschaft der dreifachen Dimension habe; drittens, daß dieses Gesetz willkürlich sei, und da Gott dafür ein anderes, zum Exempel des umgekehrten dreifachen Verhältnisses, hätte wählen können; daß endlich viertens aus einem andern Gesetze auch eine Ausdehnung von andern Eigenschaften und Abmessungen geflossen wäre. Eine Wissenschaft von allen diesen möglichen Raumesarten wäre unfehlbar die höchste Geometrie, die ein endlicher Verstand unternehmen könnte... Wenn es möglich ist, daß es Ausdehnungen von andern Abmessungen gebe, so ist es auch sehr wahrscheinlich, daß sie Gott wirklich irgendwo angebracht hat.

Denn seine Werke haben alle die Gr¨oße und Mannigfaltigkeit, ist, daß es die sie nur fassen k¨onnen. (Thoughts on the True Estimation of Living Forces§10–11: Accordingly, I am of the opinion that substances in the existing world, of which we are a part, have essential forces of such a kind that they propagate their effects in union with each 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 God could have chosen another, e.g., the inverse-cube, relation; fourthly, and finally, that an extension with different properties and dimensions would also have resulted from a different law. A science of all these possible kinds of space would undoubtedly be the highest geometry that a finite understanding could undertake... If it is possible that there are extensions of different dimensions, then it is also very probable that God has really produced them somewhere. 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 to extend it if necessary, are the strong moments of man’s self-conscience unfolding.

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root is just the class [x] in the field of residue classes modulop(x), so reducing the extraction of roots to the problem of decomposing polynomials into irreducible factors.

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 other theorem; but, after applying it to prove D’Alembert’s and Hamilton-Cayley theorems (p. 76 and p. 59), and to solve the millennial questions on ruler-and-compass constructions (p. 78), we leave to the reader the consideration that the adjectivesmere andsterile deserve.

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

if_ix_i for some functionsf_i of classC^m−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, using the relative version along the diagonal: any functionf ∈ C^m(U×U), vanishing on the diagonal, isf =P

ifi(yi−xi) for some functions fi ∈ C^m−1(U ×U).

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

Finally, we introduce topological vector spaces and we prove that C^m(U) is complete with the 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 constructions that will pervade the remaining chapters.

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

2. Localization of modules, proving that most properties of modules are local¹², in the sense that 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 introduce the concepts ofgeometric property (stable under changes k→K of the base field) andlocal property (valid whenever it holds after some base changek→K).

4. Categories and functors, aiming to give a first rigorous approximation to the fundamental concepts ofstructure, natural and canonical.

5. Points of SpecAwith coordinates in a fieldK are viewed as morphisms SpecK →SpecAand, in general, Grothendieck’s representability theorem gives necessary and sufficient conditions for a functor to be a functor of points.

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

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

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6 INTRODUCTION

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

The central result is Galois theorem, stated as an antiequivalence of the category¹³ of k- algebras trivial on a given Galois extensionk→Lof Galois groupGwith the category of finite G-sets. The proof¹⁴ directly follows from the fundamental exact sequencek→A⇒A⊗_kAand the stability of the algebra of invariants A^G under base changes, (A^G)⊗_kL= (A⊗_kL)^G.

We also introduce the Frob¨enius automorphism of a polynomial q(x)∈Z[x] at a primep, so relating the global properties ofq(x) to the behavior of its reductions ¯q(x)∈Fp[x].

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

Projective Geometry: We introduce the projective spaceP(E) attached to a vector spaceE, but only as a mere set, without elucidating the underlying structure. To figure out the implicit structure 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 of automorphisms of such elusive structure: an isomorphism of a k-vector spaceE into ak⁰-vector space E⁰ clearly should be defined as a group isomorphism f:E → E⁰ such that f(λe) = σ(λ)f(e) for some ring isomorphism σ:k → k⁰; hence the group of automorphisms of P(E) is the group P Sl(E) of all bijections P(E) → P(E) represented by a semilinear automorphism E →E.

Klein, in his foundational Erlangen Programme, states that the structure of the projective space is given by the action of the group P Gl(E) of all projectivities¹⁵ on P(E), and that in general any action of a group G on a set defines a geometry: concepts of the geometry being just G-invariant concepts, statements being relations between concepts, and theorems being true relations. Considering different subgroups of P Gl(E), in this course we also study affine geometry, Euclidean geometry and Non-Euclidean geometries. In this course we show that the structure of the projective spaces of dimension ≥ 2 is captured by the lattice of linear subvarieties¹⁶; but to figure out the structure of the real projective line as a ringed space (so grounding projective geometry on non-algebraically closed fields) was a first success of the theory of schemes.

On the other hand, any concept of a geometry gives rise to a classification problem, considered as the determination of the set of orbits of the induced G-action. In this course we study the projective and affine classifications of quadrics (reduced to the linear classification of symmetric metrics), the projective classification of projectivities (reduced to the linear classification of endomorphims) and the projective classification of pencils of quadrics (reduced to the classification of metrics on modules over the finite k-algebras k[x]/(p(x)) considered in the companion 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 domainA, using the crucial fact that B =A/pⁿA is an injective B-module when p is irreducible, a result that follows directly from the Ideal Criterion studied in Algebra II.

It is worthy of attention that, in the former courseLinear Algebra, the characteristic polynomial of an endomorphism was introduced in a non-intrinsic way, defining it in a base and checking

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

14Two alternative proofs are sketched in exercises 105 and 106, p. 488. The first one reduces the proof to the simple casek=L, once the exactness properties of the involved functors are checked; while the second one uses Grothendieck’s characterization of the forgetful functor on the category of finiteG-sets.

15so that the group of automorphisms ofP(E) is just the normalizer ofP Gl(E) in the group of all bijections P(E)→P(E), which may be shown (ex. 9, p. 493) to beP Sl(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 ofP(E) given in p. 322.

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that it does not depend on the base. Now we introduce the K-group to give Grothendieck’s astonishing definition of the characteristic polynomial: it is the universal additive function on the finitely generated torsionk[x]-modules.

Finally, we study semisimple rings and the Brauer group, obtaining that the quaternions are the 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 totally analogous to the Galois theory of separable finite k-algebras and that, when considering the induced morphism SpecB → SpecA, it is totally analogous to the theory of coverings of a topological space, so obtaining the definition of the fundamental groupπ1(SpecA, p) of SpecA at a geometric pointp: Spec ¯k→SpecA.

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

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

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

and it is clear that any closed setY in a topological spaceXis the zero-set of a unique continuous functionf_Y :X → K, so that the lattice A(X) of all closed sets in a topological space X may be viewed as a ”ring” of continuous functions,A(X) =C(X,K), the addition corresponding to the 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, and they 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 sets A(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 theory of polyhedra.

17In factA(X) is not a true ring, since the existence of opposite element fails, but it comes with the extra properties 1 +a= 1 anda² =a. We name themlattice semirings. In a suggestive way, Connes uses to say that semirings where 1 + 1 = 1 have characteristic 1, so thatKis a semifield of characteristic 1.

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8 INTRODUCTION

3. Typically the Krull dimension ofA(X) is infinite, but Sancho realized that the minimal Krull dimension of a base¹⁸B ⊆A(X) provides a good definition for the dimension of a topological space X, since it coincides with Grothendieck’s dimension (the maximal length of a chain of irreducible closed sets) when X is a noetherian space, and with the inductive dimension and Lebesgue cover dimension whenX is a separable metric space.

Finally we study uniform spaces, and the theory of coveringsX →Sis developed parallel to the Galois theory of fields and rings, taking advantage of the existence of a universal covering with Galois group π1(S, p).

Analysis III: First we study partitions of unity and we show how a smooth manifold X may be reconstructed from the ring of smooth functions C^∞(X), the basic facts being the natural homeomorphism X = Spec_RC^∞(X) and the coincidence of C^∞(U) with the ring of fractions of C^∞(X) with respect to the multiplicative system of all smooth functions without zeros in the open 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 the continuous and differentiable dependence on the initial conditions when considering adequate functional spaces.

Then we study Pfaff systems, the central result being a geometrically obvious proof of a projection theorem given necessary and sufficient condition for a Pfaff system P on a smooth manifold X to be projectable by a regular projectionπ:X →Y, in the sense thatP =π^∗Qfor some Pfaff system Q on Y. As a direct consequence we obtain Frob¨enius theorem (involutive distributions are integrable) and Darboux’s classification of germs of 1-forms.

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

Finally, we introduce Riemann surfaces as ringed spaces. It is easy to prove that locally any non constant analytic morphism isu=zⁿfor some exponentn≥1 and, as a direct consequence of 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 spacesX locally isomorphic to open sets in Euclidean spaces with the sheaf of smooth functions (so eliminating the cumbersome atlases) and submanifolds as topological subspaces Y ,→ X such that (Y,C_Y^∞) is a smooth manifold, where C_Y^∞ is the sheaf of restrictions of smooth functions on X. Then we develop 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 of null curvature is locally Euclidean, and riemannian manifolds, proving that any simply connected and complete riemannian surface of constant curvature is isometric to the Euclidean plane, the hyperbolic 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 classification of abelian Lie groups.

Algebraic Geometry I: This chapter may be considered the hearth of these notes. We introduce sheaves and we define the cohomology groups H^p(X,F) using Godement’s resolution.

Then we prove a cohomological bound theorem

H^p(X,F) = 0, p >dimX,

18a subring whose complements form a base of open sets ofX in the usual sense.

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9

whenX is a noetherian space, and the acyclicity theorem for arbitrary rings, H^p(SpecA,Mf) = 0, p >0.

Now, after a massive use of the spectrum of a ring in the coursesAlgebra II, Topology and Commutative Algebra, and the comprehension of smooth manifolds and Riemann surfaces as ringed spaces inDifferential Geometry I and Analysis III, we may naturally introduce schemes as ringed spaces locally isomorphic to (SpecA,Ae). The major example of scheme in this course will be the Riemann variety of a finite extension ofk(t), the Riemann variety ofk(t) itself being the projective lineP1 over the field k.

First we calculate the cohomology groups of the structural sheaf of P1, projecting it onto a finite space with two closed points and a dense point, and it readily follows the determination of the cohomology groups of the line sheavesO_P₁(n). Then we prove the fundamental finiteness theorem for coherent sheaves, dimH^p(X,M) <∞, projecting the curve X onto P1, since the direct image preserves cohomology.

Now the weak Riemann-Roch theorem for line sheavesL_D is immediate, dimH⁰(X, L_D)−dimH¹(X, L_D) = 1−g+ degD.

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

dimH¹(X, L_D) = dim HomO_X(L_D, L_K),

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

H¹(X,M)^∗ = HomO_X(M,D_X)

for some quasi-coherent sheafD_X, and the problem is to determine D_X.

It is easy to show that D_X 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 laborious local calculation of the conductor of a projectionX→P1, while the more natural second proof uses the diagonal embedding X → X×_k X and the stability of the cohomology of coherent sheaves under base changes (so pointing that the theory of curves is naturally entangled with the theory of higher dimensional varieties).

Finally we introduce the projective spectrum of a graded ring, and determine the cohomology groups of the line sheavesO_P_d(n) over the projective space Pd= Projk[x0, . . . , x_d].

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

Then we introduce the inverse image and the cup product, and we prove some fundamental results: calculation of the cohomology of the fibres, theorem of base change, finiteness theorem, universal coefficients formula, K¨unneth theorem, and the classification of line sheaves. We also use local cohomology groups to define the cohomology class of a closed submanifold and to develop a topological intersection theory.

Now we get down to the duality theorem. So as to include non locally Euclidean spaces, we do not consider the dual of the last cohomology groupH_cⁿ(X,F), but of a complex determining all the cohomology groups with compact supports. Again the representability theorem gives the existence of a dualizing complexD_X^• such that

RHom(RΓ_c(F),Z) =RHom(F,D_X^•)

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10 INTRODUCTION

and in this case it is not hard to determine it when X is a topological manifold of dimensionn:

the dualizing complex D_X^• has a unique non-zero cohomology sheaf¹⁹ TX, locally constant and placed at degree −n, so that Hom_Z(H_cⁿ(X,F),Z) = Hom(F,TX).

Then we determine the cohomology groups of a projective bundle, a result grounding the theory 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 sheaf cohomology to prove the uniformization theorem: any simply connected Riemann surface is the complex plane, the complex projective line or the unit disk.

Then we present a brief introduction to Fr´echet spaces and the duality theory of locally convex 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 have finite 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 complete and non singular algebraic curves over the field C, studied in the companion course Algebraic Geometry I. Hence, most of the results there obtained for complete non singular curves over arbitrary fields may be extended to compact Riemann surfaces, in particular the fundamental Riemann-Roch theorem. However, we show that now the residue theorem provides a very simple and 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 differential forms with values in a vector bundle (or a locally free sheaf, the point of view adopted in this chapter) and the differential calculus with such forms requires a linear connection on the vector bundle. We develop this fruitful differential calculus up to the point of obtaining Bianchi’s identities and Cartan’s structure equations. We also determine the Chern classes of a complex vector bundle in terms of the curvature 2-form of any linear connection.

We present the variational calculus, obtaining the fundamental Poincar´e-Cartan form and Noether’s theorem on the infinitesimal symmetries of a variational problem.

Finally, since most bundles used in Geometry and Physics are natural bundles, we include the Galois theory of natural bundles (of a given finite order). The statements are parallel to those 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 theorem on regular rings and some characterizations of Cohen-Macaulay rings. Then, after a brief study of quasi-coherent sheaves (including Deligne’s formula and the base change theorem) we get down 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 of vector bundles with values in the gradedK-theoryGK^•(X). Then we prove the fundamental result: theK-theory is the universal multiplicative²⁰ cohomology theory.

Now, if a cohomology theoryA(X) follows the additive lawc1(L⊗L⁰) =c1(L) +c1(L⁰), then we may modify the direct image in A(X)⊗Q with an exponential, so that it follows the multiplicative law and we have a functorial ring morphism K(X) → A(X)⊗Q preserving the new direct image. This is just the Riemann-Roch-Grothendieck theorem.

19that would be obtained in case of dualizing the last cohomology group, as inAlgebraic Geometry I.

20in the sense that Chern classes of line sheaves follow the lawc1(L⊗L⁰) =c1(L) +c1(L⁰)−c1(L)c1(L⁰) of the multiplicative group. Remark that (1−x)(1−y) = 1−(x+y−xy).

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11

A direct consequence, in the complex case, is the existence of a natural ring morphism GK^•(X)⊗Q → H^2•(X_an,Q) preserving inverse and direct images, hence Chern classes.

Since GK⁰(SpecC)⊗Q = H⁰(pt,Q) = Q, the algebraic and topological definitions of any numerical cohomological invariant coincide. For example, ifX is a projective smooth variety, the topological and algebraic self-intersection numbers of the diagonal coincide,

P

i(−1)ⁱdim_QHⁱ(Xan,Q) =P

p,q(−1)^p+qdim_CH^p(X,Ω^q_X).

2. Given a projective morphismf:X →S, the representability theorem gives the existence of a dualizing complexD_X/S such that for any quasi-coherent O_X-moduleM,

RHomO_S(Rf∗(M),O_S) =RHomO_X(M,D_X/S)

and again the crux is to calculate it in the case of a smooth morphism; to prove that it is the highest exterior power of the sheaf of differentials,D_X/S 'Ω^d_X/S.

Using the Koszul complex, we may see that the dualizing sheaf of a closed regular embedding Y → X of codimension d is just Λ^dN_{Y /X}, where N_{Y /X} stands for the normal bundle. Now the calculation of the dualizing sheaf of a smooth projective morphism X → S is a simple question. Just consider the composition

X −−−→^∆ X×_SX −−−−→^π¹ X

of the diagonal embedding with the first projection (obviously it is the identity), and the normal bundleN to the diagonal embedding (which is dual to the sheaf of differentials Ω_X/S by the very definition). We have

O_X =D_X/X =D_X/X×X ⊗∆^∗D_X_×X/X = Λ^dN⊗∆^∗(π₁^∗D_X/S) = Λ^dN ⊗ D_X/S

and we are done: D_X/S = (Λ^dN)^∗ = Ω^d_X/S. Once the question is placed in the relative and general 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 at a pointx, 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 atxof the global dualizing complex. In the crucial case of a smooth varietyX, we prove it determining the local cohomology groupsHx^p(X,O_X).

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

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

21So, even if the spectrum SpecA of a ringAis defined to be the set of all prime ideals ofA, the statements 0∈ x ∈SpecA and 1∈/ x ∈ SpecA are non-sense: points of SpecA are not prime ideals ofA, but naturally correspond to prime ideals. And the elements of the quotient set are not equivalence classes, but naturally correspond to equivalence classes.

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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 a tensor field T in the direction of a vector field D is denoted D^∇T, instead of the usual ∇_DT; and the Lie derivative D^LT, instead of the usualL_DT.

SometimesT_p^q denotes a tensor of type (p, q); soR_2,2denotes the Riemann-Christoffel tensor, R2 denotes the Ricci tensor, a p-form is Ωp, a metric isS2, and so on.

When a fixed mapf:X→Y is unambiguously understood, andZ ⊆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 ofZ. The one point set is denoted by ∗.

Acknowledgements: My debt and gratitude to my teachers at the University of Salamanca in the seventies, mainly D. Juan Bautista Sancho Guimerá and his collaborators D. Antonio Pérez-Rendón Collantes, D. Pedro Luis Garc´ıa Pérez, D. Cristóbal Garc´ıa-Loygorri y Urzaiz, D.

Jesús Muñoz D´ıaz and D. Jaime Muñoz Masqué.

Also my grateful acknowledgement to many companions who have written notes on these courses, with many improvements of their own, since I have made good use of them: Daniel Hernández, 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.

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Chapter 1

Analysis I

1.1 Integer and Rational Numbers

Definitions:A relation ≡on a set X is an equivalence relationif 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.

Theequivalence class of x∈X is ¯x= [x] ={y ∈X:x≡y}.

A subset C⊆X is an equivalence class when C= [x] for somex∈X.

Thequotient 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 mapπ:X→X/≡,π(x) = [x], is thecanonical projection.

Theorem:InX/≡, only equivalent elements are identified; [x] = [y]⇔x≡y. Proof: If [x] = [y], then y∈[y] = [x], andx≡y.

Conversely, ifx≡y, since≡is reflexive, it is enough to show that [y]⊆[x].

Now, if z∈[y], then y≡z; hencex≡zand z∈[x].

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

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

Construction ofZ: LetN={0,1,2,3, . . .}be the set of naturalnumbers. The set of integer numbersZis defined to be the quotient set ofN×Nby the equivalence relation

(m, n)≡(m⁰, n⁰) when m+n⁰ =m⁰+n,

andm−n denotes the class of (m, n). Any natural number ndefines an integer numbern−0, and so we may identifyNwith a subset of Z={. . . ,−2,−1,0,1,2, . . .}.

The sumand theproductof 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, ifa=m⁰−n⁰, thenm+n⁰ =m⁰+n, and

m+n⁰+r+s=m⁰+n+r+s

(m+n⁰)r+ (m⁰+n)s= (m⁰+n)r+ (m+n⁰)s 13

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14 CHAPTER 1. ANALYSIS I

and we also have a+b=m⁰+r−(n⁰+s),ab=m⁰r+n⁰s−(n⁰r+m⁰s).

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 numbern≥2. Thecongruencerelation,a≡b(mod. n) when b−a∈nZ, is an equivalence relation onZ, and [a] =a+nZ={a+cn:c∈Z}.

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

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

(a, s)≡(b, t) when at=bs,

and â_s denotes the class of (a, s). Each integeradefines a rational number â₁ and we may identify Z with a subset ofQ. The sumand theproductofq = â_s and r= ^b_t are defined to be

q+r= at+bs

st , q·r = ab st and both are well-defined. In fact, ifq = ^a_s0⁰, then as⁰ =a⁰s, so that

(at+bs)s⁰t=a⁰tst+bss⁰t= (a⁰t+bs⁰)st, abs⁰t=a⁰bst,

and we also have q+r= ^a⁰^t+bs_s0t ⁰,qr= ^a_s⁰0^bt.

We say that a rational number is ≥0 when it may be represented as a quotient _mⁿ of two natural numbers. Ifq, r∈Q, we putq ≤r when r−q≥0.

So we reduce statements onQto statements onZ. The theory of rational numbers is just a part 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: LetX,Y be two sets. Amap f:X→Y assigns to any element x∈X a unique element f(x)∈Y, and thecompositionwith another map g:Y →Z is

g◦f:X→Z , (g◦f)(x) =g f(x) . IfA⊆X, we putf(A) ={f(a) :a∈A} ⊆Y.

IfB ⊆Y, we putB∩X:=f⁻¹(B) ={x∈X:f(x)∈B} ⊆X.

A mapf:X→Y isinjectiveiff(x) =f(x⁰) ⇒ x=x⁰,surjective whenf(X) =Y, and bijective if it is injective and surjective: any element y ∈Y is the image of a unique element f⁻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 X and Y have the same cardinaland 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 becountable.

Schr¨oder-Bernstein Theorem: If |X| ≤ |Y|and |Y| ≤ |X|, then |X|=|Y|.

Notes for a Licenciatura