Commutative algebra: Constructive methods.Finite projective modules

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projective modules

Henri Lombardi, Claude Quitté

To cite this version:

Henri Lombardi, Claude Quitté. Commutative algebra: Constructive methods.Finite projective modules. 20, 2015, Algebra and Applications, 978-94-017-9943-0. �10.1007/978-94-017-9944-7�. �hal- 01682492�

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Henri Lombardi & Claude Quitté

Commutative algebra:

Constructive methods

Finite projective modules

Course and exercises

English translation by Tania K. Roblot

Last corrections December 2017, see page viii

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Henri Lombardi. Maître de Conférences at the Université de Franche- Comté. His research focuses on constructive mathematics, real algebra and algorithmic complexity. He is one of the founders of the international group M.A.P. (Mathematics, Algorithms, Proofs), created in 2003: see the site http://map.disi.unige.it/

[email protected] http://hlombardi.free.fr

Claude Quitté. Maître de Conférences at the Université de Poitiers. His research focuses on effective commutative algebra and computer algebra.

[email protected]

Mathematics Subject Classification (2010) – Primary: 13 Commutative Algebra.

– Secondary:

03F Proof theory and constructive mathematics.

06D Distributive lattices.

14Q Computational aspects of algebraic geometry.

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Preface of the French edition

This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules, which constitutes the algebraic version of the vector bundles in differential geometry.

We adopt the constructive point of view, with which all existence theorems have an explicit algorithmic content. In particular, when a theorem affirms the existence of an object – the solution of a problem – a construction algorithm of the object can always be extracted from the given proof.

We revisit with a new and often simplifying eye several abstract classical theories. In particular, we review theories which did not have any algorithmic content in their general natural framework, such as Galois theory, the Dedekind rings, the finitely generated projective modules or the Krull dimension.

Constructive algebra is actually an old discipline, developed among others by Gauss and Kronecker. We are in line with the modern “bible” on the subject, which is the book by Ray Mines, Fred Richman and Wim Ruitenburg,A Course in Constructive Algebra, published in 1988. We will cite it in abbreviated form [MRR].

This work corresponds to an MSc graduate level, at least up to Chapter XIV, but only requires as prerequisites the basic notions concerning group theory, linear algebra over fields, determinants, modules over commutative rings, as well as the definition of quotient and localized rings. A familiarity with polynomial rings, the arithmetic properties ofZand Euclidian rings is also desirable.

Finally, note that we consider the exercises and problems (a little over 320 in total) as an essential part of the book.

We will try to publish the maximum amount of missing solutions, as well as additional exercises on the web page of one of the authors:

http://hlombardi.free.fr/publis/LivresBrochures.html

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Acknowledgements.

We would like to thank all the colleagues who encouraged us in our project, gave us some truly helpful assistance or provided us with valuable information. Especially MariEmi Alonso, Thierry Coquand, Gema Díaz-Toca, Lionel Ducos, M’hammed El Kahoui, Marco Fontana, Sarah Glaz, Laureano González-Vega, Emmanuel Hallouin, Hervé Perdry, Jean-Claude Raoult, Fred Richman, Marie-Françoise Roy, Peter Schuster and Ihsen Yengui. Last but not least, a special mention for our L^ATEX expert, François Pétiard.

Finally, we could not forget to mention the Centre International de Recherches Mathématiques à Luminy and the Mathematisches Forschungsinstitut Ober- wolfach, who welcomed us for research visits during the preparation of this book, offering us invaluable working conditions.

Henri Lombardi, Claude Quitté August 2011

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Preface of the English edition

In this edition, we have corrected the errors that we either found ourselves or that were signalled to us.

We have added some exercise solutions as well as some additional content.

Most of that additional content is corrections of exercises, or new exercises or problems.

The additions within the course are the following. A paragraph on the null tensors added as the end of Section IV -4. The paragraph on the quotients of flat modules at the end of Section VIII -1 has been fleshed out. We have added Sections 8 and 9 in Chapter XV devoted to the local-global principles.

None of the numbering has changed, except for the local-global principle XII-7.13 which has become XII -7.14.

There are now 297 exercises and 42 problems.

Any useful precisions are on the site:

http://hlombardi.free.fr/publis/LivresBrochures.html

Acknowledgements.

We cannot thank Tania K. Roblot enough for the work achieved translating the book into English.

Henri Lombardi, Claude Quitté May 2014 – vii –

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This is the web updated version of the book

Except for the corrections indicated below, it is the same text as the one of the printed book. The unique structural modifications concern the table of contents: the general table of contents is shortened, and there is a detailed table of contents at the beginning of each chapter.

Corrections to the printed book (december 2017)

Solution of Problem 3 in Chapter XII: page 733 replace “all nonzero” by

“not all zero”.

Chapter XIII. Exercise 17 item3. The solution is changed.

In Section XV-9, the proof of Lemma 9.3 is not correct. It is necessary to give directly a proof of the (a, b,(ab)) trick for depth 2, allowing us to prove the concrete local-global principle. So 9.3, 9.4, 9.5 and 9.6 become 9.6, 9.3, 9.4 and 9.5.

More details on http://hlombardi.free.fr/publis/LivresBrochures.

html

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As indicated in the preface, we adopt the constructive method, with which all existence theorems have an explicit algorithmic content. Constructive mathematics can be seen as the most theoretical branch of Computer Algebra, which handles mathematics which “run on a computer.” Our course is nevertheless distinguishable from usual Computer Algebra courses in two key aspects.

First of all, our algorithms are often only implied, underlying the proof, and are in no way optimized for the fastest execution, as one might expect when aiming for an efficient implementation.

Second, our theoretical approach is entirely constructive, whereas Computer Algebra courses typically have little concern for this issue. The philosophy here is then not, as is customary “black or white, the good cat is one that catches the mouse”² but rather follows “Truth includes not only the result

1The official translation by John W. Bolduc (1963) is as follows: “As for me, I would propose that we be guided by the following rules:

1. Never consider any objects but those capable of being defined in a finite number of words;

2. Never lose sight of the fact that every proposition concerning infinity must be the translation, the precise statement of propositions concerning the finite;

3. Avoid nonpredicative classifications and definitions.”

2Chinese proverb.

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but also the path to it. The investigation of truth must itself be true; true investigation is developed truth, the dispersed elements of which are brought together in the result.”³

We often speak of two points of view on a given subject: classical and constructive. In particular, we have marked with a star the statements (theorems, lemmas, . . . ) which are true in classical mathematics, but for which we do not give a constructive proof and which often cannot have one. These “starred” statements will then likely never be implemented on a machine, but are often useful as intuition guides, and to at least link with the usual presentations written in the style of classical mathematics.

As for the definitions, we generally first give a constructive variant, even if it means showing the equivalence with the usual definition in the context of classical mathematics.

The reader will notice that in the “starred” proofs we freely use Zorn’s lemma and the Law of Excluded Middle (LEM)⁴, whereas the other proofs always have a direct translation into an algorithm.

Constructive algebra is actually an old discipline, developed by Gauss and Kronecker, among others. As also specified in the preface, we are in line with the modern “bible” on the subject, which is the book by Ray Mines, Fred Richman and Wim Ruitenburg,A Course in Constructive Algebra, published in 1988. We will cite it in abbreviated form [MRR]. Our work is however self-contained and we do not demand [MRR] as a prerequisite. The books on constructive mathematics by Harold M. Edwards [Edwards89, Edwards05]

and the one of Ihsen Yengui [Yengui] are also recommended.

The work’s content

We begin with a brief commentary on the choices that have been made regarding the covered themes.

The theory of finitely generated projective modules is one of the unifying themes of this work. We see this theory in abstract form as an algebraic theory of vector bundles, and in concrete form as that of idempotent matrices.

The comparison of the two views is sketched in the introductory chapter.

The theory of finitely generated projective modules itself is treated in Chapters V (first properties), VI (algebras which are finitely generated

3Karl Marx, Comments on the latest Prussian censorship instruction, 1843 (cited by Georges Perec inLes Choses); transcribed here as by Sally Ryan onhttp://www.

marxists.org/archive/marx/works/1842/02/10.htm.

4The Law of the Excluded Middle states thatP∨ ¬P is true for every propositionP.

This principle is accepted in classical mathematics. See page xxvii for a first explanation regarding the refusal ofLEMin constructive mathematics.

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Foreword xvii

projective modules), X (rank theory and examples), XIV (Serre’s Splitting Off theorem) and XVI (extended finitely generated projective modules).

Another unifying theme is provided by local-global principles, as in [Kunz]

for example. It is a highly efficient conceptual framework, even though it is a little vague. From a constructive point of view, we replace the localization at an arbitrary prime ideal with a finite number of localizations at comaximal monoids. The notions which respect the local-global principle are considered “good notions,” in the sense that they are ready for the passage of commutative rings to Grothendieck schemes, which we will unfortunately be unable to address due to the restricted size of this book.

Finally, one last recurrent theme is found in the method, quite common in computer algebra, calledthe lazy evaluation, or in its most advanced form, the dynamic evaluation method. This method is necessary if one wants to set up an algorithmic processing of the questions which a priori require the solution to a factorization problem. This method has also led to the development of the local-global constructive machinery found in Chapters IV and XV, as well as the constructive theory of the Krull dimension (Chapter XIII), with important applications in the last chapters.

We now proceed to a more detailed description of the contents of the book.

In Chapter I, we explain the close relationship that can be established between the notions of vector bundles in differential geometry and of finitely generated projective modules in commutative algebra. This is part of the general algebraization process in mathematics, a process that can often simplify, abstract and generalize surprisingly well concepts from particular theories.

Chapter II is devoted to systems of linear equations over a commutative ring, treated as elementary. It requires almost no theoretical apparatus, apart from the question of localization at a monoid, of which we give a reminder in Section II -1. We then get to our subject matter by putting in place the concrete local-global principle for solving systems of linear equations (Section II -2), a simple and effective tool that will be repeated and varied constantly. From a constructive point of view, solving systems of linear equations immediately renders as central the concept of coherent rings that we treat in Section II -3. Coherent rings are those for which we have a minimal grip on the solution of homogeneous systems of linear equations. Very surprisingly, this concept does not appear in the classical commutative algebra treatises. That is because in general the concept is completely obscured by that of a Noetherian ring. This obscuration does not occur in constructive mathematics where Noetherianity does not necessarily imply coherence. We develop in Section II -4 the question of finite products of rings, with the notion of a fundamental system of orthogonal idempotents

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and the Chinese Remainder theorem. The long Section II -5 is devoted to many variations on the theme of determinants. Finally, Section II -6 returns to the basic local-global principle in a slightly more general version devoted to exact sequences of modules.

Chapter III develops the method of indeterminate coefficients, first developed by Gauss. Numerous theorems of existence in commutative algebra rely on “algebraic identities under conditions” and thus on memberships g∈ hf1, . . . , f_siin a ring Z[c₁, . . . , c_r, X₁, . . . , X_n], where the X_i’s are the variables and thecj’s are the parameters of the theorem under consideration.

In this sense, we can consider that commutative algebra is a vast theory of algebraic identities, which finds its natural framework in the method of indeterminate coefficients, i.e. the method in which the parameters of the given problem are taken as indeterminates. In that assurance we are, to the extent our powers allow to, systematically “chasing algebraic identities.”

This is the case not only in the “purely computational” Chapters II and III, but throughout the book. In short, rather than simply assert in the context of an existence theorem “there is an algebraic identity which certifies this existence,” we have tried each time to give the algebraic identity itself.

Chapter III can be considered as a basic algebra course with 19th cen- tury methods. Sections III -1, III -2 and III -3 provide certain generalities about polynomials, featuring in particular the algorithm for partial factorization, the “theory of algebraic identities” (which explains the method of indeterminate coefficients), the elementary symmetric polynomials, the Dedekind-Mertens lemma and the Kronecker’s theorem. The last two results are basic tools which give precise information on the coefficients of the product of two polynomials; they are often used in the rest of this manuscript. Section III -4 introduces the universal splitting algebra of a monic polynomial over an arbitrary commutative ring, which is an efficient substitute for the field of the roots of a polynomial over a field. Section III -5 is devoted to the discriminant and explains in what precise sense a generic matrix is diagonalizable. With these tools in hand, we can treat the basic Galois theory in Section III -6. The elementary theory of elimination via the resultant is given in Section III -7. We can then give the basics of algebraic number theory with the theorem of unique decomposition into prime factors for a finitely generated ideal of a number field (Section III -8). Section III -9 shows Hilbert’s Nullstellensatz as an application of the resultant. Finally, Section III -10 on Newton’s method in algebra closes this chapter.

Chapter IV is devoted to the study of the elementary properties of finitely presented modules. These modules play a role for rings similar to that played by finite dimensional vector spaces for fields: the theory of finitely presented modules is a more abstract, and often profitable, way to address

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Foreword xix

the issue of systems of linear equations. Sections IV -1 to IV -4 show the basic stability properties as well as the important example of the ideal of a zero for a polynomial system (on an arbitrary commutative ring). We then focus on the classification problem of finitely presented modules over a given ring. Working towards principal ideal domains (PID), for which the classification problem is completely solved (Section IV -7), we will encounter pp-rings (Section IV -6), which are the rings where the annihilator of an element is always generated by an idempotent. This will be the opportunity to develop anelementary local-global machinerywhich conveniently reformulates a constructively established result for integral rings into the analogous result for pp-rings. This proof-rewriting machinery is elementary as it is founded on the decomposition of a ring into a finite product of rings. The interesting thing is that this decomposition is obtained via a rereading of the constructive proof written in the integral case; here we see that in constructive mathematics the proof is often even more important than the result.

Similarly, we have anelementary local-global machinery which conveniently reformulates a constructively established result for discrete fields into the analogous result for reduced zero-dimensional rings (Section IV -8). The zero-dimensional rings elementarily defined here constitute an important intermediate step to generalize specific results regarding discrete fields to arbitrary commutative rings: they are a key tool of commutative algebra.

Classically, these appear in the literature in their Noetherian form, i.e. that of Artinian rings. Section IV -9 introduces very important invariants: the Fitting ideals of a finitely presented module. Finally, Section IV -10 applies this notion to introduce the resultant ideal of a finitely generated ideal over a polynomial ring when the ideal contains a monic polynomial, and to prove a theorem of algebraic elimination over an arbitrary ring.

Chapter V is a first approach to the theory of finitely generated projective modules. Sections V -2 to V -5 state basic properties along with the important example of the zero-dimensional rings. Section V -6 states the local structure theorem: a module is finitely generated projective if and only if it becomes free after localization at suitable comaximal elements. Its constructive proof is a rereading of a result established in Chapter II for “well conditioned” systems of linear equations (Theorem II -5.26). Section V -7 develops the example of the locally cyclic projective modules. Section V -8 introduces the determinant of an endomorphism of a finitely generated projective module. This renders the decomposition of such a module into a direct sum of its components of constant rank accessible. Finally, Section V -9, which we were not too sure where to place in this work, hosts some additional considerations on properties of finite character, a concept introduced in Chapter II to discuss the connections between concrete local-global principles and abstract local-global principles.

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Chapter VI is essentially devoted to algebras which are finitely generated projective modules over their base rings. We call these strictly finite algebras.

When applied to commutative rings, they constitute a natural generalization of the concept of a finite algebra over a field. The icing on the cake being the important case of Galois algebras, which generalize Galoisian extensions of discrete fields to commutative rings.

Section VI -1 treats the case where a base ring is a discrete field. It provides the constructive versions of the structure theorem obtained in classical mathematics. The case of étale algebras (when the discriminant is invertible) is particularly enlightening. We discover that the classical theorems always implicitly assume that we know how to factorize the separable polynomials over the base field. The constructive proof of the primitive element theorem VI -1.9 is significant for its deviation from the classical proof. Section VI -2 applies the previous results to complete the basic Galois theory started in Section III -6 by characterizing the Galoisian extensions of discrete fields like the étale and normal extensions. Section VI -3 is a brief introduction to finitely presented algebras, by focusing on integral algebras⁵, with a weak Nullstellensatz and the Lying Over lemma. Section VI -4 introduces strictly finite algebras over an arbitrary ring. In Sections VI -5 and VI -6, the related concepts of a strictly étale algebra and of a separable algebra are introduced. These generalize the concept of an étale algebra over a discrete field. In Section VI -7, we constructively present the basics of the theory of Galois algebras for commutative rings. It is in fact an Artin-Galois theory since it adopts the approach Artin had developed for the case of fields, starting directly from a finite group of automorphisms of a field, the base field appearing only as a byproduct of subsequent constructions.

In Chapter VII, the dynamic method – a cornerstone of modern methods in constructive algebra – is implemented to deal with the field of roots of a polynomial and the Galois theory in the separable case. From a constructive point of view, we need to use the dynamic method when we do not know how to factorize the polynomials over the base field.

For training purposes, Section VII -1 begins by establishing results in a constructive form for the Nullstellensatz when we do not know how to factorize the polynomials over the base field. General considerations on the dynamic method are developed in Section VII -2. More details on the course of the festivities are given in the introduction of the chapter.

Chapter VIII is a brief introduction to flat modules and to flat and faithfully flat algebras. Intuitively speaking, anA-algebra Bis flat when the homogeneous systems of linear equations overAhave “no more” solutions

5By “integral algebra” we meanan algebra that is integral on its base ring, not to be confused with an algebra that is an integral ring.

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Foreword xxi

inB than in A, and it is faithfully flat if this statement is also true for nonhomogeneous systems of linear equations. These crucial notions of commutative algebra were introduced by Serre in [169, GAGA,1956]. We will only state the truly fundamental results. This is also when we will introduce the concepts of a pf-ring (i.e. a ring whose principal ideals are flat), of a torsion-free module (for an arbitrary ring), of an arithmetic ring and of a Prüfer ring. As always, we focus on the local-global principle when it applies.

Chapter IX discusses local rings and some generalizations. Section IX -1 introduces the constructive terminology for some common classical concepts, including the important concept of a Jacobson radical. A related concept is that of a residually zero-dimensional ring (a ringAsuch thatA/RadAis zero-dimensional). It is a robust concept, which never uses maximal ideals, and most of the theorems in the literature regarding semi-local rings (in classical mathematics they are the rings which only have a finite number of maximal ideals) apply to residually zero-dimensional rings. Section IX -2 lists some results which show that on a local ring we reduce the solution of particular problems to the case of fields. Sections IX -3 and IX -4 establish, based on geometric examples (i.e. regarding the study of polynomial systems), a link between the notion of a local study in the topological intuitive sense and the study of certain localizations of rings (in the case of polynomial systems over a discrete field these localizations are local rings). In particular we introduce the notions of tangent and cotangent spaces at zero of a polynomial system. Section IX -5 is a brief study of decomposable rings, including the particular case from classical mathematics of decomposed rings (finite products of local rings), which play an important role in the theory of Henselian local rings. Finally, Section IX -6 treats the notion of a local-global ring, which generalizes both the concept of local rings and that of zero-dimensional rings. These rings verify very strong local-global properties; e.g. the projective modules of constant rank are always free.

Moreover, the class of local-global rings is stable under integral extensions.

Chapter X continues the study of finitely generated projective modules started in Chapter V. In Section X -1, we return to the question of the characterization of finitely generated projective modules as locally free modules, i.e. of the local structure theorem. We give a matrix version of it (Theorem X -1.7), which summarizes and clarifies the different statements of the theorem. Section X -2 is devoted to the ring of ranks over A. In classical mathematics, the rank of a finitely generated projective module is defined as a locally constant function in the Zariski spectrum. We give here an elementary theory of the rank which does not call upon prime ideals. In Section X -3, we provide some simple applications of the local structure theorem. Section X -4 introduces Grassmannians. In Section X -5,

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we introduce the general problem of the complete classification of the finitely generated projective modules over a fixed ringA. This classification is a fundamental and difficult problem, which does not have a general algorithmic solution. Section X -6 presents a nontrivial example for which this classification can be obtained.

Chapter XI is devoted to distributive lattices and lattice ordered groups (l-groups). The first two sections describe these algebraic structures along with their basic properties. These structures are important in commutative algebra for several reasons.

First, the divisibility theory has as its “ideal model” the natural numbers’

divisibility theory. The structure of the multiplicative monoid (N^∗,×,1) makes it the positive part of anl-group. In commutative algebra, this can be generalized in two possible ways. The first generalization is the theory of integral rings whose finitely generated ideals form a distributive lattice, called Prüfer domains, which we will study in Chapter XII; their nonzero finitely generated ideals form the positive part of anl-group. The second is the theory of gcd rings that we study in Section XI -3. Let us notify the first appearance of the Krull dimension 61 in Theorem XI -3.12: an integral gcd ring with dimension61 is a Bézout ring.

Secondly, the distributive lattices act as the constructive counterpart of various spectral spaces which have emerged as powerful tools of abstract algebra. The relationship between distributive lattices and spectral spaces will be discussed in Section XIII -1. In Section XI -4, we set up the Zariski lattice of a commutative ringA, which is the constructive counterpart of the famous Zariski spectrum. Our goal here is to establish a parallel between the construction of the zero-dimensional reduced closure of a ring (denoted byA^•) and that of the Boolean algebra generated by a distributive lattice (which is the subject of Theorem XI -4.26). The objectA^• constructed as above essentially contains the same information as the product of the rings Frac(A/p) for all prime idealsp ofA(⁶). This result is closely related to the fact that the Zariski lattice ofA^• is the Boolean algebra generated by the Zariski lattice ofA.

A third reason to be interested in distributive lattices is constructive logic (or intuitionistic logic). In this logic, the set of truth values of classical logic, that is the two-element Boolean algebra {True,False}, is replaced by a quite mysterious distributive lattice. Constructive logic is informally discussed in the Annex. In Section XI -5, we set up the tools that provide a framework for a formal algebraic study of constructive logic: entailment relations and Heyting algebras. In addition, entailment relations and Heyting

6This product is not accessible in constructive mathematics,A^•is its perfectly effective constructive substitute.

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Foreword xxiii

algebras have their own use in the general study of distributive lattices.

For example, the Zariski lattice of a coherent Noetherian ring is a Heyting algebra (Proposition XIII -6.9).

Chapter XII deals with arithmetic rings, Prüfer rings and Dedekind rings.

Arithmetic rings are rings for which the lattice of finitely generated ideals is distributive. A Prüfer ring is a reduced arithmetic ring and is characterized by the fact that all of its ideals are flat. A coherent Prüfer ring is the same thing as an arithmetic pp-ring. It is characterized by the fact that its finitely generated ideals are projective. A Dedekind ring is a Noetherian and strongly discrete coherent Prüfer ring (in classical mathematics, withLEM, every ring is strongly discrete and every Noetherian ring is coherent). These rings first appeared as the rings of integers of number fields. The paradigm in the integral case is the unique decomposition into prime factors of any nonzero finitely generated ideal. The general arithmetic properties of finitely generated ideals are mostly verified by all the arithmetic rings. For the most subtle properties concerning the factorizations of the finitely generated ideals, and in particular the decomposition into prime factors, a Noetherian assumption, or at least a dimension61 assumption, is essential. In this chapter, we wanted to show the progression of the properties satisfied by the rings as we strengthen the assumptions from the arithmetic rings to the total factorization Dedekind rings. We focus on the simple algorithmic character of the definitions in the constructive framework. Certain properties only depend on dimension 6 1, and we wanted to do justice to pp-rings of dimension at most 1. We also carried out a more progressive and more elegant study of the problem of the decomposition into prime factors than in the presentations which allow LEM. For example, Theorems XII -4.10 and XII -7.12 provide precise constructive versions of the theorem concerning the normal finite extensions of Dedekind rings, with or without the total factorization property.

The chapter begins with a few epistemological remarks on the intrinsic interest of addressing the factorization problems with the partial factorization theorem rather than the total factorization one. To get a good idea of how things unfold, simply refer to the table of contents at the beginning of the chapter on page 677 and to the table of theorems on page 984.

Chapter XIII is devoted to the Krull dimension of commutative rings, of their morphisms and of distributive lattices, and to the valuative dimension of commutative rings.

Several important notions of dimension in classical commutative algebra are dimensions of spectral spaces. These very peculiar topological spaces have the property of being fully described (at least in classical mathematics) by their compact-open subspaces, which form a distributive lattice. It so

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happens that the corresponding distributive lattice generally has a simple interpretation, without any recourse to spectral spaces. In 1974, Joyal showed how to constructively define the Krull dimension of a distributive lattice. Since this auspicious day, the theory of dimension which seemed bathed in ethereal spaces – that are invisible when you do not trust the axiom of choice – has become (at least in principle) an elementary theory, without any further mysteries.

Section XIII -1 describes the approach of the Krull dimension in classical mathematics. It also explains how to interpret the Krull dimension of such a space in terms of the distributive lattice of its compact-open subspaces.

Section XIII -2 states the constructive definition of the Krull dimension of a commutative ring, denoted byKdimA, and draws some consequences.

Section XIII -3 states some more advanced properties, in particular the local- global principle and the closed covering principle for the Krull dimension.

Section XIII -4 deals with the Krull dimension of integral extensions and Section XIII -5 that of geometric rings (corresponding to polynomial systems) on the discrete fields. Section XIII -6 states the constructive definition of the Krull dimension of a distributive lattice and shows that the Krull dimension of a commutative ring and that of its Zariski lattice coincide. Section XIII -7 is devoted to the dimension of the morphisms between commutative rings.

The definition uses the reduced zero-dimensional closure of the source ring of the morphism. To prove the formula which defines the upper bound of KdimBfromKdimAandKdimρ(when we have a morphismρ:A→B), we must introduce the minimal pp-closure of a commutative ring. This object is a constructive counterpart of the product of all theA/p, whenp ranges over the minimal prime ideals of A. Section XIII -8 introduces the valuative dimension of a commutative ring and in particular uses this concept to prove the following important result: for a nonzero arithmetic ringA, we haveKdimA[X1, . . . , Xn] =n+KdimA. Section XIII -9 states constructive versions of the Going up and Going down Theorems.

In Chapter XIV, titled Number of generators of a module, we establish the elementary, non-Noetherian and constructive versions of the “great”

theorems of commutative algebra, their original form due to Kronecker, Bass, Serre, Forster and Swan. These results relate to the number of radical generators of a finitely generated ideal, the number of generators of a module, the possibility of producing a free submodule as a direct summand in a module, and the possibility to simplifying isomorphisms, in the following way: ifM⊕N 'M⁰⊕N thenM 'M⁰. They involve the Krull dimension or other, more sophisticated dimensions introduced by R. Heitmann as well as by the authors of this work and T. Coquand.

Section XIV -1 is devoted to Kronecker’s Theorem and its extensions (the most advanced, non-Noetherian, is due to R. Heitmann [99]). Kronecker’s

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Foreword xxv

Theorem is usually stated in the following form: an algebraic variety in Cⁿ can always be defined by n+ 1 equations. The form due to Heitmann is that in a ring of Krull dimension less than or equal ton, for all finitely generated ideala there exists an ideal bgenerated by at most n+ 1 elements of a such that √

b = √

a. The proof also gives Bass’ stable range Theorem. The latter theorem was improved by involving “better”

dimensions than the Krull dimension. This is the subject of Section XIV -2 where theHeitmann dimensionis defined, discovered while carefully reading Heitmann’s proofs (Heitmann uses another dimension, a priori a little worse, which we also explain in constructive terms). In Section XIV -3, we explain which matrix properties of a ring allow Serre’s Splitting Off theorem, Forster-Swan’s theorem (controlling the number of generators of a finitely generated module according to the local number of generators) and Bass’

simplification theorem. Section XIV -4 introduces the concepts of support (a mapping from a ring to a distributive lattice satisfying certain axioms) and ofn-stability. The latter was defined by Thierry Coquand, after having analyzed one of Bass’ proofs which establishes that the finitely generated projective modules over a ringV[X], whereVis a valuation ring of finite Krull dimension, are free. In the final section, we prove that the crucial matrix property introduced in Section XIV -3 is satisfied, on one hand by the n-stable rings, and on the other by the rings of Heitmann dimension< n.

Chapter XV is devoted to the local-global principle and its variants. Sec- tion XV -1 introduces the notion of the covering of a monoid by a finite family of monoids, which generalizes the notion of comaximal monoids. The covering Lemma XV -1.5 will be decisive in Section XV -5. Section XV -2 states some concrete local-global principles. This is to say that some properties are globally true as soon as they are locally true. Here, “locally” is meant in the constructive sense: after localization at a finite number of comaximal monoids. Most of the results have been established in the previous chapters. Grouping them shows the very broad scope of these principles.

Section XV -3 restates some of these principles as abstract local-global principles. Here, “locally” is meant in the abstract sense: after localization at any arbitrary prime ideal. We are mainly interested in comparing the abstract principles and the corresponding concrete local-global principles.

Section XV -4 explains the construction of “global” objects from objects of the same type defined only locally, as is usual in differential geometry.

It is the impossibility of this construction when seeking to glue certain rings together which is at the root of Grothendieck schemes. In this sense, Sections XV -2 and XV -4 constitute the basis from which we can develop the theory of schemes in a completely constructive framework.

The following sections are of a different nature. Methodologically, they are devoted to the decryption of different variations of the local-global principle

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in classical mathematics. For example, the localization at every prime ideal, the passage to the quotient by every maximal ideal or the localization at every minimal prime ideal, each of which applies in particular situations.

Such a decryption certainly presents a confusing character insofar as it takes as its starting point a classical proof that uses theorems in due and proper form, but where the constructive decryption of this proof is not only given by the use of constructive theorems in due and proper form. One must also look at what the classical proof does with its purely ideal objects (e.g.

maximal ideals) to understand how it gives us the means to construct a finite number of elements that will be involved in a constructive theorem (e.g. a concrete local-global principle ) to reach the desired result. Decrypting such a proof we use the general dynamic method presented in Chapter VII. We thus describelocal-global machineriesthat are significantly less elementary than those in Chapter IV: the basic constructive local-global machinery

“with prime ideals” (Section XV -5), the constructive local-global machinery

“with maximal ideals” (Section XV -6) and the constructive local-global machinery “with minimal prime ideals” (Section XV -7). By carrying out

“Poincaré’s program” used as an epigraph for this foreword, our local-global machineries take into account an essential remark made by Lakatos that the most interesting and robust thing in a theorem is always its proof, even if it can be criticized in some respects (see [Lakatos]).

In Sections XV -8 and XV -9, we examine to what extent certain local- global principles remain valid when we replace in the statements the lists of comaximal elements by lists of depth>1 or of depth>2.

In Chapter XVI, we treat the question of finitely generated projective modules over rings of polynomials. The decisive question is to establish for which classes of rings the finitely generated projective modules over a polynomial ring are derived by scalar extension of a finitely generated projective module over the ring itself (possibly by putting certain restrictions on the considered finitely generated projective modules or on the number of variables in the polynomial ring). Some generalities on the extended modules are given in Section XVI -1. The case of the projective modules of constant rank 1, which is fully clarified by Traverso-Swan-Coquand’s theorem, is dealt with in Section XVI -2. Coquand’s constructive proof uses the constructive local-global machinery with minimal prime ideals in a crucial way. Section XVI -3 deals with Quillen and Vaserstein’s patching theorems, which state that certain objects are obtained by scalar extension (from the base ring to the polynomial ring) if and only if this property is locally satisfied. We also have a sort of converse to Quillen’s patching due to Roitman, in a constructive form. Section XVI -4 is devoted to Horrocks’ theorems. The constructive proof of Horrocks’ global theorem is obtained from the proof of Horrocks’ local theorem by using the basic

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Foreword xxvii

local-global machinery and concluding with Quillen’s constructive patching.

Section XVI -5 gives several constructive proofs of Quillen-Suslin’s theorem (the finitely generated projective modules over a polynomial ring on a discrete field are free) founded on different classical proofs. Section XVI -6 establishes Lequain-Simis’ theorem (the finitely generated projective modules over a polynomial ring on an arithmetic ring are extended). The proof uses the dynamic method presented in Chapter VII. This allows us to establish Yengui’s induction theorem, a constructive variation of Lequain- Simis’ induction.

In Chapter XVII, we prove “Suslin’s Stability Theorem” in the special case of discrete fields. Here also, we use the basic local-global machinery presented in Chapter XV to obtain a constructive proof.

The Annex describes a Bishop style constructive set theory. It can be seen as an introduction to constructive logic. In it we explain the Brouwer- Heyting-Kolmogorov semantic for connectives and quantifiers. We discuss certain weak forms of LEMalong with several problematic principles in constructive mathematics.

Some epistemological remarks

In this work, we hope to show that classical commutative algebra books such as [Eisenbud], [Kunz], [Lafon & Marot], [Matsumura], [Glaz], [Kaplansky], [Atiyah & Macdonald], [Northcott], [Gilmer], [Lam06] (which we highly rec- ommend), or even [Bourbaki] and the remarkable work available on the web [Stacks-Project], could be entirely rewritten from a constructive point of view, dissipating the veil of mystery which surrounds the nonexplicit existence theorems of classical mathematics. Naturally, we hope that the readers will take advantage of our work to take a fresh look at the classical Com- puter Algebra books like, for instance, [Cox, Little & O’Shea], [COCOA], [Elkadi & Mourrain], [von zur Gathen & Gerhard], [Mora], [TAPAS]

or [SINGULAR].

Since we want an algorithmic processing of commutative algebra we cannot use all the tricks that arise from the systematic use of Zorn’s Lemma and the Law of Excluded Middle in classical mathematics. Undoubtedly, the reader understands that it is difficult to implement Zorn’s lemma in Computer Algebra. The refusal of LEM, however, must seem harder to stomach. It is simply a practical observation on our part. If in a classical proof there is a reasoning that leads to a computation in the form “ifxis invertible, do this, otherwise do that,” then, clearly, it directly translates into an algorithm only when there is an invertibility test for the ring in question. It is in stressing this difficulty, which we must constantly work around, that we are