Introduction

In the first part of this introduction, we give details on the contents of the present volume; in the second, on the whole of the “Séminaire de Géométrie Algébrique du Bois-Marie”, of which the present volume is the first tome.

1

The present volume presents the foundations of a theory of the fundamental group in algebraic geometry, from the “Kroneckerian” point of view that makes it possible to treat on the same footing the case of an algebraic variety in the usual sense and, for example, that of the ring of integers of a number field. This point of view can be expressed satisfactorily only in the language of schemes, and we shall freely use this language, as well as the principal results set out in the first three chapters of the Éléments de Géométrie Algébrique by J. Dieudonné and A. Grothendieck, cited below as EGA. The study of the present volume of the “Séminaire de Géométrie Algébrique du Bois-Marie” requires no other knowledge of algebraic geometry, and can therefore serve as an introduction to current techniques in algebraic geometry for a reader wishing to become familiar with them.

Exposés I through XI of this book are a textual reproduction, practically unchanged, of the mimeographed notes of the oral seminar, which were distributed by the Institut des Hautes Études Scientifiques.¹ We have limited ourselves to adding a few footnotes to the original text, correcting a few typographical errors, and making one terminological adjustment: in particular, the term “simple morphism” was in the meantime replaced by “smooth morphism”, which does not give rise to the same confusions.

Exposés I through IV present the local notions of étale morphism and smooth morphism; they make little use of the language of schemes, set out in Chapter I of the Éléments.² Exposé V presents the axiomatic description of the fundamental group of a scheme, useful even in the classical case where the scheme reduces to the spectrum of a field, where one finds a very convenient reformulation of the usual Galois theory. Exposés VI and VIII present descent theory, which has taken on growing importance in algebraic geometry in recent years and could render analogous services in analytic geometry and topology. It should be noted that Exposé VII had not been written up, and that its substance is incorporated into a work of J. Giraud, Méthode de la Descente, Bulletin de la Société Mathématique de France, Mémoire 2, 1964, viii + 150 pages.

In Exposé IX, one studies more specifically the descent of étale morphisms, obtaining a systematic approach to Van Kampen type theorems for the fundamental group, which appear here as simple translations of descent theorems. It is essentially a method for computing the fundamental group of a connected scheme X, equipped with a surjective and proper morphism, say X′ → X, in terms of the fundamental groups of the connected components of X′ and of the fiber products X′ ×_X X′, X′ ×_X X′ ×_X X′, and of the homomorphisms induced between these groups by the canonical simplicial morphisms between the preceding schemes. Exposé X gives the theory of specialization of the fundamental group for a proper and smooth morphism; its most striking result consists in the determination, up to a small ambiguity, of the fundamental group of a smooth algebraic curve in characteristic p > 0, thanks to the result known by transcendental methods in characteristic zero. Exposé XI gives some examples and complements, making explicit in cohomological form Kummer’s theory of coverings and Artin-Schreier’s.

For other comments on the text, see the “Foreword” to the multigraphed version, which follows the present Introduction.

Since this Seminar was written in 1961, M. Artin and I have developed the language of the étale topology and a corresponding cohomological theory, set out in SGA 4, “Cohomologie étale des schémas”, of the Séminaire de Géométrie Algébrique, to appear in the same series as the present volume. This language, and the results already available in it, provide a particularly flexible tool for the study of the fundamental group, making it possible to understand better, and to go beyond, some of the results set out here. The theory of the fundamental group should therefore be taken up again entirely from this point of view; in fact all the key results already appear in that work.

This was what had been planned for the chapter of the Éléments devoted to the fundamental group, which was also to contain several other developments that could not find a place here, relying on the technique of resolution of singularities: the computation of the “local fundamental group” of a complete local ring in terms of a suitable resolution of the singularities of that ring; local and global Künneth formulas for the fundamental group without a properness hypothesis (cf. Exposé XIII); and M. Artin’s results on the comparison of the local fundamental groups of an excellent henselian local ring and of its completion (SGA 4 XIX). Let us also point out the need to develop a theory of the fundamental group of a topos, which will encompass at once the ordinary topological theory, its semi-simplicial version, the “profinite” variant developed in Exposé V of the present volume, and the slightly more general pro-discrete variant of SGA 3 X 7, adapted to the case of schemes that are non-normal and not unibranch.

While awaiting a complete recasting of the theory in this spirit, Exposé XIII by Mme Raynaud, using the language and results of SGA 4, is intended to show the use that can be made of it in a few typical questions, especially by generalizing some results of Exposé X to non-proper relative schemes. In particular, it gives the structure of the prime-to-p fundamental group of a non-complete algebraic curve in arbitrary characteristic, which I had announced in 1959 but for which no proof had been published to date.

Despite these many gaps and imperfections, or as others would say because of these gaps and imperfections, I think the present volume may be useful to the reader who wishes to become familiar with the theory of the fundamental group, and also as a reference work, while awaiting the writing and publication of a text escaping the criticisms I have just enumerated.

2

The present volume is tome 1 of the “Séminaire de Géométrie Algébrique du Bois-Marie”, whose following volumes are planned to appear in the same series. The aim of the Séminaire, parallel to the treatise Éléments de Géométrie Algébrique by J. Dieudonné and A. Grothendieck, is to lay the foundations of algebraic geometry according to the points of view of the latter work. The standard reference for all volumes of the Séminaire consists of Chapters I, II, and III of the Éléments de Géométrie Algébrique, cited as EGA I, II, and III; the reader is assumed to possess the background in commutative algebra and homological algebra implied by those chapters.³ In addition, in each volume of the Séminaire, reference will be made freely, as needed, to earlier volumes of the same Séminaire, or to other published or soon-to-appear chapters of the Éléments.

Each part of the Séminaire is centered on a main subject, indicated in the title of the corresponding volume or volumes; the oral seminar generally covers one academic year, sometimes more. The exposés within each part of the Séminaire are generally in close logical dependence on one another; by contrast, the different parts of the Séminaire are, to a large extent, logically independent of one another. Thus the part “Group Schemes” is almost entirely independent of the two parts of the Séminaire that chronologically precede it, although it frequently appeals to results of EGA IV. Here is the list of the parts of the Séminaire that are to appear shortly, cited below as SGA 1 through SGA 7:

• SGA 1. Étale coverings and the fundamental group, 1960 and 1961.
• SGA 2. Local cohomology of coherent sheaves and local and global Lefschetz theorems, 1961/62.
• SGA 3. Group schemes, 1963 and 1964, three volumes, in collaboration with M. Demazure.
• SGA 4. Theory of topoi and étale cohomology of schemes, 1963/64, three volumes, in collaboration with M. Artin and J. L. Verdier.
• SGA 5. ℓ-adic cohomology and L-functions, 1964 and 1965, two volumes.
• SGA 6. Intersection theory and the Riemann-Roch theorem, 1966/67, two volumes, in collaboration with P. Berthelot and L. Illusie.
• SGA 7. Local monodromy groups in algebraic geometry.

Three of these partial seminars were directed in collaboration with other mathematicians, who will appear as co-authors on the covers of the corresponding volumes. As for the other active participants in the Séminaire, whose role, both editorial and mathematical, has grown from year to year, each participant’s name appears at the head of the exposés for which he is responsible as lecturer or writer, and the list of those appearing in a given volume is indicated on that volume’s flyleaf.

It is appropriate to give a few details on the relation between the Séminaire and the Éléments. The latter were intended in principle to give an overall account of the notions and techniques judged most fundamental in algebraic geometry, as those notions and techniques themselves emerge through the natural play of demands of logical and aesthetic coherence. From this viewpoint, it was natural to consider the Séminaire as a preliminary version of the Éléments, destined sooner or later to be absorbed almost entirely into them. This process had already begun to some extent several years ago, since Exposés I through IV of the present volume SGA 1 are entirely encompassed by EGA IV, and Exposés VI through VIII were to be so within a few years in EGA VI.

However, as the work of building undertaken in the Éléments and in the Séminaire develops, and as the overall proportions become clearer, the initial principle, according to which the Séminaire would constitute only a preliminary and provisional version, appears less and less realistic, for reasons including the limits prudently imposed by nature on the length of human life. Given the care generally taken in writing the different parts of the Séminaire, there will doubtless be reason to take up such a part again in the Éléments, or in treatises that might take over from them, only when later progress permits very substantial improvements, at the cost of fairly deep modifications. This is already the case for the present seminar SGA 1, as said above, and also for SGA 2, thanks to recent results of Mme Raynaud. By contrast, nothing at present indicates that this will be so in the near future for any of the parts cited above, SGA 3 through SGA 7.

References inside the “Séminaire de Géométrie Algébrique du Bois Marie” are given as follows. An internal reference to one of the parts SGA 1 through SGA 7 of the Séminaire is given in the style III 9.7, where the numeral III denotes the number of the exposé, which appears at the top of each page of the exposé in question, and 9.7 denotes the number of the statement, definition, remark, or similar item inside that exposé. If needed, longer decimal numbers may be used, for example 9.7.1 and 9.7.2 to designate the various steps in the proof of Proposition 9.7. The reference III 9 denotes paragraph 9 of Exposé III. The number of the exposé is omitted for references internal to an exposé. For a reference to another part of the Séminaire, the same sigla are used, but preceded by the mention of the SGA part in question, for example SGA 1 III 9.7. Similarly, the reference EGA IV 11.5.7 means: Éléments de Géométrie Algébrique, Chapter IV, statement, definition, etc. 11.5.7; here, the first Arabic numeral again denotes the paragraph number. Apart from these conventions, in force throughout the SGA, the bibliography for an exposé will generally be gathered at its end, and inside the exposé it will be referred to by numbers in square brackets, such as [3], according to custom.

Finally, for the reader’s convenience, whenever it seems necessary, we shall append to the end of SGA volumes an index of notation and a terminological index containing, where appropriate, an English translation of the French terms used.

I wish to add an extra-mathematical comment to this introduction. In November 1969 I learned that the Institut des Hautes Études Scientifiques, where I had been a professor essentially since its founding, had for three years been receiving subsidies from the Ministry of the Armed Forces. Already as a beginning researcher I had found extremely regrettable the lack of scruple shown by most scientists in accepting collaboration in one form or another with military apparatuses. My motivations at that time were essentially moral in nature, and hence not very likely to be taken seriously. Today they acquire a new force and a new dimension, given the danger of destruction of the human species with which we are threatened by the proliferation of military apparatuses and of the means of mass destruction at their disposal.

I have explained myself elsewhere in more detail on these questions, much more important than the advancement of any science, mathematics included; one may for example consult on this subject G. Edwards’s article in number 1 of the journal Survivre (August 1970), summarizing a more detailed exposition of these questions that I had given elsewhere. Thus I found myself working for three years in an institution while it was taking part, unbeknownst to me, in a mode of financing that I consider immoral and dangerous.⁴ Being at present alone in holding this opinion among my colleagues at the IHES, which has doomed to failure my efforts to obtain the removal of military subsidies from the IHES budget, I have taken the necessary decision and leave the IHES on September 30, 1970, and likewise suspend all scientific collaboration with this institution as long as it continues to accept such subsidies.

I have asked M. Motchane, director of the IHES, that from October 1, 1970 the IHES abstain from distributing mathematical texts of which I am the author, or which form part of the Séminaire de Géométrie Algébrique du Bois Marie. As was said above, distribution of this seminar will be carried out by the Julius Springer publishing house, in the Lecture Notes series. I am happy to thank Springer and Mr. K. Peters here for the effective and courteous help they gave me in making this publication possible, in particular by taking charge of the typing for photo-offset of the new exposés added to old seminars, and of the missing exposés in incomplete seminars.

I also thank Mr. J. P. Delale, who took on the thankless task of compiling the index of notation and the terminological index.

Massy, August 1970.