Foreword
Each written exposé gives the substance of several consecutive oral lectures. It did not seem useful to specify their dates.
Exposé VII, to which reference is made several times in the course of Exposé VIII, was not written up by the lecturer. In the oral lectures he had limited himself to sketching the language of descent in general categories, taking a strictly utilitarian point of view and without entering into the logical difficulties raised by this language. It became clear that a correct exposition of this language would exceed the limits of the present notes, if only by its length. For a fully formed exposition of descent theory, I refer to an article in preparation by Jean Giraud. Pending its publication,1 I think an attentive reader will have no difficulty supplying by his own means the phantom references in Exposé VIII.
Other oral exposés, placed after Exposé XI and alluded to at certain points in the text, were likewise not written up, and were intended to form the substance of Exposé XII and Exposé XIII. The first of these oral exposés took up, in the framework of schemes and analytic spaces with nilpotent elements as introduced in the Cartan Seminar 1960/61, the construction of the analytic space associated to a prescheme locally of finite type over a complete valued field k, the GAGA-type theorems in the case where k is the field of complex numbers, and the application to the comparison of the fundamental group defined by transcendental methods with the fundamental group studied in these notes; compare A. Grothendieck, Fondements de la Géométrie Algébrique, Séminaire Bourbaki no. 190, page 10, December 1959.
The last oral exposés sketched the generalization of the methods developed in the text for the study of coverings admitting tame ramification, and of the structure of the fundamental group of a complete curve deprived of a finite number of points; compare the same source, no. 182, page 27, Theorem 14. These exposés introduce no essentially new idea, which is why it did not seem indispensable to give them a formal written version before the appearance of the corresponding chapters of the Éléments de Géométrie Algébrique.2
By contrast, Lefschetz-type theorems for the fundamental group and the Picard group, both locally and globally, were the subject of a separate seminar in 1962, which has been completely written up and is available to users.3 Let us point out that the results developed both in the present Seminar and in that of 1962 will be used in an essential way in the publication of several key results in the étale cohomology of preschemes, which will be the subject of a seminar, conducted by M. Artin and myself, in 1963/64 and currently in preparation.4
Exposés I through IV, essentially local and very elementary in nature, will be entirely absorbed by Chapter IV of the Éléments de Géométrie Algébrique, whose first part is in press and will doubtless be published toward the end of 1964. They may nevertheless be useful to a reader who wishes to acquaint himself with the essential properties of smooth, étale, or flat morphisms before entering the arcana of a systematic treatise. As for the other exposés, they will be absorbed into Chapter VIII5 of the Éléments, whose publication can hardly be contemplated before several years.
Bures, June 1963.
Now published: J. Giraud, Méthodes de la descente, Mémoire no. 2 of the Société mathématique de France, 1964.
They are included in the present volume in Exposé XII by Mme Raynaud, with a proof different from the original proof presented in the oral seminar; cf. the introduction.
Cohomologie étale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), published by North-Holland Publishing Company.
Cohomologie étale des schémas (cited as SGA 4), to appear in this same series.
In fact, because of a change in the initially planned outline of the Éléments, the study of the fundamental group is postponed there to a chapter later than the one just indicated. Compare the introduction preceding the present Foreword.