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Revêtements Étales et Groupe Fondamental (SGA 1)

A. Grothendieck, editor
Société Mathématique de France
Publication Date: 
Number of Pages: 
Documents Mathématiques
[Reviewed by
Fernando Q. Gouvêa
, on

This is the first volume in the third printed incarnation of Alexandre Grothendieck's famous Séminaire de Géométrie Algébrique. Reset by a team of volunteers and with a few notes added to update the material, the new edition is valuable both as a historical document and as a reference for students of algebraic geometry. 

In the early 1960s, Grothendieck and a large group of collaborators set out to remake the foundations of algebraic geometry. With Jean Dieudonné, he started writing the Éléments de Géométrie Algébrique, later known as EGA, in which they planned to completely recast the basics of the subject. Simultaneously, Grothendieck started an algebraic geometry seminar at the Institut des Hautes Études Scientifiques (IHES) in which he and others could develop the new ideas further. The published reports from the seminar made up the several "SGA" volumes. The stated goal was to eventually incorporate all or most of SGA into EGA.

The plan was extremely ambitious, and, as is well known, it was never actually completed. Four (of ten originally planned) parts of EGA were eventually published (only one in book format), and seven parts of SGA (several of which had multiple volumes).

SGA was originally published and distributed by IHES, with this first volume, containing material presented at the seminar in 1960 and 1961, appearing in 1963. In the late 1960s, Grothendieck resigned from IHES and prohibited it from having anything further to do with the work. As a result, the SGA volumes were reprinted (with only minor updates) in Springer's Lecture Notes in Mathematics series. At that point, they were still photo-reproductions from typed notes, not very easy to read or handle.

The new edition of SGA 1 is elegantly done in TeX, with a few added notes by Michel Raynaud updating the material when necessary. (Large chunks of the material in SGA 1 were superseded by EGA IV; this had already been noted in the 1970 reprint and is not belabored in the new version.) The page numbers from the original edition are reproduced in the margins, a nice touch that historians will value. The book still retains much of its character as seminar notes; for example, there are no notes from the seventh seminar, and this is simply noted in the table of contents as "VII: n'existe pas."

SGA 1 includes the 1970 introduction to the whole series, outlining Grothendieck's ambitious plans, listing all of the SGA volumes, and explaining his break with IHES. The main topic of the seminar is the theory of étale coverings and the related notion of the (algebraic) fundamental group. This material has, of course, been treated elsewhere since, but this remains a useful reference. Many thanks are due to the SMF for bringing these volumes back; let's hope they stay the course and get through the whole series! Even more, let's hope they do EGA also!

Note: SGA 2 is also available in the same series.

Fernando Q. Gouvêa is looking forward to retiring his old copies of the SGA volumes.

  • Morphismes étales
  • Morphismes lisses: généralités, propriétés différentielles
  • Morphismes lisses: propriétés de prolongement
  • Morphismes plats
  • Le groupe fondamental: généralités
  • Catégories fibrées et descente
  • n'existe pas
  • Descente fidèlement plate
  • Descente des morphismes étales. Application au groupe fondamental
  • Théorie de la spécialisation du groupe fondamental
  • Exemples et compléments
  • Géométrie algébrique et géométrie analytique
  • Propreté cohomologique des faisceaux d'ensembles et des faisceaux de groupes non commutatifs
  • Index terminologique
  • Index des notations