Éléments de géométrie algébrique

Éléments de géométrie algébrique

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Author	Alexander Grothendieck and Jean Dieudonné
Language	French
Subject	Algebraic geometry
Publisher	Institut des Hautes Études Scientifiques
Publication date	1960–1967

The Éléments de géométrie algébrique ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiques. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation stone and basic reference of modern algebraic geometry.

Editions

Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the Séminaire de géométrie algébrique (known as SGA). Indeed, as explained by Grothendieck in the preface of the published version of SGA, by 1970 it had become clear that incorporating all of the planned material in EGA would require significant changes in the earlier chapters already published, and that therefore the prospects of completing EGA in the near term were limited. An obvious example is provided by derived categories, which became an indispensable tool in the later SGA volumes, was not yet used in EGA III as the theory was not yet developed at the time. Considerable effort was therefore spent to bring the published SGA volumes to a high degree of completeness and rigour.

Grothendieck nevertheless wrote a revised version of EGA I which was published by Springer-Verlag. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functors. The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters.

Grothendieck's EGA 5 which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website. Monografie Matematyczne in Poland has accepted this volume for publication but the editing process is quite slow at this time 2010. James Milne has preserved some of the original Grothendieck notes and a translation of them into English. They may be available from his websites connected with the University of Michigan in Ann Arbor.

Chapters

The following table lays out the original and revised plan of the treatise and indicates where (in SGA or elsewhere) the topics intended for the later, unpublished chapters were treated by Grothendieck and his collaborators.

#	First edition	Second edition	Comments
I	Le langage des schémas	Le langage des schémas	Second edition brings in certain schemes representing functors such as Grassmannians, presumably from intended Chapter V of the first edition. In addition, the contents of Section 1 of Chapter IV of first edition was moved to Chapter I in the second edition.
II	Étude globale élémentaire de quelques classes de morphismes	Étude globale élémentaire de quelques classes de morphismes	First edition complete, second edition did not appear.
III	Étude cohomologique des faisceaux cohérents	Cohomologie des Faisceaux algébriques cohérents. Applications.	First edition complete except for last four sections, intended for publication after Chapter IV: elementary projective duality, local cohomology and its relation to projective cohomology, and Picard groups (all but projective duality treated in SGA 2).
IV	Étude locale des schémas et des morphismes de schémas	Étude locale des schémas et des morphismes de schémas	First edition essentially complete; some changes made in last sections; the section on hyperplane sections made into the new Chapter V of second edition (draft exists)
V	Procédés élémentaires de construction de schémas	Complements sur les morphismes projectifs	Did not appear. Some elementary constructions of schemes apparently intended for first edition appear in Chapter I of second edition. The existing draft of Chapter V corresponds to the second edition plan. It includes also expanded treatment of some material from SGA 7.
VI	Technique de descente. Méthode générale de construction des schémas	Techniques de construction de schémas	Did not appear. Descent theory and related construction techniques treated by Grothendieck in FGA.
VII	Schémas de groupes, espaces fibrés principaux	Schémas en groupes, espaces fibrés principaux	Did not appear. Treated in detail in SGA 3.
VIII	Étude différentielle des espaces fibrés	Le schéma de Picard	Did not appear. Material apparently intended for first edition can be found in SGA 3, Picard scheme is treated in FGA.
IX	Le groupe fondamental	Le groupe fondamental	Did not appear. Treated in detail in SGA 1.
X	Résidus et dualité	Résidus et dualité	Did not appear. Treated in detail in Hartshorne's edition of Grothendieck's notes "Residues and duality"
XI	Théorie d'intersection, classes de Chern, théorème de Riemann-Roch	Théorie d'intersection, classes de Chern, théorème de Riemann-Roch	Did not appear. Treated in detail in SGA 6.
XII	Schémas abéliens et schémas de Picard	Cohomologie étale des schémas	Did not appear. Étale cohomology treated in detail in SGA 4, SGA 5.
XIII	Cohomologie de Weil	none	Intended to cover étale cohomology in the first edition.

In addition to the actual chapters, an extensive "Chapter 0" on various preliminaries was divided between the volumes in which the treatise appeared. Topics treated range from category theory, sheaf theory and general topology to commutative algebra and homological algebra. The longest part of Chapter 0, attached to Chapter IV, is more than 200 pages.

Grothendieck never gave permission for the 2nd edition of EGA I to be republished, so copies are rare but found in many libraries. The work on EGA was finally disrupted by Grothendieck's departure first from IHÉS in 1970 and soon afterwards from the mathematical establishment altogether. Grothendieck's incomplete notes on EGA V can be found at [1].

In historical terms, the development of the EGA approach set the seal on the application of sheaf theory to algebraic geometry, set in motion by Serre's basic paper FAC. It also contained the first complete exposition of the algebraic approach to differential calculus, via principal parts. The foundational unification it proposed (see for example unifying theories in mathematics) has stood the test of time.

EGA has been scanned by NUMDAM and is available at [2] under "Publications mathématiques de l'IHÉS", volumes 4, 8, 11, 17, 20, 24, 28 and 32.

External links

• Scanned copies and partial English translations: Mathematical Texts
• Detailed table of contents: [3]