Éléments de géométrie algébrique

The Éléments de géométrie algébrique ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, are an unfinished 1500-page 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.

The table of contents is as follows:

I. Le langage des schémas ("The language of schemes").

II. Étude globale élémentaire de quelques classes de morphismes ("Global elementary study of certain classes of morphisms").

III. Étude cohomologique des faisceaux cohérents ("Cohomological study of coherent sheaves").

IV. Étude locale des schémas et des morphismes de schémas ("Local study of schemes and morphisms of schemes").

Initially thirteen sections were planned. Much of the material which would have been found in the following sections 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 incorporation 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 Science+Business Media|Springer-Verlag]]. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functors. Grothendieck never gave permission for this volume to be republished, so copies are very rare but found in many libraries. The work on EGA was finally disrupted by Grothendieck's departure first from IHES 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.

Bibliographic information

• Grothendieck, Alexandre; Jean Dieudonné (1971). Éléments de géométrie algébrique, 2nd edition (in French), Berlin; New York: [[Springer Science+Business Media|Springer-Verlag]]. ISBN 3540051139. 
• Grothendieck, Alexandre; Jean Dieudonné (1960). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas". Publications Mathématiques de l'IHÉS 4: 5-228. 
• Grothendieck, Alexandre; Jean Dieudonné (1961). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS 8: 5-222. 
• Grothendieck, Alexandre; Jean Dieudonné (1961). "Eléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS 11: 5-167. 
• Grothendieck, Alexandre; Jean Dieudonné (1963). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Seconde partie". Publications Mathématiques de l'IHÉS 17: 5-91. 
• Grothendieck, Alexandre; Jean Dieudonné (1964). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS 20: 5-259. 
• Grothendieck, Alexandre; Jean Dieudonné (1965). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Seconde partie". Publications Mathématiques de l'IHÉS 24: 5-231. 
• Grothendieck, Alexandre; Jean Dieudonné (1966). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS 28: 5-255. 
• Grothendieck, Alexandre; Jean Dieudonné (1967). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS 32: 5-361. 

External links

• Scanned copies and partial English translations: Mathematical Texts

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