Éléments de géométrie algébrique

Infobox Book
name = Éléments de géométrie algébrique


author = Alexander Grothendieck and Jean Dieudonné
language = French
subject = Algebraic geometry
publisher = Institut des Hautes Études Scientifiques
pub_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.

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 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-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.

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original and revised plan of the treatise and indicates where (in SGA or elsewhere) the topics intended for the later, unpublished chapters were treated by Grothendieck and his collaborators.

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 [http://www.grothendieckcircle.org/] .

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 [http://www.numdam.org] under "Publications mathématiques de l'IHÉS", volumes 4, 8, 11, 17, 20, 24, 28 and 32.

Bibliographic information

*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1971
title = Éléments de géométrie algébrique
edition = 2nd edition
publisher = Springer-Verlag
location = Berlin; New York
language = French
id = ISBN 3540051139
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1960
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : I. Le langage des schémas | journal = Publications Mathématiques de l'IHÉS
volume = 4
pages = 5–228
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1960__4_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1961
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes | journal = Publications Mathématiques de l'IHÉS
volume = 8
pages = 5–222
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1961
title = Eléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie
journal = Publications Mathématiques de l'IHÉS
volume = 11
pages = 5–167
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__11_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1963
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Seconde partie
journal = Publications Mathématiques de l'IHÉS
volume = 17
pages = 5–91
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1963__17_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1964
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Première partie
journal = Publications Mathématiques de l'IHÉS
volume = 20
pages = 5–259
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1964__20_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1965
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Seconde partie
journal = Publications Mathématiques de l'IHÉS
volume = 24
pages = 5–231
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1965__24_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1966
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie
journal = Publications Mathématiques de l'IHÉS
volume = 28
pages = 5–255
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_
*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1967
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie
journal = Publications Mathématiques de l'IHÉS
volume = 32
pages = 5–361
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1967__32_

External links

*Scanned copies and partial English translations: [http://www.math.jussieu.fr/~leila/grothendieckcircle/mathtexts.php Mathematical Texts]
*Detailed table of contents: [http://www.dma.ens.fr/~madore/ega-toc.pdf]