nLab EGA

This entry is about the text

• Alexander Grothendieck, Éléments de géométrie algébrique (numdam)

EGA

EGA stands for Éléments de géométrie algébrique (Elements of algebraic geometry), which was written by Alexandre Grothendieck and co-edited with Jean Dieudonné. These volumes (and a list is given below) were among his many works attempting to build the foundations for algebraic geometry in the language of schemes. They were preceded by FGA and were followed up by the SGA, more details of these can be found below.

The published part of EGA is in Publ. IHÉS, now free online at numdam (detailed links to chapters, and their contents will be added here later, see also below for a list of the volumes in the series). We plan to list here the grand plan and some remarks and links.

List of volumes of EGA

Note that there is a chapter 0 that continues at the beginning of each chapter, establishing preliminaries from topology, category theory, commutative algebra, homological algebra, etc.

• I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 5–228. Ch. 0.§1–7 ; Ch I.§1–10.

• II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 5–222. Ch. II.§1–8.

• III_1. Étude cohomologique des faisceaux cohérents (Première partie) Inst. Hautes Études Sci. Publ. Math. 11 (1961), 5–167. Ch. 0.§8–13 ; Ch.III.§1–5.

• III_2. Étude cohomologique des faisceaux cohérents (Seconde partie) Inst. Hautes Études Sci. Publ. Math. 17 (1963), 5–91. Ch. III .§6–7.

• IV_1. Étude locale des schémas et des morphismes de schémas (Première partie) Inst. Hautes Études Sci. Publ. Math. 20 (1964), 5–259. Ch.0.§14–23 ; Ch. IV.§1.

• IV_2. Étude locale des schémas et des morphismes de schémas (Seconde partie) Inst. Hautes Études Sci. Publ. Math. 24 (1965), 5–231. Ch. IV2.§2–7 .

• IV_3. Étude locale des schémas et des morphismes de schémas (Troisième partie) Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255. Ch. IV.§8–15 (numdam:PMIHES_1966__28__5_0/)

• IV_4. Étude locale des schémas et des morphismes de schémas (Quatrième partie) Inst. Hautes Études Sci. Publ. Math. 32 (1967), 5–361. Ch.IV.§16–21 (numdam:PMIHES_1967__32__5_0)

There was a second edition of EGA I:

• I. Le langage des schémas. (2nd ed.) Grundlehren der Mathematischen Wissenschaften (1971). 166. Springer-Verlag.

In this edition, the term “prescheme” (now outdated but used in the first edition for what we now call schemes) was replaced by “scheme”, and the term “scheme” became separated scheme. Also, there was a transition to the functor of points approach to scheme theory.

EGA V and beyond

EGA was never completed. The listed volumes I-IV are just a part of the original plan. Grothendieck outlined what was meant to be in chapters V-VII, at least, and some handwritten prenotes existed for a small part of those. See e.g. these prenotes for some parts of EGA V.

Wikipedia lists the titles of planned chapters I-XIII.

The other series:

FGA (Fondements de la Géometrie Algébrique)

The earliest of the series is FGA written alone at the end of the 1950s; it is sort of quick outline of the theory and some deep results with proofs at the beginning of the period of the development of scheme theory?. A modern version of the material in FGA appeared as a book FGA explained (ICTP, Trieste 2003–2005). The original is available here.

SGA (Séminaire de Géométrie Algébrique du Bois Marie)

(Bois Marie is the name of an area of woodland at Bures-sur-Yvette, where the IHÉS is situated.)

SGA consists of the writeups of seminars at IHÉS lead by Grothendieck; the work of the members of the seminar under the guidance of Grothendieck are included. In $n$lab we have pages on SGA1, and SGA4.

The Wikipedia entry lists all of the seminars.

Pierre Deligne edited a supplementary volume SGA $4\frac{1}{2}$ comprisings parts of SGA5?, his own articles proving new results and some substantially rewritten and expanded material exposed in less satisfactory form in SGA5 (these results were used for his proof of the Weil conjectures which appeared before SGA5).

An ongoing project aims to retype the SGA volumes in TeX. So far the following volumes have been retyped:

• SGA 1 - Edited by B. Edixhoven, 2004

• SGA 2 - Edited by Y. Laszlo, 2004

• SGA 3 - Edited by P. Gilles and P. Polo, 2011

• SGA 4 - Edited by Y. Laszlo.

  Part 1,

  Part 2,

  Part 3.

• SGA $4{1\over2}$ - Edited by Daniel Miller.

See also:

• SGA 4 Edited by F. Orgogozo

References

There are Wikipedia entries

• Wikipedia, Éléments de géométrie algébrique

• Wikipedia, Séminaire de Géométrie Algébrique du Bois Marie

• Wikipedia, Fondements de la Géometrie Algébrique

EGA and the other published texts of Grothendieck are available from

• Grothendieck Circle?, Scans of published texts.

and some of the EGA and SGA links are at this html.

David Madore has prepared a detailed table of contents of EGA:

• Detailed table of contents of EGA

A translated table of contents has been prepared by Mark Haiman, available at

• Mark Haiman, English translation of table of contents of EGA

along with synopsises, in English, of many sections at

• Mark Haiman, English synopsises of most of EGA I and II

There is an ongoing open-source project to translate EGA into English:

• English translation of EGA I, and partial translations of the rest

Daniel Murfet has written notes for parts of EGA, see

• Daniel Murfet, The Rising Sea

Project of classification and transcription

• SGA, EGA, FGA By Mateo Carmona

For an essay on content of EGA (about 50 years after the publication) see

• U. Görtz, Classics revisited: Éléments de Géométrie Algébrique, Jahresbericht der Deutschen Mathematiker-Vereinigung 120, 235–290 (2018) doi

category: reference