Éléments de géométrie algébrique
Encyclopedia
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.
, 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 functor
s. 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.
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.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
") by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
(assisted by Jean Dieudonné
Jean Dieudonné
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
), or EGA for short, is a rigorous treatise, in French, on algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiques
Institut des Hautes Études Scientifiques
The Institut des Hautes Études Scientifiques is a French institute supporting advanced research in mathematics and theoretical physics...
. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, 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 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 categoriesDerived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
, 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
Springer Science+Business Media
- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functor
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
s. 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 Grassmannian Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological... s, presumably from intended Chapter V of the first 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 Fondements de la Géometrie Algébrique FGA, or Fondements de la Géometrie Algébrique, is a book that collected together seminar notes of Alexander Grothendieck. It is animportant source for his pioneering work on scheme theory, which laid foundations for algebraic geometry in its modern technical developments.The title is a translation... . |
| 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 Fondements de la Géometrie Algébrique FGA, or Fondements de la Géometrie Algébrique, is a book that collected together seminar notes of Alexander Grothendieck. It is animportant source for his pioneering work on scheme theory, which laid foundations for algebraic geometry in its modern technical developments.The title is a translation... . |
| 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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, sheaf theory
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
and general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
to commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
and homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
. 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
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
'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
Unifying theories in mathematics
There have been several attempts in history to reach a unified theory of mathematics. Some of the greatest mathematicians have expressed views that the whole subject should be fitted into one theory.-Historical perspective:...
) 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.
External links
- Scanned copies and partial English translations: Mathematical Texts
- Detailed table of contents: http://www.dma.ens.fr/~madore/ega-toc.pdf