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 attempted to establish 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. Some 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. Grothendieck's incomplete notes on EGA V can be found at [1].

Grothendieck later 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 prescheme", and heavily emphasizes the use of representable functors. Grothendieck never gave permission for this volume to be republished, so copies are very rare. Nevertheless, it may be found in many libraries.

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.

External link

A scanned copy of the EGA can be found at the NUMDAM archive, under "Publications mathématiques de l'IHÉS" (volumes 4, 8, 11, 17, 20, 24, 28 and 32).