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. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... Jean-Alexandre-Eugène Dieudonné (July 1, 1906 - November 29, 1992) was a French mathematician, known 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 mathematics, particularly in... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Fascicles are sections of a book, usually a reference work, that because of its length, is issued in parts so that the information may be made available to the public as soon as possible rather than waiting years or decades to complete the entire work. ... 1960 was a leap year starting on Friday (link will take you to calendar). ... 1967 was a common year starting on Sunday (the link is to a full 1967 calendar). ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

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] (http://www.math.jussieu.fr/~leila/mathtexts.php). In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn. ...

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. The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...

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. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... In mathematics, there have been many attempts down the centuries to unify the whole subject. ...

External link

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