Eléments de géométrie algébrique

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The Éléments de géométrie algébrique ("Elements of Algebraic Geometry ", EGA for short ) by Alexander Grothendieck (with the help of Jean Dieudonné ) are an approximately 1,800-page unfinished treatise on algebraic geometry, which was divided into eight parts (fascicles) gradually between 1960 and was published in 1967. Grothendieck tried to systematically rewrite algebraic geometry by reversing the then common method of changing from a variety to a ring , and constructing a variety from a ring . This is the basic idea behind the construction of the so-called schemes .

Contents of the published chapters:

  • I. Le langage des schémas ("The language of the schemes").
  • II. Étude global élémentaire de quelques classes de morphismes ("Global properties of some classes of morphisms").
  • III. Étude cohomologique des faisceaux cohérents ("Study of the coherence of coherent sheaves"). In two volumes.
  • IV. Étude locale des schémas et des morphismes de schémas (“Local properties of schemas and schema morphisms”). In four volumes.

Initially, 13 chapters were planned (as in Euclid's Elements , which also included 13 chapters and became a basic structure for mathematics par excellence). Originally he wanted to plan sections on the fundamental group, residuals, cuts, category theory and a little homotopy, and according to Grothendieck's optimistic assessment, three to four years were estimated. Grothendieck worked on some of the topics with his students in the Séminaire de géométrie algébrique du Bois Marie and elsewhere (for example, a book on residuals and duality by Robin Hartshorne was created from a correspondence with Grothendieck).

The titles of the other planned but not realized chapters were:

  • V. Procédés élémentaires de construction des schémas.
  • VI. Techniques de descente. Méthode générale de construction des schémas.
  • VII. Schémas de groupes, espaces fibrés principaux.
  • VIII. Étude différentielle des espaces fibers.
  • IX. Le groupe fondamental.
  • X. Résidus et dualité.
  • XI. Théories d'intersection, classes de Chern, théorème de Riemann-Roch.
  • XII. Schémas abéliens et schémas de Picard.
  • XIII. Cohomology de Weil.

There is also a chapter 0 with preliminaries, which was spread over different chapters.

The first draft mostly came from Grothendieck and was worked out by Dieudonné. Within his strictly regulated training schedule, Grothendieck worked regularly on EGA in the 1960s. According to Luc Illusie , the main part of Dieudonné was the section in EGA IV differential calculus in positive characteristics with complete local rings. He also sees EGA primarily as a reference work, but it is indispensable for students, as it contains details that are otherwise not found in the literature. The effort to explain apparently "trivial" facts precisely and completely was typical for Grothendieck and also led to the many backlinks in the book - after a malicious remark by Illusie, this also served to make the content understandable to Dieudonné. Dieudonné had come to IHES with Grothendieck in 1959, whereby its director Motchane wanted him above all, Dieudonné insisted on Grothendieck, because at that time it was already clear that Grothendieck was the rising “star” of algebraic geometry at that time, with a place for them Elaboration of his monumental program of the re-establishment of the algebraic geometry (at a university this was not so easy, since he was stateless and uncompromisingly represented his views). Both also knew each other from Nicolas Bourbaki's circle , in which Dieudonné also mostly took over the final editing. Grothendieck left Bourbaki prematurely in 1960 because he wanted to pursue his own program. The EGA title is based on the Éléments de Mathématique by Nicolas Bourbaki.

In the early 2000s, IHES was still selling over a hundred copies a year.

Grothendieck himself reports on the creation in his Récoltes et Sémailles from 1986.

literature

  • A. Grothendieck, J. Dieudonné: Éléments de géométrie algébrique. Publications mathématiques de l'IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960–1967)
A scanned copy can be found in the NUMDAM archive:
    • Volume I , pub. Math. IHES 4, 1960, pp. 4-228, Chapter 0, 1-7, Chapter I, 1-10
    • Volume II , pub. Math. IHES 8, 1961, pp. 5-22, Chapter II, 1-8
    • Volume III, 1 , pub. Math. IHES 11, 1961, pp. 5-167, Chapter 0, 8-13, Chapter III, 1-5
    • Volume III, 2 , pub. Math. IHES 17, 1963, pp. 5–91, Chapter III, 6-7,
    • Volume IV, 1 , pub. Math. IHES 20, 1964, pp. 5-259, Chapter 0, 14-23, Chapter IV, 1
    • Volume IV, 2 , pub. Math. IHES 24, 1965, pp. 5-231, Chapter IV, 2-7
    • Volume IV, 3 , pub. Math. IHES 28, 1966, pp. 5-255, Chapter IV, 8-15
    • Volume IV, 4 , Publ Math. IHES 32, 1967, pp. 5-361, Chapter IV, 16-21
  • A. Grothendieck, J. Dieudonné: éléments de géométrie algébrique I . Basics of mathematical Sciences. Springer-Verlag, 1971, ISBN 3-540-05113-9
Second, revised edition of the first chapter. The structure differs from that of the first edition, and schemes are no longer assumed to be separate ; the word pre-schema is omitted.
  • Luc Illusie et al. a., Reminiscences of Grothendieck and his school, Notices AMS, October 2010, digitized

Web links

Individual evidence

  1. ↑ Allyn Jackson, Comme de Lachapelle Néant, Notices AMS, Oct. 2004, p 1050. He quotes Grothendieck according to a letter to Jean-Pierre Serre in August 1959th
  2. ↑ In Mathematical Reviews the first volume was listed without Dieudonné, but Grothendieck and Dieudonné always quote themselves together, and the 1971 edition also lists both. Dino Lorenzini, How should one cite the Èlèments de Géometrie Algébrique?, Pdf
  3. ^ Illusie, Notices AMS, October 2010, p. 1115