Fondements de la Géometrie Algébrique (FGA) is a book that collected together seminar notes of Alexander Grothendieck. It is an important source for his pioneering work on scheme theory, which laid foundations for algebraic geometry in its modern technical developments. The title is a translation of the title of André Weil's book Foundations of Algebraic Geometry. It contained material on descent theory, and existence theorems including that for the Hilbert scheme. The Technique de descente et théorèmes d'existence en géometrie algébrique is one series of seminars within FGA.

Like the bulk of Grothendieck's work of the IHÉS period, duplicated notes were circulated, but the publication was not as a conventional book.

Contents

These are Séminaire Bourbaki notes, by number, from the years 1957 to 1962.

• Fondements de la géométrie algébrique. Commentaires [Séminaire Bourbaki, t. 14, 1961/62, Complément];
• Théorème de dualité pour les faisceaux algébriques cohérents [Séminaire Bourbaki, t. 9, 1956/57, no. 149]; (coherent duality)
• Géométrie formelle et géométrie algébrique [Séminaire Bourbaki, t. 11, 1958/59, no. 182]; (formal geometry)
• Technique de descente et théorèmes d'existence en géométrie algébrique. I-VI
  • I. Généralités. Descente par morphismes fidèlement plats [Séminaire Bourbaki, t. 12, 1959/60, no. 190];
  • II. Le théorème d'existence en théorie formelle des modules [Séminaire Bourbaki, t. 12, 1959/60, no. 195];
  • III. Préschémas quotients [Séminaire Bourbaki, t. 13, 1960/61, no. 212];
  • IV. Les schémas de Hilbert [Séminaire Bourbaki, t. 13, 1960/61, no. 221];
  • V. Les schémas de Picard. Théorèmes d'existence [Séminaire Bourbaki, t. 14, 1961/62, no. 232];
  • VI. Les schémas de Picard. Propriétés générales [Séminaire Bourbaki, t. 14, 1961/62, no. 236]

See also

• Éléments de géométrie algébrique
• Séminaire de Géométrie Algébrique du Bois Marie

References

• Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo (2005), Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3541-8, MR 2222646
• Fantechi, Barbara; Göttsche, Lothar (2005), "Local properties and Hilbert schemes of points", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 139–178, MR 2223408
• Grothendieck, Alexander (1962), Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957--1962.] (PDF), Paris: Secrétariat Mathématique, MR 0146040
• Illusie, Luc (2005), "Grothendieck's existence theorem in formal geometry", Fundamental algebraic geometry (PDF), Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 179–233, MR 2223409, archived from the original (PDF) on 2006-12-08, retrieved 2006-09-27
• Kleiman, Steven L. (2005), "The Picard scheme", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 235–321, arXiv:math/0504020, Bibcode:2005math......4020K, MR 2223410
• Nitsure, Nitin (2005), "Construction of Hilbert and Quot schemes", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 105–137, arXiv:math/0504590, Bibcode:2005math......4590N, MR 2223407
• Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 1–104, arXiv:math/0412512, Bibcode:2004math.....12512V, MR 2223406

• SGA, EGA, FGA By Mateo Carmona