Séminaire de géométrie algébrique du Bois Marie

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The Séminaire de géométrie algébrique du Bois Marie (SGA) was an influential mathematical seminar in which Alexander Grothendieck developed his theoretical structures for algebraic geometry and arithmetic geometry from May 1959 to 1969 and which took place at IHES (and initially until 1962 in Paris at the Fondation Thiers).

The name Bois Marie refers to a small forest near the IHES in Bures-sur-Yvette , a Paris suburb.

Active participants in the seminar were, among others, Pierre Berthelot , Luc Illusie , Michael Artin , Jean-Louis Verdier , Michel Raynaud , Michèle Raynaud , Jacob Murre , Michel Demazure , Jean Giraud , Pierre Deligne , Pierre Gabriel , André Neron , Pierre Cartier , Pierre Samuel .

Bulk

The staff and Grothendieck has elaborated lectures at the seminar were beside the éléments de géométrie algébrique (EGA 1960 to 1967) and Grothendieck's contributions in the Séminaire Nicolas Bourbaki (FGA, Fonda Apartments de la Géométrie algébrique, and other), the main source from which the Most mathematicians use modern algebraic and arithmetic geometry (as well as new concepts in homological algebra and topology such as the topos theory) according to the von Grothendieck school (today's standard approach) with the category-theoretical foundation of mathematics, the central concept of the sheaf and the schema (both not originally by Grothendieck, but taken up and further developed by Grothendieck) and learned the various cohomology concepts introduced (Ètale cohomology, crystalline cohomology). During this time Grothendieck also developed the first beginnings of the theory of motives as a hypothetical underlying unifying cohomology theory, but this is not dealt with in the SGA.

In the 1960s under Grothendieck, the seminar made IHES one of the world's major centers for mathematics. The model was the famous seminar of Henri Cartan in Paris, in which Grothendieck attended from 1949 until he left for Nancy. As in his own later seminar, it was a research seminar in which a particular topic was followed for a year and the lectures were written down and published. Many young mathematicians from Paris (and established mathematicians such as Grothendieck's friend Jean-Pierre Serre , Pierre Cartier, Claude Chevalley , Pierre Samuel, Georges Poitou and Jean Dieudonné) were drawn there. Often their dissertations resulted from the seminar work. The doctorates were formally carried out at one of the Paris universities, as the IHES, as a private institute, could not award any degrees. It was also used by foreign mathematicians such as Michael Artin, Hyman Bass , Barry Mazur (1962/63), David Mumford (spring 1968), Jacques Tits (from 1965) and Deligne (from early 1965) from Belgium, John T. Tate , Steven Kleiman , Daniel Quillen (1968/69), Marvin Jay Greenberg and Nicholas Katz (1968/69) ( he interacted with others like Robin Hartshorne through correspondence in the 1960s). The seminar building had large viewing windows and was located in a forest-like park. The discussions were very intense, but Grothendieck conducted them in a very targeted manner. Previous knowledge of the seminar participants was not necessarily required, as Grothendieck patiently instructed promising students on the subject. It didn't bother him that "stupid questions" were asked as long as he saw potential. He gave such a student parts of the seminar notes to work on, then went through them carefully and line by line very carefully for himself (he demanded that only one page was written with a large gap so that he could write his numerous corrections on it) and discussed it Then the result with the processor for one or more invitations to his home. The points of criticism ranged from the use of commas and choice of words to the complete revision of individual sections. At the time of Illusie (1963 to 1965) the seminar took place on Tuesdays from 2.15pm and lasted one and a half hours. Then there was tea. Most of the lectures were by Grothendieck, who also did an enormous amount of work outside of the seminar and divided his time very precisely. He encouraged students to take notes, which he collected from time to time, and issued preliminary drafts for lectures. His lectures were described as very clear and structured, with patient explaining details. Mumford described his blackboard writing as the most elegant, precise and fastest he had ever seen in a lecture.

EGA was written with the experienced textbook author and Bourbakist Jean Dieudonné - both had been at IHES since March 1959 - and was planned as a much more extensive basic project, but then got stuck in the fourth chapter (after 1800 pages). SGA, on the other hand, dealt with the current research of the seminar and in the end came up with over 6100 printed pages.

The first versions were published by IHES during the seminar and were often revised several times. With the exception of Volume 2, all of them subsequently appeared in the Lecture Notes in Mathematics series by Springer Verlag .

A Grothendieck project ( Bas Edixhoven et al.) Began in the 1990s to reissue the volumes, which are often in poor print and typesetting quality and are faulty, some of which are out of print. Originally they were written with an electric typewriter and the formulas used by hand. In the project, the text was set in latex and minor errors were corrected. In addition, the originals were in French and not English, the scientific language that most mathematicians are familiar with. The Société Mathématique de France wanted to print it, but in the end Grothendieck, who had withdrawn completely into a small Pyrenean village in the 1990s, refused to give his consent (after his death it was agreed to bypass this). Scanned versions ( William A. Stein et al.) Were and are accessible online. The newly set volume 1 appeared in 2004, volume 2 in 2005 in the ArXiv preprint archive. The work on Volume 3 (Philippe Gille, Patrick Polo) and Volume 4 (Y. Laszlo) were also completed.

Not all topics of his seminars and lectures were published in the SGA, for example there was Dix Exposés sur la cohomologie des schémas (written down by Kleiman after John Coates gave up after the exact correction procedure usual for Grothendieck was unnerved).

Single volumes

The translation of the title is in brackets, but the volumes also contain various other topics.

  • SGA 1 ( Étale Fundamental Group ): Alexander Grothendieck, Michèle Raynaud: Revêtements étales et groupe fondamental, Documents Mathématiques (Paris) 3, Paris: Société Mathématique de France, and Lecture Notes in Mathematics 447, Springer Verlag 1971 (447 pages), Arxiv ( Editor Bas Edixhoven, notes by Michel Raynaud). The seminar in 1960/61 (still in Paris).
  • SGA 2 (local coherence of coherent sheaves and local and global Lefschetz theorems): Alexander Grothendieck, Michèle Raynaud: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, Documents Mathématiques (Paris), 4, Paris: Société Mathématique de France, and North Holland / Masson 1968 (287 pages), Arxiv (edited by Yves Laszlo)
  • SGA 3 (group schemes): Alexander Grothendieck, Michel Demazure: Séminaire de Géométrie Algébrique du Bois Marie - 1962–64 - Schémas en groupes, Volume 1, Lecture Notes in mathematics 151, Springer Verlag (564 pages), Volume 2, Lecture Notes in Mathematics 152 (654 pages), Volume 3, Lecture Notes in Mathematics 153 (529 pages), 1970. Employees mentioned by name were Michael Artin, José Bertin, Pierre Gabriel, Michel Raynaud, Jean-Pierre Serre. Demazure received his doctorate from 1962 to 1964 at Grothendieck, after which he was in Strasbourg and he also considers SGA 3 to be slightly outside the main line of SGA.
  • SGA 4 ( Topos - Theory and Étale Cohomology of Schemas): Alexander Grothendieck, Michael Artin, Jean-Louis Verdier: Séminaire de Géométrie Algébrique du Bois Marie - 1963–64 - Théorie des topos et cohomologie étale des schémas, Volume 1, Lecture notes in mathematics 269, 1972 (525 pages), Volume 2, Lecture Notes in mathematics 270 (418 pages), Volume 3, Lecture Notes in Mathematics 305 (640 pages), Springer Verlag 1972. Grothendieck had Michael Artin in 1961 at his lectures at the Harvard University and both developed the concept of Ètale cohomology in dialogue. The 1963/64 seminar that they led with Verdier was dedicated to the subject. Pierre Deligne, Bernard Saint-Donat and the fictional Nicholas Bourbaki were mentioned as employees.
  • SGA 4½ (Étale Kohomologie): Pierre Deligne: Séminaire de Géométrie Algébrique du Bois Marie - Cohomologie étale, Lecture Notes in mathematics 569, Springer 1977 (312 pages) In the preface from 1976 Deligne writes that the book is the understanding of L-adic cohomology for The aim is to facilitate non-specialists without resorting to SGA 4 and 5, but after these seminars and later, and it also presents new results. Jean-François Boutot (lecture elaboration from Arcata), Grothendieck, Verdier (from his book on derived categories) and Illusie (from which Deligne presents results) are mentioned by name for contributions. In it Deligne rewrote some of the evidence of the results in SGA 5, which he used for his proof of the Weil conjectures .
  • SGA 5 ( L-adic cohomology and L-functions ): Alexander Grothendieck: Séminaire de Géométrie Algébrique du Bois Marie - 1965–66 - Cohomologie l-adique et Fonctions L, Lecture Notes in mathematics 589, Springer Verlag 1977 (484 pages). Employees mentioned by name were Luc Illusie, who came to the seminar from Paris in 1964 as a topology expert, Jean-Pierre Serre , Ionel Bucur, Christian Houzel , Jean Pierre Joanolou and Pierre Gabriel.
  • SGA 6 (Theory of Sections and theorem by Riemann-Roch ): Alexander Grothendieck, Pierre Berthelot, Luc Illusie: Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in mathematics 225 , Springer Verlag 589 (700 pages) Daniel Ferrand, Jean Pierre Joanolou, Olli Jussila, Steven Kleiman , Michel Raynaud, Jean-Pierre Serre are named.
  • SGA 7 (monodromic groups and algebraic geometry): Alexander Grothendieck: Séminaire de Géométrie Algébrique du Bois Marie - 1967–69 - Groupes de monodromie en géométrie algébrique, Volume 1, Lecture Notes in mathematics 288 (523 pages), Volume 2, by Pierre Deligne , Nicholas Katz, Lecture Notes in Mathematics 340 (438 pages), 1973. Michel Raynaud and Dock Sang Rim are named in volume 1 (Grothendieck was still in charge of this part, but he was still very active in the seminar in 1968/69) , the second part only mentions Deligne and Katz. The latter came to IHES as a post-doc in 1968 and was commissioned by Grothendieck, who then came to IHES about once a week to receive visitors, to give lectures on Lefschetz pencils. In Recolts et Semailles (14.1.3), Grothendieck later strongly criticized this division of SGA 7 (he had the impression Deligne wanted to exclude him from the second part, even though, in his own words, he had also made significant contributions there), and also of SGA 4 1/2 as Delignes new version of SGA 5 (Recolts et Semailles, 02/14/11).

literature

  • Allyn Jackson, Comme Appelé du Néant - As if summoned from the void: the life of Alexandre Grothendieck, Part 1, Notices AMS, October 2004, digitized
  • Luc Illusie et al. a., Reminiscences of Grothendieck and his school, Notices AMS, October 2010, digitized
  • Michael Artin et al. a .: Alexandre Grothendieck 1928–2014, Part 1, Notices AMS, March 2016, digitized
  • Pierre Cartier: Alexander Grothendieck. A country known only by name, Inference, Volume 1, October 15, 2014, online

Web links

References and comments

  1. ↑ He planned a series of lectures in the Bourbaki seminary, but this was rejected as too long
  2. ↑ Description of the collaboration of Luc Illusie in Allyn Jackson, Notices AMS, October 2004
  3. ↑ Luc Illusie, Reminiscences of Grothendieck and his school, Notices AMS, October 2010, p. 1106
  4. ↑ Mumford in: Mumford, Jackson, Artin, Tate (Eds.), Alexandre Grothendieck 1928–2014, Part 1, Notices AMS, April 2016, p. 408
  5. ↑ Grothendieck was also a member, but left Bourbaki in 1960 because of divergent views
  6. ↑ Gille, Polo, SGA III
  7. ↑ See links in nLab to EGA, SGA
  8. ^ Illusie, Notices AMS, October 2010, p. 1110
  9. ↑ More details on this and the background to the individual parts of the SGA can be found in the memoirs of Luc Illusie, Notices AMS, October 2010
  10. ↑ As mentioned by name in SGA 2 as the author of some exposés
  11. ↑ Notices AMS, March 2016, p. 250
  12. ^ Illusie, Notices AMS, October 2010, p. 1109