Grothendieck's Séminaire de géométrie algébrique

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In mathematics, Alexander Grothendieck's Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois Marie, the small wood on the estate in Bures-sur-Yvette where the IHÉS is located). The seminar notes were eventually published in twelve volumes, almost all in the Springer Lecture Notes in Mathematics series.

1 Style
2 Lack of peer review
3 First publication
4 Series scope
5 Re-publishing SGA
6 Bibliographic information
7 External links

[edit] Style

The material is hard to read, for a number of reasons. More elementary or foundational parts were relegated to the EGA series of Grothendieck and Jean Dieudonné, causing long strings of logical dependencies in the statements. The style is very abstract and assumes that intensive use of category theory ideas is within the reader's comfort zone. An attempt was made to achieve very general statements (in particular to remove finiteness conditions, such as Noetherian hypotheses, considered 'parasitic').

The geometric motivations were known to the participants, certainly, but are not easy to connect to the words on the page. Overall, innovation was taking place on a grand scale, but only the experts could see how to localise it and apply it to problem solving.

[edit] Lack of peer review

The material was not refereed in the conventional sense. This led to a discreet controversy, after the ultimate proof of the Weil conjectures was completed by Pierre Deligne. He received a Fields Medal, but only after a delay of one occasion; it was argued in the IMU committee that the proof depended on material in SGA7 that had not been subject to the normal peer review process.

[edit] First publication

The original notes to SGA were published in fascicles by the IHÉS, most of which went through two or three revisions. These were published as the seminar proceeded, beginning in the early 60's and continuing through most of the decade. They can still be found in large math libraries, but distribution was limited. In the early 70's, the original seminar notes were comprehensively revised and rewritten to take into account later developments. SGA 4½ was newly written for this revision; it contains significant simplifications of many of the proofs in SGA 4. The revised notes were then published, mostly by Springer in its Lecture Notes in Mathematics series.

As has been his habit, Grothendieck refused permission for republication. While these later revisions were more widely distributed than the original fascicles, they are still uncommon outside of libraries.

Usually "SGA" is taken to refer to the later, revised editions and not to the original fascicles.

[edit] Series scope

The subdivisions of the series were these:

SGA1 Revêtements étales et groupe fondamental (Etale coverings and the fundamental group)
SGA2 Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (Local cohomology of coherent sheaves and global and local Lefschetz theorems)
SGA3 Schémas en groupes (Group schemes)
SGA4 Théorie des topos et cohomologie étale des schémas (Topos theory and etale cohomology)
SGA4½ Cohomologie étale (Etale cohomology)
SGA5 Cohomologie l-adique et fonctions L (l-adic cohomology and L-functions)
SGA6 Théorie des intersections et théorème de Riemann-Roch (Intersection theory and the Riemann-Roch theorem) (Lecture Notes in Mathematics 225, 1971).
SGA7 Groupes de monodromie en géometrie algébrique (Monodromy groups in algebraic geometry) (Lecture Notes in Mathematics 288, 340, 1972/3).

[edit] Re-publishing SGA

In the 1990s it became obvious that the lack of availability of the SGA was becoming more and more of a problem to researchers and graduate students in algebraic geometry: not only are the copies in book form too few for the growing number of researchers, but they are also difficult to read because of the way they are typeset (on an electric typewriter, with mathematical formulae written by hand). Thus, under the impetus of various mathematicians from several countries, a project was formed of re-publishing SGA in a more widely-available electronic format and using LaTeX for typesetting; also, various notes are to be added to correct for minor mistakes or obscurities. The result should be published by the Société Mathématique de France. Legal permission to reprint the works was obtained from every author except Alexander Grothendieck himself, who cannot be contacted; it was decided to proceed without his explicit agreement on the grounds that his refusal for the SGA to be re-published by Springer-Verlag was an objection against Springer and not one of principle.

As a first step, the entire work was scanned and made available on-line (see the links section below) by Frank Calegari, Jim Borger and William Stein of Harvard University. The job of typesetting the text anew and proofreading it was then distributed among dozens of volunteers (most of them junior French mathematicians, because of the required fluency in French and knowledge of algebraic geometry), starting with SGA1 in late 2001.

The coordinating editor for the work on SGA1 was Bas Edixhoven from the University of Rennes: the first version was available on the arXiv.org e-print archive on June 20, 2002, and the proof-read version was uploaded on January 4, 2004, and later published in book form by the Société Mathématique de France. Work on SGA2 was started in 2004 with Yves Laszlo as coordinating editor. The LaTeX source file is available on the arXiv.org e-print archive; SGA2 appeared in print in late 2005 by the Société Mathématique de France (see http://smf.emath.fr/Publications/DocumentsMathematiques/).

[edit] Bibliographic information

Grothendieck, Alexandre; Michèle Raynaud [1971] (2003). Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3), Updated edition (in French), Société Mathématique de France, xviii+327. ISBN 2-85629-141-4.
Grothendieck, Alexandre (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Lecture notes in mathematics 224) (in French). Berlin; New York: Springer-Verlag, xxii+447. ISBN 3540056149.
Grothendieck, Alexandre; Michèle Raynaud [1968] (2005). Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2) (Documents Mathématiques 4), Updated edition (in French), Société Mathématique de France, x+208. ISBN 2-85629-169-4.
Grothendieck, Alexandre (1968). Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2) (Advanced Studies in Pure Mathematics 2) (in French). Amsterdam: North-Holland Publishing Company, vii+287.
Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics 151) (in French). Berlin; New York: Springer-Verlag, xv+564.
Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 2 (Lecture notes in mathematics 152) (in French). Berlin; New York: Springer-Verlag, ix+654.
Demazure, Michel; Alexandre Grothendieck, eds. (1970). Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 3 (Lecture notes in mathematics 153) (in French). Berlin; New York: Springer-Verlag, vii+529.
Artin, Michael; Alexandre Grothendieck, Jean-Louis Verdier, eds. (1972). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag, xix+525.
Artin, Michael; Alexandre Grothendieck, Jean-Louis Verdier, eds. (1972). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 2 (Lecture notes in mathematics 270) (in French). Berlin; New York: Springer-Verlag, iv+418.
Artin, Michael; Alexandre Grothendieck, Jean-Louis Verdier, eds. (1972). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 3 (Lecture notes in mathematics 305) (in French). Berlin; New York: Springer-Verlag, vi+640.
Deligne, Pierre; ed. (1977). Séminaire de Géométrie Algébrique du Bois Marie - Cohomologie étale - (SGA 4½) (Lecture notes in mathematics 569) (in French). Berlin; New York: Springer-Verlag, iv+312.
Grothendieck, Alexandre (1977). Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5) (Lecture notes in mathematics 589) (in French). Berlin; New York: Springer-Verlag, xii+484. ISBN 3540082484.
Berthelot, Pierre; Alexandre Grothendieck, Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag, xii+700.
Grothendieck, Alexandre (1972). Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1 (Lecture notes in mathematics 288) (in French). Berlin; New York: Springer-Verlag, viii+523.
Deligne, Pierre; Nicholas Katz, eds. (1973). Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 2 (Lecture notes in mathematics 340) (in French). Berlin; New York: Springer-Verlag, x+438.