Grothendieck's Séminaire de géométrie algébrique

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 the work done in 1960–1969 in the eponymous seminar held at the IHÉS near Paris (the seminar was also known as Séminaire de géométrie algébrique du Bois Marie after the small wood on the estate in Bures-sur-Yvette where the IHÉS is located as of 1962). The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series.

tyle

The material has a reputation of being 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 makes heavy use of category theory. Moreover, an attempt was made to achieve maximally general statements, while assuming that the reader is aware of the motivations and concrete examples.

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.

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 late 60's and early 70's, the original seminar notes were comprehensively revised and rewritten to take into account later developments. In addition, a new volume, SGA 4½, was compiled by Pierre Deligne and published in 1977; it contains simplified and new results by Deligne within the scope of SGA4 as well as some material from SGA5, which had not yet appeared at that time. The revised notes, except for SGA2, were published by Springer in its "Lecture Notes in Mathematics" series.

After a dispute with Springer, Grothendieck refused the permission for reprints of the series. While these later revisions were more widely distributed than the original fascicles, they are still uncommon outside of libraries.

References to SGA typically mean the later, revised editions and not the original fascicles; some of the originals were labelled by capital letters, thus for example S.G.A.D. = SGA3 and S.G.A.A. = SGA4.

eries titles

The volumes of the SGA series are the following:

*SGA1 "Revêtements étales et groupe fondamental, 1960–1961" (Etale coverings and the fundamental group), Lecture Notes in Mathematics 224, 1971
*SGA2 "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, 1961–1962" (Local cohomology of coherent sheaves and global and local Lefschetz theorems), North Holland 1968
*SGA3 "Schémas en groupes, 1962–1964" (Group schemes), Lecture Notes in Mathematics 151, 152 and 153, 1970
*SGA4 "Théorie des topos et cohomologie étale des schémas, 1963–1964" (Topos theory and etale cohomology), Lecture Notes in Mathematics 269, 270 and 305, 1972/3
*SGA4½ "Cohomologie étale" (Etale cohomology), Lecture Notes in Mathematics 569, 1977
*SGA5 "Cohomologie l-adique et fonctions L, 1965–1966" ("l"-adic cohomology and "L"-functions), Lecture Notes in Mathematics 589, 1977
*SGA6 "Théorie des intersections et théorème de Riemann-Roch, 1966–1967" (Intersection theory and the Riemann-Roch theorem) (Lecture Notes in Mathematics 225, 1971), Lecture Notes in Mathematics 225, 1971
*SGA7 "Groupes de monodromie en géometrie algébrique, 1967–1969" (Monodromy groups in algebraic geometry), Lecture Notes in Mathematics 288 and 340, 1972/3.

SGA 4½ does not correspond to any of the actual seminars. It is a compilation by Pierre Deligne of some survey articles, new results within the scope of SGA4, and finally material from SGA5. The publication of SGA5 was delayed until 1977.

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. 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/).

Bibliographic information

*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Michèle Raynaud
title = Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Documents Mathématiques 3)
origyear = 1971
edition = Updated edition
year = 2003
publisher = Société Mathématique de France
language = French
pages = xviii+327
id = ISBN 2-85629-141-4
*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
title = Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1) (Lecture notes in mathematics 224)
year=1971
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = xxii+447
id = ISBN 3540056149
*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Michèle Raynaud
title = 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)
origyear = 1968
edition = Updated edition
year = 2005
publisher = Société Mathématique de France
language = French
pages = x+208
id = ISBN 2-85629-169-4
*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
title = 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)
year = 1968
publisher = North-Holland Publishing Company
location = Amsterdam
language = French
pages = vii+287
*cite book
last = Demazure
first = Michel
authorlink = Michel Demazure
coauthors = Alexandre Grothendieck, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics 151)
year = 1970
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = xv+564
*cite book
last = Demazure
first = Michel
authorlink = Michel Demazure
coauthors = Alexandre Grothendieck, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 2 (Lecture notes in mathematics 152)
year = 1970
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = ix+654
*cite book
last = Demazure
first = Michel
authorlink = Michel Demazure
coauthors = Alexandre Grothendieck, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 3 (Lecture notes in mathematics 153)
year = 1970
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = vii+529
*cite book
first = Michael
last = Artin
authorlink = Michael Artin
coauthors = Alexandre Grothendieck, Jean-Louis Verdier, eds.
title = 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)
year = 1972
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = xix+525
*cite book
first = Michael
last = Artin
authorlink = Michael Artin
coauthors = Alexandre Grothendieck, Jean-Louis Verdier, eds.
title = 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)
year = 1972
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = iv+418
*cite book
first = Michael
last = Artin
authorlink = Michael Artin
coauthors = Alexandre Grothendieck, Jean-Louis Verdier, eds.
title = 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)
year = 1972
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = vi+640
*cite book
first = Pierre
last = Deligne
authorlink = Pierre Deligne
coauthors = ed.
title = Séminaire de Géométrie Algébrique du Bois Marie - Cohomologie étale - (SGA 4½) (Lecture notes in mathematics "569")
year = 1977
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = iv+312
*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
title = Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5) (Lecture notes in mathematics 589)
year = 1977
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = xii+484
id = ISBN 3540082484
*cite book
last = Berthelot
first = Pierre
authorlink = Pierre Berthelot (mathematician)
coauthors = Alexandre Grothendieck, Luc Illusie, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225)
year = 1971
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = xii+700
*cite book
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
title = Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1 (Lecture notes in mathematics 288)
year = 1972
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = viii+523
*cite book
last = Deligne
first = Pierre
authorlink = Pierre Deligne
coauthors = Nicholas Katz, eds.
title = Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 2 (Lecture notes in mathematics 340)
year = 1973
publisher = Springer-Verlag
location = Berlin; New York
language = French
pages = x+438

External links

* [http://modular.fas.harvard.edu/sga/sga/ Scanned version of SGA] (1, 2, 3, 4, 4½, 5, 6 and 7)
* [http://www.arxiv.org/abs/math.AG/0206203 Re-edited version of SGA1]
* [http://www.math.polytechnique.fr/~laszlo/sga2/sga2-smf.pdf Re-edited version of SGA2 -- Laszlo's page]
* [http://www.arxiv.org/abs/math.AG/0511279 Re-edited version of SGA2 -- in the arXiv]