Grothendieck's Séminaire de géométrie algébrique
Encyclopedia
In mathematics
, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck
. It was a unique phenomenon of research and publication outside of the main mathematical journal
s that ran from 1960 to 1969 at the IHÉS near Paris. (The name came from the small wood on the estate in Bures-sur-Yvette
where the IHÉS was located from 1962.) The seminar notes were eventually published in twelve volumes, all except one in the Springer
Lecture Notes in Mathematics series.
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.
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.
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.
, 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 University of Leiden (at the time 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/).
Laszlo has also edited SGA4 and recently Philippe Gille and Patrick Polo have uploaded TeXed version of SGA3. In January 2010, however, Grothendieck requested that work cease on republishing SGA, and all SGA4 material at Laszlo's page has been removed.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
. It was a unique phenomenon of research and publication outside of the main mathematical journal
Mathematical journal
A mathematics journal is a scientific journal which publishes exclusively mathematics papers. A practical definition of the current state of mathematics, as a research field, is that it consists of theorems with proofs published in a reputable mathematics journal, and which usually have passed...
s that ran from 1960 to 1969 at the IHÉS near Paris. (The name came from the small wood on the estate in Bures-sur-Yvette
Bures-sur-Yvette
Bures-sur-Yvette is a commune in the Essonne department in Île-de-France in northern France.Inhabitants of Bures-sur-Yvette are known as Buressois.-Geography:...
where the IHÉS was located from 1962.) The seminar notes were eventually published in twelve volumes, all except one in the Springer
Springer Science+Business Media
- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
Lecture Notes in Mathematics series.
Style
The material has a reputation of being hard to read for a number of reasons. More elementary or foundational parts were relegated to the EGAÉléments de géométrie algébrique
The Éléments de géométrie algébrique by Alexander Grothendieck , or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published from 1960 through 1967 by the Institut des Hautes Études Scientifiques...
series of Grothendieck and Jean Dieudonné
Jean Dieudonné
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
, causing long strings of logical dependencies in the statements. The style is very abstract and makes heavy use of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. Moreover, an attempt was made to achieve maximally general statements, while assuming that the reader is aware of the motivations and concrete examples.
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 DelignePierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
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
Springer Science+Business Media
- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
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.
Series 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 cohomologyLocal cohomologyIn mathematics, local cohomology is a chapter of homological algebra and sheaf theory introduced into algebraic geometry by Alexander Grothendieck. He developed it in seminars in 1961 at Harvard University, and 1961-2 at IHES. It was later written up as SGA2...
of coherent sheaves and global and local Lefschetz theorems), North Holland 1968 - SGA3 Schémas en groupes, 1962–1964 (Group schemeGroup schemeIn mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
s), Lecture Notes in Mathematics 151, 152 and 153, 1970 - SGA4 Théorie des topos et cohomologie étale des schémas, 1963–1964 (Topos theory and etale cohomologyÉtale cohomologyIn mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
), 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
- SGA7 Groupes de monodromie en géométrie algébrique, 1967–1969 (MonodromyMonodromyIn mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
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
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
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 typewriterIBM Selectric typewriter
The IBM Selectric typewriter was a highly successful model line of electric typewriters introduced by IBM on July 31, 1961.Instead of the "basket" of individual typebars that swung up to strike the ribbon and page in a traditional typewriter, the Selectric had a type element that rotated and...
, 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
LaTeX
LaTeX is a document markup language and document preparation system for the TeX typesetting program. Within the typesetting system, its name is styled as . The term LaTeX refers only to the language in which documents are written, not to the editor used to write those documents. In order to...
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
Société Mathématique de France
The Société Mathématique de France is the main professional society of French mathematicians.The society was founded in 1872 by Émile Lemoine and is one of the oldest mathematical societies in existence...
. 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 University of Leiden (at the time 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
Yves Laszlo
Yves Laszlo is a French mathematician working in the Université Paris-Sud XI. He specializes in algebraic geometry.- External links :*...
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/).
Laszlo has also edited SGA4 and recently Philippe Gille and Patrick Polo have uploaded TeXed version of SGA3. In January 2010, however, Grothendieck requested that work cease on republishing SGA, and all SGA4 material at Laszlo's page has been removed.
External links
- Scanned version of SGA (1, 2, 3, 4, 4½, 5, 6 and 7)