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 around 15 volumes, almost all in the Springer Lecture Notes in Mathematics series.

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.

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.

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-Verlag in its Lecture Notes in Mathematics series. As usual, Grothendieck refused to allow republication, and while these later revisions were more widely distributed than the original fasicules, they are still uncommon outside of libraries. Usually "SGA" refers to the later revised editions and not to the original fascicles.

The subdivisions of the series were these:

• SGA1 Revetements etales 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 Schemas en groups (Group schemes)
• SGA4 Theorie des topos et cohomologie etale des schemas (Topos theory and etale cohomology)
• SGA4½ Cohomologie etale (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).

External links

• Scanned version of SGA (http://modular.fas.harvard.edu/sga/sga/) (1, 2, 3, 4, 4½, 5, 6 and 7)
• Re-typeset version of SGA1 (http://www.arxiv.org/abs/math.AG/0206203)