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 work done starting from 1960 and centred on the IHES near Paris (the official title was the seminar of Bois Marie, the small wood on the estate in Bures-sur-Yvette where the IHES 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 subdivisions of the series were these:

• SGA1 Theory of the fundamental group in algebraic geometry, deformation theory
• SGA2 Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux
• SGA3 Group schemes
• SGA4 Topos theory and etale cohomology
• SGA4.5 (Interpolation later) Shortened proofs of some of the main results on étale cohomology
• SGA5 Cohomologie l-adique et fonctions L
• SGA6 Théorie des intersections et théorème de Riemann-Roch Intersection theory, Riemann-Roch theorem (Lecture Notes in Mathematics 225, 1971).
• SGA7 Groupes de monodromie en géometrie algébrique Monodromy groups (Lecture Notes in Mathematics 288, 340, 1972/3).