Missing exposés in SGA 5, and the composition of the SGA's

Question

Over the past couple of years I had to look in SGA for various results, and I can't but marvel at how poorly constructed it is. In SGA1 exposé VII "n'existe pas", SGA 1 references higher SGA's, and so forth.

My first, albeit less urgent, question is: how were these composed, and why the many missing exposés and references to future manuscripts? Were all SGA's written simultaneously? Are the missing exposés a product of type-writing it (so changing the table of contents is too much of a bother?). Were they going to write those exposés later but never did? Why didn't they?

My second question is specifically regarding the missing exposés in SGA5. In the introduction to expose XIII in SGA1 it says that it generalizes the results of SGA5 II — one of the missing exposés! What happened there? Is there a more appropriate reference for what SGA1 exposé XIII generalizes?

Answer 1

This answer addresses your second question, regarding the specific case of SGA 5. In short: the contents of SGA 5 Exposé II can be found in Deligne's "SGA 4 1/2". Also, see my other answer about the original Exposé.

According to Grothendieck's "Récoltes et Semailles", SGA 5 was totally butchered by Illusie, in a combined effort with Deligne so that it looked useless in comparison with SGA 4 1/2 (which wasn't a true seminar, and stole some of the missing exposés from SGA 5). This is also the reason why SGA 5 was the last to be published.

About the lost exposés:

• There was a group of introductory exposés about the relations of SGA 5 to other contexts and about the philosophy of six operations. Illusie had them, but he sent them to Grothendieck and I don't know where they ended up. I personally think they are in the Grothendieck Archives (Cote nº33) You can read more about them here.
• Exposé II, as I said earlier, was reworked by Deligne and included in his volume about Étale Cohomology. The original exposé is kept at the IHÉS and can be read online (see my other answer).
• Exposé IV, about "The cohomology class associated with a cycle", was going to be redacted by Deligne, who instead included it in SGA 4 1/2, chapter 4. This theme also included an étale version of homology, with a formalism about the homology class associated to a cycle. According to RéS, these ideas were published by Verdier in an article with the same name. You can read it here: Verdier, Jean-Louis, Classe d’homologie associee à un cycle, Astérisque 36–37, 101–151 (1976). ZBL0346.14005.
• Exposé IX was about Serre-Swan modules and was published elsewhere by Serre (Linear Representations of Finite Groups). This one was published by the IHÉS and can be found in some libraries.
• Exposé XI was called "Computation of local terms" or something like that, and was substituted by exposé III-b. Apparently, Bucur wrote it and sent it to Grothendieck, who somehow lost it.
• There was also a last exposé about open problems, which Grothendieck talks about in RéS and which notably contained a conjectural "discrete Riemann–Roch theorem", later referred to as "Grothendieck-Deligne conjecture" and studied by MacPherson. He also mentioned a trace formula modulo $p$, which was treated by Deligne in SGA 4 1/2.

Answer 2

"It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris.[…] The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series". SGA, Wikipedia.

In regard of SGA1 exposé VII "n'existe pas":

"Ayant à rédiger l'exposé VII du Séminaire de Géométrie Algébrique de l'IHES de 1961, qui devait être consacré au formalisme général de la théorie de la descente, nous avons été conduit, et même poussé, à y inclure un certain nombre de considérations qui n'ont qu'un rapport lointain avec la Géométrie. C'est une des raisons pour lesquelles ce travail est présenté de manière autonome, l'autre étant sa date d'achèvement. La Géométrie Algébrique moderne faisant grand usage des techniques de descente, il est bon qu'elles soient exposées en détail. C'est aussi la raison pour laquelle nous ne donnons que peu d'exemples: on en trouvera dans le séminaire susnommé et dans celui de 1963". Méthode de la descente, J. Giraud.

I see the SGAs as divergent in nature with a complex structure not intended to be a single publication. For example, M. Demazure says "actually SGA3 was a parenthesis and did not really belong to the SGA mainstream" and that could be said with the majority of the SGAs, descent (SGA I) could be re-studied with topos theory (SGA IV) and also Galois theory, multigaloisian topos, stacks…. SGA must be seen as an unfinished work and not (as the inclusion of SGA IV 1/2 suggests) a completed work. A work that requires left aside concepts and approaches for the more natural ones, always searching the harmony. I think what is lost in SGA (due to its titanic dimension) is what Pursuing stacks accomplishes, namely, the process of writing a result that lately will be ameliorated (or "naturalized", "harmonized" …) divergently.

Answer 3

I've found Illusie's redaction of Grothendieck's original notes for SGA 5 Exp II:

Formules de Künneth pour la cohomologie à supports propres, par L. Illusie (d'aprés des papiers secrets de A. Grothendieck).

It's a very heavy file (1.6 GB). You can instead look at each page by clicking on the jpgs of this page.