# Has the mathematical content of Grothendieck's “Récoltes et Semailles” been used?

This question is partly motivated by this one.

## Motivation

Grothendieck's "Récoltes et Semailles" has been cited on various occasions on this forum. See for instance the answers to this question or this one. However, these citations reflect only one aspect of "Récoltes et Semailles", namely the nontechnical reflexion about Mathematics and mathematical activity. Putting aside the wonderful "Clef du Yin et du Yang", whis is a great reading almost unrelated to Mathematics, I remember reading in "Récoltes et Semailles" a bunch of technical mathematical reflexions, almost all of which were above my head due to my having but a smattering of algebraic geometry. However, I recall for instance reading Grothendieck's opinion that standard conjectures were false, and claiming he had in mind a few related conjectures (which he doesn't state precisely) which might turn out to be the right ones. I still don't even know what the standard conjectures state and thus didn't understand anything, but I know many people are working hard to prove these conjectures. I've thus often wondered what was the value of Grothendieck's mathematical statements (which are not limited to standard conjectures) in "Récoltes et Semailles".

The questions I'd like to ask here are the following:

Have the mathematical parts of "Récoltes et Semailles" proved influential? If so, is there any written evidence of it, or any account of the development of the mathematical ideas that Grothendieck has expressed in this text? If the answer to the first question is negative, what are the difficulties involved in implementing Grothendieck's ideas?

## Idle thoughts

In the latter case, I could come up with some possible explanations:
1. Those who could have developed and spread these ideas didn't read "Récoltes et Semailles" seriously and thus nobody was aware of their existence.
2. Those people took the mathematical content seriously but it was beyond anyone's reach to understand what Grothendieck was trying to get at because of the idiosyncratic writing style.
Should one of these two suppositions be backed by evidence, I'd appreciate a factual answer.
3. The ideas were already outdated or have been proven wrong.
If this is the case, I'd appreciate a reference.

## Epanorthosis

Given that "Pursuing Stacks" and "Les Dérivateurs" were written approximately in 1983 and 1990 respectively and have proved influential (see Maltsiniotis's page for the latter text, somewhat less known), I would be surprised should Grothendieck's mathematical ideas expressed around 1985 be worthless.

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4. Récoltes et Semailles is written in a French far beyound the reach of an average mathematician, who may be able to understand a proof but not a heuristical treatise with philosophical and spiritual undercurrents. –  darij grinberg Dec 8 '10 at 10:52
I am still not sure I interpret you question correctly, so I will refrain from editing it myself. But a few more changes possible: (a) the word "influential" is, to a certain extent, subjective. For a specialist in PDEs, certainly the answer is a resounding no. If you are seeking reference request as Thierry interprets, then it would be much better to use a neutral word like "implemented" or "used". That way the question becomes purely factual. (b) If it is a reference request, please tag it as such. –  Willie Wong Dec 8 '10 at 12:53
In view of your comment, I'd add (c) a better way to ask your third question is "If the answer to the first is negative, what are the mathematical difficulties involved in implementing G's ideas?" That was the question is targeted and no-longer open to Idle Speculation. (You may also want to remove or rephrase that section in your question.) –  Willie Wong Dec 8 '10 at 12:56
Perhaps a natural pre-question is "What are the mathematical ideas in R&S that do not appear elsewhere in Grothendieck's work?" –  j.c. Dec 9 '10 at 14:43
Dear jc: To answer this question, one would have to have read not only "Pursuing Stacks" and "Les Dérivateurs", but also the letters Grothendieck have sent to various mathematicians and which may have influenced them. Therefore I thought I had better ask the question the way I did, but sure enough I wish someone could answer yours. –  Jonathan Chiche Dec 9 '10 at 21:19

I would dare to say that yes, R&S has proved influential in the mathematical sense. At least it made Grothendieck's "Esquisse d'un programme" more visible and it is clear that the topics there like anabelian geometry or the new foundations for homotopical algebra have been two avenues of research of great interest recently. As for "tame topology" my impression is that he topic has not taken off, but I may be wrong about this.[Edit: I am wrong: see Thierry Zell comment after this.]

Also it is clear that motives have won a renewed interest since the 90's and the importance of his visions about this (though perhaps not specific details) is amply explained in R&S.

On another front he has expressed the interest in D-modules as a central topic in the cohomology of algebraic varieties together with the philosophy of "six operations" and "cohomological coefficients" that has produced a lot of results and extensions, including, for instance, $p$-adic and logarithmic versions.

A topic that perhaps has not been so intensely pursued is his point of view on the cohomology of singular spaces. According to R&S, there should be a theory o crystals and a theory of co-crystals over any (reasonable) scheme. With smoothness assumptions (over a regular base, say) they should agree (a sort of "Poincaré duality") but on the general case there should be a relationship (related to the nature of singularity). This ideas are presented in a series of footnotes in the 4th part of R&S.

It seems to me that this line of research has not been pursued, mainly for two reasons. Grothendieck himself expressed the possibility of using resolution of singularity and simplicial techniques (or variants) to study the cohomology of a singular variety reducing it to its resolution and resolution of certain open subsets. This was accomplished successfully by Deligne in his "Theorie de Hodge". However the lack of advance in the characteristic $p$ case gives sense to R&S approach, but it seems that mathematicians have other priorities. On the other hand, the big panoply of new objects (algebraic spaces, stacks, derived algebro-geometric objects) possibly has drained people from working on this questions.

Another topic from R&S that has not been addressed is: What is the correct definition of D-module (or crystal) over $\mathrm{Spec}(\mathbb{Z})$? I have no doubt that this is a really hard question to tackle. The advances so far have been small and using a great deal of machinery, I am thinking on the various generalizations of De Rham-Witt theory to mixed characteristic situations.

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"As for "tame topology" my impression is that he topic has not taken off". Some of the inspirations for the theory of o-minimal structures can be traced back to Grothendieck's ideas; early trailblazers like Lou van den Dries and Angus MacIntyre are on the record about this. Now, is the theory developing the way Grothendieck envisioned, that's another story, but the topic has definitely taken off. –  Thierry Zell Dec 9 '10 at 14:03
Thanks Thiery for correcting my misconception. It would be interesting to know Grothendieck's ideas in greater detail. No doubt the emphasis would be in scheme-like categorical constructions and "devissage" style properties. Are you familiar with the work of his students from Montpellier, such as Ladegallerie, Magloire and others? –  Leo Alonso Dec 9 '10 at 15:20
Leo: I am not familiar with the Montpellier people. To expand on my last comment, the theory of o-minimal structures has been very successful, especially in the constructions of new and unexpectedly tame structures (and applications). But the transfer results between tame theories that are sketched in "Esquisse" are not really there yet, and I'm not sure if it's because it's too early or because it's not really what researchers in the field are after. (Moreover, there are some negative results, since there is no largest tame category available, but even partial results don't seem here yet.) –  Thierry Zell Dec 9 '10 at 17:18
Dear Leo: Thank you very much for this informative answer, which makes clear that some mathematicians have studied the mathematical part in R&S. Given that there are some texts developing these ideas, do you know if any of them explicitly mentions R&S as a mathematical source? As regards the development of homotopical algebra, I'd rather incline to think that it stems from "Pursuing Stacks" and "Les Dérivateurs" rather than R&S, since I've never come across R&S as a reference in texts developing homotopical algebra à la Grothendieck. But I've come across a very small portion of them only. –  Jonathan Chiche Dec 9 '10 at 21:38
Jonathan: There are a few transcriptions of parts of R&S at the beginning of some papers about motives, like a leiv-motiv. I vaguely recall also to have spotted R&S in some bibliographies but I can't recall any example now. The controversial aspect of Grothendieck's opinions has had the effect of making the math in R&S less visible and people less inclined to cite it in their publications. –  Leo Alonso Dec 10 '10 at 10:45

Begging your pardon for indulging in some personal history (perhaps personal propaganda), I will explain how I ended up applying R\'ecollte et Semaille. I do apologize in advance for interpreting the question in such a self-centered fashion!

I didn't come anywhere near to reading the whole thing, but I did spend many hours dipping into various portions while I was a graduate student. Serge Lang had put his copy into the mathematics library at Yale, a very cozy place then for hiding among the shelves and getting lost in thoughts or words. Even the bits I read of course were hard going. However, one thing was clear even to my superficial understanding: Grothendieck, at that point, was dissatisfied with motives. Even though I wasn't knowledgeable enough to have an opinion about the social commentary in the book, I did wonder quite a bit if some of the discontent could have a purely mathematical source.

A clue came shortly afterwards, when I heard from Faltings Grothendieck's ideas on anabelian geometry. I still recall my initial reaction to the section conjecture: `Surely there are more splittings than points!' to which Faltings replied with a characteristically brief question:' Why?' Now I don't remember if it's in R&S as well, but I did read somewhere or hear from someone that Grothendieck had been somewhat pleased that the proof of the Mordell conjecture came from outside of the French school. Again, I have no opinion about the social aspect of such a sentiment (assuming the story true), but it is interesting to speculate on the mathematical context.

There were in Orsay and Paris some tremendously powerful people in arithmetic geometry. Szpiro, meanwhile, had a very lively interest in the Mordell conjecture, as you can see from his writings and seminars in the late 70's and early 80's. But somehow, the whole thing didn't come together. One suspects that the habits of the Grothendieck school, whereby the six operations had to be established first in every situation where a problem seemed worth solving, could be enormously helpful in some situations, and limiting in some others. In fact, my impression is that Grothendieck's discussion of the operations in R&S has an ironical tinge. [This could well be a misunderstanding due to faulty French or faulty memory.] Years later, I had an informative conversation with Jim McClure at Purdue on the demise of sheaf theory in topology. [The situation has changed since then.] But already in the 80's, I did come to realize that the motivic machinery didn't fit in very well with homotopy theory.

To summarize, I'm suggesting that the mathematical content of Grothendieck's strong objection to motives was inextricably linked with his ideas on homotopy theory as appeared in 'Pursuing Stacks' and the anabelian letter to Faltings, and catalyzed by his realization that the motivic philosophy had been of limited use (maybe even a bit of an obstruction) in the proof of the Mordell conjecture. More precisely, motives were inadequate for the study of points (the most basic maps between schemes!) in any non-abelian setting, but Faltings' pragmatic approach using all kinds of Archimedean techniques may not have been quite Grothendieck's style either. Hence, arithmetic homotopy theory.

Correct or not, this overall impression was what I came away with from the reading of R&S and my conversations with Faltings, and it became quite natural to start thinking about a workable approach to Diophantine geometry that used homotopy groups. Since I'm rather afraid of extremes, it was pleasant to find out eventually that one had to go back and find some middle ground between the anabelian and motivic philosophies to get definite results.

This is perhaps mostly a story about inspiration and inference, but I can't help feeling like I did apply R&S in some small way. (For a bit of an update, see my paper with Coates here.)

Added, 14 December: I've thought about this question on and off since posting, and now I'm quite curious about the bit of R&S I was referring to, but I no longer have access to the book. So I wonder if someone knowledgeable could be troubled to give a brief summary of what it is Grothendieck really says there about the six operations. I do remember there was a lot, and this is a question of mathematical interest.

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Thanks for the story and the links! I too sense an ironic undertone in R&S-parts on the six operations. Would it be ok to say that the motive of anabelian studies is to see, how far "arithmetics" and "topology" coincide? In view of the speculations on "fractional motives", which pop up sometimes as maybe-spaces with non-integer dimensions: Could something anabelian/homotopic connect with that? –  Thomas Riepe Dec 10 '10 at 17:30
I can't say I understand much of this, but I really enjoyed it. Thanks for posting it. –  Deane Yang Dec 10 '10 at 22:34
Deane: I'm glad you enjoyed it. Happy holidays and all that to you and your family. Thomas: I don't know much about fractional motives. I wouldn't quite put matters the way you did, but I have no real objection to that formulation either. –  Minhyong Kim Dec 11 '10 at 12:19
Dear Minhyong: I don't know much of the underlying Mathematics either but I enjoyed it too, thanks a lot for sharing your experience. Now I've got two answers quite different in nature and in tone and since I'm equally pleased with both of them perhaps I should just let them benefit the community and don't decide to officially accept one of them. –  Jonathan Chiche Dec 11 '10 at 14:18
I honestly liked it! Mathematics is not just about the problems, it is also about the (eventual) solutions, and the ones who solve them of course! And the hierarchy goes on... solutions lead to more problems... until such time that all complexities are simplified... –  Jose Arnaldo Dris Dec 11 '10 at 14:39