Excerpts from the Grothendieck-Serre Correspondence

Like many fellow mathematicians, I was very sad to hear the news that Alexander Grothendieck passed away yesterday.  The word “genius” is overused; or rather, does not possess sufficiently fine gradations.  I know quite a few mathematical geniuses, but Grothendieck was a singularity.  His ideas were so original, so profound, and so revolutionary – and he had so many of them! – that I will not even attempt to summarize his contributions to mathematics here.  Rather, I thought that I would share some of my favorite passages from the fascinating Grothendieck-Serre Correspondence, published in a bilingual edition by the AMS and SMF.   They illuminate in brief flashes what made Grothendieck so extraordinary — but also human.  They also illustrate how influential Serre was on Grothendieck’s mathematical development.  Before I begin, here is a quote from another wonderful book, Alexander Grothendieck: A Mathematical Portrait, edited by Leila Schneps:

…the features which constitute in some sense his personal mathematical signature… are very familiar to those who know Grothendieck’s work: the search for maximum generality, the focus on the harmonious aspects of structure, the lack of interest in special cases, the transfer of attention from objects themselves to morphisms between them, and—perhaps most appealingly—Grothendieck’s unique approach to difficulties that consisted in turning them, somehow, upside down, and making them into the actual central point and object of study, an attitude which has the power to subtly change them from annoying obstacles into valuable tools that actually help solve problems and prove theorems… Of course, Grothendieck also possessed tremendous technical prowess, not even to mention a capacity for work that led him to concentrate on mathematics for upward of sixteen hours a day in his prime, but those are not the elements that characterize the magic in his style. Rather, it was the absolute simplicity (in his own words, “nobody before me had stooped so low”) and the total freshness and fearlessness of his vision, seemingly unaffected by long-established views and vantage points, that made Grothendieck who he was.

Excerpts from the correspondence:

Serre to Grothendieck, July 13, 1955

“Your paper on homological Algebra was read carefully and converted everyone (even Dieudonne, who seems to be completely functorised) to your point of view… We would really like you to come to the Bourbaki meeting in October, if possible… I don’t remember exactly what the program is to be… and I don’t think you will find it particularly interesting.  But one is not in Bourbaki for fun, as Dieudonne never stops repeating…”

Grothendieck to Serre, December 15, 1955

“Thinking  a bit about your duality theorem, I notice that its general form is almost obvious, and in fact I just checked that (for a projective space) it is implicitly contained in your theorem giving the $T^q(M)$ in terms of Exts.  (I have the impression, you bastard, that $\S{3}$ and $4$ in your Chap. 3 could be done without any computation.)..”

Serre to Grothendieck, December 22, 1955

“I find your formula $H^{n-p}(X,F)' = {\rm Ext}^p_{\mathcal O}(X,F,\Omega^n)$ very exciting, as I am quite convinced that it is the right way to state the duality theorem in both the analytic case and the algebraic case (the more important one – for me!).”

Grothendieck to Serre, January 12, 1956

“I am glad you liked [my] Ext formula… Unfortunately the role played by projective spaces in all this still seems rather excessive.  I feel like looking into whether one doesn’t get something for ‘regular arithmetic varieties’ which are ‘complete’ (i.e. obtained by gluing together spectra of regular rings).  But to start with, do you have any idea of what complete really means in this context?… I would also like to look at whether one can’t state a duality theorem for projective varieties which may have singularities, and whether a more general and technical statement might not actually be simpler to prove than the one we are considering now.”

Serre to Grothendieck, March 14, 1956

“Thank you for your letter.  I am a bit panic-stricken by this flood of cohomology, but have borne up courageously.  Your spectral sequence seems reasonable to me (I thought I had shown that it was wrong in a special case, but I was mistaken, on the contrary it works remarkably well).”

Grothendieck to Serre, November 13, 1956

“I have read the statements in Weil’s books on abelian varieties, in the hope that the proofs might have been improved since he wrote them; they are really discouraging in Weil, and on top of that his language disgusts me.”

Serre to Grothendieck, November 17, 1956

“On the subject of Weil’s book on abelian varieties: I hardly like its style any more than you do.  But it has to be admitted that certain results in the book (the most important ones) appear to be accessible only by his methods (by which I mean the use of generic points, generic divisors, etc.): this holds in particular for the construction of the Jacobian, which is a little masterpiece of juggling with generic points.  I will be curious to see how Chevalley goes about doing this (assuming he deals with it in his seminar).”

Grothendieck to Serre, November 22, 1956

“I am sad to hear that one cannot present Weil’s results without juggling with generic points; as a matter of fact, the unbridled abuse of generic points necessarily hides the few situations in which their use is truly essential, such as the proof that every endomorphism of the Jacobian comes from a correspondence.  As Chevalley says, one feels frustrated when faced with a proof like that one.”

Grothendieck to Serre, November 5, 1958

“Otherwise, I am working quite a lot on the book [EGA] with Dieudonne.  Everything is going very well for schemes, particularly for all questions of the type: if something is given, or holds… over a generic point, then it holds over a non-empty open set… There is one hitch: I foresee that I will have to redo everything to deal with gluing spectra of complete ${\mathfrak m}$-adic rings (instead of merely spectra of discrete rings).”

Grothendieck to Serre, August 18, 1959

“Tate has written to me about his elliptic curves stuff, and has asked me if I had any ideas for a global definition of analytic varieties over complete valuation fields.  I must admit that I have absolutely not understood why his results might suggest the existence of such a definition, and I remain skeptical.  Nor do I have the impression of having understood his theorem at all; it does nothing more than exhibit, via brute formulas, a certain isomorphism of analytic groups; one could conceive that other equally explicit formulas might give another one which would be no worse than his (until proof to the contrary!)…” [Footnote (MB): Despite the skepticism expressed in this letter, Grothendieck soon changed his mind.  John Tate told me at the 2007 Arizona Winter School that he feels Grothendieck does not get enough credit for foundations of (global) rigid analytic geometry.  In Tate’s foundational paper on the subject, Section 10 begins “In this section we follow fully and faithfully a plan furnished by Grothendieck.”]

Grothendieck to Serre, August 18, 1959

“Next year, I hope to get a satisfactory theory of the fundamental group, and finish up the writing of chapters IV, V, VI, VII (the last one being the fundamental group) at the same time as categories.  In two years, residues, duality, intersections, Chern, and Riemann-Roch.  In three years, Weil cohomology, and a little homotopy, God willing.  In between, I don’t know when, the ‘big existence theorem’ with Picard etc., some algebraic curves, and abelian schemes.  Unless there are unexpected difficulties or I get bogged down, the multiplodocus should be ready in 3 years time, or 4 at the outside.  We will then be able to start doing algebraic geometry!” [Footnote (MB): It is amazing that in 1956 Grothendieck had not yet formulated the notion of a scheme, but by 1959 he was planning out Chapters IV-VII of EGA and beyond.]

Grothendieck to Serre, October 1, 1961

“The mathematical atmosphere at Harvard is absolutely terrific, a real breath of fresh air compared with Paris which is gloomier every year.  There is a good number of intelligent students here, who are beginning to be familiar with the language of schemes, and ask for nothing more than to work on interesting problems, of which there is obviously no lack.  I am even selling (the little I know of) Weil sheaves and Weil cohomology with the greatest of ease, including to Tate, who has just seriously started work on the “global” analytic structures which have been bothering him for two years…  On this topic, it seems more and more obvious to me that the concept of a formal scheme will need to be completely reworked, together with that of an analytic space (or a “rigid-analytic” space as Tate and I call his “global” structures), in order to unite them into a single framework, which has yet to be found.” [Footnote (MB): This unified framework was later worked out by Grothendieck’s former student Raynaud.]

Serre to Grothendieck, October 26, 1961

“[On the subject of] Bourbaki’s Commutative Algebra.  You are very harsh on Valuations!  I persist nonetheless in keeping them, for several reasons, of which the first is practical: $n$ people have sweated over them, there is nothing wrong with the result, and it should not be thrown out without very serious reasons (which you do not have)… Even an unrepentant Noetherian needs discrete valuations and their extensions; in fact, Tate, Dwork, and all the p-adic people will tell you that one cannot restrict oneself to the discrete case and the rank 1 case is indispensible; Noetherian methods then become a burden, and one understands much better if one considers the general case and not only the rank 1 case.”

Grothendieck to Serre, October 31, 1961

“Your argument in favor of valuations is pretty funny… The principle generally respected by Bourbaki is rather that there should be very good reasons for including a huge mess, especially in a central position; the fact that $n$ people have sweated over it is certainly not a good reason, since these $n$ people had no idea of the role of the mess in commutative algebra, but had simply received an order to figure out, Bourbakically, some stuff that they unfortunately did not bother to examine critically as part of the whole… The p-adic analysts do not care any more than the algebraic geometers (or even Zariski himself, I have the impression, as he seems disenchanted with his former loves, who still cause Our Master to swoon) for endless scales and arpeggios on compositions of valuations, baroque ordered groups, full subgroups of the above and whatever.  The scales deserve at most to adorn Bourbaki’s exercise section, as long as no one uses them.”

Grothendieck to Serre, August 16, 1964

“Moreover, this question is related to the following one, which is probably far out of reach.  Let $k$ be a field, which for the sake of argument is algebraically closed, and let $L(k)$ be the ‘K group’ defined by schemes of finite type over $k$ with relations coming from decompositions into pieces (the initial L is of course suggested by the link with L-functions).  Let $M(k)$ be the ‘K group’ defined by ‘motives’ over $k$.  I will say that something is a ‘motive’ over $k$ if it looks like the $\ell$-adic cohomology group of an algebraic scheme over $k$, but is considered as being independent of $\ell$, with its ‘integral structure’, or let us say for the moment its ‘${\mathbf Q}$‘-structure, coming from the theory of algebraic cycles.  The sad truth is that for the moment I do not know how to define the abelian category of motives, even though I am beginning to have a rather precise yoga for this category…”

Serre to Grothendieck, August ? 1964

“I have received your long letter.  Unfortunately I have few (or no) comments to make on the idea of a ‘motive’ and the underlying metaphysics; roughly speaking, I think as you do that zeta functions (or cohomology with Galois action) reflect the scheme one is studying very faithfully.  From there to precise conjectures…”

Grothendieck to Serre, October 30, 1964

“I am glad Tate and Mumford have gotten rid of the screwed-up conjecture on algebraic cycles on abelian varieties [Footnote (Serre): the one that said that the ${\mathbf Q}$-algebra of cycle classes is generated by its degree 1 elements, i.e. by the divisor classes.]; as for the Hodge conjectures, I now hold them to be an article of faith on par with the Tate conjectures; indeed, they are too intimately linked for me to be able to believe that one could be wrong without the other: in other words, they must both be true.”

Serre to Grothendieck, November 8, 1964

“It is difficult to reply to your infinite letters: they would require infinite answers, which presupposes I have understood what you are talking about.  As this is not the case, I will restrict myself to answering the trivial questions…”

Grothendieck to Serre, August 27, 1965

“I am still turning algebraic cycle classes every which way in my head; from a technical point of view, I now see things more clearly, but the final breakthrough is still missing.  As I know you are allergic to cohomology [Footnote (Serre): Allergy, certainly not.  Indigestion, perhaps?], I would like to show you the two key conjectures, which should be proved purely geometrically, i.e. without reference to cohomology.” [Footnote (Serre): This is a reference to the “standard conjectures”.]

Serre to Grothendieck, July 23, 1985

“Thank you very much for the first chapters of “Recoltes et Semailles” which you sent me.  They arrived just before I went to Luminy… and I took advantage of the trip and my stay there to look at them, in some detail… As I told you on the phone (which damaged me in your eyes) I am sad that you should be so bitter about Deligne, who is one of the most honest mathematicians I know — and one of those who cares most about you.  I will not try to change your mind on this subject (nor indeed on any other subject): I know too well the strength and rigidity of your convictions.  This is probably what I find most painful in your text.  That, and the general tone of recrimination towards both yourself and your former students.  Moreover, you must have unconsciously recognized this recriminatory tone, since you tried to exorcise it…

–The fact that your work was not continued by your former students.  You are right: they did not continue.  This is hardly surprising: it was you who had a global vision of the project, not they (except Deligne, of course). They preferred to do other things.  I do not see why you should reproach them with this.

As for Deligne, he moved little by little towards the questions that go beyond the framework of algebraic geometry: modular forms, representations, the Langlands program.  And he applied his deep understanding of algebraic geometry (including “motives”) to various questions… Why should he not have used the yoga of “motives”?  You introduced it, everyone knows that, and everyone has the right to use it — provided one carefully distinguishes what is conjectural (and perhaps even false, until proof of the contrary) from what can be proved.  For example, I found very beautiful what Deligne does in LN 900 (the text you reject with horror…) to get around the problem of Hodge cycles and obtain highly useful results nevertheless (on $\ell$-adic representations, for example).  I know that the very idea of “getting around a problem” is foreign to you — and maybe that is what shocks you the most in Deligne’s work.  (Another example: in his proof of the Weil conjectures, he “gets around” the “standard conjectures” — this shocks you, but delights me).

(As a matter of fact, despite what you said… my way of thinking is not very different — depth excepted — from Deligne’s  And it is also quite distant from yours — which, moreover, explains why we complemented each other so well for 10 or 15 years, as you say very nicely in your first chapter.)”

Serre to Grothendieck, February 8, 1986

“One thing strikes me in the texts that I have seen: you are surprised and indignant that your former students did not continue the work which you had undertaken and largely completed.  But you do not ask the most obvious question, the one every reader expects you to answer:

why did you yourself abandon the work in question?”

Grothendieck to Serre, January 25, 1987 (Grothendieck’s last letter to Serre)

“My research is taking me farther and farther from what is generally considered as “scientific” work (not that I have the impression of any real “break” in the ardor and the spirit I put into my work)– and anyway, it would be entirely useless for me to tell you about it, even briefly.  I will talk about it, however, for those who might be interested (if any…).  I have taken the decision to retire next year (I will then be sixty), in order to feel freer to pursue my research in a direction which does not fit into any “discipline” of recognized usefulness to society and fundable as such, and in which I will be the only one to go.  My interest in mathematics is not dead, but I doubt I will have the spare time to write up the few grand sketches I had still intended to write.”