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L uc Illusie, an emeritus professor at theUniversité Paris-Sud, was a student ofAlexander Grothendieck. On the after-noon of Tuesday, January 30, 2007, Illusiemet with University of Chicago mathe-maticians Alexander Beilinson, Spencer Bloch, andVladimir Drinfeld, as well as a few other guests,at Beilinson’s home in Chicago. Illusie chatted bythe ﬁreside, recalling memories of his days withGrothendieck. What follows is a corrected andedited version of a transcript prepared by ThanosPapaïoannou, Keerthi Madapusi Sampath, andVadim Vologodsky.

Transcript:

Reminiscences ofGrothendieck and HisSchoolLuc Illusie, with Alexander Beilinson, Spencer Bloch,Vladimir Drinfeld, et al.

Luc Illusie, an emeritus professor at theUniversit Paris-Sud, was a student ofAlexander Grothendieck. On the after-noon of Tuesday, January 30, 2007, Illusiemet with University of Chicago mathe-

maticians Alexander Beilinson, Spencer Bloch, andVladimir Drinfeld, as well as a few other guests,at Beilinsons home in Chicago. Illusie chatted bythe fireside, recalling memories of his days withGrothendieck. What follows is a corrected andedited version of a transcript prepared by ThanosPapaoannou, Keerthi Madapusi Sampath, andVadim Vologodsky.

At the IHSIllusie: I began attending Grothendiecks semi-nars at the IHS [Institut des Hautes tudesScientifiques] in 1964 for the first part of SGA5 (19641965).1 The second part was in 19651966.The seminar was on Tuesdays. It started at 2:15and lasted one hour and a half. After that we hadtea. Most of the talks were given by Grothendieck.Usually, he had pre-notes prepared over the sum-mer or before, and he would give them to thepotential speakers. Among his many students hedistributed the exposs, and also he asked hisstudents to write down notes. The first time Isaw him I was scared. It was in 1964. I had beenintroduced to him through Cartan, who said, Forwhat youre doing, you should meet Grothendieck.

Luc Illusie is professor emeritus of mathematics at theUniversit Paris-Sud. His email address is [email protected] Alexander Beilinson is the David andMary Winton Green University Professor at the Universityof Chicago. His email address is [email protected] Spencer Bloch is R. M. Hutchins Distinguished Ser-vice Professor Emeritus at the University of Chicago. Hisemail address is [email protected] VladimirDrinfeld is professor of mathematics at the University ofChicago. His email address is [email protected]

I was indeed looking for an Atiyah-Singer indexformula in a relative situation. A relative situationis of course in Grothendiecks style, so Cartanimmediately saw the point. I was doing somethingwith Hilbert bundles, complexes of Hilbert bundleswith finite cohomology, and he said, It reminds meof something done by Grothendieck, you shoulddiscuss it with him. I was introduced to him bythe Chinese mathematician Shih Weishu. He was inPrinceton at the time of the Cartan-Schwartz semi-nar on the Atiyah-Singer formula; there had been aparallel seminar, directed by Palais. We had workedtogether a little bit on some characteristic classes.And then he visited the IHS. He was friendly withGrothendieck and proposed to introduce me.

So, one day at two oclock I went to meetGrothendieck at the IHS, at his office, which isnow, I think, one of the offices of the secretaries.The meeting was in the sitting room which wasadjacent to it. I tried to explain what I was do-ing. Then Grothendieck abruptly showed me somenave commutative diagram and said, Its not lead-ing anywhere. Let me explain to you some ideas Ihave. Then he made a long speech about finitenessconditions in derived categories. I didnt know any-thing about derived categories! Its not complexesof Hilbert bundles you should consider. Instead,you should work with ringed spaces and pseu-docoherent complexes of finite tor-dimension.(laughter)It looked very complicated. But whathe explained to me then eventually proved usefulin defining what I wanted. I took notes but couldntunderstand much.

I knew no algebraic geometry at the time. Yethe said, In the fall I am starting a seminar,

1Cohomologie `-adique et fonctions L, Sminaire deGomtrie Algbrique du Bois-Marie 1965/66, dirig parA. Grothendieck, Lecture Notes in Math. 589, Springer-Verlag, 1977.

1106 Notices of the AMS Volume 57, Number 9

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Alexander Grothendieck around 1965.

a continuation of SGA 4,2 which was not calledSGA 4, it was SGAA, the Sminaire de gomtriealgbrique avec Artin. He said, It will be on localduality. Next year we will reach `-adic cohomology,trace formulas, L-functions. I said, Well, I willattend, but I dont know if Ill be able to follow.He said, But in fact I want you to write down thenotes of the first expos. However, he gave me nopre-notes. I went to the first talk.

He spoke with great energy at the board buttaking care to recall all the necessary material. Hewas very precise. The presentation was so neatthat even I, who knew nothing of the topic, couldunderstand the formal structure. It was going fastbut so clearly that I could take notes. He startedby briefly recalling global duality, the formalismof f ! and f!. By that time, I had learned a littlebit of the language of derived categories, so I wasnot so afraid of distinguished triangles and thingslike that. Then he moved to dualizing complexes,which was much harder. After a month, I wrotedown notes. I was very anxious when I gave them tohim. They were about fifty pages. For Grothendieckit was a reasonable length. Once, Houzel, who hadbeen my teaching assistant at the cole Normale,at the end of the seminar said to Grothendieck,I have written something Id like to give you. Itwas something on analytic geometry, about tenpages. Grothendieck said, When you have writtenfifty pages, then come back (laughter)Anyway,the length was reasonable, but I was still veryanxious. One reason is that, meanwhile, I hadwritten some notes about my idea on complexes ofHilbert bundles. I had a final version which seemedto me to be good. Grothendieck said, Maybe Illhave a look at that. So I gave them to him. Nottoo long afterward, Grothendieck came to me and

2Thorie des topos et cohomologie tale des schmas,Sminaire de gomtrie algbrique du Bois-Marie 1963-64,dirig par M. Artin, A. Grothendieck, J.-L. Verdier, Lec-ture Notes in Math. 269, 270, 305, Springer-Verlag, 1972,1973.

said, I have a few comments on your text. Couldyou please come to my place, I will explain themto you.

At Grothendiecks PlaceWhen I met him, to my surprise, my text wasblackened with penciled annotations. I thought itwas in final form, but everything had to be changed.In fact, he was right all the time, even for questionsof French language. He proposed modifications inthe style, the organization, everything. So, for myexpos on local duality, I was very afraid. However,a month later or so, he said, Ive read your notes.They are okay, but I have a few comments, so couldyou please come to my place again? That was thebeginning of a series of visits to his place. At thetime he lived at Bures-sur-Yvette, rue de Moulon,in a little white pavilion, with a ground floor andone story. His office there was austere and coldin the winter. He had a portrait of his father inpencil, and also on the table there was the mortuarymask of his mother. Behind his desk he had filingcabinets. When he wanted some document, hewould just turn back and find it in no time. Hewas well organized. We sat together and discussedhis remarks on my redactions. We started at twoand worked until maybe four oclock, then he said,Maybe we could take a break. Sometimes we tooka walk, sometimes we had tea. After that we cameback and worked again. Then we had dinner aroundseven, with his wife, his daughter, and his two sons.The dinner didnt last long. Afterward we met againin his office, and he liked to explain some maths tome. I remember, one day, he gave me a course onthe theory of the fundamental group from severalviewpoints, the topological approach, the scheme-theoretic one (with the enlarged fundamental groupof SGA 3), the topos-theoretic one. I tried to catchup, but it was hard.

He was improvising, in his fast and eleganthandwriting. He said that he couldnt think withoutwriting. I, myself, would find it more convenientfirst to close my eyes and think, or maybe just liedown, but he could not think this way, he had totake a sheet of paper, and he started writing. Hewrote X S, passing the pen several times on it,you see, until the characters and arrow becamevery thick. He somehow enjoyed the sight of theseobjects. We usually finished at half past eleven,then he walked with me to the station, and I tookthe last train back to Paris. All afternoons at hisplace were like that.

Walks in the WoodsAmong the people coming to the seminar, I re-member Berthelot, Cartier, Chevalley, Demazure,Dieudonn, Giraud, Jouanolou, Nron, Poitou, Ray-naud and his wife Michle, Samuel, Serre, Verdier.Of course we also had foreign visitors, some for

October 2010 Notices of the AMS 1107

long periods (Tits; Deligne, who attended the sem-inars since 1965; Tate; and later Kleiman, Katz,Quillen). Then we had tea at four in the draw-ing room of the IHS. That was a place to meetand discuss. Another one was the lunch at theIHS, to which I decided to come after some time.There you could find Grothendieck, Serre, Tate dis-cussing motives and other topics that passed wellover my head. SGA 6,3 the seminar on Riemann-Roch, started in 1966. A little before, Grothendiecksaid to Berthelot and me, You should give thetalks. He handed me some pre-notes on finitenessconditions in derived categories and on K-groups.So Berthelot and I gave several talks, and we wrotedown notes. In this time, we usually met for lunch,and after lunchthat was very niceGrothendieckwould take us for a walk in the woods of the IHSand just casually explain to us what he had beenthinking about, what hed been reading. I remem-ber, once he said, Im reading Manins paper onformal groups4 and I think I understand what hesdoing. I think one should introduce the notion ofslope, and Newton polygon, then he explainedto us the idea that the Newton polygon shouldrise under specialization, and for the first timehe envisioned the notion of crystal. Then at thesame time, maybe, or a little later, he wrote hisfamous letter to Tate: Un cristal possde deuxproprits caractristiques : la rigidit, et la facultde crotre, dans un voisinage appropri. Il y a descristaux de toute espce de substance: des cristauxde soude, de soufre, de modules, danneaux, deschmas relatifs, etc. (A crystal possesses twocharacteristic properties: rigidity, and the abilityto grow in an appropriate neighborhood. There arecrystals of all kinds of substances: sodium, sulfur,modules, rings, relative schemes, etc.)

KnnethBloch: What about you? What about your part? Youmust have been thinking about your thesis.

Illusie: It was not working so well, I must say.Grothendieck had proposed to me some problems,of course. He said, The second part of EGA III5 isreally lousy, there are a dozen spectral sequencesabutting to the cohomology of a fiber product. Itsa mess, so, please, clean this up by introducingderived categories, write the Knneth formula inthe general framework of derived categories. I

3Thorie des intersections et thorme de Riemann-Roch, Sminaire de Gomtrie Algbrique du Bois-Marie1966/67, dirig par P. Berthelot, A. Grothendieck, L. Illusie,Lecture Notes in Math. 225, Springer-Verlag, 1971.4Yu. I. Manin, Theory of commutative formal groups overfields of finite characteristic, Uspehi Mat. Nauk. 18 (1963),no. 6 (114), 390. (Russian)5lments de Gomtrie Algbrique, par A. Grothendieck,rdigs avec la collaboration de J. Dieudonn, Pub. Math.IHS 4, 8, 11, 17, 20, 24, 28, 32, and Grundlehren 166,Springer-Verlag, 1971.

thought about that and was fairly rapidly stuck. Ofcourse, I could write some formula, but only in thetor-independent situation. Im not sure that thereis even now in the literature a nice general formulain the non-tor-independent situation.6 For this youneed homotopical algebra.

You have two rings, and you have to take thederived tensor product of the rings; what you get isan object in the derived category of simplicial rings,or you can view it as a differential graded algebra inthe characteristic 0 case, but the material was notavailable at the time. In the tor-independent case,the usual tensor product is good. In the generalone I was stuck.

SGA 6I was therefore happy to work with Grothendieckand Berthelot on SGA 6. At the time you didnt haveto finish your thesis in three years. The completionof a thse dtat could take seven, eight years. Sothe pressure was not so great. The seminar, SGA 6,went well, we eventually proved a Riemann-Rochtheorem in a quite general context, and Berthelotand I were quite happy. I remember that we tried toimitate Grothendiecks style. When Grothendieckhanded me his notes on the finiteness conditionsin derived categories, I said, This is only over apoint. We should do that in a fibered category oversome topos (laughter). It was a little nave, but,anyway, it proved to be the right generalization.

Drinfeld: What is written in the final version ofSGA 6? Is it in this generality?

Illusie: Yes, of course.Drinfeld: So, it was your suggestion, not

Grothendiecks.Illusie: Yes.Drinfeld: Did he approve it?Illusie:Of course, he liked it. As for Berthelot, he

brought original contributions to theK-theory part.Grothendieck had calculated the K0 of a projectivebundle. We did not call it K0 at the time; there werea K made with vector bundles and a K made withcoherent sheaves, which are now denoted K0 andK0. Grothendieck had proved that the K0 of a pro-jective bundle P over X is generated over K0(X) bythe class of OP(1). But he was not happy with that.He said, Sometimes youre not in a quasi-projectivesituation, you dont have any global resolutions forcoherent sheaves. Then its better to work with theK-group defined using perfect complexes. How-ever, he didnt know how to prove the similar resultfor this other K group. Berthelot thought about theproblem, and, adapting to complexes some con-structions of Proj made in EGA II for modules, hesolved it. He showed that to Grothendieck and thenGrothendieck told me, Berthelot est encore plus

6This issue is discussed again in the section under theheading Cartier, Quillen.

1108 Notices of the AMS Volume 57, Number 9

fonctoris que moi!7(laughter). Grothendieckhad given us detailed notes on lambda opera-tions, which he had written before 1960. Berthelotdiscussed them in his exposs and solved severalquestions that Grothendieck had not thought aboutat the time.

Bloch: Why did you choose this topic? Therewas this earlier paper, by Borel and Serre, basedon Grothendiecks ideas about Riemann-Roch. Imsure he wasnt happy with that!

Illusie: Grothendieck wanted a relative formulaover a general base and for fairly general mor-phisms (locally complete intersection morphisms).Also, he didnt want to move cycles. He preferredto do intersection theory using K-groups.

Bloch: But he didnt forget his program of tryingto prove the Weil conjectures?

SGA 7Illusie: No, but he had several irons in the fire. In19671968 and 19681969, there was another sem-inar, SGA 7,8 about monodromy, vanishing cycles,the R and R functors, cycle classes, Lefschetzpencils. Certainly he had already thought about theformalism of nearby cycles a few years before. Also,he had read Milnors book on singularities of hy-persurfaces. Milnor had calculated some examplesand observed that for these all the eigenvalues ofthe monodromy of the cohomology of what we nowcall the Milnor fiber of an isolated singularity areroots of unity. Milnor conjectured that that was al-ways the case, that the action was quasi-unipotent.Then Grothendieck said, What are the tools atour disposal? Hironakas resolution. But then youleave the world of isolated singularities, you canno longer take Milnor fibers, you need a suitableglobal object. Then he realized that the complexof vanishing cycles that he had defined was whathe wanted. Using resolution of singularities, he cal-culated, in the case of quasi-semistable reduction(with some multiplicities), the vanishing cycles,and then the solution came out quite easily incharacteristic zero. He also obtained an arithmeticproof in the general case: he found this marvelousargument showing that when the residue field ofyour local field is not so big, in the sense that nofinite extension of it contains all roots of unity oforder a power of `, then `-adic representationsare quasi-unipotent. He decided to make a seminaron that, and that was this magnificent seminar,SGA 7. Its in it that Deligne gave his beautifulexposs on the Picard-Lefschetz formula (at therequest of Grothendieck, who couldnt understand

7Berthelot is still more functorized than I am!8Groupes de monodromie en gomtrie algbrique, Smi-naire de Gomtrie Algbrique du Bois-Marie 19671969,I dirig par A. Grothendieck, II par P. Deligne et N. Katz,Lecture Notes in Math. 288, 340, Springer-Verlag 1972,1973.

Lefschetzs arguments) and Katz his marvelouslectures on Lefschetz pencils.

Cotangent Complex and DeformationsHowever, my thesis was still empty, I had justattended SGA 7, written up no notes. I had givenup long ago this question on Knneth formulas.I had published a little paper in Topology on fi-nite group actions and Chern numbers, but thatwas not much. One day, Grothendieck came tome and said, I have a few questions for you ondeformations. So we met on one afternoon, andhe proposed several problems on deformationswith similar answers: deformations of modules,groups, schemes, morphisms of schemes, etc. Ev-ery time the answer involved an object he hadrecently constructed, the cotangent complex. Inhis work with Dieudonn in EGA IV, there appearsa differential invariant of a morphism, called themodule of imperfection. Grothendieck realized that1 and the module of imperfection were in factthe cohomology objects of a finer invariant in thederived category, a complex of length one, which hecalled the cotangent complex. He wrote this up inhis Lecture Notes, Catgories cofibres additives etcomplexe cotangent relatif (SLN 79). Grothendieckobserved that to get to the obstructions, whichinvolved H2 groups, his theory was probably in-sufficient, because a composition of morphismsdidnt give rise to a nice distinguished trianglefor his cotangent complexes. It happens that atthe same time, independently, Quillen had beenworking on homotopical algebra and, a little later,had constructed, in the affine case, a chain com-plex of infinite length, which had Grothendieckscomplex as a truncation, and which behaved wellwith respect to composition of morphisms. Inde-pendently, too, Michel Andr had defined similarinvariants. I got interested in their work and re-alized that in Andrs construction, the classicallemma of Whitehead, which played a key role, couldeasily be sheafified. In a few months, I obtained themain results of my thesis, except for deformationof group schemes, which came much later (thecommutative ones required much more work).

After May 1968In May 1968 Grothendieck was seduced by theleftist ideology. He admired Maos thought and theCultural Revolution. He had also started thinkingabout other topics: physics (he told me he hadbeen reading books by Feynman), then biology(especially embryology). I have the impression thatfrom that time, mathematics was slowly driftingaway from his main focus of interest, though hewas still very active (e.g., the second part of SGA 7was in 19681969). He had contemplated giving aseminar on abelian schemes after that but finallydecided to go on studying Dieudonns theory for

October 2010 Notices of the AMS 1109

p-divisible groups, in the continuation of his workon crystalline cohomology.

His lectures on this (in 1966) had been writtenup by Coates and Jussila, and he let Berthelotdevelop a full-fledged theory. One can regret hedidnt give a seminar on abelian schemes. Imsure it would have produced a beautiful, unifiedpresentation of the theory, much better than thescattered references we can find in the literature.In 1970 he left the IHS and founded the ecologicalgroup Survivre (renamed later Survivre et Vivre).At the Nice congress, he was doing propagandafor it, offering documents taken out of a smallcardboard suitcase. He was gradually consideringmathematics as not being worthy of being studied,in view of the more urgent problems of the survivalof the human species. He carelessly dispatchedaround him many of his documents (papers, privatenotes, etc.). Yet, in 19701971 he gave a beautifulcourse (together with a seminar) at the Collge deFrance on Barsotti-Tate groups and lectured laterin Montreal on the same topic.

Working with GrothendieckMany people were afraid of discussing withGrothendieck, but, in fact, it was not so difficult.For example, I could call him anytime, providedthat it was not before noon, because he would getup at that time. He worked late in the night. I couldask him any question, and he would very kindlyexplain to me what he knew about the problem.Sometimes, he had afterthoughts. He would thenwrite me a letter with some complements. He wasvery friendly with me. But some students werenot so happy. I remember Lucile Bgueri-Poitou,who had asked for a topic for her thesis fromGrothendieck. It was a bit like with my Knnethformula. I think he proposed to her to write downthe theory of coherent morphisms for toposes,finiteness conditions in toposes. That was hardand thankless, things didnt go well, and sheeventually decided to stop working with him.Years later she wrote a thesis solving a totallydifferent question of his.9 He was more successfulwith Mme Raynaud, who produced a beautifulthesis.10

I said that when I handed him some notes,he would correct them heavily and suggest manymodifications. I liked it because his remarks werealmost always quite up to the point, and I washappy to improve my writing. But some didntlike it, some thought that what they had writtenwas good and there was no need to improve it.

9L. Bgueri, Dualit sur un corps local corps rsiduel al-gbriquement clos, Mm. Soc. Math. France (N. S.) 1980/81,n. 4, 121 pp.10M. Raynaud, Thormes de Lefschetz en cohomologiecohrente et en cohomologie tale, Bull. Soc. Math. France,Mm. n. 41, Supplment au Bull. Soc. Math. France, t. 103,1975, 176 pp.

Grothendieck gave a series of lectures on motivesat the IHS. One part was about the standardconjectures. He asked John Coates to write downnotes. Coates did it, but the same thing happened:they were returned to him with many corrections.Coates was discouraged and quit. Eventually, it wasKleiman who wrote down the notes in Dix exposssur la cohomologie des schmas.11

Drinfeld: But its not so good for many people,giving a thesis on coherent morphisms of toposes;its bad for most students.

Illusie: I think these were good topics forGrothendieck himself.

Drinfeld: Yes, sure.Illusie: But not for students. Similarly with

Monique Hakim, Relative schemes over toposes. Iam afraid this book12 was not such a success.

Unknown: But the logicians like it very much.Illusie: I heard from Deligne that there were

problems in some parts.13 Anyway, she was not sohappy with this topic, and she did quite differentmathematics afterward. I think that Raynaud alsodidnt like the topic that Grothendieck had givenhim. But he found another one by himself.14 Thatimpressed Grothendieck, as well as the fact thatRaynaud was able to understand Nrons construc-tion of Nron models. Grothendieck of course hadquite brilliantly used the universal property ofNron models in his exposs in SGA 7, but he couldnot grasp Nrons construction.

VerdierFor Verdier its a different story. I rememberGrothendieck had a great admiration for Verdier.He admired what we now call the Lefschetz-Verdiertrace formula and Verdiers idea of defining f ! firstas a formal adjoint, and then calculating it later.

Bloch: I thought, maybe, that was Delignes idea.Illusie: No, it was Verdiers. But Deligne in

the context of coherent sheaves used this ideaafterward. Deligne was happy to somehow killthree hundred pages of Hartshornes seminar ineighteen pages. (laughter)

Drinfeld: Which pages do you mean?

11S. Kleiman, Algebraic cycles and the Weil conjectures,in Dix exposs sur la cohomologie des schmas, A.Grothendieck and N. Kuiper, eds., North Holland Pub. Co.,Masson et Co., 1968, 359-386.12M. Hakim, Topos annels et schmas relatifs, Ergebnisseder Mathematik und ihrer Grentzgebiete, Bd 64, Springer-Verlag, 1972.13Added in April 2010: Deligne doesnt think there wasanything wrong but remembers that the objects shedefined over analytic spaces were not the desired ones.14M. Raynaud, Faisceaux amples sur les schmas engroupes et les espaces homognes, Lecture Notes in Math.119, Springer-Verlag, 1970.

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Illusie: In the appendix to Hartshornes seminarResidues and Duality,15 I say Hartshornes semi-nar, but in fact it was Grothendiecks seminar.Pre-notes had been written up by Grothendieck.Hartshorne gave the seminar from these.

Coming back to Verdier, who had written sucha nice fascicule de rsultats on triangulated andderived categories,16 one can ask why he did notembark on writing a full account. In the late 1960sand early 1970s, Verdier got interested in otherthings, analytic geometry, differential equations,etc. When Verdier died in 1989, I gave a talk on hiswork, at a celebration for him in his memory, and Ihad to understand this issue: Why didnt he publishhis thesis? He had written some summary, but not afull text. Probably one of the main reasons is simplythat in the redaction of his manuscript he had notyet treated derived functors. He had discussedtriangulated categories, the formalism of derivedcategories, the formalism of localization, but notyet derived functors.17 At the time he was alreadytoo busy with other things. And presumably he didnot want to publish a book on derived categorieswithout derived functors. Its certainly a pity.18

Drinfeld: And the Astrisque volume, how muchdoes it correspond to?

Illusie: It corresponds to what Verdier hadwritten, up to derived functors.19 This volume isquite useful, I think, but for derived functors, youhave to look at other places.20

Filtered Derived CategoriesDrinfeld: Did the notion of differential gradedcategory ever appear in Verdiers work? Anotherpotential source of dissatisfaction with derivedcategories was that the cones were defined onlyup to isomorphism; there are many natural con-structions which do not work naturally in derivedcategories as defined by Verdier. Then you needdifferential graded categories or go to stable cat-egories, but these formally have been developedonly recently. In hindsight, the idea of the differen-tial graded category seems very natural. Did youhave this idea in the discussion of the derivedcategory?

15R. Hartshorne, Residues and Duality, Lecture Notes inMath. 20, Springer-Verlag, 1966.16Catgories drives, Quelques rsultats (tat 0) in [SGA4 1/2, Cohomologie tale, par P. Deligne, Lecture Notes inMath. 569, Springer-Verlag, 1977], pp. 266316.17Derived functors were defined and studied in the abovementioned fascicule de rsultats, II 2.18Added in April 2010: According to Deligne, Verdier wasalso plagued by sign problems, for which he had not founda satisfactory treatment.19J.-L. Verdier, Des catgories drives des catgoriesabliennes, dit par G. Maltsiniotis, Astrisque 239(1996).20E.g., in Delignes expos XVII in SGA 4, where a betterdefinition of derived functors is given.

Illusie: Quillen found that differential gradedalgebras would give you a similar but in generalinequivalent category to the derived category de-fined by simplicial algebras, but this was donein the late 1960s or early 1970s and did not ap-pear in discussions with Grothendieck. However,I know the story about the filtered derived cate-gory. Grothendieck thought that if you have anendomorphism of a triangle of perfect complexes,then the trace of the middle part should be thesum of the traces of the right-hand side and theleft-hand side. In SGA 5, when he discussed traces,he explained that on the board. One of the personsattending the seminar was Daniel Ferrand. At thetime, nobody saw any problem with that, it was sonatural. But then Grothendieck gave Ferrand the

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task of writing the construc-tion of the determinant ofa perfect complex. This isa higher invariant than thetrace. Ferrand was stuck atone point. When he looked atthe weaker version, he real-ized that he could not showthat the trace of the middlepart was the sum of the twoextremes, and then he built asimple counterexample. Theproblem was: How can we re-store that? The person who atthe time could repair anything that went wrongwas Deligne. So, we asked Deligne. Deligne came upwith the construction of a category of true triangles,finer than usual triangles, obtained by a certainprocess of localization, from pairs of a complexand a subcomplex. In my thesis I wanted to defineChern classes, using an Atiyah extension. I neededsome additivity of Chern classes, hence additiv-ity of traces, and algebraic complements; I alsoneeded tensor products, which increase lengths offiltrations. So I thought: why not just take filteredobjects and localize with respect to maps induc-ing quasi-isomorphisms on the associated gradedobjects? It was very natural. So I wrote it up in mythesis, and everybody was happy. At the time, onlyfinite filtrations were considered.

Drinfeld: So it is written in your Springer Lec-ture Notes volumes on cotangent complex anddeformations?

Illusie: Yes, in SLN 239, Chapter V. Delignescategory of true triangles was just DF [0,1], thefiltered derived category with filtrations of length 1.That was the beginning of the theory. However,Grothendieck said, In triangulated categories wehave the octahedron axiom, what will replace that infiltered derived categories? Maybe the situation isnot yet fully understood today. Once, Grothendiecktold me, it must have been in 1969: We havethe K-groups defined by vector bundles, but wecould take vector bundles with a filtration of

October 2010 Notices of the AMS 1111

length one (with quotient a vector bundle), vectorbundles with filtrations of length 2, length n, withassociated graded still vector bundles. Then youhave operations such as forgetting a step of thefiltration, or taking a quotient by one step. Thisway you get some simplicial structure which shoulddeserve to be studied and could yield interestinghomotopy invariants.

Independently, Quillen had worked out the Q-construction, which is a substitute for the filtrationapproach. But, I think, if Grothendieck had hadmore time to think about it, he would have definedthe higher K-groups.

Drinfeld: But this approach looks more likeWaldhausens one.

Illusie: Yes, of course.Drinfeld: Which appeared much later.Illusie: Yes.

Cartier, QuillenDrinfeld: During the SGA 6 seminar, was it knownthat the -operations have something to do withWitt rings?

Illusie: Yes. In fact, I think that G. M. Bergmansappendix to Mumfords book on surfaces21 wasalready available at that moment.

Drinfeld: Are there -operations in this appen-dix?

Illusie:No, but I gave a talk in Bures on universalWitt rings and lambda operations. I remember Iwas going to the Arbeitstagung in Bonn. Havingmissed the night train I took an early morning train.Surprise: Serre and I were in the same compart-ment. I told him about the talk I had to prepare,and he generously helped me. During the wholetrip, he improvised in a brilliant way, explainingto me several beautiful formulas, involving theArtin-Hasse exponential and other miracles of Wittvectors. This was discussed toward the end of theSGA 6 seminar, in June 1967. I wonder, Cartierstheory should have existed at the time. Tapis deCartier, I think, existed.

Drinfeld: What is Tapis de Cartier?Illusie: Tapis de Cartier was how Grothendieck

called Cartiers theory of formal groups. Tapis (=rug) was a (slightly derogatory) expression usedby some Bourbaki members, comparing those whoadvocated for a theory to rug merchants.

Bloch: But still, if you look back, Cartier made alot of contributions.

Illusie: Yes, Cartiers theory is powerful andhad a strong impact later. But I dont think thatGrothendieck used much of it. On the other hand, atthe time, Grothendieck was impressed by Quillen,who had brilliant new ideas on many topics. Aboutthe cotangent complex, I dont remember well now,

21D. Mumford, Lectures on curves on an algebraic surface.With a section by G. M. Bergman. Annals of MathematicsStudies, No. 59, Princeton University Press, Princeton, N.J.1966.

but Quillen had a way of calculating the Exti of thecotangent complex andO as the cohomology of thestructural sheaf of a certain site, which looked likethe crystalline site, but with the arrows reversed.That surprised Grothendieck.

Unknown: Apparently, this idea was rediscov-ered later by Gaitsgory.22

Bloch: In Quillens notes on the cotangent com-plex it was the first time Id ever seen a derivedtensor product over a derived tensor product.

Illusie: Yes, in the relation between the (de-rived) self-intersection complex and the cotangentcomplex.

Bloch: I think it was something like B LBLAB B. Iremember studying for days, puzzling over exactlywhat that meant.

Illusie: But when I said I couldnt do my Knnethformula, one reason was that such an object didntexist at the time.

Drinfeld: I am afraid that even now it doesntexist in the literature (although it may exist insomebodys head). I needed the derived tensorproduct of algebras over a ring a few years agowhen I worked on the article on DG categories. I wasunable either to find this notion in the literatureor to define it neatly. So I had to write somethingpretty ugly.

Grothendiecks TastesIllusie: I realize I didnt say much aboutGrothendiecks tastes. For example, do youknow the piece of music he would like most?

Bloch: Did he like music at all?Illusie: Grothendieck had a very strong feeling

for music. He liked Bach, and his most belovedpieces were the last quartets by Beethoven.

Also, do you know what his favorite tree was? Heliked nature, and there was one tree he liked morethan the others. It was the olive tree, a modest tree,but which lives long, is very sturdy, is full of sunand life. He was very fond of the olive tree.

In fact, he always liked the south very much,long before he went to Montpellier. He had been amember of the Bourbaki group, and he had visitedLa Messuguire, where some congresses were held.

He tried to get me to go to that place, but it didntwork out. It is a beautiful estate on the heightsabove Cannes. You have Grasse a little higher, andstill a little higher you have a small village calledCabris, where there is this estate, with eucalyptustrees, olive trees, pine trees, and a magnificent view.He liked it very much. He had a fancy for this sortof landscape.

Drinfeld: Do you know what Grothendiecksfavorite books were? You mentioned his favoritemusic

22D. Gaitsgory, Grothendieck topologies and deformationtheory II. Compositio Math. 106 (1997), no. 3, 321348.

1112 Notices of the AMS Volume 57, Number 9

Illusie: I dont remember. I think he didnt readmuch. There are only twenty-four hours in a day

Automorphic Forms, Stable Homotopy,Anabelian GeometryIllusie: In retrospect, I find it strange that repre-sentation theory and automorphic forms theorywere progressing well in the 1960s but somehowignored in Bures-sur-Yvette. Grothendieck knewalgebraic groups quite well.

Bloch: Well, as you said, there are only twenty-four hours in a day.

Illusie: Yes, but he might have constructed the`-adic representations associated with modularforms like Deligne did, but he didnt. He reallywas very interested in arithmetic, but maybe thecomputational aspect of it was not so appealing tohim. I dont know.

He liked putting different pieces of mathemat-ics together: geometry, analysis, topology soautomorphic forms should have appealed to him.But for some reason he didnt get interested inthat at the time. I think the junction betweenGrothendieck and Langlands was realized onlyin 1972 at Antwerp. Serre had given a courseon Weils theorem in 19671968. But after 1968Grothendieck had other interests. And before 1967things were not ripe. Im not sure.

Beilinson: What about stable homotopy theory?Illusie: Of course Grothendieck was interested

in loop spaces, iterated loop spaces; n-categories,n-stacks were at the back of his mind, but he didntwork it out at the time.

Beilinson: When did it actually come about?Picard category is probably about 1966.

Illusie: Yes, it was related to what he did withthe cotangent complex. He conceived the notionof Picard category at that time, and then Delignesheafified it into Picard stacks.

Beilinson: And higher stacks?Illusie: He had thought about the problem,

but its only long afterwards that he wrote hismanuscript Pursuing stacks. Also,pi1(P1{0,1,})was always at the back of his mind. He wasfascinated by the Galois action, and I rememberonce he had thought about possible connectionswith that and Fermats problem. Already in the1960s he had some ideas about anabelian geometry.

MotivesIllusie: I regret that he was not allowed to speak onmotives at the Bourbaki seminar. He asked for sixor seven exposs, and the organizers considered itwas too much.

Bloch: It was kind of unique then; nobody elsewas lecturing on their own work.

Illusie: Yes, but you see, FGA (Fondements de laGomtrie Algbrique) consists of several exposs.He was thinking of doing for motives what he haddone for the Picard scheme, the Hilbert scheme, etc.

There are also three exposs on the Brauer groupwhich are important and useful, but seven exposson motives would have been even more interesting.However, I dont think they would have containedthings which have not been worked out by now.

Weil and GrothendieckBloch: I once asked Weil about nineteenth-centurynumber theory and whether he thought that therewere any ideas there that had not yet been workedout. He said, No. (laughter )

Illusie: I discussed with Serre what hethought were the respective merits of Weiland Grothendieck. Serre places Weil higher. Butthough Weils contributions are fantastic, I myselfthink Grothendiecks work is still greater.

Drinfeld: But it was Weil who revived the theoryof modular forms in his famous article.23 ProbablyGrothendieck couldnt have done it.

Illusie: Yes, this is certainly a great contribution.As for Weils books, Foundations of Algebraic Ge-ometry is hard to read. Serre the other day told methat Weil was unable to prove theorem A for affinevarieties in his language. And even Weils book onKhler varieties,24 I find it a little heavy.

Bloch: That book in particular was very influen-tial.

Grothendiecks StyleIllusie: Yes, but Im not so fond of Weils style.Grothendiecks style had some defects also. Onethat was barely perceptible at the beginning and be-came enormous later is his habit of afterthoughtsand footnotes. Rcoltes et Semailles is incredible inthis respect. So many, so long footnotes! Alreadyin his beautiful letter to Atiyah on de Rham coho-mology there are many footnotes, which containsome of the most important things.

Bloch:Oh, I remember seeing photocopies, earlyphotocopies, when photocopy machines didntwork all that well. He would type a letter and thenadd handwritten comments which were illegible.

Illusie: Well, I was used to his handwriting, so Icould understand.

Bloch: We would sit around and puzzleIllusie: To him no statement was ever the best

one. He could always find something better, moregeneral or more flexible. Working on a problem,he said he had to sleep with it for some time. Heliked mechanisms that had oil in them. For thisyou had to do scales, exercises (like a pianist),consider special cases, functoriality. At the endyou obtained a formalism amenable to dvissage.

23A. Weil, ber die Bestimmung Dirichletscher Reihendurch Funktionalgleichungen, Math. Ann. 168 (1967),149156.24A. Weil, Introduction ltude des varits khlriennes.Publications de lInstitut de Mathmatique de lUniversitde Nancago, VI. Actualits Sci. Ind. no. 1267, Hermann,Paris, 1958.

October 2010 Notices of the AMS 1113

I think one reason why Grothendieck, afterSerres talk at the Chevalley seminar in 1958, wasconfident that tale localization would give thecorrect H i s is that once you had the correctcohomology of curves, then by fibration in curvesand dvissage you should also reach the higherH i s.

I think he was the first one to write a mapvertically instead of from left to right.25

Drinfeld: It was he who put the X over S. Beforethat X was on the left and S was on the right.

Illusie: Yes. He was thinking over a base. Thebase could be a scheme, a topos, anything. The basehad no special properties. Its the relative situationthat was important. Thats why he wanted to getrid of Noetherian assumptions.

Bloch: And I remember, in the early daysschemes, morphisms were separated, but thenthey became quasi-separated.

Commutative AlgebraIllusie: At the time of Weil, you looked at fields,and then valuations, and then valuation rings, andnormal rings. Rings were usually supposed to benormal. Grothendieck thought it was ridiculousto make such systematic restrictions from thebeginning. When defining SpecA, A should be anycommutative ring.

Drinfeld: Sorry, but how did people treat thenodal curve if the rings were supposed to benormal? Non-normal varieties appear

Illusie: Of course, but they often looked atthe normalization. Grothendieck was aware ofthe importance of normality, and I think Serrescriterion of normality was one of the motivationsfor his theory of depth and local cohomology.

Bloch: I wonder whether today such a style ofmathematics could exist.

Illusie: Voevodskys work is fairly general. Sev-eral people tried to imitate Grothendieck, but Imafraid that what they did never reached that oilycharacter dear to Grothendieck.

But it is not to say that Grothendieck was nothappy to study objects having rich structures.As for EGA IV, it is of course a masterpiece oflocal algebra, a domain in which he was extremelystrong. We owe a lot to EGA IV, though maybesome rewriting could be possible now, using thecotangent complex.

Relative StatementsIllusie: Certainly were now so used to puttingsome problem into relative form that we forgethow revolutionary it was at the time. Hirzebruchsproof of Riemann-Roch is very complicated, while

25Added in April 2010: Cartier observes that vertical lineshad been commonly used to denote field extensions sincelong ago, especially in the German school.

the proof of the relative version, Grothendieck-Riemann-Roch, is so easy, with the problem shiftedto the case of an immersion. This was fantastic.26

Grothendieck was the father of K-theory, cer-tainly. But it was Serres idea to look at . I thinkthe people in the olden days, they had no ideaof the right generalization of Riemann-Roch forcurves. For surfaces, both sides of the formulawere quite intricate. Its Serre who realized thatthe Euler-Poincar characteristic, the alternate sumof the dimensions of the H i(O) or the H i(E) wasthe invariant you should look for. That was in theearly 1950s. And then Grothendieck saw that theuniversal was in the K-group

The Thse dtatDrinfeld: So when Grothendieck chose problemsfor his students he didnt care very much aboutthe problem being solvable.

Illusie: Of course, he cared about the problem,and when he didnt know how to solve it, he left itto his students. The thses dtat were like that

Drinfeld: And how many years did it take towrite the thesis? For example, how many years didyou spend? You had to change the subject onceor twice, and then in between you worked on SGA,which had nothing to do with the thesis. It wasvery helpful for humanity and very good practicefor you, but it had nothing to do with your thesis.So how many years did you spend?

Illusie: I started working on the cotangent com-plex in the end of 1967, and the whole thing wasfinished in two years, somehow.

Drinfeld: But before this, there were some at-tempts which were not so successful due to thenature of the problem. When did you begin workingon your thesis? As far as I understand, even nowthe standard amount of time in the U.S. is fiveyears.

Illusie: In fact, I did it in two years, essentially.In 1968 I sent a letter to Quillen sketching what Ihad done. He said, Its fine. And then I wrote upmy thesis very quickly.

Drinfeld: Were you a graduate student beforethat (when you began attending Grothendiecksseminar)?

Illusie: I was at the CNRS [Centre National de laRecherche Scientifique].

Drinfeld: Oh, you were alreadyIllusie: Yes, it was like paradise. You entered the

cole Normale Drinfeld: Yes, sure, I understand.Illusie: Then you worked reasonably well so

Cartan spotted you, saying, Well, this student

26Added in April 2010: as Deligne observes, equallyrevolutionaryand intimately linked to the relativeviewpointwas Grothendiecks idea of thinking of ascheme in terms of the functor it represents, thus recov-ering a geometric language somewhat concealed in theringed spaces approach.

1114 Notices of the AMS Volume 57, Number 9

should go to the CNRS. Once at the CNRS, youwere there for the rest of your life. Which is notquite true. A position at the CNRS at that time wasnot one of fonctionnaire (civil servant). But as Iwas not idle, my contract was renewed from yearto year.

Of course, we were maybe fifteen people at thecole Normale doing mathematics, and there werenot that many positions at the CNRS. Others couldget positions as assistants, which were not sogood as the CNRS, but still reasonable.

Drinfeld: And did somebody tell you from timeto time that it is time to finish your thesis?

Illusie: Well, after seven years, it could becomea problem. As I had started at the CNRS in 1963,and had finished my thesis by 1970, I was safe.

Drinfeld: And the fact that you spent sevenyears didnt diminish your chances for futureemployment?

Illusie: No. From 1963 to 1969 I was attachde recherche, then, from 1969 to 1973, chargde recherche, and promoted matre de recherchein 1973 (the equivalent of directeur de deuximeclasse today). Nowadays if a student after five yearshas not defended his thesis, its a problem.

Drinfeld: What has changed?Illusie: The thse dtat was suppressed, re-

placed by the standard thesis, following the Amer-ican model.

Drinfeld: I see.Illusie: Typically, a student has three years to

finish his thesis. After three years, the fellowshipends, and he has to find a position somewhere,either a permanent one or a temporary one (likeATER = attach denseignement et de recherche, ora postdoc).

For a few years we had a transitional system withthe nouvelle thse (new thesis), similar to the thesiswe have now, followed by the thse dtat. Now thethse dtat is replaced by the habilitation. Its notthe same kind of thing. Its a set of papers that youpresent at the defense. You need the habilitationfor applying for a position of professor.

Grothendieck TodayUnknown: Maybe you told me, but where isGrothendieck now? Nobody knows?

Illusie: Maybe some people know. I myself dontknow.

Bloch: If we were to go to Google and type inGrothendieck

Illusie: Wed find the Grothendieck site.Bloch: Yes, the website. He has a web topos 27

Unknown: What happened to his son? Did hebecome a mathematician?

Illusie: He has four sons. I heard the last onestudied at Harvard.

27Grothendieck Circle.

EGABloch: You cant tell a student now to go to EGAand learn algebraic geometry

Illusie: Actually, students want to read EGA.They understand that for specific questions theyhave to go to this place, the only place where theycan find a satisfactory answer. You have to givethem the key to enter there, explain to them thebasic language. And then they usually prefer EGAto other expository books. Of course, EGA or SGAare more like dictionaries than books you couldread from A to Z.

Bloch: One thing that always drove me crazyabout EGA was the excessive back referencing.I mean there would be a sentence and then aseven-digit number

Illusie: No Youre exaggerating.Bloch: You never knew whether behind the

veiled curtain was something very interesting thatyou should search back in a different volume tofind; or whether in fact it was just referring tosomething that was completely obvious and youdidnt need to

Illusie: That was one principle of Grothendieck:every assertion should be justified, either by areference or by a proof. Even a trivial one. Hehated such phrases as Its easy to see, Its easilychecked. When he was writing EGA, you see, hewas in unknown territory. Though he had a cleargeneral picture, it was easy to go astray. Thatspartly why he wanted a justification for everything.He also wanted Dieudonn to be able to understand!

Drinfeld:What was Dieudonns contribution tothe EGA?

Illusie: He did rewriting, filling in details,adding complements, polishing the proofs. ButGrothendiecks first drafts (tat 000), some ofwhich I have seen, were already quite elaborate.Nowadays you have such efficient TEX systems,manuscripts look very nice. In Grothendiecks timethe presentation was not so beautiful, maybe, butDieudonn-Grothendiecks manuscripts were stillfantastic.

I think Dieudonns most important contribu-tion was on the part of EGA IV dealing withdifferential calculus in positive characteristic, withcomplete local rings, which is basic in the theoryof excellent rings.

Also, Grothendieck was not thrifty. He thoughtthat some complements, even if they were notimmediately useful, could prove important laterand therefore should not be removed. He wantedto see all the facets of a theory.

Unknown: When Grothendieck started workingon EGA, did he already have a vision of what wouldcome later, tale cohomology Did he have inmind some applications?

Illusie: The plan he gives for EGA in the firstedition of EGA I (in 1960) amply shows the visionhe had at that time.

October 2010 Notices of the AMS 1115

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Reminiscences of Grothendieck and His School Luc Illusie, with Alexander Beilinson, Spencer Bloch, Vladimir Drinfeld, et al. L uc Illusie, an emeritus professor at the Université Paris-Sud, was a student of Alexander Grothendieck. On the after- noon of Tuesday, January 30, 2007, Illusie met with University of Chicago mathe- maticians Alexander Beilinson, Spencer Bloch, and Vladimir Drinfeld, as well as a few other guests, at Beilinson’s home in Chicago. Illusie chatted by the ﬁreside, recalling memories of his days with Grothendieck. What follows is a corrected and edited version of a transcript prepared by Thanos Papaïoannou, Keerthi Madapusi Sampath, and Vadim Vologodsky. At the IHÉS Illusie: I began attending Grothendieck’s semi- nars at the IHÉS [Institut des Hautes Études Scientiﬁques] in 1964 for the ﬁrst part of SGA 5 (1964–1965). 1 The second part was in 1965–1966. The seminar was on Tuesdays. It started at 2:15 and lasted one hour and a half. After that we had tea. Most of the talks were given by Grothendieck. Usually, he had pre-notes prepared over the sum- mer or before, and he would give them to the potential speakers. Among his many students he distributed the exposés, and also he asked his students to write down notes. The ﬁrst time I saw him I was scared. It was in 1964. I had been introduced to him through Cartan, who said, “For what you’re doing, you should meet Grothendieck.” Luc Illusie is professor emeritus of mathematics at the Université Paris-Sud. His email address is [email protected] math.u-psud.fr. Alexander Beilinson is the David and Mary Winton Green University Professor at the University of Chicago. His email address is [email protected] edu. Spencer Bloch is R. M. Hutchins Distinguished Ser- vice Professor Emeritus at the University of Chicago. His email address is [email protected]. Vladimir Drinfeld is professor of mathematics at the University of Chicago. His email address is [email protected] uchicago.edu. I was indeed looking for an Atiyah-Singer index formula in a relative situation. A relative situation is of course in Grothendieck’s style, so Cartan immediately saw the point. I was doing something with Hilbert bundles, complexes of Hilbert bundles with ﬁnite cohomology, and he said, “It reminds me of something done by Grothendieck, you should discuss it with him.” I was introduced to him by the Chinese mathematician Shih Weishu. He was in Princeton at the time of the Cartan-Schwartz semi- nar on the Atiyah-Singer formula; there had been a parallel seminar, directed by Palais. We had worked together a little bit on some characteristic classes. And then he visited the IHÉS. He was friendly with Grothendieck and proposed to introduce me. So, one day at two o’clock I went to meet Grothendieck at the IHÉS, at his oﬃce, which is now, I think, one of the oﬃces of the secretaries. The meeting was in the sitting room which was adjacent to it. I tried to explain what I was do- ing. Then Grothendieck abruptly showed me some naïve commutative diagram and said, “It’s not lead- ing anywhere. Let me explain to you some ideas I have.” Then he made a long speech about ﬁniteness conditions in derived categories. I didn’t know any- thing about derived categories! “It’s not complexes of Hilbert bundles you should consider. Instead, you should work with ringed spaces and pseu- docoherent complexes of ﬁnite tor-dimension.” …(laughter)…It looked very complicated. But what he explained to me then eventually proved useful in deﬁning what I wanted. I took notes but couldn’t understand much. I knew no algebraic geometry at the time. Yet he said, “In the fall I am starting a seminar, 1 Cohomologie ‘-adique et fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie 1965/66, dirigé par A. Grothendieck, Lecture Notes in Math. 589, Springer- Verlag, 1977. 1106 Notices of the AMS Volume 57, Number 9

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