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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.

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

Luc 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.

At the IHÉSIllusie: I began attending Grothendieck’s semi-nars at the IHÉS [Institut des Hautes ÉtudesScientifiques] in 1964 for the first part of SGA5 (1964–1965).1 The second part was in 1965–1966.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 exposés, 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 you’re doing, you should meet Grothendieck.”

Luc Illusie is professor emeritus of mathematics at theUniversité 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 Grothendieck’s 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 IHÉS. He was friendly withGrothendieck and proposed to introduce me.

So, one day at two o’clock I went to meetGrothendieck at the IHÉS, 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 somenaïve commutative diagram and said, “It’s not lead-ing anywhere. Let me explain to you some ideas Ihave.” Then he made a long speech about finitenessconditions in derived categories. I didn’t know any-thing about derived categories! “It’s 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 couldn’tunderstand much.

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

1Cohomologie `-adique et fonctions L, Séminaire deGéométrie Algébrique du Bois-Marie 1965/66, dirigé parA. Grothendieck, Lecture Notes in Math. 589, Springer-Verlag, 1977.

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Alexander Grothendieck around 1965.

a continuation of SGA 4”,2 which was not called“SGA 4”, it was “SGAA”, the “Séminaire de géométriealgébrique 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 don’t know if I’ll be able to follow.”He said, “But in fact I want you to write down thenotes of the first exposé.” 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 École Normale,at the end of the seminar said to Grothendieck,“I have written something I’d 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 I’llhave a look at that.” So I gave them to him. Nottoo long afterward, Grothendieck came to me and

2Théorie des topos et cohomologie étale des schémas,Séminaire de géométrie algébrique du Bois-Marie 1963-64,dirigé 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 Grothendieck’s 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 myexposé on local duality, I was very afraid. However,a month later or so, he said, “I’ve 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 o’clock, 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 didn’t 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 couldn’t 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,Dieudonné, Giraud, Jouanolou, Néron, Poitou, Ray-naud and his wife Michèle, Samuel, Serre, Verdier.Of course we also had foreign visitors, some for

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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 IHÉS. That was a place to meetand discuss. Another one was the lunch at theIHÉS, 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 lunch—that was very nice—Grothendieckwould take us for a walk in the woods of the IHÉSand just casually explain to us what he had beenthinking about, what he’d been reading. I remem-ber, once he said, “I’m reading Manin’s paper onformal groups4 and I think I understand what he’sdoing. 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 possède deuxpropriétés caractéristiques : la rigidité, et la facultéde croître, dans un voisinage approprié. Il y a descristaux de toute espèce de substance: des cristauxde soude, de soufre, de modules, d’anneaux, deschémas 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.”)

KünnethBloch: 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. It’sa mess, so, please, clean this up by introducingderived categories, write the Künneth formula inthe general framework of derived categories.” I

3Théorie des intersections et théorème de Riemann-Roch, Séminaire de Géométrie Algébrique du Bois-Marie1966/67, dirigé 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), 3–90. (Russian)5Éléments de Géométrie Algébrique, par A. Grothendieck,rédigés avec la collaboration de J. Dieudonné, Pub. Math.IHÉS 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. I’m 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 didn’t haveto finish your thesis in three years. The completionof a thèse d’État 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 Grothendieck’s 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 naïve, 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

Grothendieck’s.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 andK′0. 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 you’re not in a quasi-projectivesituation, you don’t have any global resolutions forcoherent sheaves. Then it’s better to work with theK-group defined using perfect complexes.” How-ever, he didn’t 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”.

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fonctorisé que moi!”7…(laughter). Grothendieckhad given us detailed notes on lambda opera-tions, which he had written before 1960. Berthelotdiscussed them in his exposés 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 Grothendieck’s ideas about Riemann-Roch. I’msure he wasn’t happy with that!

Illusie: Grothendieck wanted a relative formulaover a general base and for fairly general mor-phisms (locally complete intersection morphisms).Also, he didn’t want to move cycles. He preferredto do intersection theory using K-groups.

Bloch: But he didn’t forget his program of tryingto prove the Weil conjectures?

SGA 7Illusie: No, but he had several irons in the fire. In1967–1968 and 1968–1969, there was another sem-inar, SGA 7,8 about monodromy, vanishing cycles,the RΨ and RΦ functors, cycle classes, Lefschetzpencils. Certainly he had already thought about theformalism of nearby cycles a few years before. Also,he had read Milnor’s 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? Hironaka’s 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. It’s in it that Deligne gave his beautifulexposés on the Picard-Lefschetz formula (at therequest of Grothendieck, who couldn’t understand

7“Berthelot is still more functorized than I am!”8Groupes de monodromie en géométrie algébrique, Sémi-naire de Géométrie Algébrique du Bois-Marie 1967–1969,I dirigé par A. Grothendieck, II par P. Deligne et N. Katz,Lecture Notes in Math. 288, 340, Springer-Verlag 1972,1973.

Lefschetz’s 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 Künneth 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 Dieudonné in EGA IV, there appearsa differential invariant of a morphism, called themodule of imperfection. Grothendieck realized thatΩ1 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, Catégories cofibrées 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 morphismsdidn’t 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 Grothendieck’scomplex as a truncation, and which behaved wellwith respect to composition of morphisms. Inde-pendently, too, Michel André had defined similarinvariants. I got interested in their work and re-alized that in André’s 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 Mao’s 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 1968–1969). He had contemplated giving aseminar on abelian schemes after that but finallydecided to go on studying Dieudonné’s theory for

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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 hedidn’t give a seminar on abelian schemes. I’msure 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 IHÉS 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 1970–1971 he gave a beautifulcourse (together with a seminar) at the Collège 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 Bégueri-Poitou,who had asked for a topic for her thesis fromGrothendieck. It was a bit like with my Künnethformula. I think he proposed to her to write downthe theory of coherent morphisms for toposes,finiteness conditions in toposes. That was hardand thankless, things didn’t 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 didn’tlike it, some thought that what they had writtenwas good and there was no need to improve it.

9L. Bégueri, Dualité sur un corps local à corps résiduel al-gébriquement clos, Mém. Soc. Math. France (N. S.) 1980/81,n. 4, 121 pp.10M. Raynaud, Théorèmes de Lefschetz en cohomologiecohérente et en cohomologie étale, Bull. Soc. Math. France,Mém. n. 41, Supplément au Bull. Soc. Math. France, t. 103,1975, 176 pp.

Grothendieck gave a series of lectures on motivesat the IHÉS. 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 exposéssur la cohomologie des schémas.11

Drinfeld: But it’s not so good for many people,giving a thesis on coherent morphisms of toposes;it’s 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 alsodidn’t 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 Néron’s construc-tion of Néron models. Grothendieck of course hadquite brilliantly used the universal property ofNéron models in his exposés in SGA 7, but he couldnot grasp Néron’s construction.

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

Bloch: I thought, maybe, that was Deligne’s idea.Illusie: No, it was Verdier’s. But Deligne in

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

Drinfeld: Which pages do you mean?

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

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Illusie: In the appendix to Hartshorne’s seminarResidues and Duality,15 I say “Hartshorne’s semi-nar”, but in fact it was Grothendieck’s 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 résultats” 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 didn’t 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. It’s certainly a pity.18

Drinfeld: And the Astérisque 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 Verdier’s 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, 1966.16Catégories dérivées, Quelques résultats (État 0) in [SGA4 1/2, Cohomologie étale, par P. Deligne, Lecture Notes inMath. 569, Springer-Verlag, 1977], pp. 266–316.17Derived functors were defined and studied in the abovementioned “fascicule de résultats”, 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 catégories dérivées des catégoriesabéliennes, édité par G. Maltsiniotis, Astérisque 239(1996).20E.g., in Deligne’s exposé 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. Deligne’scategory 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 likeWaldhausen’s one.

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

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

Illusie: Yes. In fact, I think that G. M. Bergman’sappendix to Mumford’s book on surfaces21 wasalready available at that moment.

Drinfeld: Are there λ-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, Cartier’stheory 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 Cartier’s 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, Cartier’s theory is powerful andhad a strong impact later. But I don’t 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 don’t 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 Quillen’s notes on the cotangent com-plex it was the first time I’d 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 ⊗LB⊗LAB B. I

remember studying for days, puzzling over exactlywhat that meant.

Illusie: But when I said I couldn’t do my Künnethformula, one reason was that such an object didn’texist at the time.

Drinfeld: I am afraid that even now it doesn’texist in the literature (although it may exist insomebody’s 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.

Grothendieck’s TastesIllusie: I realize I didn’t say much aboutGrothendieck’s 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 Messuguière, where some congresses were held.

He tried to get me to go to that place, but it didn’twork 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 Grothendieck’sfavorite books were? You mentioned his favoritemusic…

22D. Gaitsgory, Grothendieck topologies and deformationtheory II. Compositio Math. 106 (1997), no. 3, 321–348.

1112 Notices of the AMS Volume 57, Number 9

Illusie: I don’t remember. I think he didn’t 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 didn’t. He reallywas very interested in arithmetic, but maybe thecomputational aspect of it was not so appealing tohim. I don’t 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 didn’t get interested inthat at the time. I think the junction betweenGrothendieck and Langlands was realized onlyin 1972 at Antwerp. Serre had given a courseon Weil’s theorem in 1967–1968. But after 1968Grothendieck had other interests. And before 1967things were not ripe. I’m 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 didn’twork 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 it’s only long afterwards that he wrote hismanuscript Pursuing stacks. Also,π1(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 Fermat’s 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 exposés, 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 laGéométrie Algébrique) consists of several exposés.He was thinking of doing for motives what he haddone for the Picard scheme, the Hilbert scheme, etc.

There are also three exposés on the Brauer groupwhich are important and useful, but seven exposéson motives would have been even more interesting.However, I don’t 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 Weil’s contributions are fantastic, I myselfthink Grothendieck’s work is still greater.

Drinfeld: But it was Weil who revived the theoryof modular forms in his famous article.23 ProbablyGrothendieck couldn’t have done it.

Illusie: Yes, this is certainly a great contribution.As for Weil’s 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 Weil’s book onKähler varieties,24 I find it a little heavy.

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

Grothendieck’s StyleIllusie: Yes, but I’m not so fond of Weil’s style.Grothendieck’s style had some defects also. Onethat was barely perceptible at the beginning and be-came enormous later is his habit of afterthoughtsand footnotes. Récoltes 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 didn’twork 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 puzzle…Illusie: 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 dévissage.

23A. Weil, Über die Bestimmung Dirichletscher Reihendurch Funktionalgleichungen, Math. Ann. 168 (1967),149–156.24A. Weil, Introduction à l’étude des variétés kählériennes.Publications de l’Institut de Mathématique de l’Universitéde Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann,Paris, 1958.

October 2010 Notices of the AMS 1113

I think one reason why Grothendieck, afterSerre’s talk at the Chevalley seminar in 1958, wasconfident that étale localization would give thecorrect H i ’s is that once you had the correctcohomology of curves, then by fibration in curvesand dévissage 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. It’s the relative situationthat was important. That’s 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 Serre’scriterion 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: Voevodsky’s work is fairly general. Sev-eral people tried to imitate Grothendieck, but I’mafraid that what they did never reached that “oily”character 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 we’re now so used to puttingsome problem into relative form that we forgethow revolutionary it was at the time. Hirzebruch’sproof 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 Serre’s idea to look at χ. 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. It’s Serre who realized thatthe Euler-Poincaré 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 χ was in the K-group…

The Thèse d’ÉtatDrinfeld: So when Grothendieck chose problemsfor his students he didn’t care very much aboutthe problem being solvable.

Illusie: Of course, he cared about the problem,and when he didn’t know how to solve it, he left itto his students. The thèses d’état 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, “It’s fine.” And then I wrote upmy thesis very quickly.

Drinfeld: Were you a graduate student beforethat (when you began attending Grothendieck’sseminar)?

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

Drinfeld: Oh, you were already…Illusie: Yes, it was like paradise. You entered the

École 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, equallyrevolutionary—and intimately linked to the relativeviewpoint—was Grothendieck’s idea of thinking of ascheme in terms of the functor it represents, thus recov-ering a geometric language somewhat concealed in theringed spaces approach.

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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 theÉcole 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 didn’t diminish your chances for futureemployment?

Illusie: No. From 1963 to 1969 I was attachéde recherche, then, from 1969 to 1973, chargéde recherche, and promoted maître de recherchein 1973 (the equivalent of directeur de deuxièmeclasse today). Nowadays if a student after five yearshas not defended his thesis, it’s a problem.

Drinfeld: What has changed…?Illusie: The thèse d’État 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 = attaché d’enseignement et de recherche, ora postdoc).

For a few years we had a transitional system withthe nouvelle thèse (new thesis), similar to the thesiswe have now, followed by the thèse d’État. Now thethèse d’État is replaced by the habilitation. It’s notthe same kind of thing. It’s 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 don’tknow.

Bloch: If we were to go to Google and type in“Grothendieck”…

Illusie: We’d 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 can’t 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… You’re 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 youdidn’t 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 “It’s easy to see,” “It’s easilychecked.” When he was writing EGA, you see, hewas in unknown territory. Though he had a cleargeneral picture, it was easy to go astray. That’spartly why he wanted a justification for everything.He also wanted Dieudonné to be able to understand!

Drinfeld: What was Dieudonné’s contribution tothe EGA?

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

I think Dieudonné’s 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, étale 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.

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