Alexandre Grothendieck 1928–2014, Part 1 , Allyn Jackson, , and , Coordinating Editors

n the eyes of many, Alexandre Grothendieck was the most original and most powerful of the twentieth century. He was also a man with many other passions who did all things his own way, Iwithout regard to convention. This is the first part of a two-part obituary; the second part will appear in the April 2016 Notices. The obituary begins here with a brief sketch of Grothendieck’s life, followed by a description of some of his most outstanding work in . After that, and continuing into the April issue, comes a set of reminiscences by some of the many who knew Grothendieck and were influenced by him.

Biographical Sketch Alexandre Grothendieck was born on March 28, 1928, in . His father, a Russian Jew named Alexander Shapiro, was a militant anarchist who devoted his life to the struggle against oppression by the powerful. His mother, Hanka Reproduced with the permissionGrothendieck of family. the Grothendieck, came from a Lutheran family in Grothendieck as a child. and was a talented writer. The upheavals of World War II, as well as the idealistic paths chosen by his parents, marked university. In 1948 he made contact with leading math- his early childhood with dislocation and deprivation. ematicians in , who recognized both his brilliance When he was five years old, his mother left him with a and his meager background. A year later, on the advice of family she knew in Hamburg, where he remained until age and André Weil, he went to the Université eleven. He was then reunited with his parents in France, Nancy, where he solved several outstanding problems but before long his father was deported to Auschwitz and in the area of topological vector spaces. He earned his perished there. doctoral degree in 1953, under the direction of Laurent By the war’s end the young Alexandre and his mother Schwartz and Jean Dieudonné. were living in , where he was able to attend the Because Grothendieck was stateless at the time, ob- taining a regular position in France was difficult. He held Michael Artin is professor of mathematics at the Massachusetts In- visiting positions in Brazil and the United States before stitute of Technology. His email address is [email protected]. returning to France in 1956, where he obtained a position Allyn Jackson is deputy editor of the Notices. Her email address is in the Centre National de la Recherche Scientifique (CNRS). [email protected]. In 1958, at the International Congress of Mathematicians David Mumford is professor of mathematics at Brown University. in Edinburgh, he gave an invited address that proved to be His email address is [email protected]. a prescient outline of many of the mathematical themes John Tate is professor of mathematics at Harvard University that would occupy him in the coming years. and University of Texas, Austin. His email address is tate@math. That same year he was approached by a French math- utexas.edu. ematician businessman, Léon Motchane, who planned to For permission to reprint this article, please contact: reprint- launch a new research institute. This was the start of the In- [email protected]. stitut des Hautes Études Scientifiques (IHES), now located DOI: http://dx.doi.org/10.1090/noti1336 in Bures sur Yvette, just outside Paris. Grothendieck and

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What differing only in their nilpotent ideals, they have the was traditionally thought to be the underlying point set same . To apply this observation, he is the case that 푆 = Spec(푘). If 푋(푆) is the set of 푆-valued considered what Weil had called a “specialization” of a points of 푋 and 푆 → 푇 is a morphism, composition defines characteristic-zero variety to characteristic 푝. Suppose, a map from 푋(푇) to 푋(푆). Thus 푆 ↦ 푋(푆) is a functor for instance, that a family 푋 of curves is given over the from the category of schemes to the category of sets. 푝-adic integers ℤ푝. Then the fibre 푋0 obtained by working Grothendieck introduced the term representable func- modulo 푝 will be a curve over the prime field 픽푝, and the tor, a functor that is isomorphic to Hom( ⋅ , 푋) for some fibre 푋휂 over the 푝-adic field ℚ푝 will have characteristic object 푋. Moreover, he insisted on the systematic use zero. In this situation one can also consider the scheme of fibred products, using them to define the concept 푛+1 푋푛 obtained by working modulo 푝 , a family of curves of relative representability. A morphism of functors 푛+1 over the ring ℤ/푝 ℤ. The schemes 푋푛 form a sequence 퐹 ⟶ 퐺 is relatively representable if, given a morphism 푋0 ⊂ 푋1 ⊂ ⋯, and they differ only in their nilpotent Hom( ⋅ , 푋) ⟶ 퐺, i.e., an element of 퐺(푋), the fibred elements. So if a covering of 푋0 is given, one can extend product 푋 ×퐺 퐹 is representable. For example, 퐹 is an it to every 푋푛. open subfunctor of 퐺 if for every such morphism, the This approach was revolutionary, though nothing fibred product is represented by an open subset of 푋. technically difficult was needed up to this point. There had been substantial work at this time defining Grothendieck’s biggest step was to go from a family varieties parametrizing certain structures; that is, their of coverings of the sequence {푋 } to a covering of the 푛 points were in one-to-one correspondence with the set of scheme 푋 itself. Once this was done, standard methods all such structures. Chow had defined a union of varieties related the covering of the curve 푋 in characteristic 푝 to 0 whose points parametrize subvarieties of a covering of the characteristic-zero curve 푋 . 휂 of given degree and dimension, Weil had defined varieties It was while studying this last step that Grothendieck whose points parametrize divisor classes of degree zero found a key Existence Theorem. To state that theorem, on a curve, and Baily had defined a variety whose points we begin with a scheme 푋 projective (or proper) over a complete local ring 푅. It might be a curve over the correspond to isomorphism classes of curves of fixed genus over the complex numbers. Grothendieck imme- ring of 푝-adic integers. Let 푅푛 denote the truncation of diately realized that in each of these constructions, one 푅 modulo a power of the maximal ideal, and let 푋푛 should look for a suitable representable functor. Instead be the corresponding truncation of 푋. The schemes 푋푛 form a sequence 푋 ⊂ 푋 ⊂ ⋯ that Grothendieck calls of Chow’s formulation, he considered subschemes of a 0 1 푛 a . Given a 푀 on 푋, one given projective space ℙ with fixed Hilbert polynomial 푃, made it into a functor by looking at flat families of sub- obtains a sequence of coherent sheaves ⋯ ⟶ 푀1 ⟶ schemes, and proved that this functor was represented 푀0 on the schemes 푋푛 by truncation: 푀푛 = 푀 ⊗푅 푅푛. 푛 The Grothendieck Existence Theorem allows one to go by a scheme that he named the Hilb푃(ℙ ). the other way. It states that there is an equivalence He described these ideas in a series of Bourbaki talks in of categories between coherent sheaves 푀 on 푋 and 1959–62 and in a seminar at Harvard in 1961. Once again, recasting old problems in their natural sequences of sheaves 푀푛 on 푋푛 such that 푀푛−1 = 푀푛 ⊗푅푛 푅푛−1. Grothendieck then stated the covering problem in more abstract settings solved old problems. Going back terms of coherent sheaves and was able to complete his to the first decades of the twentieth century, a central proof. problem in the theory of algebraic surfaces 퐹 over the com- Grothendieck’s Existence Theorem is a cornerstone of plex numbers had been showing that the irregularity that 1 modern , and the categorical properties we now call dim 퐻 (풪퐹) was the dimension of the Picard that are necessary for a theorem of that type are still not variety that classifies topologically trivial divisor classes. understood. This had been proven by complex analytic methods by Poincaré, but despite multiple attempts by Enriques and Functors and the Hilbert, Picard, and Moduli Schemes Severi, had not been proven algebraically. Grothendieck’s Prior to Grothendieck’s work, both Weil and Zariski had approach was to define a Picard scheme whose 푆-valued 1 struggled with deciding what should be called the points points correspond to the set of line bundles on 퐹 × 푆. 2 1 of a variety when it was defined over a nonalgebraically Taking 푆 = Spec 푘[푥]/(푥 ), he saw that 퐻 (풪퐹) was the closed ground field 푘: should these be the maximal tangent space to the Picard scheme at the origin. Thus ideals in their affine coordinate rings, or should they be the old problem became: show that the Picard scheme is the solutions of the defining equations in the algebraic reduced, i.e. has no nilpotent elements in its structure closure 푘? And they needed some concept of generic sheaf. But the Picard scheme is a group, and in charac- points; they were first defined by van der Waerden in his teristic zero algebraic groups have an exponential map, classical series of papers on algebraic geometry, in the hence no nilpotents. In characteristic 푝 this need not be same way as Weil and Zariski. This confusion disappeared true, and life is richer. when Grothendieck took the radical step of defining two sorts of points on a scheme 푋: on the one hand, all prime 1Technical point: the line bundles should be trivialized on {푥} × 푆 ideals in the affine coordinate rings of 푋 became the for some rational point 푥 ∈ 퐹.

March 2016 Notices of the AMS 245 In the case of moduli spaces, the functorial approach local isotriviality. A bundle 퐵 over a variety 푋 is locally first solved their local structure using the idea of pro- isotrivial if for every point 푝 of 푋 there is a finite covering representing a functor, 퐹: fix an element 푎 of 퐹(Spec(푘)) 푈′ of a Zariski open neighborhood 푈 of 푝 such that the and seek a complete local ring whose Spec defines the pullback of 퐵 to 푈′ is trivial. Moreover, Kawada and Tate subfunctor of 퐹 of all nilpotent extensions of 푎. Criteria had shown that one could recover the groups for prorepresentability were established by Grothendieck, of a curve in terms of the cohomology of its fundamental Lichtenbaum, and Schlessinger, and for moduli in partic- group. ular this led to the concept of the due M. Artin took up this idea in 1961 when Grothendieck to Grothendieck and Illusie. visited Harvard. Using unramified coverings that were not The global theory of the , however, went finite, i.e. all étale maps, he succeeded in showing that, in two directions. One sought quasi-projective moduli over the complex numbers, one did indeed obtain the schemes and was pursued by Mumford. Grothendieck’s same cohomology with torsion coefficients as with the idea, however, was to find simple general properties of a classical . In retrospect, the étale topology was a functor that characterized those that were representable, natural thing to try, since it is stronger than the Zariski solving special cases like moduli as a corollary. But topology and weaker than the classical topology. It wasn’t Hironaka found a simple 3-dimensional scheme with an at all obvious at the time, because the étale topology isn’t involution whose quotient by this involution fails to be a topology in the usual sense. Open sets are replaced by a scheme; hence schemes themselves need to be further étale maps, which aren’t mapped injectively to the base generalized if there is to be a nice characterization of space. The thought that one could do sheaf theory in such the functors they represent. This led to the concept of a setting was novel. And one needs to work with torsion an , a more general type of object. A coefficients to have a reasonable theory. Cohomology remarkable “approximation” theorem discovered by M. with nontorsion coefficients, which is needed for the Artin in 1969 led to his characterization of the functors Fixed Point Theorem, is defined by an as represented by these spaces in 1971, fully vindicating ℓ-adic cohomology. Grothendieck’s vision. Then Grothendieck proved a series of theorems, no- tably the Proper Base Change Theorem, which allows one Étale Cohomology to control the cohomology of varieties by induction on Interest in a cohomology theory for varieties in charac- the dimension, using successive fibrations and beginning teristic 푝 was stimulated by André Weil’s talk in 1954 at with the known case of dimension 1. The Proper Base the International Congress of Mathematicians (see also Change Theorem concerns a proper map 푋 ⟶ 푆 and a his earlier paper “Numbers of Solutions of Equations in point 푠 of 푆. The theorem asserts that the cohomology 푞 Finite Fields” (AMS Bulletin, vol. 55 (1949), 497–508)). In of the fibre 푋푠 over 푠, 퐻 (푋푠, 퐴), is isomorphic to the 푞 ′ ′ this talk, he compared analytic and algebraic methods limit of the cohomology 퐻 (푋 , 퐴) of pullbacks 푋 of 푋 ′ in algebraic geometry. The problem of defining cohomol- to the étale neighborhoods 푆 of 푠. To prove the theorem, ogy algebraically hadn’t attracted much interest before, Grothendieck adapted a method that had been introduced because the classical topology was available for varieties by Serre. Artin, Grothendieck, and Verdier developed the over the complex numbers. But the culmination of Weil’s full theory jointly at the IHES in 1963–64. talk was his explanation that, because rational points on Grothendieck then defined 퐿-series for cohomology of a variety 푉 over a finite field were the fixed points of arbitrary constructible sheaves. This allowed him in 1964 a Frobenius automorphism, one might be able to count to prove rationality of 퐿-series and to find a functional them by the Lefschetz Fixed Point Formula, which asserts equation, using the Base Change Theorem and Verdier’s that the number of fixed points of an automorphism 휑 duality theorem to reduce to the case of dimension 1. The 푖 ∗ Riemann Hypothesis for varieties over finite fields was is equal to the alternating sum ∑푖(−1) Trace퐻푖(푉)(휑 ) of traces of the maps induced by 휑 on the cohomology. proved by Deligne in 1974. However, a definition of the cohomology groups was required, and the was useless for this. That a definition should exist with the properties Weil predicted became known as the . Grothendieck As I Knew Him There was no problem with cohomology in dimension My first encounter with the whirlwind that was 1 1, because 퐻 (푉, ℤ/푛) can be constructed from the group Grothendieck occurred at the very first, and very of 푛-torsion divisor classes. Therefore the cohomology of small, Arbeitstagung in July 1957. I have vivid curves was understood. In fact, Weil’s conjectures were memories of Grothendieck talking for hours every day, based on the known case of curves, for which the zeta expounding his new K-theory generalization of the function had been analyzed and for which the analogue Hirzebruch-Riemann-Roch Theorem (HRR). According to of the Riemann Hypothesis had been proved by E. Artin, Don Zagier, Arbeitstagung records show that he spoke H. Hasse, and Weil himself. for a total of twelve hours spread over four days. It was Grothendieck’s idea for defining cohomology was to replace open sets of a topology by unramified coverings Michael Atiyah is Honorary Professor at the University of Edin- of Zariski open sets. There were some hints that this burgh and Fellow of Trinity College, Cambridge. His email address might work. Previously, Serre had defined what he called is [email protected].

246 Notices of the AMS Volume 63, Number 3 an exhilarating experience: brilliant ideas, delivered with verve and conviction. Fortunately I was young at the time, almost exactly the same age as Grothendieck, and so able to absorb and eventually utilize his great work. In retrospect we can see that he was the right man at the right time. Serre had laid the new foundations of algebraic geometry, including sheaf coho- Grothendieck, mology, and Hirze- standing on the bruch had developed the full cohomologi- shoulders of cal formalism based on the Chern classes, Bourbaki, looked which he and Borel had streamlined. To to the future… many it seemed that Photo courtesy of theHirzebruch. estate of Friedrich HRR was the culmina- Grothendieck (left) and Michael Atiyah, on a boat trip tion of centuries of algebraic geometry, the pinnacle of during the Arbeitstagung, around 1961. the subject. But Grothendieck, standing on the shoul- ders of Bourbaki, looked to the future, where abstract structural ideas of universality and functoriality would I continued to meet Grothendieck frequently in subse- become dominant. His introduction and development of quent years in Bonn, Paris, and elsewhere, and we had 퐾-theory rested on his mastery of friendly relations. He liked one of my early papers, which and his technical virtuosity, which steamrollered its way derived Chern classes in a sheaf-theoretic framework, through where mere mortals feared to tread. The outcome, based on what became known as “the Atiyah class”. On the Grothendieck-Riemann-Roch Theorem, was a brilliant the other hand, he rather dismissed the Atiyah-Bott fixed functorialization of HRR, which reduced the proof to an point formula, which led to the Hermann Weyl formula for exercise left to Borel and Serre! the characters of representations of compact Lie groups, This great triumph, following his earlier work in as a routine consequence of his general theories. Tech- nically he was right, but neither he nor anyone else had , established Grothendieck as a mathe- ever made the connection with the Weyl formula. matician and led to his receiving a (protesting These two reactions to my own work are illuminating. the Soviet regime, he famously did not attend the 1966 He was impressed by my early paper because it was not Moscow Congress, where the medal was awarded). His part of his general theory, but the Atiyah-Bott result, new philosophy attracted a host of disciples, who to- which I consider much more significant, was only part of gether developed grand new ideas beyond my powers to his big machine and hence not surprising or interesting. describe. There are two episodes in my memory that deserve For me personally his 퐾-theory, together with more to be recorded. The first occurred on one of the famous topological ideas germinating in subsequent Arbeitstagun- boat trips on the Rhine, which were central to the gen, led in the end to topological 퐾-theory as developed by Bonn Arbeitstagung. Grothendieck and I were sitting Hirzebruch and me, resting on the famous Bott periodic- together on a bench on the upper deck, and he had ity theorems. Subsequently, through ideas of Quillen and his feet up on the opposite bench. A sailor came up others, algebraic 퐾-theory emerged as a major framework and told him, quite reasonably, to take his feet off that linked topology, algebraic geometry, and number the bench. Grothendieck literally dug his heels in and theory in a deep and beautiful way with great promise refused. The sailor returned with a senior officer who and daunting problems for the future. This is part of the repeated the request, but Grothendieck again refused. Grothendieck legacy. This process then escalated right to the top. The captain The first Arbeitstagung also had an educational aspect came and threatened to return the boat to harbour, for me. At the Institute for Advanced Study in Princeton and it took all Hirzebruch’s diplomatic skills to prevent in the fall of 1959, Saturday mornings were devoted to a a major international incident. This story shows how detailed technical seminar run by Borel, Serre, and Tate, uncompromising Grothendieck could be in his personal which expounded the algebraic foundations of schemes life and parallels I think his uncompromising attitude in à la Grothendieck. I was a diligent student and learned mathematics. The difference is that in mathematics he enough to deliver a short course of was, in the main, successful, but in the real world his lectures in Oxford, which ended up as my joint textbook uncompromising nature led inevitably to disaster and with Ian Macdonald. It was not quite a best seller, but read tragedy. by students worldwide, mainly because of its slim size My second personal recollection is of Grothendieck and affordability. It also gave the mistaken impression confiding to me that, when he was forty, he would quit that I was an expert on commutative algebra, and I still mathematics and become a businessman. He sounded get emails asking me tricky questions on the subject! quite serious, though I took it with a grain of salt. In fact

March 2016 Notices of the AMS 247 he did essentially leave mathematics around that age, and Pierre Cartier he became an unconventional businessman, operating not in the narrow mercantile world but, as befitted such Some Youth Recollections about Grothendieck a visionary man, on the grand scale of world affairs. The scientific birth of Grothendieck occurred in October Unfortunately the talent that had stood him in good 1948 at age twenty. After getting his licence degree stead in the academic world of mathematics was totally (equivalent to a BS) from the University of Montpellier, he inadequate or inappropriate in the broader world. The obtained a fellowship for doctoral studies in Paris. This compromises that make politics the “art of the possible” year was the beginning of the famous Cartan Seminar. were anathema to Grothendieck. Grothendieck attended it but was not really attracted. He He was a tragic figure in the Shakespearian mould, then moved to Nancy to begin his work on functional the hero who is undone by his own internal failings. analysis, leading to his famous thesis. The very characteristics that made Grothendieck a great My scientific birth occurred in October 1950, when I mathematician, with enormous influence, were also those was accepted as a student at the École Normale Supérieure. that unfitted him for the very different role that he chose I was really eager to learn everything, and there I started for himself in later life. a lifelong interest in and homological Hyman Bass algebra, joined with a lasting friendship with H. Cartan and S. Eilenberg. During this time, Grothendieck’s fame at Nancy de- Bearing Witness to Grothendieck veloped rapidly, and even in Paris (!!) we took notice of Grothendieck had a big, but mostly indirect, impact on my it. I don’t remember exactly when he and I met for the mathematical life. I had only limited personal contact with first time, probably around 1953, at the occasion of some him, but during the late 1960s I was a fairly close witness 2 Bourbaki Seminar. My first acquaintance with his work to the fundamental transformation of algebraic geometry came through L. Schwartz. When Schwartz left Nancy for that he led and inspired. He was a visionary, bigger-than- Paris, we had another mathematical father (the first one life figure. Though prodigiously creative, his massive was H. Cartan). He was very famous for his invention of agenda needed distributed effort, and his stable enlisted “distributions” and taught functional analysis to an enthu- some of the best young mathematical talent in France— siastic following (J.-L. Lions, B. Malgrange, A. Martineau, Verdier, Raynaud, Illusie, Demazure,…—with whom I had F. Bruhat, me). His first seminar in Paris was devoted to closer contact. My main intermediary and mentor in that Grothendieck’s thesis, and I participated actively, taking environment was Serre, another universal mathematician a special interest in the “theorem of kernels” and the but of a totally different style and accessibility. If Serre topological version of Künneth’s theorem. Two rather was a Mozart, Grothendieck was a Wagner. Serre seemed unexpected developments came from Grothendieck’s the- to know the most significant and strategic problems to be sis. First, in France, there was a fruitful collaboration addressed across a broad expanse of mathematics, and between H. Cartan, J.-P. Serre, and L. Schwartz using he had an uncanny sense of exactly where to productively deep analytical methods to put the finishing touch on direct the attention of other individual mathematicians, the cohomology theory of complex-analytic functions. of whatever stature, myself included. Then, on the other side, Gelfand, in the then-Soviet Union, The Grothendieck seminar at IHES, though small in used topological tensor products and nuclear spaces for numbers, was intense, almost operatic. On one occasion, applications to probability theory (Minlos’s theorem and Cartier was presenting and struggling with Grothendieck’s random distributions) and mathematical (quan- questioning of the proof of a lemma. At one point, tum field theory). It would be interesting to trace the Grothendieck said, “Si tu n’as pas ça, tu n’as rien!” I transition in Grothendieck’s work from functional analy- remember feeling that the events of this period were sis to algebraic geometry. I plan to develop this some day, an important human as well as mathematical story and but this is not the proper place. that it was sad that there was no historian with the tech- The period in which we were very close is approximately nical competence to capture its intellectual and human from 1955 to 1961, and there Bourbaki plays a major role. dimensions in depth. I vividly remember one of our first encounters, which Grothendieck’s influence on my own work began with took place at the Institut Henri Poincaré. It was in March the exposition, by Borel and Serre, of Grothendieck’s proof 1955 at the Bourbaki seminar after a special lecture that of his generalized Riemann-Roch Theorem. This seminal Grothendieck gave about convexity inequalities. He told paper sowed the birth of both topological (Atiyah and me: “Very soon, both of us will join Bourbaki.” I began Hirzebruch) and algebraic 퐾-theory. The latter occupied more than two decades of my ensuing work, mostly at Pierre Cartier is an emeritus research professor at the Cen- the periphery of the Grothendieck revolution. tre National de la Recherche Scientifique, a visitor at the Institut des Hautes Études Scientifiques, and an associate mem- ber of the University of Paris-Diderot. His email address is Hyman Bass is a professor in the School of Education and the [email protected]. Samuel Eilenberg Distinguished University Professor in the De- 2At the time, this was the general meeting, three times a year, of partment of Mathematics at the University of Michigan. His email all French mathematicians. The French Mathematical Society was address is [email protected]. then a charming sleeping old lady!

248 Notices of the AMS Volume 63, Number 3 regularly attending Bourbaki meetings in June 1955. in the 1970s Grothendieck referred Grothendieck joined soon and participated actively from repeatedly to those books. Légaut He was 1956 to 1960. In June 1955 one of the most interesting had left mathematics to raise sheep pieces to read during our meeting was a first draft of and became the guru of a kind always a his famous Tôhoku paper, where he gives a new birth of phalanstery, long before the to homological algebra. One of the major challenges at wave of hippy communes. With the rebel. the time, especially after the appearance of Serre’s paper proper instructions of H. Cartan, “Faisceaux algébriques cohérents” in 1955, was to devise Grothendieck and I walked a long way together to visit a theory of valid for the most general this guru. On the way, he confessed to me that mathe- topological spaces (especially not Hausdorff and not matics was 99 percent labor and 1 percent excitement locally compact). What was required was the construction and that he wanted to leave mathematics to write novels of an injective resolution, but no one knew how to make and poetry. Which he did in the end! This was around it for sheaves.3 In what later became his favorite method, the time of his mother’s death, and it is known that his 7 Grothendieck solved the problem from above: looking for mother wanted to be a writer. At one of our meetings, he the axiomatic properties required of a category to admit brought his mother, who remained shy. injective resolutions, and then checking that the category During a Bourbaki meeting in the summer of 1960, of sheaves on the most general satisfied there was a clash between Weil and Grothendieck. It these properties. started rather unexpectedly during our reading of a Let us come back to Bourbaki. There was a turning point, report by Grothendieck about differential calculus. Weil a change of generation. The so-called first part (in six made one of his familiar unpleasant remarks that no one series) was devoted to the foundations, basing everything took seriously, except Grothendieck, who immediately on and the pervasive notions of isomorphisms left the room and did not come back for a couple of days. and structures. At that time, the publication of this first Both were uneasy characters, and we didn’t understand part was well under way,4 but what should come after? what was especially at stake. Despite diplomatic efforts Among many other projects, it was felt that geometry— of S. Lang and J. Tate, Grothendieck didn’t reconsider his both differential geometry, heritage of Elie Cartan,5 and self-imposed exile from Bourbaki. algebraic geometry, dear to Chevalley, Lang, Samuel, I would need much more space to tell the long tale of Serre, and Weil—was a cornerstone. We wanted to give the political activities of Grothendieck in the 1970s. He a unified presentation of all kinds of manifolds, and was always a dissident among the dissidents (think of there were three competing proposals: ringed spaces the ). Even if your political line was rather (Cartan, Serre), local categories of “charts” (Eilenberg), close to his own, it was often a painful experience to and a more algebraic version of differential calculus (Weil, be on his side, because he wanted to refuse any kind of Godement, Grothendieck). None was finally accepted, but compromise—and this was the way he always lived his Grothendieck used them all in his theory of schemes. life. He was always a rebel. Let me add a few personal recollections. All these summer meetings of Bourbaki took place in the Alps, 6 first near Die in Etablissement Thermo-résineux de The first time I attended Grothendieck’s seminar, early Salières-les-Bains, a kind of elaborate sauna, then in in 1965, I followed his lecture tenuously. I knew what Pelvoux-le-Poët, in a quiet inn in the mountains. In Die, I cohomology groups were but could not understand the remember the late arrival of Grothendieck; having missed expression “objet de cohomologie,” which kept recurring. an appointment with Serre, who wanted to bring both of After the lecture, I asked him what it meant. Very gently, them by car, he missed another appointment with us for he explained that if in an the composite a night train, then took the wrong train and ended up 푓푔 is zero, the kernel of 푓, divided by the image of 푔, is hitchhiking from Valence to Die! Serre was not especially the cohomology object. happy. Another time, he handed me a document to read, I view his tolerance of what appeared to be crass where, between the pages, was a letter (in German!) from ignorance and his lack of condescension as typical of an unhappy Brazilian girlfriend. him. It encouraged me to not refrain from asking “stupid” I remember a less exotic event. In the vicinity of Die, questions. deep in the mountains, lived Marcel Légaut, who was an He taught me my trade by asking me to write up, using old friend of H. Cartan and A. Weil. Weil’s autobiography his notes, the talks XVII and XVIII of SGA4. By “trade” refers to Légaut as an author of “works of piety,” and I mean both a feeling for the cohomology of algebraic varieties and how to write. My first draft was returned 3After the publication of Grothendieck’s Tôhoku paper, Gode- with comments and injunctions: “never use both sides of a ment gave an elementary construction of injective sheaves in his page,” “keep ample empty space between lines,” as well as well-known textbook. 4 A number of years were still required to finish it, revise it, and 7She wrote a kind of autobiography entitled Eine Frau (A Woman), produce a so-called “final version”. in German. 5 The then-deceased father of Henri Cartan. Pierre Deligne is professor emeritus of mathematics at the In- 6Site of the family summer house of H. Cartan, where he was the stitute for Advanced Study in Princeton. His email address is regular organ player in the Huguenot church. [email protected].

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Grothendieck Strasbourg, to relation of my University the at “alien”. an was he different; Grothendieck essentially workers. seemed harder always faster, brighter, were they production the in him assisted SGA3. of I thesis and my (1962–64), was completed Grothendieck adviser and service, instead France military to of jee” returned years sub I two After the “jay jay”. pronouncing and sub following primitive, “jee by of talk very listeners was the a English lost gave My I I. I EGA the of where at manuscript Study Princeton Advanced in Institute the for at 1959–60 seminar a remember year I and academic college, graduate the spent the I obviously and (new, geometry algebraic infancy “new” the the enthusiasm of with watching talks, his ( hard-to- followed and “PTT” heavy two and his ( through “EVT” thirtieth monographs, him my read met and first twentieth I my year. between past, common a Amicalement,—October l’autre. et l’un impliquent nous f“’ne ’ur”isedo h biu tu e deux” les “tous obvious the of instead l’autre” et “l’un of si ehad he if as abstraction h rnhverb French The enthere. been Rereading of view his with ease at really felt never I say must I professor a became I 1964, fall in thesis, my After n h tog“impliquent”: strong the And eesof levels climbed already happily [He] rdistnoil topologiques tensoriels Produits éotse Semailles et Récoltes impliquer ekyhl-a itn thsside his at sitting half-day weekly togadilmntn tws The was. how it illuminating know and “advice” strong his from bene- fited having of experience unique sIbte arc ob ue as sure, be I to who fabric fabric, same career, I—better the my as of made in were felt met mathematicians have as great him I other view did not did I I already had there. he been if as abstraction of levels way climbing happily they along, the came as things by (saw!) amazed discovered he was I never I’ll forget. something com- is or sheets mon parallel on scribbling and hs h hr ihm the me with share who Those a eudrto ntowy:sim- ways: two in understood be can sae etrestopologiques vectoriels Espaces 8 e,Ia ipiae”by “implicated” am I yes, oue6,Nme 3 Number 63, Volume n lohscorrespon- his also and ,adthen and ), ) mathematics about? Of course, I am really pleased when thesis for publication using that language. Moreover, I see (or in a few cases contribute to) the perfection of a that foundation showed that I had discovered a very general tool, but the pleasure is much greater when I see natural, useful functor (Grothendieck later incorporated what such tools say in specific situations, where there is that functor as part of his general theory of “descent”). not enough room for those tools (size, dimension,…) or While teaching at Berkeley, I heard that Grothendieck when they collide. I remember Robert Steinberg saying, “It would visit Harvard in fall 1961, so I obtained a fellowship is a pity there are so few simple Lie groups and that most to learn from him there and also at the IHES in Paris in of them are classical.” He would have been happy (and so spring 1962. would I) had the number of exceptional Lie groups been My first impression on seeing Grothendieck lecture larger! I think this pleasure in exceptions was foreign to was that he had been transported from an advanced alien Grothendieck. civilization in some distant solar system to visit ours in The second component centers on the question of “com- order to speed up our intellectual evolution. His shaven putation”, which takes a large place in Grothendieck’s head, his rapid, intense, commanding mode of speaking, correspondence with (and controversy about) Ronald plus the new concepts and generality of his view of Brown. I always liked to compute (I even spent the sum- algebraic geometry conveyed that impression. mer of 1955, before entering École Normale Supérieure, in I recall a lecture he gave at Harvard about Hilbert the first computer company in France). For me, a complete schemes, at the end of which he suddenly announced that mathematical theory should lead to effective computa- he could develop a certain topic much more generally. tions. Grothendieck did not like computations (and hated Professor , who was in the audience, stopped computers!). He wrote to Brown: “The question you raised Grothendieck from speaking overtime, asking him to ‘how can such a formulation lead to computations’ doesn’t “please exercise a little self-control.” bother me in the least!” It is striking to compare this to In Paris I attended Grothendieck’s lectures that were what Voevodsky, whom I see as Grothendieck’s true con- later published as SGA. The lectures were overwhelming, tinuator, wrote thirty years later: “It soon became clear and I was also somewhat intimidated by his forceful per- that the only real long-term solution to the problems sonality. that I encountered is to start using computers in the Nevertheless, during an in- verification of mathematical reasoning.” termission in one of his What I have written might give a wrong impression and I told him presentations, I approached hide how much I owe to Grothendieck—as well as to Serre about an him and attempted to in- and Tits—and how intellectually enriched I have been by formally chat with him. I him. One cannot get rid of the “Grothendieck way”. For excellent told him about an excellent years, when I was stuck while struggling with a problem, symphony concert I had at- I used to ask myself, what would Grothendieck say? Most symphony I tended the night before that certainly: If you just had stated the problem in the right had cost me only a few way, you’d see the answer in the question. had attended francs. His firm response If there is something like a “space of mathematics”, was, “Ah, but it also cost you I see Grothendieck as an extremal point, and maybe that cost me your time!” Grothendieck ev- so extremal as to be felt outside. In the “space-time only a few idently worked so hard on of mathematics” there was a time interval in which I mathematics that he spent came into contact with that extremal point. That was a francs. His firm very little time on anything crucial period of my life, when I was, to use his wording, else. “implicated” by him and with him. response…“Ah, Feeling utterly out of place attempting to relate to Marvin Jay Greenberg but it also cost such a formidable person, I was subsequently surprised you your time!” and elated when he invited Memories of Alexandre Grothendieck me to dine with him and his My 1959 thesis, which proved a conjecture in arithmetic wife at his home. It was a working-class, unpretentious algebraic geometry by Serge Lang, introduced a technique abode. His wife was busy caring for their young baby. that seemed complicated. When I learned, from notes Grothendieck wasted very little time making small talk. by Dieudonné, about Grothendieck’s new foundation for With paper and pen at hand, he spent nearly the entire algebraic geometry based on schemes, I rewrote my time sketching ways to use the functor I had found. I couldn’t follow what he was suggesting. He also urged me Marvin Jay Greenberg is professor emeritus of mathematics at to work on presenting, within the framework of schemes, the University of California at Santa Cruz. He is the author of A. Neron’s important minimal models theory, which had Euclidean and Non-Euclidean : Development and His- tory (4th edition 2008, W. H. Freeman, NY, NY) and of the Lester been written in the old language of Weil’s foundations. I Ford Award-winning article ”Old and New Results in the Foun- did begin studying Neron’s publications. Three years later dations of Elementary Euclidean and Non-Euclidean Geometries” I was able to push through a little of what Grothendieck (Amer. Math. Monthly) March, 2010, 198–219. His email address had suggested. With the help of Michael Artin, I took is [email protected]. one result in Neron’s work, expressed it in the language

March 2016 Notices of the AMS 251 of schemes, and proved a new version of it in much seminar. My job was to digest them, fill in details, give the greater generality. Grothendieck arranged to have this seminar talks, and then write up the notes. This was quite work appear in the Publications IHES. a challenge, as it included the first occurrence of Verdier’s I had no further direct interaction with Grothendieck theory of the and Grothendieck’s use after that publication, but other connections to him did of it in developing the duality theory for a morphism of arise. For example, I taught a course at Crown College, schemes. I regard this period as my “apprenticeship” with UC Santa Cruz, called The Quest for Enlightenment, in Grothendieck. We had a constant interchange of letters, part presenting the teachings of J. Krishnamurti. Many as I sent him drafts of the seminar talks, and he returned years later Grothendieck, in his Récoltes et Semailles, listed them covered with red ink. In this way I learned the craft Krishnamurti as one of eighteen enlightened masters of of exposition in his style. After the lecture notes were our age. published (Residues and Duality, Springer Lecture Notes Grothendieck’s copious output and originality in math- 20, 1966), I did not see him so often. But some time later ematics demonstrated a level of intellectual achievement he did ask me, “Well, those lecture notes were a good I never imagined was possible by one man. I will forever rough account, but when are you going to write the book be grateful that he took a little time to kindly inspire me on duality?” I did not answer that because I was already to contribute a bit. There seems to be a consensus that moving in other directions. Grothendieck went mad in his later years. I strongly dis- The last time I saw Grothendieck in person was in agree with that consensus. It is the madness of ordinary Kingston, Ontario, in 1971. He had so withdrawn from society that eventually drives geniuses like Grothendieck engagement in mathematics that he devoted equal time (and more recently Grigory Perelman) to withdraw. in his talks to his new brainchild “Survivre”. I could appreciate the sincerity of his beliefs but felt he was hopelessly naïve about political action. When I finished my book Algebraic Geometry in 1977, Reflections on Grothendieck which is basically an introduction to Grothendieck’s way of thinking using schemes and cohomology, I sent him a After majoring in math at Harvard, I spent a year at the copy together with a note of thanks and appreciation for École Normale Supérieure in Paris. I had courses with all that I had learned from him. He sent a polite card in Cartan, Serre, and Chevalley and learned some general reply, saying, “It looks like a nice book. Perhaps if one topology and sheaf theory. After becoming a graduate day I again teach a course on algebraic geometry, I will student at Princeton, I started reading Serre’s article look at the inside.” “Faisceaux algébriques cohérents” and thought I would Near the beginning of his rambling reflections Récoltes like to study algebraic geometry. At that time there was et Semailles, Grothendieck mentions “les héritiers et le no algebraic geometry at Princeton, so the fall of 1961 bâtisseur,” the heirs and the constructor. As an heir of the found me back at Harvard, and there was Grothendieck. master builder Grothendieck, I am now happily inhabiting He gave a lecture course on local properties of mor- several of the rooms he built and using his tools to refine phisms, which later became part of EGA IV, and he my understanding of classical geometry. I owe him the gave two seminars, one on and one on inspiration for my life work. construction techniques—the Hilbert scheme, the Picard scheme, and so forth. I could see that his was “the right way” to do things and jumped headlong into his world. In 1963 I finished my thesis, which was on the connected- Alexander Grothendieck was a professor at the IHÉS ness of the Hilbert scheme. While Grothendieck was not from 1959 until 1970. In the seminars he led—the famous my official advisor, nor did I discuss the work in progress SGA—a team of students coalesced around him, exploring with him, I am sure it was the stimulating atmosphere of the new territories that “the Master” had discovered. We discovery he created that provided the context for me to were many, coming from various corners of the world, to be able to do this work. participate in this adventure, which constituted a sort of I sent a draft copy of my thesis to Grothendieck. He golden age of algebraic geometry. responded with a long letter, containing a few sentences The seminars took place at the IHÉS on Tuesday of appreciation for the result and then many pages of afternoons and spread out over a year or two. They further questions about the Hilbert scheme, most of were held in a former music pavilion that had been which are still unanswered today. Each new result he Luc Illusie is professeur honoraire at Université de Paris–Sud. His encountered gave rise to a myriad of further questions to email address is [email protected]. investigate. This is a slightly edited translation of a piece, “Grothendieck A couple of years later I offered to run a seminar at était d’un dynamisme impressionnant”, which appeared in Harvard on his theory of duality, which he had hinted at CNRS, Le journal (https://lejournal.cnrs.fr/billets/ in his ICM talk in 1958 but had not yet developed. He grothendieck-etait-dun-dynamisme-impressionnant), and agreed and sent me about 250 pages of “prenotes” for the is an excerpt of a longer article “Alexandre Grothendieck, magicien des foncteurs”, which appeared in the online publi- Robin Hartshorne is professor emeritus of mathematics at the cation of the Mathematical Institute of the CNRS (www.cnrs.fr/ University of California, Berkeley. His email address is robin@ insmi/spip.php?article1093). The Notices thanks the editors math.berkeley.edu. of both publications for permission to include this piece here.

252 Notices of the AMS Volume 63, Number 3 transformed into a library and lecture hall, with large Nicholas Katz picture windows onto the Bois-Marie park. Occasion- ally before the lectures, the Master took us for a There is no need, I hope, to discuss the mathemat- walk in the woods to tell us about his latest ideas. ical achievements and the mathematical vision of The seminars were mainly about his Grothendieck. What is perhaps less known to people who own work. There were also related did not interact with Grothendieck personally was his [He] was results, of which he sometimes en- incredible charisma. We thought of him (as he did of always trusted the exposition to students himself, as he says in Récoltes et Semailles) as the boss or colleagues. For instance, he asked (patron) of a construction site (chantier). When he asked clear and Deligne, in the seminar SGA7, to someone to carry out some work that would be part transpose into the setting of étale of this, the person asked felt that he or she had been methodical. cohomology the classical Picard- honored to have been asked, was proud to have been Lefschetz formula, whose proof he asked, and was delighted to undertake the task at hand No black confessed to me he had not under- (which might take many years to complete). Combined stood. This étale analogue was later with this charisma, Grothendieck had an uncanny sense boxes, no to play a key role in the proof, of whom to ask to do what. One sees this in looking at by Deligne, of the Riemann hy- the long list of people whose work became an essential sketches. pothesis over finite fields. At the part of Grothendieck’s chantier. blackboard Grothendieck had an impressive dynamism but was always clear and methodical. No black boxes, no Steven L. Kleiman sketches—everything was explained in detail. Occasion- The first time I saw Grothendieck was in September 1961 ally he omitted a verification that he considered purely at Harvard. I was an eager new graduate student; he, routine (but that could turn out to be more delicate than a second-time visitor teaching a course. He started by it had appeared). After the lecture, the audience went explaining he’d cover some preliminaries to appear in [4]: to have tea in the administrative building. This was an the course would be elementary; the prerequisites, just opportunity to discuss various points from the seminar the basics. and exchange ideas. Soon I found my three terms of graduate alge- Grothendieck liked to ask his students to write up his bra as an MIT undergraduate hadn’t prepared me for lectures. In this way, they learned their craft. When it Grothendieck’s course. So I dropped it and skipped his came to editing, he was tough and demanding. My typed two weekly seminars developing the Picard scheme and manuscripts, which might reach fifty pages, would be local cohomology. I believe he assigned no homework and blackened all over with his critiques and suggestions. I gave no exams in the course; at the end he unexpectedly remember reviewing them one by one over the course collected the notebooks of the registered students and of long afternoons at his home. The results had to assigned grades. be presented in their natural framework, which usually Grothendieck began each meeting of the course by meant the most general possible. Everything had to be erasing the board and then writing 푋 ⟶푓 푌 vertically. One proved. Phrases like “it is clear” and “one easily sees” were time before Grothendieck arrived, John Fogarty, another banished. We discussed the mathematics point by point, graduate student, erased the board and wrote 푋 ⟶푓 푌 but also punctuation and the order of words in a sentence. vertically. When Grothendieck arrived, he looked at the Length was not an issue. If a digression looked interesting, board and silently erased it. Then he began his lecture, it was welcome. Very often we were not finished before 8 writing 푋 ⟶푓 푌 vertically. o’clock in the evening. He would then invite me to share Fogarty had considerable skill as a caricature artist. a simple dinner with his wife, Mireille, and their children. One day he drew a large, lovingly detailed cartoon on the After the meal, as a form of recreation, he would explain blackboard in the common room. It showed a side view of to me bits of mathematics he had been thinking about Grothendieck with a quiver of arrows on his back, looking lately. He would improvise on a white sheet of paper, ahead where he’d written 푋 ⟶푓 푌 vertically. with his large pen, in his fine and rapid hand, stopping Thus Fogarty satirized one of Grothendieck’s signature occasionally at a certain symbol to once again run his insights: it pays off in better understanding and in greater pen over it in delight. I can hear his sweet and melodious flexibility to generalize absolute properties of objects 푋 voice, punctuated from time to time with a sudden “Ah!” to relative properties of maps 푋 ⟶푓 푌. when an objection came to mind. Then he would see me Grothendieck’s paper [2] was highly regarded in the off at the station, where I would take the last train back student Algebraic Geometry Seminar, which I joined in to Paris. fall 1962. Grothendieck had upgraded sheaf cohomology: he found enough injectives to resolve sheaves and yield

Nicholas Katz is professor of mathematics at Princeton University. His email address is [email protected]. Steven L. Kleiman is emeritus professor of mathematics at the Massachusetts Institute of Technology. His email address is [email protected].

March 2016 Notices of the AMS 253 their higher cohomology groups as derived functors. Thus he demoted Čech cohomology: taken as the definition in Jean-Pierre Serre’s [9], it became just a computational device. Later Grothendieck went further. He generalized the very notion of topology! In an open covering of 푈 by 푈푖 for 푖 ∈ 퐼, the maps 푈푖 → 푈 needn’t be inclusions, just members of a suitable class. For example, they could be étale, his generalization of the local isomorphisms of analytic spaces. Michael Artin began the systematic development of in a Harvard seminar in spring 1962. I didn’t attend but did lecture from [1] in a student seminar at Woods Hole, July 1964. The ensuing work of Grothendieck and collaborators (especially Artin, Jean- Louis Verdier, and Pierre Deligne) culminated in the resolution of André Weil’s celebrated conjectures, just as Photo courtesy of the estate of Friedrich Hirzebruch. Grothendieck [3, p. 104] had predicted. I learned more of Grothendieck’s innovations in David Grothendieck around 1965. Mumford’s course, spring 1964, published as [7]. It was devoted to Grothendieck’s proof of completeness of the characteristic system of a good complete algebraic system said, “I would appreciate knowing a simple proof,” of the of curves on a smooth projective complex surface 퐹. It key ingredient, the Nakai–Moishezon criterion. is the first algebraic proof of a theorem with a long, rich, I sent Grothendieck a reprint of my first paper [5], and colorful history (see [6]). where I had simplified Nakai’s proof and extended it to “The key,” Mumford wrote on p. viii, “… is the a nonprojective 푛-fold as announced by Moishezon. He systematic use of nilpotent elements” to handle higher- replied with more comments, and in a PS he gave his order infinitesimal deformations. That’s another of opinion on the history of the development of the crite- Grothendieck’s signature insights. Yet another is to use rion. The body was typewritten, but the PS, handwritten, flatness to formalize the notion of algebraic system. showing he’d thought more about it and really wanted Moreover, Grothendieck proved a complete algebraic everyone to receive proper credit, not because it’s due, system is parameterized by a component 퐻 of his Hilbert but to indicate how ideas develop. scheme of 퐹; namely, 퐻 classifies all systems via maps Grothendieck’s letters show impressive clarity and into 퐻. thoroughness. They pose questions, indicating a wish to Grothendieck showed the Theorem of Completeness continue the discussion. They suggest being generous simply provides conditions for 퐻 to be smooth at a given with ideas while acknowledging their provenance. His point. To prove the conditions work, he used his Picard letters are complimentary and encouraging. This is the scheme 푃 and the map 퐻 → 푃. In an ingenious sense, 푃 way to do collaborative mathematics! classifies families of line bundles: its functor of points, that Grothendieck agreed to supervise my NATO postdoc is, functor of maps 푇 → 푃, isn’t equal to the naive functor 1966–67, and I returned to his institute, the IHES, the of line bundles on 푇×퐹, but rather to its associated sheaf summers of 1968 and 1969 and the spring of 1970. In [6], in the étale Grothendieck topology. I discussed my mathematical experiences. Mumford sent a preliminary copy of [7] to Grothendieck, Socially, Grothendieck had me and others over to his who commented in a letter [8, pp. 693–6] dated August house for dinner several times. The last time, in spring 31, 1964. Mumford’s numerical characterization of good 1970, I brought my new wife. Beverly remembers “a feeling systems reminded Grothendieck about his conjectural of trepidation, as he was a living legend. However, the numerical theory of ampleness. In particular, on an 푛-fold, minute we entered his home, it was apparent that he just as on a surface, a divisor should be pseudoample if was an exceptional person, gracious and attentive. Not it meets every curve nonnegatively. More generally, the for an instant did I feel my deficiency in mathematics ample divisors should form the interior of the polar cone and French was something that even occurred to him. His of the numerical cone of curves. In [8, p. 701], Mumford genuine interest and participation in conversation, the replied he didn’t know if the conjectures are true, even general atmosphere of inclusion, is something I’ve always on a 3-fold, but he’d ask me. remembered.” Shortly afterwards, I proved Grothendieck’s first con- Spring 1970 was hard on Grothendieck, as his era at jecture; in January, the second. Then I used the second the IHES ended. Outwardly, he didn’t show his feelings, to prove Chevalley’s conjecture: a complete smooth vari- but people did talk about what was happening. I never ety is projective if any finitely many points lie in some saw him again and heard from him only once more when affine subset. In April, Mumford suggested I write to he sent me his four volumes of Récoltes et Semailles, Grothendieck. Grothendieck replied with comments and with this inscription opposite a picture of himself as

254 Notices of the AMS Volume 63, Number 3 an adorable six(?)-year-old: “To Steven Kleiman, with my friendly regards, Oct. 1985, Alexander Grothendieck.” References [1] Michael Artin, Grothendieck , 1962 (English). [2] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119–221 (French). MR0102537 (21 #1328) [3] , The cohomology theory of abstract algebraic varieties, Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge Univ. Press, New York, 1960, pp. 103–118. MR0130879 (24 #A733) [4] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR0173675 (30 #3885) [5] Steven Kleiman, A note on the Nakai-Moisezon test for am- pleness of a divisor, Amer. J. Math. 87 (January 1965), 221–226. MR0173677 (30 #3887) [6] Steven L. Kleiman, The Picard scheme, Alexandre Grothendieck: A mathematical portrait, Int. Press, Somerville, MA, 2014, pp. 35–74. MR3287693 [7] David Mumford, Lectures on Curves on an Algebraic Sur- face, with a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, NJ, 1966. MR0209285 (35 #187) [8] , Selected Papers, Volume II, On algebraic geometry, in- cluding correspondence with Grothendieck, edited by Ching-Li Chai, Amnon Neeman and Takahiro Shiota, Springer, New York, 2010. MR2741810 (2011m:14001) [9] Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR0068874 (16,953c)

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