Alexandre Grothendieck 1928–2014, Part 1 Michael Artin, Allyn Jackson, David Mumford, and John Tate, Coordinating Editors

n the eyes of many, Alexandre Grothendieck was the most original and most powerful mathematician 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 mathematics. After that, and continuing into the April issue, comes a set of reminiscences by some of the many mathematicians who knew Grothendieck and were influenced by him. the permission of the Biographical Sketch Alexandre Grothendieck was born on March 28, 1928, in Berlin. His father, a Russian Jew named Alexander Shapiro, was a militant anarchist who devoted his life to the struggle Reproduced with against oppression by the powerful. His mother, Hanka Grothendieck family. Grothendieck, came from a Lutheran family in Hamburg 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 Paris, 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 Henri Cartan 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 Montpellier, 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 fundamental group. 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 projective space 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 formal scheme. Given a coherent sheaf 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 Hilbert scheme 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 algebraic geometry, 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 cohomology 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 cotangent complex 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 moduli space, 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 topology. 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 algebraic space, 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 inverse limit 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 Zariski topology was useless for this. Michael Atiyah That a definition should exist with the properties Weil predicted became known as the Weil Conjectures. 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, Bonn 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 of the estate of Friedrich on the Chern classes, Bourbaki, looked which he and Borel had streamlined. To to the future… many it seemed that Photo courtesy Hirzebruch. 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 homological algebra 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 functional analysis, established Grothendieck as a mathe- ever made the connection with the Weyl formula. matician and led to his receiving a Fields Medal (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 commutative algebra 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 algebraic topology 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 physics (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 sheaf cohomology 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 topological space 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 set theory 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 Vietnam War). 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, Pierre Deligne 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 abelian category 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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Photo courtesy of the estate of Friedrich Hirzebruch. rtedekin Grothendieck [email protected]. ouea mg from image an use To In to not learned also have I example, his and him From What Master. my was he that fortunate extremely feel I mdel rtflfrhshligm obcm a become to me helping his for grateful deeply am I éotse Semailles et Récoltes a tteÉoePltcnqefo 1976–1999 from Polytechnique École the at was 1965. ntefis ae poieapooof photo a opposite page, first the On . éotse Semailles et Récoltes rtedekciiie me criticized Grothendieck tta time that at , oie fteAMS the of Notices rgt n,t hs fu fteyugrgeneration). younger the of us of those to one, “right” choice the and “past/present”, balanced the “reflection”, etiswy(culySA a aetei n did and parenthesis a was SGA3 (actually way its went of meaning dual the from weighed, carefully was word (both). 95 lxnr rtedek”A sa ihhm every him, with usual As Grothendieck.” Alexandre 1985, osipiun ’ne ’ur.Amicalement,—October l’autre. et l’un impliquent nous l s“ml” si AipisB,o s“mlct” si Ais “A in as “implicate”, as or B”, B”. against implies crime “A the my in in implicated as of “imply”, is as all, component ply after first What, the question: 8 the find on I centers Serre, uneasiness with dence components. it two understand are I I There words. him, now. into with better uneasiness contact this had put I not when could time the At mathematics. was new I and interests and new responsibilities. with mainstream), joined Orsay, I in SGA later Paris-Sud years Université Two the it. follow to to unable geographically belong really distance, not the seminar with SGA The and weakened. 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 deux” les “tous obvious the of instead l’autre” et “l’un of as abstraction h 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 “impliquent”: strong the And eesof levels climbed already happily fh had he if [He] verb 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 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 hr ihm the me with share ,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 Oscar Zariski, 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 Geometries: 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 derived category 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 Robin Hartshorne 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 local cohomology 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 Luc Illusie 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 Grothendieck topology in a Harvard seminar in spring 1962. I didn’t attend but did lecture from [1] in a student of the estate of Friedrich Hirzebruch. 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 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 Topologies, 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)

March 2016 Notices of the AMS 255 How Grothendieck Simplified Algebraic Geometry of the IHES. Photo courtesy Colin McLarty

256 Notices of the AMS Volume 63, Number 3 The idea of scheme is childishly simple—so simple, The Beginnings of Cohomology so humble, no one before me dreamt of stooping Surfaces with holes are not just an amusing so low.…It grew by itself from the sole demands pastime but of quite fundamental importance for of simplicity and internal coherence. the theory of equations. (Atiyah [4, p. 293]) [A. Grothendieck, Récoltes et Semailles (R&S), pp. P32, P28] The Cauchy Integral Theorem says integrating a holo- morphic form 휔 over the complete boundary of any Algebraic geometry has never been really simple. It region of a Riemann surface gives 0. To see its signifi- was not simple before or after David Hilbert recast it in cance look at two closed curves 퐶 on Riemann surfaces his algebra, nor when André Weil brought it into number theory. Grothendieck made key ideas simpler. His schemes that are not complete boundaries. Each surrounds a hole, give a bare minimal definition of space just glimpsed as and each has ∫퐶 휔 ≠ 0 for some holomorphic 휔. early as Emmy Noether. His derived functor cohomology Cutting the torus along the dotted curve 퐶1 around the pares insights going back to Bernhard Riemann down to center hole of the torus gives a tube, and the single curve an agile form suited to étale cohomology. To be clear, étale 퐶1 can only bound one end. cohomology was no simplification of anything. It was a ...... radically new idea, made feasible by these simplifications...... ⋆ .... Grothendieck got this heritage at one remove from ...... the original sources, largely from Jean-Pierre Serre in ...... 퐶1 퐶2 ...... ⋆ shared pursuit of the Weil conjectures. Both Weil and ...... Serre drew deeply and directly on the entire heritage. The original ideas lie that close to Grothendieck’s swift The punctured sphere on the right has stars depicting reformulations. punctures, i.e. holes. The regions on either side of 퐶2 are unbounded at the punctures. Generality As the Superficial Aspect Riemann used this to calculate integrals. Any curves 퐶 Grothendieck’s famous penchant for generality is not and 퐶′ surrounding just the same holes the same number enough to explain his results or his influence. Raoul of times have ∫퐶 휔 = ∫퐶′ 휔 for all holomorphic 휔. That Bott put it better fifty-four years ago describing the is because 퐶 and the reversal of 퐶′ form the complete Grothendieck-Riemann-Roch theorem. boundary of a kind of collar avoiding those holes. So Riemann-Roch has been a mainstay of analysis for ∫퐶 휔 − ∫퐶′ 휔 = 0. one hundred fifty years, showing how the topology of a Modern cohomology sees holes as obstructions to Riemann surface affects analysis on it. Mathematicians solving equations. Given 휔 and a path 푃∶ [0, 1] → 푆 it from Richard Dedekind to Weil generalized it to curves would be great to calculate the integral ∫ 휔 by finding a over any field in place of the complex numbers. This 푃 function 푓 with d푓 = 휔, so ∫ 휔 = 푓(푃 ) − 푓(푃 ). Clearly, makes theorems of arithmetic follow from topological 푃 1 0 there is not always such a function, since that would and analytic reasoning over the field 픽 of integers 푝 imply ∫ 휔 = 0 for every closed curve 푃. But Cauchy, modulo a prime 푝. Friedrich Hirzebruch generalized the 푃 complex version to work in all dimensions. Riemann, and others saw that if 푈 ⊂ 푆 surrounds no Grothendieck proved it for all dimensions over all holes there are functions 푓푈 with d푓푈 = 휔 all over 푈. fields, which was already a feat, and he went further in Holes are obstructions to patching local solutions 푓푈 into a signature way. Beyond single varieties he proved it for one solution of d푓 = 휔 all over 푆. suitably continuous families of varieties. Thus: This concept has been generalized to algebra and Grothendieck has generalized the theorem to the number theory: point where not only is it more generally applica- Indeed one now instinctively assumes that all ble than Hirzebruch’s version but it depends on obstructions are best described in terms of a simpler and more natural proof. (Bott [6]) cohomology groups. [32, p. 103] This was the first concrete triumph for his new cohomol- ogy and nascent scheme theory. Recognizing that many Cohomology Groups mathematicians distrust generality, he later wrote: With homologies, terms compose according to I prefer to accent “unity” rather than “general- the rules of ordinary addition. [24, pp. 449-50] ity.” But for me these are two aspects of one Poincaré defined addition for curves so that 퐶 + 퐶′ is quest. Unity represents the profound aspect, and the union of 퐶 and 퐶′, while −퐶 corresponds to reversing generality the superficial. [16, p. PU 25] the direction of 퐶. Thus for every form 휔: Colin McLarty is Truman P. Hardy Professor of Philosophy and professor of mathematics, Case Western Reserve University. His ∫ 휔 = ∫ 휔 + ∫ 휔 and ∫ 휔 = − ∫ 휔. email address is [email protected]. 퐶+퐶′ 퐶 퐶′ −퐶 퐶

For permission to reprint this article, please contact: reprint- When curves 퐶1, … , 퐶푘 form the complete boundary [email protected]. of some region, then Poincaré writes 훴푘퐶푘 ∼ 0 and says DOI: http://dx.doi.org/10.1090/noti1338 their sum is homologous to 0. In these terms the Cauchy

March 2016 Notices of the AMS 257 Integral Theorem says concisely: So topologists wrote H푖(푆) for the 푖th cohomology group of space 푆 and left the coefficient group implicit in the

If 훴푘퐶푘 ∼ 0, then ∫ 휔 = 0 context. 훴푘퐶푘 In contrast group theorists wrote H푖(퐺, 퐴) for the for all holomorphic forms 휔. 푖th cohomology of 퐺 with coefficients in 퐴, because Poincaré generalized this idea of homology to higher many kinds of coefficients were used and they were as dimensions as the basis of his analysis situs, today called interesting as the group 퐺. For example, the famed Hilbert topology of manifolds. Theorem 90 became Notably, Poincaré published two proofs of Poincaré H1(Gal(퐿/푘), 퐿×) ≅ {0}. duality using different definitions. His first statement of it was false. His proof mixed wild non sequiturs The Galois group Gal(퐿/푘) of a Galois field extension 퐿/푘 and astonishing insights. For topological manifolds 푀 has trivial 1-dimensional cohomology with coefficients in × of any dimension 푛 and any 0 ≤ 푖 ≤ 푛, there is a the multiplicative group 퐿 of all nonzero 푥 ∈ 퐿. Olga tight relation between the 푖-dimensional submanifolds Taussky[33, p. 807] illustrates Theorem 90 by using it on of 푀 and the (푛 − 푖)-dimensional. This relation is hardly the Gaussian numbers ℚ[푖] to show every Pythagorean expressible without using homology, and Poincaré had to triple of integers has the form revise his first definition to get it right. Even the second 푚2 − 푛2, 2푚푛, 푚2 + 푛2. version relied on overly optimistic assumptions about triangulated manifolds. Trivial cohomology means there is no obstruction to Topologists spent decades clarifying his definitions solving certain problems, so Theorem 90 shows that and theorems and getting new results in the process. some problems on the field 퐿 have solutions. Algebraic relations of H1(Gal(퐿/푘), 퐿×) to other cohomology groups They defined homology groups H푖(푆) for every space 푆 imply solutions to other problems. Of course Theorem 90 and every dimension 푖 ∈ ℕ. In each H1 the group addition is Poincaré’s addition of curves modulo the homology was invented to solve lots of problems decades before relations 훴푘퐶푘 = 0. They also defined related cohomology group cohomology appeared. Cohomology organized and 푖 extended these uses so well that Emil Artin and John Tate groups H (푆) such that Poincaré duality says H푖(푀) is isomorphic to H푛−푖(푀) for every compact orientable made it basic to class field theory. 푛-dimensional manifold 푀. Also in the 1940s topologists adopted sheaves of Poincaré’s two definitions of homology split into many coefficients. A sheaf of Abelian groups ℱ on a space using simplices or open covers or differential forms or 푆 assigns Abelian groups ℱ(푈) to open subsets 푈 ⊆ 푆 metrics, bringing us to the year 1939: and homomorphisms ℱ(푈) → ℱ(푉) to subset inclusions Algebraic topology is growing and solving prob- 푉 ⊆ 푈. So the sheaf of holomorphic functions 풪푀 assigns lems, but nontopologists are very skeptical. At the additive group 풪푀(푈) of holomorphic functions on 푈 to each open subset 푈 ⊆ 푀 of a complex manifold. Harvard, Tucker or perhaps Steenrod gave an 푖 expert lecture on cell complexes and their homol- Cohomology groups like H (푀, 풪푀) began to organize ogy, after which one distinguished member of the complex analysis. audience was heard to remark that this subject Leaders in these fields saw cohomology as a unified had reached such algebraic complication that it idea, but the technical definitions varied widely. In the was not likely to go any further. (MacLane [21, Séminaire Henri Cartan speakers Cartan, Eilenberg, and p. 133] Serre organized it all around resolutions. A resolution of an Abelian group 퐴 (or module or sheaf) is an exact sequence of homomorphisms, meaning the image of each Variable Coefficients and Exact Sequences homomorphism is the kernel of the next: / / / / In his Kansas article (1955) and Tôhoku article {0} 퐴 퐼0 퐼1 …. (1957) Grothendieck showed that given any cate- gory of sheaves a notion of cohomology groups It quickly follows that many sequences of cohomology results. (Deligne [10, p. 16]) groups are also exact. That proof rests on the Snake Lemma immortalized by Hollywood in a scene widely Algebraic complication went much further. Methods available online: “A clear proof is given by Jill Clayburgh in topology converged with methods in Galois theory at the beginning of the movie It’s My Turn” [35, p. 11]. and led to defining cohomology for groups as well as Fitting a cohomology group H푖(푋, ℱ) into the right for topological spaces. In the process, what had been a exact sequence might show H푖(푋, ℱ) ≅ {0}, so the ob- technicality to Poincaré became central to cohomology, 푖 namely, the choice of coefficients. Certainly he and others structions measured by H (푋, ℱ) do not exist. Or it may prove some isomorphism, H푖(푋, ℱ) ≅ H푘(푌, 풢). Then the used integers, rational numbers or reals or integers 푖 modulo 2 as coefficients: obstructions measured by H (푋, ℱ) correspond exactly to those measured by H푘(푌, 풢). 푎1퐶1 + ⋯ + 푎푚퐶푚 푎푖 ∈ ℤ or ℚ or ℝ or ℤ/2ℤ. Group cohomology uses resolution by injective mod- But only these few closely related kinds of coefficients ules 퐼푖. A module 퐼 over a ring 푅 is injective if for were used, chosen for convenience for a given calculation. every 푅-module inclusion 푗∶ 푁 ↣ 푀 and homomorphism

258 Notices of the AMS Volume 63, Number 3 푓∶ 푁→퐼 there is some 푔∶ 푀→퐼 with 푓 = 푔푗. The diagram They were wrong on both counts. These axioms also sim- is simple: plified proofs of already-known theorems. Most especially / 푗 / they subsumed many useful spectral sequences (not all) 푁 N 푀 NN under the Grothendieck spectral sequence so simple as to NNN NN 푔 be Exercise A.3.50 of (Serre [11, p. 683]). NNN 푓 NN'  Early editions of Serge Lang’s Algebra gave the Abelian 퐼 category axioms with a famous exercise: “Take any book This works because every 푅-module 퐴 over any ring 푅 on homological algebra, and prove all the theorems embeds in some injective 푅-module. No one believed without looking at the proofs given in that book” [20, p. anything this simple would work for sheaves. Sheaf coho- 105]. He dropped that when homological algebra books mology was defined only for sufficiently regular spaces all began using axiomatic proofs themselves, even if their using various, more complicated topological substitutes theorems are stated only for modules. David Eisenbud, for injectives. Grothendieck found an unprecedented for example, says his proofs for modules “generalize with proof that sheaves on all topological spaces have injec- just a little effort to [any] nice Abelian category” [11, tive embeddings. The same proof later worked for sheaves p. 620]. on any Grothendieck topology. Injective resolutions in any Abelian category give derived functor cohomology of that category. This was obviously general beyond any proportion to the then- Tôhoku known cases. Grothendieck was sure it was the right Consider the set of all sheaves on a given topolog- generality: For a cohomological solution to any prob- ical space or, if you like, the prodigious arsenal lem, notably the Weil conjectures, find the right Abelian of all “meter sticks” that measure the space. We category. consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so The Weil Conjectures to speak “right in front of your nose”; that is what This truly revolutionary idea thrilled the mathe- we call the structure of a “category.” [16, p. P38] maticians of the time, as I can testify at first hand. We will not fully define sheaves, let alone spectral [30, p. 525] sequences and other “drawings (called “diagrams”) full of The Weil Conjectures relat- arrows covering the blackboard” which “totally escaped” ing arithmetic to topology were Grothendieck at the time of the Séminaire Cartan [R&S, immediately recognized as a The p. 19]. We will see why Grothendieck wrote to Serre on huge achievement. Weil knew conjectures February 18, 1955: “I am rid of my horror of spectral that just conceiving them was sequences” [7, p. 7]. a great moment in his career. were too The Séminaire Cartan emphasized how few specifics The cases he proved were im- about groups or modules go into the basic theorems. pressive. The conjectures were beautiful not Those theorems only use diagrams of homomorphisms. too beautiful not to be true and For example, the sum 퐴 + 퐵 of Abelian groups 퐴, 퐵 can yet nearly impossible to state to be true and be defined, uniquely up to isomorphism, by the facts that fully. it has homomorphisms 푖퐴 ∶ 퐴 → 퐴 + 퐵 and 푖퐵 ∶ 퐵 → 퐴 + 퐵 Weil [37] presents the yet nearly and any two homomorphisms 푓∶ 퐴→퐶 and 푔∶ 퐵→퐶 give topology using the nineteenth- impossible to a unique 푢∶ 퐴 + 퐵→퐶 with 푓 = 푢푖퐴 and 푔 = 푢푖퐵: century terminology of Betti numbers. But he was an estab- 푖퐴 / o 푖퐵 state fully. 퐴 FF 퐴 + 퐵 y 퐵 lished expert on cohomology FF yy 푓 FF 푢 yy 푔 and in conversations: FF yy F"  y| y At that time, Weil was explaining things in terms of 퐶 cohomology and Lefschetz’s fixed point formula The same diagram defines sums of modules or of sheaves [yet he] did not want to predict [this could actually of Abelian groups. work]. Indeed, in 1949–50, nobody thought that Grothendieck [13, p. 127] took the basic patterns used it could be possible. (Serre quoted in [22, p. 305].) by the Séminaire Cartan as his Abelian category axioms. Lefschetz used cohomology, relying on the continuity He added a further axiom, AB5, on infinite colimits. of manifolds, to count fixed points 푥 = 푓(푥) of continuous Theorem 2.2.2 says if an Abelian category satisfies AB5 functions 푓∶ 푀→푀 on manifolds. Weil’s conjectures deal plus a set-theoretic axiom, then every object in that with spaces defined over finite fields. No known version category embeds in an injective object. These axioms of those was continuous. Neither Weil nor anyone knew taken from module categories obviously hold as well for what might work. Grothendieck says: sheaves of Abelian groups on any topological space, so Serre explained the Weil conjectures to me in the conclusion applies. cohomological terms around 1955 and only in People who thought this was just a technical result on these terms could they possibly “hook” me. No sheaves found the tools disproportionate to the product. one had any idea how to define such a cohomology

March 2016 Notices of the AMS 259 and I am not sure anyone but Serre and I, not subvarieties 푉′ ⊆ 푉 of a variety 푉 purely in terms even Weil if that is possible, was deeply convinced of fields of rational functions. Raynaud [25] gives an such a thing must exist. [R&S, p. 840] excellent overview. We list three key topics. (1) Weil has generic points. Indeed, a variety defined Exactly What Scheme Theory Simplified by polynomials over a field 푘 has infinitely many Kronecker was, in fact, attempting to describe and generic points with coordinates transcendental to initiate a new branch of mathematics, which over 푘, all conjugate to each other by Galois would contain both number-theory and algebraic actions over 푘. geometry as special cases. (Weil [38, p. 90]) (2) Weil defines abstract algebraic varieties by data Riemann’s treatment of complex curves left much to telling how to patch together varieties defined by geometric intuition. So Dedekind and Weber [8, p. 181] equations. But these do not exist as single spaces. proved a Riemann-Roch theorem from “a simple yet rig- They only exist as sets of concrete varieties plus orous and fully general viewpoint,” over any algebraically patching data. closed field 푘 containing the rational numbers. They note (3) Weil could not define a variety over the integers, 푘 can be the field of algebraic numbers. They saw this though he could systematically relate varieties bears on arithmetic as well as on analysis and saw all too over ℚ to others over the fields 픽푝. well that their result is “very difficult in exposition and expression” [8, p. 235]. Serre Varieties and Coherent Sheaves Meromorphic functions on any compact Riemann sur- Then Serre [27] temporarily put generic points and non- face 푆 form a field 푀(푆) of transcendence degree 1 over closed fields aside to describe the first really penetrating the complex numbers ℂ. Each point 푝 ∈ 푆 determines a cohomology of algebraic varieties: function 푒푝 from 푀(푆) to ℂ + {∞}: namely 푒푝(푓) = 푓(푝) This rests on the use of the famous Zariski topol- when 푓 is defined at 푝, and 푒푝(푓) = ∞ when 푓 has a pole ogy, in which the closed sets are the algebraic at 푝. Then, if we ignore sums ∞ + ∞: sub-varieties. The remarkable fact that this coarse topology could actually be put to genuine mathe- 푒푝(푓 + 푔) = 푒푝(푓) + 푒푝(푔), matical use was first demonstrated by Serre and 푒푝(푓 ⋅ 푔) = 푒푝(푓) ⋅ 푒푝(푔), it has produced a revolution in language and 1 1 techniques. (Atiyah [3, p. 66]) 푒푝( ) = . 푛 푓 푒푝(푓) Say a naive variety over any field 푘 is a subset 푉 ⊆ 푘 Dedekind and Weber define a general field of algebraic defined by finitely many polynomials 푝푖(푥1, … , 푥푛) over 푘: 푛 functions as any transcendence degree 1 extension 퐿/푘 of 푉 = { ⃗푥∈ 푘 | 푝1(⃗푥)= ⋯ = 푝ℎ(⃗푥)= 0 }. any algebraically closed field 푘. They define a point 푝 of 퐿 They form the closed sets of a topology on 푘푛 called to be any function 푒 from 퐿 to 푘 + {∞} satisfying those 푝 the Zariski topology. Even their infinite intersections are equations. Their Riemann-Roch theorem treats 퐿 as if it defined by finitely many polynomials, since the polyno- were 푀(푆) for some Riemann surface. mial ring 푘[푥 , … , 푥 ] is Noetherian. Also, each inherits Kronecker [19] achieved some “algebraic geometry over 1 푛 a Zariski topology where the closed sets are the subsets an absolutely algebraic ground-field” [38, p. 92]. These 푉′ ⊆ 푉 defined by further equations. fields are the finite extensions of ℚ or of finite fields 픽푝. These are very coarse topologies. The Zariski closed They are not algebraically closed. He aimed at “algebraic subsets of any field 푘 are the zero-sets of polynomials geometry over the integers” where one variety could be over 푘: that is, the finite subsets and all of 푘. defined over all these fields at once, but this was far too Each naive variety has a structure sheaf 풪푉 which difficult at the time [38, p. 95]. assigns to every Zariski open 푈 ⊆ 푉 the ring of regular Italian algebraic geometers relied on an idea of generic functions on 푈. Omitting important details: points of a complex variety 푉, which are ordinary complex points 푝 ∈ 푉 with no apparent special properties [26]. For 푓(⃗푥) 풪푉(푈) = { such that when ⃗푥∈ 푈 then 푔(⃗푥)≠ 0 }. example, they are not points of singularity. Noether and 푔(⃗푥) Bartel van der Waerden gave abstract generic points which A Serre variety is a topological space 푇 plus a sheaf actually have only those properties common to all points 풪푇 which is locally isomorphic to the structure sheaf of 푉. Van der Waerden [34] made these rigorous but not of a naive variety. Compare the sheaf of holomorphic so usable as Weil would want. Oscar Zariski, trained in functions 풪푀 of a complex manifold. The sheaf apparatus Italy, worked with Noether in Princeton, and later with lets Serre actually paste varieties together on compatible Weil, to give algebraic geometry a rigorous algebraic basis patches, just as patches of differentiable manifolds are [23, p. 56]. pasted together. Weil could not do this with his abstract Weil’s bravura Foundations of Algebraic Geometry [36] varieties. combined all these methods into the most complicated Certain sheaves related to the structure sheaves 풪푇 foundation for algebraic geometry ever. To handle vari- are called coherent. Serre makes them the coefficient eties of all dimensions over arbitrary fields 푘, he uses sheaves of a cohomology theory widely used today with algebraically closed field extensions 퐿/푘 of infinite tran- schemes. The close tie of coherent sheaves to structure scendence degree. He defines not only points but also sheaves makes this cohomology unsuitable for the Weil

260 Notices of the AMS Volume 63, Number 3 conjectures. When variety (or scheme) 푉 is defined over a modulo a prime 푝, such as 푥 = 2 and 푦 = 6 in the finite finite field 픽푝, its coherent cohomology is defined modulo field 픽13, are not complex numbers. And solutions modulo 푝 and can count fixed points of maps 푉→푉 only modulo one prime are different from those modulo another. All 푝. Still: these solutions are organized in the single scheme The principal, and perhaps only, external inspira- Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). tion for the sudden vigorous launch of scheme theory in 1958 was Serre’s (1955) article known The coordinate functions are simply integer polynomi- 2 2 by the acronym FAC. [R&S, p. P28] als modulo 푥 + 푦 − 1. The nonzero prime ideals are not simple at all. They correspond to solutions of this equa- Schemes tion in all the absolutely algebraic fields by which Weil explicated Kronecker’s goal, including all finite fields. In- The point, grosso modo, was to rid algebraic geom- deed, the closest Grothendieck comes to defining schemes etry of parasitic hypotheses encumbering it: base in Récoltes et Semailles is to call a scheme a “magic fan” fields, irreducibility, finiteness conditions. (Serre (éventail magique) folding together varieties defined over [29, p. 201]) all these fields (p. P32). This is algebraic geometry over Schemes overtly simplify algebraic geometry. Where the integers. earlier geometers used complicated extensions of alge- Now consider the ideal (푥2 + 푦2 − 1) consisting of braically closed fields, scheme theorists use any ring. all polynomial multiples of 푥2 + 푦2 − 1 in ℤ[푥, 푦]. It is Polynomial equations are replaced by ring elements. prime, so it is a point of Spec(ℤ[푥, 푦]). And schemes Generic points become prime ideals. The more intri- are not Hausdorff spaces: their points are generally not cate concepts come back in when needed, which is fairly closed in the Zariski topology. The closure of this point is often, but not always and not from the start. Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). This ideal is the generic point In fact this perspective goes back to unpublished work of the closed subscheme by Noether, van der Waerden, and Wolfgang Krull. Prior to Grothendieck: Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)) ↣ Spec(ℤ[푥, 푦]). The person who was closest to scheme-thinking The irreducible closed subschemes of any scheme are, (in the affine case) was Krull (around 1930). He roughly speaking, given by equations in the coordinate used systematically the localization process, and ring, and each has exactly one generic point. proved most of the nontrivial theorems in Com- In the ring ℤ[푥, 푦]/(푥2 + 푦2 − 1) the ideal (푥2 + 푦2 − 1) mutative Algebra. (Serre, email of 21/06/2004, appears as the zero ideal, since in this ring 푥2 + 푦2 − Serre’s parentheses) 1 = 0. So the zero ideal is the generic point for the Grothendieck made it work. He made every ring 푅 whole scheme Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). What happens the coordinate ring of a scheme Spec(푅) called the at this generic point also happens almost everywhere spectrum of 푅. The points are the prime ideals of 푅, on Spec(ℤ[푥, 푦]/(푥2 + 푦2 − 1)). Generic points like this and the scheme has a structure sheaf 풪푅 on the Zariski achieve what earlier algebraic geometers sought from topology for those points, like the structure sheaf on their attempts. a Serre variety. It follows that the continuous structure Schemes vindicate more classical intuitions as well. preserving maps from Spec(푅) to another affine scheme Ancient Greek geometers debated whether a tangent line Spec(퐴) correspond exactly to the ring homomorphisms meets a curve in something more than a point. Scheme in the other direction: theory says yes: a tangency is an infinitesimal segment 푓 / Spec(푓) / around a point. 퐴 푅 Spec(푅) Spec(퐴) . The contact of the parabola 푦 = 푥2 with the 푥-axis The points can be quite intricate: “When one has to 푦 = 0 in ℝ2 is plainly given by 푥2 = 0. As a variety it would construct a scheme one generally does not begin with the just be the one point space {0}, but it gives a nontrivial set of points” [10, p. 12]. scheme Spec(ℝ[푥]/(푥2)). The coordinate functions are For example, the ring ℝ[푥] of real polynomials in real polynomials modulo 푥2 or, in other words, real linear one variable is the natural coordinate ring for the real polynomials 푎 + 푏푥. line, so the spectrum Spec(ℝ[푥]) is the scheme of the Intuitively Spec(ℝ[푥]/(푥2)) is an infinitesimal line seg- real line. Each nonzero prime ideal is generated by a ment containing 0 but no other point. This segment is monic irreducible real polynomial. Those polynomials big enough that a function 푎 + 푏푥 on it has a slope 푏 are 푥 − 푎 for 푎 ∈ ℝ and 푥2 − 2푏푥 + 푐 for 푏, 푐 ∈ ℝ with but is too small to admit a second derivative. Intuitively 푏2 < 푐. The first kind correspond to ordinary points 푥 = 푎 a scheme map 푣 from Spec(ℝ[푥]/(푥2) to any scheme 푆 of the real line. The second kind correspond to pairs is an infinitesimal line segment in 푆, i.e. a tangent vector of conjugate complex roots 푏 ± √푏2 − 푐. The scheme with base point 푣(0) ∈ 푆. Spec(ℝ[푥]) automatically includes both real and complex Grothendieck’s signature method, called the relative points, with the nuance that a single complex point is a viewpoint, also reflects classical ideas. Earlier geometers conjugate pair of complex roots. would speak of, for example, 푥2 + 푡 ⋅ 푦2 = 1 as a quadratic A polynomial equation like 푥2 + 푦2 = 1 has many kinds equation in 푥, 푦 with parameter 푡. So it defines a conic of solutions. One could think of rational and algebraic section 퐸푡 which is an ellipse or a hyperbola or a pair of solutions as kinds of complex solutions. But solutions lines depending on the parameter. More deeply, this is

March 2016 Notices of the AMS 261 a cubic equation in 푥, 푦, 푡 defining a surface 퐸 bundling information and express operations which earlier geome- together all the curves 퐸푡. Over the real numbers this ters had only begun to explore. Grothendieck and Jean gives a map of varieties Dieudonné took this as a major advantage of scheme / theory: 퐸={⟨푥, 푦, 푡⟩∈ℝ3|푥2 +푡⋅푦2 = 1} ℝ ⟨푥, 푦, 푡⟩↦푡 . The idea of “variation” of base ring which we intro- Each curve 퐸 is the fiber of this map over its parameter 푡 duce gets easy mathematical expression thanks to 푡 ∈ ℝ. Classical geometers used the continuity of the the functorial language whose absence no doubt family of curves 퐸 bundled into surface 퐸 but generally 푡 explains the timidity of earlier attempts. [17, p. 6] left the cubic surface implicit as they spoke of the variable quadratic curve 퐸푡. Grothendieck used rigorous means to treat a scheme Étale Cohomology map 푓∶ 푋→푆 as a single scheme simpler than either one of In the Séminaire Chevalley of April 21, 1958, Serre pre- 1 푋 and 푆. He calls 푓 a relative scheme and treats it roughly sented new 1-dimensional cohomology groups H̃ (X, G) 1 suitable for the Weil conjectures: “At the end of the as the single fiber 푋푝 ⊂ 푋 over some indeterminate 푝 ∈ 푆. Grothendieck had this viewpoint even before he had oral presentation Grothendieck said this would give the schemes: Weil cohomology in all dimensions! I found this very Certainly we’re now so used to putting some optimistic” [31, p. 255]. That September Serre wrote: problem into relative form that we forget how One may ask if it is possible to define higher revolutionary it was at the time. Hirzebruch’s cohomology groups H̃ 푞(X, G)…in all dimensions. proof of Riemann-Roch is very complicated, while Grothendieck (unpublished) has shown it is, and the proof of the relative version, Grothendieck- it seems that when G is finite these furnish Riemann-Roch, is so easy, with the problem “the true cohomology” needed to prove the Weil shifted to the case of an immersion. This was conjectures. On this see the introduction to [14]. fantastic. [18, p. 1114] [28, p. 12] What Hirzebruch proved for complex varieties Grothen- Grothendieck later described that unpublished work dieck proved for suitable maps 푓∶ 푋→푆 of varieties over of 1958, saying, “The two key ideas crucial in launching any field 푘. Among other advantages this allows reducing and developing the new geometry were those of scheme the proof to the case of maps 푓, called immersions, with and of topos. They appeared almost simultaneously and simple fibers. The method relies on base change transforming a in close symbiosis.” Specifically he framed “the notion relative scheme 푓∶ 푋→푆 on one base 푆 to some 푓′ ∶ 푋′ → 푆′ of site, the technical, provisional version of the crucial on some related base 푆′. Fibers themselves are an example. notion of topos” [R&S, pp. P31 and P23n]. But before Given 푓∶ 푋→푆 each point 푝 ∈ 푆 is defined over some field pursuing this idea into higher-dimensional cohomology 푘, and 푝 ∈ 푆 amounts to a scheme map 푝∶ Spec(푘) → 푆. he used Serre’s idea to define the fundamental group of The fiber 푋푝 is intuitively the part of 푋 lying over 푝 and a variety or scheme in a close analogy with Galois theory. is precisely the relative scheme 푋푝 → Spec(푘) given by Notice that Zariski topology registers punctures much pullback: more directly than it registers holes like those through / 푋푝 푋 the center of the torus or inside the tube.

푓 ⋆푃   / ⋆ Spec(푘) 푝 푆 Other examples of base change include extending a ⋆ scheme 푓∶ 푌 → Spec(ℝ) defined over the real numbers into one 푓′ ∶ 푌′ → Spec(ℂ) over the complex numbers by Zariski closed subsets are (locally) the zero-sets of pullback along the unique scheme map from Spec(ℂ) to polynomials, so a nonempty Zariski open subset of Spec(ℝ): the torus is the torus minus finitely many punctures // 푌′ 푌 (possibly none). Such a subset might or might not be punctured at some point 푃 itself, so the Zariski opens 푓′ 푓   themselves distinguish between having and not having / Spec(ℂ) Spec(ℝ) that puncture. But every nonempty Zariski open subset surrounds the hole through the torus center and the Other changes of base go along scheme maps 푆′ → 푆 one through the torus tube. These subsets by themselves between schemes 푆, 푆′ taken as parameter spaces for cannot distinguish between having and not having those serious geometric constructions. Each is just a pullback holes. Coherent cohomology registers those holes by in the sense of category theory, yet they encode intricate using coherent sheaves, which cannot work for the Weil 1In 1942 Oscar Zariski urged something like this to Weil [23, conjectures, as noted above. p. 70]. Weil took the idea much further without finally making it So Serre used many-sheeted covers. Consider two a working method [38, p. 91ff]. different 2-sheeted covers of one torus 푇. Let torus 푇′ be

262 Notices of the AMS Volume 63, Number 3 ′ twice as long as 푇, with the same tube diameter. Wrap 푇 set of étale maps 푈푖 → 푆 such that the union of all the twice around 푇 along the tube: images is the whole of 푆. Sites today are often called Grothendieck topologies, and this site may be called the ″ 푇 étale topology on 푆. There are two basic ways to solve a problem locally in the étale topology on 푆. You could solve it on each of a set of Zariski open subsets of 푆 whose union is 푆, or you could solve it in a separable algebraic extension wrap 2× | of the coordinate ring of 푆. The first gives an actual, ′ ↓ vertically 푇 global solution if the local solutions agree wherever they wrap 2× horizontally overlap. The second gives a global solution if the local ⟶ 푇 solution is Galois invariant—like first factoring a real polynomial over the complex numbers, then showing the ″ Let torus 푇 be as long as 푇 with twice the tube diameter. factors are actually real. Étale cohomology would measure Wrap it twice around the tube. The difference between obstructions to patching actual solutions together from these two covers, and both of them from 푇 itself, reflects combinations of such local solutions. the two holes in 푇. In 1961 Michael Artin proved the first higher- Riemann created Riemann surfaces as analogues to dimensional geometric theorem in étale cohomology [1, number fields. As ℚ[√2] is a degree 2 field extension of p. 359]. According to David Mumford this was that the rational numbers ℚ, so 푇′ → 푇 is a degree 2 cover of the plane with origin deleted has nontrivial H3; in the 푇. As ℚ[√2]/ℚ has a two element Galois group where context of étale cohomology that means the coordinate the nonidentity element interchanges √2 with −√2, so plane punctured at the origin, 푘2 − {0}, for any field of 푇′ → 푇 has a two-element symmetry group over 푇 where coordinates 푘. Weil’s conjectures suggest that, when 푘 the nonidentity symmetry interchanges the two sheets of is absolutely algebraic, this cohomology should largely 푇′ over 푇. agree with the classical cohomology of the complex case Serre consciously extended Riemann’s analogy to a ℂ2 − ⟨0, 0⟩. That space is topologically ℝ4 punctured at far-reaching identity. He gave a purely algebraic defini- its origin. It has the classical cohomology of the 3-sphere 3 tion of unramified covers 푆′ → 푆 which has the Riemann 푆3, and that is nontrivial in H . So Artin’s result needed surfaces above as special cases, as well as Galois field to hold in any Weil cohomology. Artin proved it does extensions, and much more. Naturally, in this generality hold in the derived functor cohomology of sheaves on some theorems and proofs are a bit technical, but over the étale site. Today this is étale cohomology. and over Serre’s unramified covers make intuitions taken In short, Artin showed the étale site yields not only from Riemann surfaces work for all these cases. Groth- some sheaf cohomology but a good usable one. Classical endieck used these to give the first useful theory of the theorems of cohomology survive with little enough change. fundamental group of a variety or a scheme, that is, the Grothendieck invited Artin to France to collaborate in the one-dimensional homotopy. He also worked with a slight seminar that created Théorie des topos et cohomologie generalization of unramified covers, called étale maps, étale [2]. The subject exploded, and we will go no further which include all algebraic Riemann covering surfaces. into it. Serre had not calculated cohomology of sheaves but Toposes are less popular than schemes or sites in ge- of isotrivial fiber spaces. Over a torus 푇 those are roughly ometry today. Deligne expresses his view with care: “The spaces mapped to 푇 which may twist around 푇 but tool of topos theory permitted the construction of étale can be untwisted by lifting to some other torus 푇‴ → 푇 cohomology” [10, p. 15]. Yet, once constructed, this coho- wrapped some number of times around each hole of 푇. mology is “so close to classical intuition” that for most While Grothendieck [12] also used fiber spaces for one- purposes one needs only some ordinary topology plus dimensional cohomology, he found his Tôhoku methods “a little faith/un peu de foi” [9, p. 5]. Grothendieck would more promising for higher dimensions. He wanted some “advise the reader nonetheless to learn the topos lan- notion of sheaf matching Serre’s idea. guage which furnishes an extremely convenient unifying During 1958 Grothendieck saw that instead of defining principle” [5, p. VII]. sheaves by using open subsets 푈 ⊆ 푆 of some space 푆, he We close with Grothendieck’s view of how schemes and could use étale maps 푈→푆 to a scheme. He published this his cohomology and toposes all came together in étale idea by spring 1961 [15, §4.8, p. 298]. Instead of inclusions cohomology, which indeed in his hands and Deligne’s 푉 ⊆ 푈 ⊆ 푆, he could use commutative triangles over 푆: gave the means to prove the Weil conjectures: The crucial thing here, from the viewpoint of the 푓 / / Weil conjectures, is that the new notion of space 푉 >> 푈 푈 ×푆 푉 푉 >> is vast enough, that we can associate to each >> scheme a “generalized space” or “topos” (called >    / the “étale topos” of the scheme in question). 푆 푈 푆 Certain “cohomology invariants” of this topos In place of intersections 푈∩푉 ⊆ 푆 he could use pullbacks (as “babyish” as can be!) seemed to have a good 푈 ×푆 푉 over 푆. Then an étale cover of a scheme 푆 is any chance of offering “what it takes” to give the

March 2016 Notices of the AMS 263 conjectures their full meaning, and (who knows!) 10. Pierre Deligne, Quelques idées maîtresses de l’œuvre de perhaps to give the means of proving them. [16, A. Grothendieck, in Matériaux pour l’Histoire des Mathéma- e p. P41] tiques au XX Siècle (Nice, 1996), Soc. Math. France, 1998, pp. 11–19. 11. David Eisenbud, Commutative Algebra, Springer-Verlag, New York, 2004. 12. Alexander Grothendieck, A General Theory of Fibre Spaces with Structure Sheaf, technical report, University of Kansas, 1955. 13. , Sur quelques points d’algèbre homologique, Tôhoku Mathematical Journal, 9:119–221, 1957. 14. , The cohomology theory of abstract algebraic va- rieties, in Proceedings of the International Congress of Mathematicians, 1958, Cambridge University Press, 1958, pp. 103–18. 15. , Revêtements Étales et Groupe Fondamental, Sémi- naire de géométrie algébrique du Bois-Marie, 1, Springer- Verlag, 1971. Generally cited as SGA 1. 16. , Récoltes et Semailles, Université des Sciences et Tech- niques du Languedoc, Montpellier, 1985–1987. Published in several successive volumes. Patricia Princehouse. 17. Alexander Grothendieck and Jean Dieudonné, Éléments de Géométrie Algébrique I, Springer-Verlag, 1971. 18. Luc Illusie, Alexander Beilinson, Spencer Bloch, Vladimir

Courtesy of Drinfeld et al., Reminiscences of Grothendieck and his Colin McLarty with a Great Pyrenees, from the region school, Notices of the Amer. Math. Soc., 57(9):1106–15, 2010. 19. Leopold Kronecker, Grundzüge einer arithmetischen theorie where Grothendieck spent much of his life. der algebraischen grössen, Crelle, Journal für die reine und angewandte Mathematik, XCII:1–122, 1882. 20. Serge Lang, Algebra, 3rd edition, Addison-Wesley, Reading, References MA, 1993. 1. Michael Artin, Interview, in Joel Segel, editor, Recountings: 21. Saunders Mac Lane, The work of Samuel Eilenberg in topol- Conversations with MIT Mathematicians, A K Peters/CRC ogy, in Alex Heller and Myles Tierney, editors, Algebra, Press, Wellesley, MA, 2009, pp. 351–74. Topology, and Category Theory: A Collection of Papers in 2. Michael Artin, Alexander Grothendieck, and Jean-Louis Honor of Samuel Eilenberg, Academic Press, New York, 1976, Verdier, Théorie des Topos et Cohomologie Etale des Sché- pp. 133–44. mas, Séminaire de géométrie algébrique du Bois-Marie, 4, 22. Colin McLarty, The rising sea: Grothendieck on simplicity Springer-Verlag, 1972, Three volumes, cited as SGA 4. and generality I, in Jeremy Gray and Karen Parshall, edi- 3. Michael Atiyah, The role of algebraic topology in mathemat- tors, Episodes in the History of Recent Algebra, American ics, Journal of the London Mathematical Society, 41:63–69, Mathematical Society, 2007, pp. 301–26, 1966. 23. Carol Parikh, The Unreal Life of Oscar Zariski, Springer- 4. , Bakerian lecture, 1975: Global geometry, Proceedings Verlag, New York, 2009. of the Royal Society of London Series A, 347(1650):291–99, 24. Henri Poincaré, Analysis situs, and five complements to it, 1976. 1895–1904, Collected in G. Darboux et al., eds., Oeuvres de 5. Pierre Berthelot, Alexander Grothendieck, and Luc Illusie, Henri Poincaré in 11 volumes, Paris: Gauthier-Villars, 1916– Théorie des intersections et théorème de Riemann-Roch, Num- 1956, Vol. VI, pp. 193–498. I quote the translation by John ber 225 in Séminaire de géométrie algébrique du Bois-Marie, Stillwell, Papers on Topology: Analysis Situs and Its Five 6, Springer-Verlag, 1971. Generally cited as SGA 6. Supplements, American Mathematical Society, 2010, p. 232. 6. Raoul Bott, Review of A. Borel and J-P. Serre, Le théorème 25. Michel Raynaud, André Weil and the foundations of algebraic de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136. geometry, Notices of the American Mathematical Society, 46: MR0116022 (22 #6817). 864–867, 1999. 7. Pierre Colmez and Jean-Pierre Serre, editors, Correspon- 26. Norbert Schappacher, A historical sketch of B. L. van der dance Grothendieck-Serre, Société Mathématique de France, Waerden’s work in algebraic geometry: 1926–1946, in Jeremy 2001. Expanded to Grothendieck-Serre Correspondence: Bilin- Gray and Karen Parshall, editors, Episodes in the History gual Edition, American Mathematical Society, and Société of Modern Algebra (1800–1950), American Mathematical Mathématique de France, 2004. Society, 2011, pp. 245–84. 8. Richard Dedekind and Heinrich Weber, Theorie der alge- 27. Jean-Pierre Serre, Faisceaux algébriques cohérents, Annals of braischen funktionen einer veränderlichen, J. Reine Angew. Mathematics, 61:197–277, 1955. Math., 92:181–290, 1882. Translated and introduced by John 28. , Espaces fibrés algébriques, in Séminaire Chevalley, Stillwell as Theory of Algebraic Functions of One Variable, chapter exposé no. 1, Secrétariat Mathématique, Institut copublication of the AMS and the London Mathematical Henri Poincaré, 1958. Society, 2012. 29. , Rapport au comité Fields sur les travaux de 9. Pierre Deligne, editor, Cohomologie Étale, Séminaire de A. Grothendieck (1965), 퐾-Theory, 3:199–204, 1989. géométrie algébrique du Bois-Marie, Springer-Verlag, 1977. 30. , André Weil: 6 May 1906–6 August 1998, Biographical Generally cited as SGA 4 1/2. This is not strictly a report on Memoirs of Fellows of the Royal Society, 45:520–29, 1999. Grothendieck’s Seminar.

264 Notices of the AMS Volume 63, Number 3 A  M  S 31. , Exposés de Séminaires 1950–1999, Société Mathéma- tique de France, 2001. 32. Peter Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, 2001. 33. Olga Taussky, Sums of squares, American Mathematical Monthly, 77:805–30, 1970. 34. Bartel L. van der Waerden, Zur Nullstellentheorie der Polynomideale, Mathematische Annalen, 96:183–208, 1926. 35. Charles Weibel, An Introduction to Homological Algebra, Algebraic Geometry II Cambridge University Press, 1994. David Mumford, Brown University, Providence, RI, and 36. André Weil, Foundations of Algebraic Geometry, American Tadao Oda, Tohoku University, Japan Mathematical Society, 1946. Several generations of students of algebraic geometry have 37. , Number of solutions of equations in finite fields, learned the subject from David Mumford’s fabled “Red Book”, Bulletin of the American Mathematical Society, 55:487–95, which contains notes of his lectures at Harvard University. Their 1949. genesis and evolution are described by Mumford in the preface: 38. , Number-theory and algebraic geometry, in Proceed- Initially, notes to the course were mimeographed and bound and sold by the Harvard mathematics department with a red cover. ings of the International Congress of Mathematicians (1950: These old notes were picked up by Springer and are now sold as Cambridge, Mass.), American Mathematical Society, 1952, pp. The Red Book of Varieties and Schemes. However, every time I 90–100. taught the course, the content changed and grew. I had aimed to eventually publish more polished notes in three volumes… This book contains what Mumford had then intended to be Volume II. It covers the material in the “Red Book” in more depth, with several topics added. Mumford has revised the notes in col- laboration with Tadao Oda. The book is a sequel to Algebraic Geometry I, published by Springer-Verlag in 1976. Hindustan Book Agency; 2015; 516 pages; Hardcover; ISBN: 978-93-80250-80-9; List US$76; AMS members US$60.80; Order code HIN/70

Operators on Hilbert Space V. S. Sunder, Institute of Mathematical Sciences, Chennai, India

This book’s principal goals are: (i) to present the spectral theorem as a statement on the existence of a unique continuous and mea- surable functional calculus, (ii) to present a proof without digress- ing into a course on the Gelfand theory of commutative Banach algebras, (iii) to introduce the reader to the basic facts concerning the various von Neumann-Schatten ideals, the compact oper- ators, the trace-class operators and all bounded operators, and finally, (iv) to serve as a primer on the theory of bounded linear operators on separable Hilbert space. Hindustan Book Agency; 2015; 110 pages; Softcover; ISBN: 978- 93-80250-74-8; List US$40; AMS members US$32; Order code HIN/69

Problems in the Theory of Modular Forms M. Ram Murty, Michael Dewar, and Hester Graves, Queen’s University, Kingston, Ontario, Canada

This book introduces the reader to the fascinating world of mod- ular forms through a problem-solving approach. As such, it can be used by undergraduate and graduate students for self-instruc- tion. The topics covered include q-series, the modular group, the upper half-plane, modular forms of level one and higher level, the Ramanujan τ-function, the Petersson inner product, Hecke operators, Dirichlet series attached to modular forms, and further special topics. It can be viewed as a gentle introduction for a deeper study of the subject. Thus, it is ideal for non-experts seeking an entry into the field. Hindustan Book Agency; 2015; 310 pages; Softcover; ISBN: 978- 93-80250-72-4; List US$58; AMS members US$46.40; Order code HIN/68

Publications of Hindustan Book Agency are distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels. Order Online: www.ams.org/bookstore Order by Phone: (800)321-4267 (U.S. & Canada), (401)455-4000 (Worldwide)

March 2016 Notices of the AMS 265 COMMUNICATION Meeting Grothendieck, 2012

Katrina Honigs

I met Alexander yard, which had many plants and terra cotta pots Grothendieck on in various degrees of wholeness, and walked up January 2, 2012. I the steps. I knocked on the door and then shouted had read a bit of his “Monsieur Grothendieck!’’ and waited, but there mathematical work was no response. and felt a sense of Suddenly I realized that a figure with a large connection with it. I white beard and a brown robe over his clothes was at a point in my had appeared utterly silently quite nearby on third year of grad- my left. In one hand, he held a short pitchfork uate school where loosely at his side. It reminded me of his doodle I was not only not of devils with pitchforks around the Grothendieck- making progress on Riemann-Roch formula. His free hand rose, bran- solving any prob- dishing an admonitory finger. “Il ne faut pas lems but misera- entrer,” he said, advancing slowly toward me. I bly unengaged by tried to form some sentences about visiting, but my work. Despite Grothendieck did not react. He continued to walk the burnout, Gro- Photo courtesy of Michael Moewe. slowly toward me, wagging his finger, telling me Katrina Honigs enjoys travel. thendieck’s work that I shouldn’t be in here disturbing him. I tried to In addition to having met remained an island give him the galettes, but he told me again to leave. Alexander Grothendieck, she of enjoyment in an Once he had seen me leave his yard, we studied also once touched the nose of otherwise feature- each other from opposite sides of the gate for a a marmot in the Swiss Alps. less sea. So when I moment. We were a similar height, and his blue Both experiences were very was in France for a eyes were alert and focused. Grothendieck asked thrilling, though in different conference, I sought me not angrily, but a bit sternly, in French how ways. him out. I knew his address and how I had gotten there. The village of He told me again that I should not have come Lasserre is small and remote. My appearance in a in and should not have disturbed him in his rental car was a strange enough event that as soon “cloˆıtre”, which reinforced the impression given by as I parked, a friendly man came out of a nearby the brown robe that he thought of himself, in some house to ask if I needed any help. It turned out sense, as a monk. When I was given the address, I Grothendieck’s house was not fifty feet away. had said I wouldn’t tell Grothendieck how I came I had purchased some galettes du roi that morn- by it, so I just watched him silently during this ing in preparation for apologizing to Grothendieck monologue, looking shocked. for entering his yard. After clearing the fence, I Then, he asked me my name and explained that stepped furtively across the slightly ramshackle he could not hear very well anymore and so I must shout into his ear. After I said my name, I started to Katrina Honigs is research assistant professor at the spell it, but he stopped me partway, since he had al- University of Utah. Her email address is honigs@math. ready recognized it: a couple of weeks before I had utah.edu. sent what I now realize was a very enthusiastic fan DOI: http://dx.doi.org/10.1090/noti1346 letter. He then switched to English and, irritably,

266 Notices of the AMS Volume 63, Number 3 asked me why I had included a French translation. to bring baked goods. They smooth everything over “To be polite? La politesse?” Nothing by him that in the American Midwest, where I’m from. I had read had been in English. But of course he I was not able to discuss math with him at all. knew English, and I had offended him a little. At one point, when I tried to make our conversa- He told me he had responded to my letter, ex- tion more detailed by writing on a piece of paper, plaining that my reasons for contacting him were he waved it away. But we had spoken more than not sufficient and that I should not visit. I felt a I had thought we might. When he came back out bit deflated but also couldn’t help but be a little of his house he presented me with a tomato and a amused. Trust a mathematician to tell me that my packet of almond paste. The tomato was large and reasons for writing were “not sufficient.” fresh and came from his garden—impressive for After Grothendieck discovered that I had not January—and he told me to eat it in good health. He received his response to my letter, he seemed to also said I should remember that it was his friend decide that this partly explained my presence. But (likely something was lost in translation). The packet he was still dissatisfied and asked again why I had of almond paste was very visited. Clearly a bit suspicious that I had some large. A kilo. “He told me unsavory motive, he said he thought my visit must After the exchange of indicate that I wanted something. I told him that gifts was over, it seemed we would maybe he didn’t realize, but he is very famous and we were finished. Gro- meet again, I just wanted to meet him. He shrugged and said thendieck wished me well, again that I didn’t have any satisfactory reason for shook my hand again, very soon, visiting, but I could tell that he was a bit amused by and, after entreating me being told that he was famous, and he relaxed a bit. once more to write a letter though not He told me that he could see from my face telling him how I came to that I didn’t have any bad intentions and that know his address, told in this life.” he would never want to harm anyone. I saw then me goodbye and walked that the pitchfork was no longer in his hand but back to his house. I said propped against the fence. If you had received my goodbye as well, but his back was already turned reply, he said, you would have understood that I to me, and I realized right after I spoke that he am not taking visitors and you should not disturb likely didn’t hear me. someone in their retirement. He expected, though, My experience of the rest of the day was odd to receive a letter from me soon explaining how I and heightened. The drive back through the coun- got his address. “C’est la moindre des choses,” he tryside. The primary colors of the public transit intoned, switching back to French for a moment. train in Toulouse. The tomato, when I ate it later He used to receive all his visitors, he said, but he that day. As the days and weeks went on, the visit had had two very bad experiences and no longer was something I reflected on with enjoyment. My did it, though he was very sorry that I came such a burnout faded, and I got more excited about my long way to not be invited in and sorry for himself work again. as well that he was not able to invite me in. He took A little while after my visit, I did write Grothen- my hand and shook it. He told me that he thought dieck again, but my letters were returned. Although we would meet again, very soon, though not in this my fantasies of having some magical conversation life. He told me he thought that he would die within about math with him had to be swept aside, I am the year, though this prediction was made with a grateful to have had the chance to meet him. practiced air that suggested this was not the first A fuller version of this essay is at Katrina Ho- nigs’s personal website, math.utah.edu/~honigs/ time he had made it. Grothendieck.pdf. After these heavy declarations, he turned his attention back to my visit. For all his bluster about not wishing to be disturbed, a part of him was curious about his visitor. How did I get here? On a train? No, in a car. Am I rich? No. Am I poor? Of course I’m poor! I’m a graduate student! I laughed, and he chuckled good-naturedly. Am I alone? Yes. Didn’t I have something for him? The bakery box reemerged, and I opened it to show him the con- tents. He looked at the pastry inside. What is it? Galettes. What? Galettes! Did you make them? No, I bought them. What? I bought them! Oh, thank you for making them. He took the box from me and said he wanted to get something for me too and then went back into his house. I was glad for my instinct

March 2016 Notices of the AMS 267