Alexander Grothendieck

How to use this course

Alexander Grothendieck changed mathematics twice. In his twenties he reshaped functional analysis. In his thirties, at the Institut des Hautes Études Scientifiques, he and his students rewrote algebraic geometry from its foundations, so that the same language covers polynomial equations over the complex numbers, over finite fields and over the integers. That language made possible the proof of the Weil conjectures, the proof of Fermat’s Last Theorem and much of modern number theory. He then left the institution, and later the mathematical community, and spent his last decades in near-solitude.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order follows his work: analysis (1949–1955), homological algebra (1955–1957), Riemann–Roch (1957), schemes and moduli (1958–1962), étale cohomology (1962–1970), and the programmes of his later writings (1980s).

Timeline

Berlin, Hamburg, the camps, 1928–1945

Alexander Grothendieck was born in Berlin on 28 March 1928. His mother, Hanka Grothendieck, came from a Lutheran family in Hamburg and wrote poetry and journalism. His father, Sascha Schapiro, was a Russian Jew from a Hasidic family who had taken part in anarchist revolts, been imprisoned, and lost an arm. In Berlin he worked as a street photographer. The boy took his mother’s surname. He had an older half-sister, Maidi.

When the Nazis came to power in 1933, both parents left for France. Hanka placed her son with a foster family, that of Wilhelm Heydorn, a former pastor who ran a small school near Hamburg. He lived there for five years and went to school in Blankenese. His parents went to Spain and fought on the Republican side in the Civil War.

In 1939, as war approached, the Heydorns could no longer keep him safely, and he was sent by train to France. When war began, he and his mother were interned as enemy aliens in the camp at Rieucros near Mende, from which he walked to the local school. His father was held at Le Vernet; in 1942 the Vichy government handed him to the Germans, and he was murdered at Auschwitz.

After Rieucros closed in 1942, Grothendieck was taken in at Le Chambon-sur-Lignon, a Protestant village on the plateau of the Haute-Loire where many Jewish children were hidden. He boarded at the Collège Cévenol and passed his baccalauréat in 1945. He later recalled that at school he wondered why the textbook definitions of length, area and volume were never made precise, and tried to make them so himself.

Montpellier, Paris, Nancy, 1945–1953

After the war he lived with his mother near Montpellier, in poverty, and studied at the university on a small grant while working in the vineyards. The teaching disappointed him. Working alone, he reconstructed a theory of measure and integration close to Lebesgue’s, without knowing that it already existed.

In 1948 he went to Paris with a letter to Henri Cartan and joined Cartan’s seminar at the École Normale Supérieure. There he met Jean-Pierre Serre, Jean Dieudonné, Laurent Schwartz and André Weil. Cartan judged that he needed a less rarefied setting, and in 1949 he moved to Nancy, where Schwartz and Dieudonné were developing the theory of topological vector spaces.

Dieudonné and Schwartz had just finished a paper on duality for certain locally convex spaces that ended with a list of open problems. In a few months Grothendieck solved them. His output in Nancy was enormous: the thesis, defended in 1953 and published by the American Mathematical Society in 1955, introduced nuclear spaces and a systematic theory of topological tensor products. He was supported by the CNRS from 1950. Because he was stateless, holding only a Nansen passport, and refused French citizenship in this period, a permanent academic post in France was hard to obtain.

São Paulo, Kansas, and the turn to geometry, 1953–1958

From 1953 to 1955 Grothendieck taught at the University of São Paulo. There he completed his work in analysis, including the Résumé de la théorie métrique des produits tensoriels topologiques (1953), which contains what is now called Grothendieck’s inequality (Lesson 04). The paper was little read for fifteen years and is now central in Banach space theory, combinatorial optimization and quantum information.

He then decided he had finished with functional analysis. During 1955–56 at the University of Kansas he turned to homological algebra and sheaf theory, the tools that Cartan, Serre and Jean Leray had built for topology and complex geometry. The result was the long paper Sur quelques points d’algèbre homologique, published in the Tôhoku Mathematical Journal in 1957 (Lesson 05).

Serre’s 1955 paper Faisceaux algébriques cohérents had shown that sheaf cohomology works for algebraic varieties with the Zariski topology. Grothendieck saw how far this could go. In 1957 he proved a sweeping generalization of the Riemann–Roch theorem and lectured on it at the first Arbeitstagung in Bonn; Armand Borel and Serre wrote up the proof (Lesson 06). At the 1958 International Congress in Edinburgh he outlined a programme to rebuild algebraic geometry and to attack the conjectures of André Weil. He was a member of the Bourbaki group in the second half of the 1950s.

Topological tensor products and nuclear spaces

For the general reader. Many objects in analysis are functions of two variables, such as the kernel of an integral operator. Grothendieck asked when such a function can be built from products of functions of one variable, and how to measure the cost. His answers produced a class of spaces, the nuclear spaces, where infinite-dimensional analysis behaves almost as in finite dimensions, and an inequality that now appears in computer science and quantum physics.

Two norms on a tensor product

Let \(E,F\) be Banach spaces. For \(u\in E\otimes F\), define

Always \(\varepsilon\le\pi\). The completions \(E\hat\otimes_\pi F\) and \(E\hat\otimes_\varepsilon F\) are the projective and injective tensor products. For example \(L^1(\mu)\hat\otimes_\pi F\cong L^1(\mu;F)\), while \(C(K)\hat\otimes_\varepsilon F\cong C(K;F)\). Grothendieck showed that, among reasonable tensor norms, these two are the largest and the smallest.

Nuclear spaces

Nuclearity explains the Schwartz kernel theorem: since \(\mathcal D(X)\hat\otimes\mathcal D(Y)\cong\mathcal D(X\times Y)\), every continuous bilinear form on test functions is given by a distribution on the product. Grothendieck also introduced the approximation property: every compact operator into the space is a norm limit of finite-rank operators. He asked whether every Banach space has it. Per Enflo answered no in 1973.

Grothendieck’s inequality

The best constant is unknown. Jean-Louis Krivine showed \(K_G\le\pi/\big(2\ln(1+\sqrt2)\big)\approx1.782\), and this bound was later shown not to be sharp; the best lower bound is about \(1.676\). In modern terms the inequality says that a semidefinite relaxation of a hard optimization problem over signs loses at most a constant factor, which is the basis of approximation algorithms for problems such as MAX-CUT. In physics it bounds how far quantum correlations can exceed classical ones.

Tôhoku: abelian categories and derived functors

For the general reader. Cohomology measures how local information fails to fit together globally. By 1955 there were several cohomology theories, each built by hand. Grothendieck found a common framework in which a cohomology theory is the systematic correction of a construction that fails to preserve exact sequences, and he showed that sheaves fit the framework. This made cohomology a tool that could be used anywhere.

Abelian categories

An abelian category is a category with a zero object, finite direct sums, kernels and cokernels, in which every monomorphism is a kernel and every epimorphism is a cokernel. Modules over a ring and sheaves of abelian groups on a space are examples. Grothendieck introduced a hierarchy of axioms; AB5 asks that arbitrary direct sums exist and that filtered colimits be exact.

Derived functors

Let \(F:\mathcal A\to\mathcal B\) be a left exact additive functor and suppose \(\mathcal A\) has enough injectives. Choose an injective resolution \(0\to A\to I^0\to I^1\to\cdots\) and define \(R^iF(A)=H^i(F(I^\bullet))\). The result is independent of the resolution, \(R^0F=F\), and a short exact sequence \(0\to A'\to A\to A''\to0\) gives a long exact sequence

Sheaf cohomology is \(H^i(X,\mathcal F)=R^i\Gamma(X,\mathcal F)\), the derived functors of global sections. Grothendieck characterized derived functors as universal \(\delta\)-functors, which made comparison theorems almost formal.

Taking \(F=f_*\) for a continuous map \(f:X\to Y\) and \(G=\Gamma(Y,-)\) gives the Leray spectral sequence \(H^p(Y,R^qf_*\mathcal F)\Rightarrow H^{p+q}(X,\mathcal F)\).

The Grothendieck–Riemann–Roch theorem and K-theory

For the general reader. The Riemann–Roch theorem counts how many independent functions with prescribed poles live on a curve. Friedrich Hirzebruch extended it in 1954 to complex manifolds of any dimension. Grothendieck’s version applies to a whole family of varieties at once, and to prove it he invented a new invariant, the group \(K\), which became the starting point of K-theory in topology and algebra.

The Grothendieck group

For a smooth quasi-projective variety \(X\), let \(K(X)\) be the free abelian group on isomorphism classes of coherent sheaves (equivalently, vector bundles) modulo \([\mathcal F]=[\mathcal F']+[\mathcal F'']\) for every exact sequence \(0\to\mathcal F'\to\mathcal F\to\mathcal F''\to0\). Tensor product makes \(K(X)\) a ring. For a projective morphism \(f:X\to Y\) there is a push-forward \(f_!\,[\mathcal F]=\sum_i(-1)^i[R^if_*\mathcal F]\), well defined because of the long exact sequence.

When \(Y\) is a point, \(f_!\) is the Euler characteristic and the theorem becomes Hirzebruch’s formula \(\chi(X,E)=\int_X\operatorname{ch}(E)\operatorname{td}(T_X)\). Proof idea. Factor \(f\) as a closed embedding \(X\hookrightarrow\mathbb P^n_Y\) followed by the projection \(\mathbb P^n_Y\to Y\). Check the projection directly from the structure of \(K(\mathbb P^n_Y)\), and reduce the embedding case, by deformation to the normal bundle, to the zero section of a vector bundle, where an explicit Koszul resolution does the work. The strategy of proving a statement for all morphisms, so that it can be factored and composed, is characteristic of Grothendieck.

Michael Atiyah and Hirzebruch adapted \(K\) to topology in 1959, and it led to the Atiyah–Singer index theorem.

Schemes

For the general reader. Classical algebraic geometry studied the solutions of polynomial equations over the complex numbers. Number theorists wanted the same geometric intuition for equations over the integers or over finite fields. Grothendieck’s schemes do this by attaching a geometric space to every commutative ring. Primes become points, and an integer becomes a function whose value at the prime \(p\) is its remainder mod \(p\).

The spectrum of a ring

For a commutative ring \(A\), the set \(\operatorname{Spec}A\) consists of the prime ideals of \(A\). Its closed sets are \(V(I)=\{\mathfrak p : \mathfrak p\supseteq I\}\). The sets \(D(f)=\{\mathfrak p : f\notin\mathfrak p\}\) form a basis of open sets, and there is a unique sheaf of rings \(\mathcal O\) with \(\mathcal O(D(f))=A_f\). The stalk at \(\mathfrak p\) is the local ring \(A_{\mathfrak p}\). A scheme is a locally ringed space covered by open sets isomorphic to such spectra.

Three features distinguish schemes from classical varieties. First, non-closed points: \((0)\in\operatorname{Spec}\mathbb Z\) is a generic point whose closure is everything. Second, nilpotents: \(\operatorname{Spec}k[\varepsilon]/(\varepsilon^2)\) is one point carrying a tangent direction, which is how deformations and infinitesimal neighbourhoods enter. Third, the base is arbitrary: a scheme over \(\operatorname{Spec}\mathbb Z\) has a fibre over every prime, its reduction mod \(p\), and one generic fibre over \(\mathbb Q\).

The Éléments de géométrie algébrique, written with Dieudonné and published by the IHÉS from 1960 to 1967, built the theory in four volumes, the first part of a much larger plan: schemes, morphisms, cohomology of coherent sheaves, and local properties of morphisms.

Spec ℤ[x]

The points of \(\operatorname{Spec}\mathbb Z[x]\) are: the generic point \((0)\); the primes \((p)\) (generic points of the fibres); the primes \((f)\) for \(f\) irreducible in \(\mathbb Z[x]\) of positive degree (generic points of horizontal curves); and the maximal ideals \((p,g)\) with \(g\) irreducible mod \(p\). The hero picture above shows this space.

Functors of points, representability and moduli

For the general reader. A good way to know an object is to know how everything else maps into it. Grothendieck treated a scheme as a rule assigning to each test space the set of maps from it. Questions such as “is there a space parametrizing all curves of a given kind?” then become precise: is a certain rule of this form? This turned the classification of geometric objects into a theory.

A functor \(F\) is representable if \(F\cong h_X\); the element of \(F(X)\) corresponding to the identity is the universal family. Examples: \(\mathbb A^1\) represents \(T\mapsto\mathcal O(T)\); \(\mathbb G_m\) represents \(T\mapsto\mathcal O(T)^\times\); \(\mathbb P^n\) represents \(T\mapsto\{\)quotients \(\mathcal O_T^{n+1}\twoheadrightarrow L\) with \(L\) a line bundle\(\}\).

The Hilbert scheme, together with the Picard scheme and the theory of descent, appeared in Grothendieck’s Bourbaki seminar talks collected as Fondements de la géométrie algébrique (FGA). Many natural moduli functors are not representable because the objects have automorphisms. Two curves of genus 1 can be locally but not globally isomorphic in a family, so no fine moduli space of elliptic curves exists. The answer was to enlarge the notion of space, first to algebraic spaces (Michael Artin) and then to stacks: Pierre Deligne and David Mumford (1969) showed that the moduli of stable curves form a smooth proper stack.

Topoi, étale cohomology and the Weil conjectures

For the general reader. In 1949 André Weil predicted that counting solutions of equations over finite fields would obey patterns governed by the topology of the corresponding complex shapes, and that a version of the Riemann hypothesis would hold. Proving this needed a cohomology theory for varieties over finite fields. The Zariski topology is too coarse for this. Grothendieck changed what a topology is: covers are allowed to be maps rather than inclusions of open sets.

Sites and topoi

A Grothendieck topology on a category specifies, for each object, which families of maps into it count as covers, subject to axioms modelled on open covers. A category with such a topology is a site; its category of sheaves is a topos. On the étale site of a scheme \(X\), objects are étale morphisms \(U\to X\) (the algebraic analogue of local homeomorphisms) and covers are jointly surjective families. Sheaf cohomology on this site, with coefficients \(\mathbb Z/\ell^n\) for a prime \(\ell\) different from the characteristic, passing to the limit and tensoring with \(\mathbb Q_\ell\), gives \(\ell\)-adic cohomology \(H^i(\bar X,\mathbb Q_\ell)\). It was developed with Michael Artin and Jean-Louis Verdier in SGA 4 (1963–64) and SGA 5.

The Weil conjectures

Let \(X\) be a smooth projective variety of dimension \(n\) over \(\mathbb F_q\), and \(N_m=\#X(\mathbb F_{q^m})\). Its zeta function is \(Z(X,t)=\exp\big(\sum_{m\ge1}N_mt^m/m\big)\).

Bernard Dwork proved rationality in 1960 by \(p\)-adic analysis. Grothendieck’s \(\ell\)-adic cohomology gives 1, 2 and 4 at once, through the Lefschetz trace formula for the Frobenius endomorphism \(F\), whose fixed points on \(X(\bar{\mathbb F}_q)\) are exactly \(X(\mathbb F_q)\):

Poincaré duality gives the functional equation. Grothendieck hoped to prove the Riemann hypothesis through his “standard conjectures” on algebraic cycles, which remain open. Deligne, his student, proved it in 1973 by a different route.

Motives, fundamental groups and dessins d’enfants

For the general reader. By the mid-1960s there were several cohomology theories for algebraic varieties, one for each prime and others besides, all giving the same numbers. Grothendieck proposed that they are shadows of one underlying object, the motive. In his later writings he turned to symmetry: how the Galois group of all algebraic numbers acts on simple drawings on surfaces.

Motives

Grothendieck built a category of pure motives over a field \(k\): objects are pairs \((X,p)\) with \(X\) smooth projective and \(p\) an idempotent algebraic correspondence, up to an adequate equivalence of cycles. Every Weil cohomology should factor through this category. He showed that if his standard conjectures hold, numerical motives form a semisimple abelian category with good properties and the Riemann hypothesis follows. Uwe Jannsen proved semisimplicity unconditionally in 1992; the standard conjectures remain open. Motives guide much of modern arithmetic geometry, including the Langlands programme.

The étale fundamental group (SGA 1)

For a connected scheme \(X\) with geometric point \(\bar x\), the finite étale covers of \(X\) form a category equivalent to finite sets with a continuous action of a profinite group \(\pi_1(X,\bar x)\). For \(X=\operatorname{Spec}k\) this is \(\operatorname{Gal}(k^{\mathrm{sep}}/k)\), so Galois theory and covering spaces become one theory. For \(X\) geometrically connected over a field \(k\),

and over \(\mathbb C\), \(\pi_1\) is the profinite completion of the topological fundamental group. The sequence gives an outer action of \(\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\) on \(\pi_1\) of \(\mathbb P^1\setminus\{0,1,\infty\}\), the starting point of the Esquisse d’un programme (1984) and of anabelian geometry.

Grothendieck’s dessin d’enfant of a Belyi map is the bipartite graph \(\beta^{-1}([0,1])\) drawn on \(X\), with black vertices over 0 and white vertices over 1. It determines \(\beta\) up to isomorphism, and its number of edges is \(\deg\beta\). Hence \(\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\) acts on these combinatorial drawings, and the action is faithful.

The IHÉS years and the break, 1958–1973

The Institut des Hautes Études Scientifiques was founded in 1958 by the businessman Léon Motchane, with Dieudonné and Grothendieck as its first permanent professors. It had no building at first; the seminar met in borrowed rooms in Paris before moving to Bures-sur-Yvette. For twelve years Grothendieck worked there at an extraordinary pace. The Séminaire de géométrie algébrique du Bois Marie (SGA 1 to SGA 7) and EGA together run to many thousands of pages.

The seminar was also a school. His doctoral students included Michel Demazure, Jean Giraud, Jean-Louis Verdier, Michel Raynaud, Pierre Deligne, Jean-Pierre Jouanolou, Luc Illusie, Pierre Berthelot and William Messing, and later in Paris and Montpellier Hoàng Xuân Sính and others. Michael Artin, Jean-Pierre Serre and many visitors took part. His correspondence with Serre, published in 2001, shows the two working in close partnership, Serre often supplying a key example or objection.

He received the Fields Medal at the 1966 International Congress in Moscow, for work that built on Weil and Oscar Zariski and transformed algebraic geometry. He did not travel to Moscow, in protest against Soviet military action in Eastern Europe; Motchane accepted the medal for him. In late 1967 he spent about a month in North Vietnam, teaching mathematics to Hanoi University staff evacuated to the countryside during the American bombing.

In 1970 he learned that part of the IHÉS budget came from the French Ministry of Defence. When his objections were not met, he resigned. That summer he founded, with colleagues, the group Survivre (later Survivre et Vivre), which campaigned against military research and for ecology, and published a bulletin. He held a temporary post at the Collège de France from 1970 to 1972, where he devoted many lectures to these questions, then a visiting post at Orsay. He became a French citizen in 1971. In 1973 Deligne completed the proof of the Weil conjectures by a method Grothendieck had not anticipated.

Montpellier, Lasserre and legacy, 1973–2014

From 1973 Grothendieck was a professor at the University of Montpellier, living in villages of the Hérault and then the Vaucluse. He taught undergraduates and supervised a few doctoral students. In the 1980s mathematics returned in a new form. À la poursuite des champs (1983), begun as a letter to Daniel Quillen, sketched higher stacks and homotopy theory; Esquisse d’un programme (1984), written for a CNRS application, set out dessins, anabelian geometry and “tame topology”; Les dérivateurs followed later in the decade. From 1984 to 1988 he held a research position at the CNRS.

Between 1983 and 1986 he wrote Récoltes et Semailles, a reflection of more than a thousand pages on his life as a mathematician, on creativity, and on what he saw as the neglect of his work and the misappropriation of his ideas by former students and colleagues. It circulated in photocopies and was published by Gallimard only in 2022.

In 1988 he retired. The Royal Swedish Academy of Sciences awarded him the Crafoord Prize jointly with Deligne; he declined it in a letter criticizing the ethics of the scientific community. In August 1991 he left his home without notice and moved to Lasserre, a village in the Ariège foothills of the Pyrenees, where he lived in near-isolation and continued to write. Before leaving, he entrusted about 20,000 pages of manuscripts to Jean Malgoire at Montpellier. In 2010 he asked that his works not be republished or distributed. He died in hospital at Saint-Girons on 13 November 2014, aged 86. After his death the Montpellier manuscripts were digitized and placed online, and the larger Lasserre archive was deposited with the Bibliothèque nationale de France.

Legacy

• Language: schemes, sheaves, derived functors, topoi, stacks and K-theory are the working language of algebraic geometry and much of number theory and topology.
• Number theory: étale cohomology underlies Deligne’s proof of the Weil conjectures, the Langlands programme, and Andrew Wiles’s proof of Fermat’s Last Theorem.
• Later programmes: higher stacks and homotopical algebra, dessins d’enfants and anabelian geometry (proved in key cases by Shinichi Mochizuki and others), and motives, still an active field.
• Analysis and beyond: nuclear spaces in distribution theory and quantum field theory; Grothendieck’s inequality in algorithms and quantum information.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Alexander Grothendieck, MacTutor History of Mathematics.
2. Grothendieck Circle, A brief timeline of the life of Alexander Grothendieck.
3. A. Jackson, Comme Appelé du Néant: As If Summoned from the Void, Part I and Part II, Notices of the AMS 51 (2004).
4. L. Illusie et al., Alexandre Grothendieck 1928–2014, Parts 1 and 2, Notices of the AMS 63 (2016).
5. University of Montpellier, In search of Alexandre Grothendieck (on the manuscript archives).
6. Mathematics Genealogy Project, Alexander Grothendieck.
7. A. Grothendieck, Récoltes et Semailles, 2 vols., Gallimard, 2022.
8. P. Colmez and J.-P. Serre (eds.), Grothendieck–Serre Correspondence, AMS/SMF, 2004.

Mathematics

1. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
2. A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953).
3. G. Pisier, Grothendieck’s theorem, past and present, arXiv:1101.4195.
4. A. Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. 9 (1957).
5. A. Borel and J.-P. Serre, Le théorème de Riemann–Roch, Bull. Soc. Math. France 86 (1958).
6. A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, Publ. Math. IHÉS 4, 8, 11, 17, 20, 24, 28, 32 (1960–67), freely available on Numdam.
7. B. Fantechi et al., Fundamental Algebraic Geometry: Grothendieck’s FGA Explained, AMS, 2005.
8. P. Deligne, La conjecture de Weil I, Publ. Math. IHÉS 43 (1974).
9. A. Grothendieck, Esquisse d’un programme (1984), in L. Schneps and P. Lochak (eds.), Geometric Galois Actions I, Cambridge, 1997.
10. D. Mumford, The Red Book of Varieties and Schemes, Springer, 1988 (for the picture of Spec ℤ[x]).
11. R. Vakil, The Rising Sea: Foundations of Algebraic Geometry, lecture notes.