Claude Chevalley

How to use this course

Claude Chevalley is less famous outside mathematics than his Bourbaki colleagues André Weil and Henri Cartan, but his name is everywhere inside it: Chevalley groups, Chevalley bases, the Chevalley–Warning theorem, the Chevalley–Eilenberg complex, Chevalley’s restriction theorem, the Jordan–Chevalley decomposition. He introduced ideles into number theory, wrote the first modern book on Lie groups, and showed that every simple Lie algebra gives a simple group over every field. That construction supplied most of the families in the classification of finite simple groups.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 and 05 cover his number theory of the 1930s. Lessons 06 to 10 follow his move into Lie theory and algebraic groups from the 1940s to the late 1950s and can be read in order as a short course on that subject.

Timeline

Johannesburg to the rue d’Ulm, 1909–1929

Claude Chevalley was born on 11 February 1909 in Johannesburg, in what was then the British Transvaal Colony, where his father, Abel Chevalley, was serving as a French diplomat. His mother was Marguerite Chevalley, née Sabatier. He was their only son. His parents were scholars as well as officials: together they compiled the Concise Oxford French Dictionary, a standard reference for English-speaking students of French for decades.

The family returned to France. Chevalley went to primary school at Chançay, a village in Indre-et-Loire, and then to the Lycée Louis-le-Grand in Paris, the school that prepared many of the best French students for the competitive entrance examinations. In 1926 he entered the École Normale Supérieure on the rue d’Ulm. Its mathematics teaching was then dominated by the old masters of analysis; he attended the lectures of Émile Picard. In 1929 he graduated and passed the agrégation, placed third.

A few years ahead of him at the École were Weil, Henri Cartan, Jean Dieudonné and Jean Delsarte, the future core of Bourbaki. They shared a frustration with French mathematics of the time. The generation that would have taught them had been decimated in the First World War, and the curriculum still centred on function theory while algebra and topology were being transformed in Germany. Chevalley turned towards the German algebraic school.

Hamburg, Marburg and the thesis, 1931–1933

In 1931–32 Chevalley studied with Emil Artin in Hamburg, then moved to Marburg, where Helmut Hasse had succeeded Kurt Hensel in 1930. These were the two leading centres of class field theory, the description of the abelian extensions of a number field in terms of data internal to that field. The theory was complete in its statements after the work of Teiji Takagi (1920) and Artin’s reciprocity law (1927), but its proofs were long and depended on analysis: Dirichlet series and their behaviour near \(s=1\).

Chevalley set himself the task of making the theory arithmetic. His thesis, Sur la théorie du corps de classes dans les corps finis et les corps locaux, defended at the University of Paris in 1933, developed class field theory for local fields directly and studied its relation with the global theory. It was published in the journal of the Faculty of Science of the University of Tokyo. The same year he married his cousin Jacqueline; the marriage lasted until 1948.

From Artin he also took a problem about finite fields. Artin had conjectured that every form over a finite field whose degree is less than its number of variables has a non-trivial zero. Chevalley proved it in a short paper published in 1936, and Ewald Warning strengthened the result at the same time (Lesson 04). The proof is a page long and is taught today in first courses in number theory.

Bourbaki and the non-conformists, 1934–1938

In December 1934 Weil gathered young mathematicians who taught the calculus course in French universities to write a modern treatise. Chevalley, aged 25, was the youngest of the founding members, alongside Weil, Cartan, Delsarte, Dieudonné and René de Possel. At the first congress, at Besse-en-Chandesse in July 1935, the group adopted the name Nicolas Bourbaki. Chevalley’s taste for complete generality and for proofs stripped of every inessential step was close to the style the group adopted. He remained an active member for many years, and his own books, on Lie groups and on algebra, read like Bourbaki volumes signed with a personal name.

In these years Chevalley was also part of the non-conformist movements of the early 1930s, small groups of young intellectuals who rejected both liberal capitalism and communism and sought a new political and social order. Pierre Cartier later wrote that he moved in avant-garde circles in politics and the arts and left pamphlets and notes on these subjects. He taught for a time at the University of Rennes.

Ideles

His mathematics in the mid-1930s concentrated on class field theory. In a paper of 1936 on infinite extensions he introduced “ideal elements” of a number field, families of local units and numbers indexed by all primes. Under the name ideles they became the standard language of the subject (Lesson 05). In 1938 he went to the Institute for Advanced Study in Princeton. He would not hold a post in France again for nineteen years.

The Chevalley–Warning theorem

For the general reader. An equation like \(x^2+y^2+z^2=0\) has the obvious solution \(x=y=z=0\). Does it have any other? Over the real numbers, no. Over a finite field, Chevalley showed in 1936 that the answer is always yes whenever there are more unknowns than the degree of the equation. The proof uses nothing more than a clever polynomial and a sum over the whole field.

Proof

Lemma. For an integer \(k\ge0\), with the convention \(0^0=1\),

For \(k=0\) the sum is \(q=0\) in \(\mathbb F_q\). For \(k\ge1\), \(\mathbb F_q^\times\) is cyclic of order \(q-1\); if \((q-1)\nmid k\), pick \(c\) with \(c^k\ne1\); multiplying the sum by \(c^k\) permutes its terms, so \((c^k-1)\sum x^k=0\). If \((q-1)\mid k\), each of the \(q-1\) non-zero terms equals 1, giving \(q-1=-1\).

Main argument. Let \(P(x)=\prod_{i=1}^r\big(1-f_i(x)^{q-1}\big)\). Since \(a^{q-1}=1\) for \(a\ne0\), \(P(x)=1\) if \(x\) is a common zero and \(P(x)=0\) otherwise. So \(N\equiv\sum_{x\in\mathbb F_q^n}P(x)\) modulo \(p\). Now \(\deg P\le(q-1)\sum\deg f_i<n(q-1)\). Each monomial \(x_1^{a_1}\cdots x_n^{a_n}\) of \(P\) has some \(a_j<q-1\), and its sum over \(\mathbb F_q^n\) factors as \(\prod_j\sum_{x_j}x_j^{a_j}\). The factor with \(a_j<q-1\) vanishes by the lemma (if \(a_j=0\) it equals \(q=0\); otherwise \((q-1)\nmid a_j\)). Hence \(\sum_xP(x)=0\) in \(\mathbb F_q\), and \(p\mid N\). If \(0\) is a zero, \(N\ge1\) and \(p\mid N\) force \(N\ge p\ge2\). \(\square\)

Consequences

A field \(K\) is quasi-algebraically closed (or \(C_1\)) if every homogeneous form of degree \(d\) in \(n>d\) variables has a non-trivial zero. Chevalley’s theorem says finite fields are \(C_1\), as Artin had conjectured. Two consequences:

• Every quadratic form in at least 3 variables over \(\mathbb F_q\) is isotropic.
• Over a \(C_1\) field the norm map of every finite extension is surjective, which implies that the Brauer group of a finite field is trivial; this recovers Wedderburn’s theorem that every finite division ring is commutative.

Ideles and class field theory

For the general reader. Class field theory describes certain extensions of a number system, such as the rationals, using only information about the system itself. Before Chevalley, this information was organized through ideals and analytic tools. He replaced it with one group, the ideles, which records a number’s behaviour at every prime at once. The theory became shorter, purely algebraic, and ready to be generalized.

Let \(K\) be a number field. For each place \(v\) (a prime ideal or an archimedean embedding up to conjugation) let \(K_v\) be the completion and, for finite \(v\), \(\mathcal O_v^\times\) its unit group. The idele group is the restricted product

topologized so that \(\prod_{v\mid\infty}K_v^\times\times\prod_{v\nmid\infty}\mathcal O_v^\times\) is open. \(K^\times\) embeds diagonally as a discrete subgroup, and the idele class group is \(C_K=\mathbb I_K/K^\times\). The map sending an idele to the fractional ideal \(\prod_{v\nmid\infty}\mathfrak p_v^{v(x_v)}\) is surjective with kernel \(\prod_{v\mid\infty}K_v^\times\times\prod_{v\nmid\infty}\mathcal O_v^\times\), so

Chevalley’s contributions

Chevalley introduced ideles in 1936 and used them in his 1940 Annals paper to give the first proof of the main theorems that avoided analysis entirely. The analytic input had been needed for the “second inequality” \([C_K:N_{L/K}C_L]\le[L:K]\) for cyclic \(L/K\); Chevalley proved it arithmetically. The idelic language also made the passage between local and global theory transparent, because the global map is assembled from local maps place by place. Weil later added the additive adeles (see the Weil course, Lesson 08), and the cohomological treatment of Hochschild, Nakayama, Artin and Tate was written entirely in this language.

Lie groups and Lie algebra cohomology

For the general reader. A Lie group is a group that is also a smooth space, like the rotations of space. Before 1946 the subject was taught mostly through local formulas, valid near the identity. Chevalley’s Theory of Lie Groups treated these groups globally, as manifolds, and became the model for every later textbook. Two years later, with Samuel Eilenberg, he turned the topology of compact groups into pure algebra.

Theory of Lie Groups I (1946)

Chevalley defined analytic manifolds intrinsically, by an atlas, and built the theory on that basis. The book proved the global correspondence between Lie algebras and groups and gave a complete treatment of compact groups, including Tannaka duality: a compact Lie group is recovered from its representations, and is the group of real points of a complex algebraic group, its complexification.

The Chevalley–Eilenberg complex (1948)

Let \(\mathfrak g\) be a Lie algebra over a field \(k\) and \(M\) a \(\mathfrak g\)-module. Put \(C^n(\mathfrak g,M)=\mathrm{Hom}_k(\Lambda^n\mathfrak g,M)\) and

Then \(d^2=0\), and \(H^n(\mathfrak g,M)\) is the cohomology of this complex. The formula is the algebraic shadow of the exterior derivative of left-invariant forms on a Lie group.

The first statement rests on averaging over the compact group, which makes every cohomology class representable by an invariant form. The vanishing of \(H^1\) is Weyl’s complete reducibility in cohomological form; the vanishing of \(H^2\) gives the Levi decomposition. Chevalley and Eilenberg’s paper was one of the starting points of homological algebra, alongside the group cohomology of Eilenberg and Saunders Mac Lane.

The Jordan–Chevalley decomposition

In the second volume (1951), on algebraic groups, Chevalley used systematically the following fact: over a perfect field, every endomorphism \(x\) of a finite-dimensional space is uniquely \(x=x_s+x_n\) with \(x_s\) semisimple, \(x_n\) nilpotent and \(x_sx_n=x_nx_s\); moreover \(x_s\) and \(x_n\) are polynomials in \(x\). In an algebraic group the multiplicative version \(g=g_sg_u\) holds, and both factors stay in the group.

Clifford algebras and spinors

For the general reader. Spinors are the objects that physicists use for electrons: they change sign under a full turn and return to themselves only after two. Élie Cartan discovered them geometrically, and Paul Dirac found them in his equation. Chevalley’s 1954 book built the whole theory algebraically, for a quadratic form over any field, so that it applies equally to physics, number theory and finite groups.

Let \(V\) be an \(n\)-dimensional vector space over a field \(k\) with a quadratic form \(Q\). The Clifford algebra is

In \(C(V,Q)\) one has \(v^2=Q(v)\) and \(uv+vu=B(u,v)\), where \(B(u,v)=Q(u+v)-Q(u)-Q(v)\). The algebra is graded by parity, \(C=C^+\oplus C^-\). It has the universal property: any linear map \(f:V\to A\) into an associative algebra with \(f(v)^2=Q(v)\cdot1\) extends uniquely to an algebra homomorphism \(C(V,Q)\to A\).

The spin group

Let \(\alpha\) be the parity automorphism and \(x\mapsto x^t\) the anti-automorphism reversing products of vectors. The Clifford group \(\Gamma\) consists of invertible \(x\) with \(\alpha(x)Vx^{-1}=V\); each such \(x\) acts on \(V\) by an orthogonal transformation, and an anisotropic vector \(v\) acts as minus the reflection in \(v^\perp\). The spinor norm \(N(x)=x^tx\) takes values in \(k^\times\). Define \(\mathrm{Spin}(Q)=\{x\in\Gamma\cap C^+:N(x)=1\}\). Then there is an exact sequence

where \(\theta\) is the spinor norm. For \(k=\mathbb R\) and \(Q\) negative definite, \(\theta\) is trivial and \(\mathrm{Spin}(n)\to\mathrm{SO}(n)\) is the connected double cover. Over a finite field of odd order, the image of \(\mathrm{Spin}\) is a subgroup \(\Omega(Q)\) of index 2 in \(\mathrm{SO}(Q)\); modulo its centre it gives, in dimension at least 5, the finite simple orthogonal groups. Chevalley’s book also treated characteristic 2, where quadratic and bilinear forms must be distinguished.

Invariants of reflection groups

For the general reader. Symmetric polynomials, such as \(x+y+z\) and \(xyz\), are those unchanged by permuting the variables. A classical theorem says every symmetric polynomial is built from a few basic ones, with no relations among them. Chevalley found the exact reason in 1955: the symmetric group is generated by reflections. Any finite group generated by reflections has the same property.

Let \(V\) be a finite-dimensional vector space over a field \(k\) of characteristic 0 and \(G\subseteq\mathrm{GL}(V)\) a finite group, acting on the polynomial ring \(S=k[V]\). A reflection is an element of finite order fixing a hyperplane pointwise.

Shephard and Todd had established the equivalence for complex reflection groups through their classification. Chevalley gave the first uniform proof that reflection groups have polynomial invariants; a general proof of the converse is due to Jean-Pierre Serre.

Proof idea

Let \(I\subseteq S\) be the ideal generated by invariants of positive degree, and choose homogeneous invariants \(f_1,\dots,f_m\) forming a minimal generating set of \(I\). One shows they generate \(S^G\) as an algebra (by induction on degree, using the averaging operator) and that they are algebraically independent. The key tool for independence is the reflection: if \(s\) fixes the hyperplane \(\ell_s=0\), then \(p-s\cdot p\) is divisible by \(\ell_s\) for every \(p\in S\). Using these divided differences, a minimal relation among the \(f_i\) would produce a relation of lower degree. Counting dimensions via the Hilbert series \(\sum\dim S^G_d\,t^d=\prod_i(1-t^{d_i})^{-1}\) and Molien’s formula \(\frac1{|G|}\sum_g\det(1-tg)^{-1}\) gives the formulas for \(\prod d_i\) and \(\sum(d_i-1)\) by comparing the poles at \(t=1\).

The restriction theorem

For \(\mathfrak{gl}_n\), this says that conjugation-invariant polynomials on matrices are exactly the symmetric functions of the eigenvalues, generated by the coefficients of the characteristic polynomial.

Chevalley groups

For the general reader. The simple Lie algebras over the complex numbers were classified by Wilhelm Killing and Élie Cartan: four infinite families and five exceptional ones. In 1955 Chevalley showed how to turn each of them into a group over any field whatsoever, including finite fields. Over finite fields this produced, in one uniform construction, most of the known finite simple groups and several new ones. The classification of finite simple groups, completed decades later, rests on his construction.

Let \(\mathfrak g\) be a complex simple Lie algebra with Cartan subalgebra \(\mathfrak h\), root system \(\Phi\) and simple roots \(\alpha_1,\dots,\alpha_\ell\). For each root \(\alpha\) let \(h_\alpha\in\mathfrak h\) be its coroot.

Since \(\mathrm{ad}\,e_\alpha\) is nilpotent, \(\exp(t\,\mathrm{ad}\,e_\alpha)=\sum_k t^k(\mathrm{ad}\,e_\alpha)^k/k!\) is a finite sum. Chevalley showed that \((\mathrm{ad}\,e_\alpha)^k/k!\) maps \(\mathfrak g_{\mathbb Z}\) into itself. So for any field \(K\) and \(t\in K\), the operator \(x_\alpha(t)=\exp(t\,\mathrm{ad}\,e_\alpha)\) makes sense on \(\mathfrak g_K=\mathfrak g_{\mathbb Z}\otimes K\).

For the classical types the construction recovers the projective special linear, symplectic and orthogonal groups studied by Camille Jordan and Leonard Dickson. Dickson had also found groups of types \(G_2\) and \(E_6\). Over finite fields, types \(F_4\), \(E_7\) and \(E_8\) were new. For \(K=\mathbb F_q\) the order of the adjoint Chevalley group is

where \(N=|\Phi^+|\), \(d_1,\dots,d_\ell\) are the degrees of the basic invariants of the Weyl group (Lesson 08), and \(d\) is the order of the centre of the corresponding simply connected group over \(\mathbb F_q\). Steinberg, Tits, Ree and Suzuki soon added the twisted groups, and the families of groups of Lie type, together with the cyclic and alternating groups and 26 sporadic groups, make up all finite simple groups.

Constructible sets and algebraic groups

For the general reader. The image of a curve or surface under a polynomial map need not be a nice set: it can miss isolated points or lines. Chevalley showed that it is always built from finitely many pieces of the simplest kind. In the same years he classified the semisimple algebraic groups over any algebraically closed field and found that the answer does not depend on the characteristic, a surprise that shaped the next half-century.

A subset of a variety (or Noetherian topological space) is constructible if it is a finite union of locally closed sets, that is, of sets \(U\cap Z\) with \(U\) open and \(Z\) closed.

Over \(\mathbb C\) this is the algebraic form of elimination of quantifiers for algebraically closed fields: a set defined by a formula with existential quantifiers over polynomial equations can be defined without them. It is used constantly in the theory of algebraic groups, for instance to show that orbits are locally closed and that the image of a homomorphism of algebraic groups is closed.

Classification of semisimple groups

In his seminar of 1956–58, Chevalley used Armand Borel’s theory of Borel subgroups and tori to classify semisimple algebraic groups over an algebraically closed field \(k\). A root datum is a quadruple \((X,\Phi,X^\vee,\Phi^\vee)\): a lattice \(X\) (the characters of a maximal torus), its dual \(X^\vee\), and finite subsets of roots and coroots satisfying the axioms of a root system.

So the list of simple algebraic groups is the same over \(\mathbb C\) as over \(\overline{\mathbb F}_p\). Michel Demazure and Grothendieck extended the result to group schemes over arbitrary base schemes in SGA 3 (1962–64): the Chevalley group schemes over \(\mathbb Z\) are the split forms from which all others are twisted.

In algebraic geometry, Chevalley’s 1943 paper on local rings developed Krull’s theory of completions, and his seminar with Henri Cartan in 1955–56 defined varieties as collections of local rings in a function field, one of the drafts that led to Grothendieck’s schemes.

America, 1938–1957

Chevalley arrived at the Institute for Advanced Study in 1938. When war broke out in September 1939 he reported to the French Embassy, but he remained in the United States for the duration of the war and joined the faculty of Princeton University.

The American years transformed his work. His 1940 paper on class field theory won the Cole Prize in Number Theory of the American Mathematical Society in 1941. His first doctoral student at Princeton, Gerhard Hochschild (1941), went on to found the cohomology of algebras and, with Tadashi Nakayama, the cohomological approach to class field theory. Oscar Goldman was also his doctoral student. Chevalley turned increasingly to Lie theory: Theory of Lie Groups I (1946) and the paper with Samuel Eilenberg on Lie algebra cohomology (1948) date from this period.

From July 1949 to June 1957 he was professor of mathematics at Columbia University in New York, where Leon Ehrenpreis was among his students, and he became an American citizen. He wrote Introduction to the Theory of Algebraic Functions of One Variable (1951), the second and third volumes of his Lie groups treatise (1951, 1955), The Algebraic Theory of Spinors (1954) and Fundamental Concepts of Algebra (1956). In 1953–54 he lectured on class field theory at Nagoya, and the Tôhoku paper on simple groups appeared in 1955.

He wished to return to France and applied for a vacant chair at the Sorbonne. The application ran into severe difficulties, and he did not obtain a post in Paris until 1957. His first marriage had ended in 1948. He later married Sylvie, a historian of the theatre. His daughter Catherine Chevalley (1951–2022) became a philosopher.

Paris, Vincennes, legacy, 1957–1984

In Paris Chevalley ran a seminar that completed the classification of semisimple algebraic groups (1956–58) and published Fondements de la géométrie algébrique (1958). He was an invited speaker at the International Congress of Mathematicians in Edinburgh in 1958, and an honorary member of the London Mathematical Society from 1967. His later doctoral students included Michel Broué, Lê Dũng Tráng and Michel André.

He supported the student movement of May 1968 and took part in founding the experimental university centre at Vincennes, where he created the mathematics department and taught. In 1970, when the University of Paris was divided, he was assigned to Paris VII. The same year he co-founded with Alexander Grothendieck and Pierre Samuel the group Survivre et vivre, an early French movement combining ecology, anti-militarism and criticism of the role of science. He was a strong player of the game of go and taught it to the poet and mathematician Jacques Roubaud. He retired in 1978 and died in Paris on 28 June 1984, aged 75.

Catherine Chevalley described his view of mathematics as the search for the reason a proof must exist. Rigour meant removing everything inessential. Pierre Cartier wrote that mathematics was the centre of his life and that he drew no line between it and the rest.

Legacy

• Number theory: ideles and the arithmetic proof of class field theory; the Chevalley–Warning theorem and quasi-algebraically closed fields.
• Lie theory: the global, manifold-based treatment of Lie groups; Lie algebra cohomology; Chevalley bases and the restriction theorem.
• Finite groups: Chevalley groups, the core of the classification of finite simple groups, and the starting point of Steinberg’s and Tits’s work.
• Algebraic groups: the characteristic-free classification by root data, the foundation of the Langlands programme’s dual groups.
• Honours: the AMS Chevalley Prize in Lie Theory, endowed by George Lusztig in 2014 and first awarded to Geordie Williamson in 2016.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Claude Chevalley, MacTutor History of Mathematics.
2. Claude Chevalley, Wikipedia; Claude Chevalley, Wikipédia (French).
3. Chevalley Claude (1909–1984), Encyclopædia Universalis.
4. J. Dieudonné and J. Tits, Claude Chevalley (1909–1984), Bull. Amer. Math. Soc. 17 (1987), 1–7.
5. Nicolas Bourbaki, Wikipedia.

Mathematics

1. C. Chevalley, Démonstration d’une hypothèse de M. Artin, Abh. Math. Sem. Hamburg 11 (1936), 73–75.
2. C. Chevalley, La théorie du corps de classes, Ann. of Math. 41 (1940), 394–418.
3. C. Chevalley, Theory of Lie Groups I, Princeton University Press, 1946.
4. C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.
5. C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, 1954.
6. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782.
7. C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. 7 (1955), 14–66.
8. C. Chevalley, Classification des groupes algébriques semi-simples (Séminaire 1956–58), Springer, 2005.
9. R. Steinberg, Lectures on Chevalley Groups, Yale, 1967; reprinted AMS University Lecture Series 66, 2016.
10. R. W. Carter, Simple Groups of Lie Type, Wiley, 1972.
11. J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Academic Press, 1967.
12. Chevalley–Warning theorem and Group of Lie type, Wikipedia.