Claude Chevalley

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Claude Chevalley (11 February 1909 - 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry and the theory of algebraic groups. He was born in Johannesburg, and died in Paris. He was a founding member of the Bourbaki group.

After graduating from the École Normale Supérieure, he studied under Emil Artin in Hamburg from 1931, a time which may have formed his mathematical taste, and where he had contact with the Japanese school in the person of Shokichi Iyanaga, and then under Helmut Hasse. One of his achievements was a step in the technical development of class field theory, removing a use of L-functions and replacing it by an algebraic method. At that time use of group cohomology was implicit, cloaked by the language of central simple algebras. In the introduction to his book Basic Number Theory, Chevalley's friend André Weil explains that the book's adoption of that road goes back to an old, unpublished manuscript of Chevalley.

Chevalley also wrote a three-volume treatment of Lie groups around 1950. A few years later he published an investigation into what are now called Chevalley groups, and for which he is most remembered. An accurate discussion of conditions of integrality in the Lie algebras of semisimple groups enabled their theory to be abstracted from the real and complex fields. As a consequence, analogues over finite fields could be defined. This was an essential stage in the theory of the finite simple groups. After Chevalley's work the distinction between 'classical groups' falling into the Dynkin diagram classification, and 'sporadic groups' which did not, became sharp enough to be useful. What are called 'twisted' groups of the classical families could be fitted into the picture.

Chevalley's theorem usually refers to his result on solubility of equations over a finite field (also called the Chevalley-Warning theorem). Another theorem concerns the "constructible" sets in algebraic geometry, i.e. those in the Boolean algebra generated by the Zariski-open and closed sets. It states that the image of such a set by a morphism of algebraic varieties is of the same type. In logicians' terms, this is an 'elimination of quantifiers'.

Chevalley in the 1950s led some Paris seminars (working groups) of major importance: the Séminaire Cartan-Chevalley of the academic year 1955/6, with Henri Cartan, and the Séminaire Chevalley of 1956/7 and 1957/8. These dealt with topics on algebraic groups and the foundations of algebraic geometry, as well as pure algebra. This was the actual point of genesis of scheme theory, in the Cartan-Chevalley seminar; though the subsequent development by Alexander Grothendieck was so rapid, thorough and inclusive that its historical tracks can appear well covered. The more specialised ideas of Serre, Chevalley, Shimura and probably others such as Kähler and Nagata were all subsumed.

In 1939 he was at Princeton University. After reporting to the French Embassy at the outbreak of World War II, he stayed in the USA, taking a position in 1947 at Columbia University, and becoming an American citizen. Subsequently, when he wanted to return to France, his difficulties in a candidature for a Sorbonne chair were described by André Weil in a polemical piece (Science Française?, in La Nouvelle N. R. F. — Chevalley was the professeur B of the piece, as confirmed in the endnote to the reprint in Weil, Oeuvres Scientifiques, tome II). He later (1957) did obtain a position at the Université de Paris VII. He had a number of distinguished students. [1]

Chevalley's works have been published in six volumes, edited by Pierre Cartier and Catherine Chevalley.

Incidentally, Chevalley discovered Zorn's Lemma before Max Zorn did himself and mentioned it to Zorn as a way to simplify a proof in one of the latter's papers. However, like many mathematical results, this lemma has been attributed to the wrong person.

[edit] Selected bibliography

  • La théorie du corps de classes, Annals of Mathematics 41 (1940), 394 – 418.
  • Theory of Lie groups, I, Princeton, Princeton University Press, 1946.
  • Sur certaines groupes simples, Tôhoku Mathematical Journal 7 (1955), 14 – 66.

[edit] See also

[edit] External links

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