Claude Chevalley - Work

Work

In his PhD thesis, Chevalley made an important contribution to 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 In the introduction to André Weil's Basic Number Theory, Weil attributed the book's adoption of that path to an unpublished manuscript by Chevalley.

Around 1950, Chevalley wrote a three-volume treatment of Lie groups. A few years later, he published the work for which he is best remembered, his investigation into what are now called Chevalley groups. Chevalley groups make up 9 of the 18 families of finite simple groups.

Chevalley's accurate discussion of integrality conditions in the Lie algebras of semisimple groups enabled abstracting their theory from the real and complex fields. As a consequence, analogues over finite fields could be defined. This was an essential stage in the evolving classification of 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" (also called the Chevalley–Warning theorem) usually refers to his result on the solubility of equations over a finite field. Another theorem of his concerns the constructible sets in algebraic geometry, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states that the image of such a set by a morphism of algebraic varieties is of the same type. Logicians call this an elimination of quantifiers.

In the 1950s, Chevalley led some Paris seminars (working groups to the English speaking world) 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 abstract algebra. The Cartan–Chevalley seminar was the genesis of scheme theory, but its subsequent development in the hands of Alexander Grothendieck was so rapid, thorough and inclusive that its historical tracks can appear well covered. Grothendieck's work subsumed the more specialised contribution of Serre, Chevalley, Goro Shimura, and others such as Erich Kähler and Masayoshi Nagata.