Algebraic Geometry between Noether and Noether - a forgotten chapter in the history of Algebraic Geometry

Gray, Jeremy

Revue d'histoire des mathématiques, Tome 3 (1997) no. 1, pp. 1-48.

Mathématiciens et historiens considèrent en général que les travaux de Kronecker et de Hilbert inaugurent la période moderne de la géométrie algébrique. Mais on a souvent envisagé les articles correspondants de Hilbert comme une reformulation de la théorie des invariants, sujet de caractère nettement plus algébrique, alors que Kronecker était présenté comme promoteur doctrinaire d'une mathématique arithmétisée, finie. À partir de là, l'attention s'est portée sur la tradition italienne, laissant dans l'oubli la voie menant à Emmy Noether.Et pourtant, il y eut un flux continu de publications, répondant aux travaux de Hilbert aussi bien que de Kronecker. Les mathématiciens hongrois Gyula (Julius) König et József Kürschák, les Français Jules Molk et Jacques Hadamard, Emmanuel Lasker et enfin le professeur de lycée anglais F.S.Macaulay, ont tous publié abondamment sur le sujet. Ces travaux sont étroitement liés à une élaboration progressive des notions d'anneau, de corps, et autres concepts connexes. L'évolution des préoccupations que manifestent ces publications fait ressortir de combien la géométrie algébrique est redevable à ses deux éminents fondateurs, et la façon dont se présentaient les rapports entre algèbre et géométrie dans la période immédiatement antérieure aux débuts de l'œuvre d'Emmy Noether.

Mathematicians and historians generally regard the modern period in algebraic geometry as starting with the work of Kronecker and Hilbert. But the relevant papers by Hilbert are often regarded as reformulating invariant theory, a much more algebraic topic, while Kronecker has been presented as the doctrinaire exponent of finite, arithmetical mathematics. Attention is then focused on the Italian tradition, leaving the path to Emmy Noether obscure and forgotten.There was, however, a steady flow of papers responding to the work of both Hilbert and Kronecker. The Hungarian mathematicians Gyula (Julius) König and József Kürschák, the French mathematicians Jules Molk and Jacques Hadamard, Emanuel Lasker and the English school teacher F.S.Macaulay all wrote extensively on the subject. This work is closely connected to a growing sophistication in the definitions of rings, fields and related concepts. The shifting emphases of their work shed light on how algebraic geometry owes much to both its distinguished founders, and how the balance was struck between algebra and geometry in the period immediately before Emmy Noether began her work.

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