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Correspondence Grothendieck-Serre. (Correspondance Grothendieck-Serre.) (French) Zbl 0986.01019
Documents Mathématiques 2. Paris: Société Mathématique de France (ISBN 2-85629-104-X/hbk). x, 288 p. (2001).
This is the publication of a greater part of the correspondence by letters between Jean-Pierre Serre (born in 1926) and Alexander Grothendieck (born in 1928), when they lived far apart (in Princeton, Harvard, etc.). The period covered is 1955 (with Grothendieck’s letter from Lawrence, Kansas) to 1969, and a few letters from 1984 to 1987. There are 47 letters by Grothendieck (one of them to Cartan), and 39 by Serre. The letters are slightly linguistically edited, and some personal comments are omitted. One can enjoy watching the intellectual exchanges between the two men, related, for instance to the field of algebraic geometry as well as the corresponding cast of characters; this can not only help a historian of mathematics but also a researcher, for these juicy letters are quite inspiring in their own right by offering diverse thought meanderings absent from finished products, such as published papers. Of course we can read here about the happenings related to Bourbaki activities as well.
The letters also offer a glimpse into the two men’s personal (academic) lives; for instance, in one of the letters of 1955, Grothendieck is inquiring with Serre about a possibility of a position for himself in relation to a hundred of new positions to be opened in France (as he had heard). He says that he is enormously interested in this, for he would rather stay in France (or perhaps in Germany or South America), rather than in the USA. Serre replies that they have no details about these positions and how many will be available for mathematicians. “In Paris, Cartan has a candidate that everybody wants: Chevalley (confidential!).” In case that he is interested in a position in a province, Grothendieck need not worry for there is a good possibility for a position in Strasbourg. Moreover, Serre understands that Spencer is recommending him for a good position at Stanford, or a place like that. Serre tells Grothendieck that Grothendieck’s paper on algebraic homology has converted the whole world to his point of view (including Dieudonné who seems to be completely functorial!).
There are humorous points, such as when Grothendieck says that he is pleased that “your formula with Ext pleases you” with Serre’s comment that “your” is an obvious error that should be replaced by “my”. An interesting point is Grothendieck’s letter from the same year (before the reviewer’s time) where he introduces the Mittag-Leffler condition, that is to be found again in his Éléments de Géometrie Algébrique, Publ. Math., Inst. Hautes Étud. Sci. 11, 349-511 (1962; Zbl 0118.36206). One finds many topics discussed, such as sheaf theory, the zeta function, Rieman-Roch, etc. The letters are teaming with crucial mathematical personalities and their results of the period (including the Russians; or a Serre’s “Izvestiya” paper) and one can follow the major players feeling as though one is participating on an equal footing.
Not knowing either Grothendieck or Serre personally, the reviewer had got an impression of Serre as an aloof royal, contrasted (or complemented) by an emotional and colorful Grothendieck. For instance in one of his letters, Grothendieck speaks of the mathematical atmosphere at Harvard as rich and full of fresh air in comparison to Paris that gets more stale every year; or his passionate letter to Cartan against military service (for mathematicians), etc. Grothendieck has retreated from the public mathematical view in 1971 and the first letter to Serre after this year is in 1984; there Grothendieck announces his “reflexion-retrospective” work “Récoltes et Semailles,” (“Harvests and Crops”).
Comments at the end of the book provide further clarifications of the letters and the references are useful. Translating the book into English (or other languages) would be a good idea and adding a (much more detailed) subject and name indexes would increase the book’s value.

01A70 Biographies, obituaries, personalia, bibliographies
14-03 History of algebraic geometry
01A60 History of mathematics in the 20th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography