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Algèbre commutative. Langages géométrique et algébrique. (Commutative algebra. Geometric and algebraic languages). Nouvelle éd., revue et corrigée par l’auteur. Nouvelle éd., revue et corrigée par l’auteur. (Algèbre commutative. Langages géométrique et algébrique.) (French) Zbl 0907.13001
Collection Enseignement des Sciences. 24. Paris: Hermann. 454 p. (1998).
J.-P. Lafon’s “Algèbre commutative. Langages géométrique et algébrique”, first published in 1977 (cf. Zbl 0369.13001) in the same series of treatises in mathematics and natural sciences, has become one of the classics in the textbook literature on this subject. Back then, it was among the very first university texts that provided students with a systematic, comprehensive account of the principles of commutative algebra in its modern setting (à la Grothendieck-Serre-Bourbaki) together with its categorical-homological framework and with its applications to algebraic number theory, algebraic geometry, local analytic geometry, and formal local geometry. In a previous textbook, entitled “Les formalismes fondamentaux de l’algèbre commutative” (1974; Zbl 0301.13001; see also the followings review), the author had developed the basic concepts and methods of categorical and homological algebra as an indispensible framework for the present text, and with regard to this very fact, both books must be seen as an organic unit. According to both the historical significance and the timeless methodical excellence of these two standard texts on commutative algebra, it is highly gratifying to see that new editions of both of them have been made available by the publisher. After about a quarter of a century since their first appearance, and in spite of the publishing of several just as masterly textbooks on advanced commutative algebra, in the meantime, J.-P. Lafon’s two related volumes have maintained, without any doubt and in an impressive manner, their unique character as well-proved, wide-spread and highly esteemed guides through commutative algebra at its advanced level.
The present new edition of J.-P. Lafon’s “Algèbre Commutative” is basically a reprint of the original one from 1977. However, the author has revised the entire, well-proved text, thereat effected several minor corrections, improved some arrangements of the material, simplified a few proofs, and added some more examples and exercises. Altogether, the text as a whole has largely been left intact and contains, now as before, the following material divided into nine chapters and one appendix:
Chapter 1: Rings and modules of fractions. Localisation and globalisation;
Chapter 2: Chain conditions and finiteness conditions (for rings and modules);
Chapter 3: Associated prime ideals and primary decomposition;
Chapter 4: Extensions of rings, algebraic and integral dependence;
Chapter 5: Elements of the theory of commutative fields;
Chapter 6: Elements of (affine and projective) algebraic geometry;
Chapter 7: Homomorphisms of rings and morphisms of algebraic sets;
Chapter 8: $$a$$-adic topologies and completion;
Chapter 9: Derivations and differentials;
Appendix: Spectral (topological) spaces, constructible topologies, and sources.
The topics of each chapter are discussed in a comprehensive, nearly exhaustive, rigorous and very elegant, Bourbaki-style manner. Numerous remarks, examples, and (challenging) exercises lead the (abstract-minded and ambitious) reader to various concrete applications and further-going results in the field. Certainly, J.-P. Lafon’s standard text on commutative algebra will be as valuable to the present (and coming) generations of students and young mathematicians as it has been to the previous ones over the past two decades.

##### MSC:
 13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory 20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory 13Bxx Commutative ring extensions and related topics 13A15 Ideals and multiplicative ideal theory in commutative rings 13Cxx Theory of modules and ideals in commutative rings 14A05 Relevant commutative algebra 12F99 Field extensions