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Commutative Algebra

By Andrea Ferretti. GSM/233, 2023, 373 pp.

Commutative algebra is a foundational subject that must be mastered by any student wishing to pursue research in number theory, algebraic geometry, representation theory, or a number of other areas. Among the many resources available for beginning students, perhaps the three most widely known are the classic textbooks by Atiyah–Macdonald, by Matsumura, and by Eisenbud. A lively discussion of the relative merits of these and other works can be found in the comments section of a MathOverflow post by Andrea Ferretti from 2010. In this post, Ferretti was seeking suggestions for a textbook on commutative algebra that was “(i) more comprehensive than Atiyah–Macdonald; (ii) more readable than Matsumura; (iii) less thick than Eisenbud.”

Ultimately, Ferretti decided to answer his own question by writing the two-volume set under review. In the introduction to Volume 1, Ferretti says that his intent is to “expand on the material in [Atiyah–Macdonald].” (Advanced topics from Matsumura or Eisenbud are covered in Volume 2; more on that later.)

Ferretti achieves this in two ways. First, the core topics from Atiyah–Macdonald (including localization, primary decomposition, and the going-up and going-down theorems) are covered at about half the pace of Atiyah–Macdonald, occupying roughly 300 pages of this 450-page book. In large part, the extra length is due to Ferretti’s extended discussions of important special cases and examples, which serve as motivation for abstract definitions.

Second, Ferretti’s book covers several topics that are beyond the scope of Atiyah–Macdonald. For instance, there is a whole chapter devoted to “Computational methods,” including Gröbner bases. This book covers a number of concepts of particular importance in number theory, such as Witt vectors and ramification. In my opinion, Volume 1 is an excellent choice for a one-semester introductory course on commutative algebra.

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Homological Methods in Commutative Algebra

By Andrea Ferretti. GSM/234, expected 2023.

Volume 2 is structured as two mini-books in one: the first four chapters are “pure” homological algebra (abelian categories, derived functors, spectral sequences); while the last six chapters are on topics in commutative algebra that require a background in homological methods.

The homological algebra part of this book is much more extensive than the corresponding parts of Matsumura or Eisenbud. Notably, the book includes a full proof of the Freyd–Mitchell embedding theorem. These chapters assume no prior contact with homological algebra, and they provide more than adequate preparation for the latter half of the book. They aren’t quite enough for a self-contained course on modern homological algebra but they can serve as a warm-up for another text that does, such as that of Gelfand–Manin.

The second part of Volume 2 covers an array of topics including flatness, Koszul complexes, Cohen–Macaulay and Gorenstein rings, and local cohomology. In my opinion, these chapters are the crowning gem of the whole two-volume set. They include proofs of some of the most significant theorems in the field, including the Quillen–Suslin theorem and Kunz’s theorem.

Together, this two-volume set provides an engaging and friendly introduction to the subject, and is a welcome addition to the literature.