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13A50 Commutative algebra -- General commutative ring theory -- Actions of groups on commutative rings; invariant theory

13Axx Commutative algebra -- General commutative ring theory

13Cxx Commutative algebra -- Theory of modules and ideals En-ligne: Springerlink | MathSciNet | Zentralblatt

Location | Call Number | Status | Date Due |
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Salle R | 11389-01 / 13 EIS (Browse Shelf) | Available |

With so many texts on commutative algebra available, one's reaction to this one might be "Why yet another one?'', and "Why is it so fat?'' The answer to the second question answers the first as well—this text has a distinctively different flavor than existing texts, both in coverage and style. Motivation and intuitive explanations appear throughout, there are many worked examples, and both text and problem sets lead up to contemporary research.

Structured historically, the introductory chapter is unusually long, and opens the book with a bang. Theorems are stated, and where possible, even at this early stage, proven. Topics such as geometric invariant theory, Hilbert functions, and free resolutions are discussed. I particularly liked the motivation of graded rings via invariant theory. (Graded rings are singled out for special attention in conjunction with many topics, a good idea since they are used so often in geometric and computational settings.) Strands are traced from their roots to current research, and even if not completely understood give reason for the technical work ahead.

This style continues. In the chapter on associated primes and primary decomposition there is a preview of local cohomology. Symbolic powers are treated with much more than the usual attention: there is an outline of the Zariski-Nagata theorem on functions vanishing to high order, and a thoroughly worked determinantal example. Flatness is motivated by flat families, perhaps an unusual topic for a beginning text, but certainly one that establishes "a view toward algebraic geometry''. Application to deformations appears in the exercises, and further application arises in the context of Gröbner bases.

The second section, dimension theory, also begins with an overview. There is a nice discussion of the geometry of systems of parameters and the Krull principal ideal theorem. The latter is illustrated by a picture, which suggests one reason for the book's length, and at the same time its pedagogical strength. Eisenbud gives pictures that illustrate the geometry of primary decomposition, normalization and gluing, deformation under various term orderings, duality in graded Artinian rings, and others. These serve to build intuition about subtle or complicated concepts. Students sometimes complain that algebra is too formal for intuition; this sort of exposition should encourage them to think about algebraic concepts in new ways. Topics in this section include special features of dimension and codimension one, the Hilbert-Samuel polynomial, Noether normalization and the last (of five!) proofs of the Nullstellensatz, elimination theory, semi-continuity of fiber dimension, differentials, and the Jacobian criterion for regularity.

A chapter devoted to Gröbner bases is novel for a text on commutative algebra, but is appropriate because of the increasing ubiquity of computational problems and methods in the field. Eisenbud describes what can (and what cannot, at least so far) be effectively computed. The chapter concludes with eight computer algebra topics designed to be explorations growing out of classes of examples.

The third major section, homological methods, has a faster pace. The chapters are vignettes of representative methods and applications, mostly in the context of regular, Gorenstein, and Cohen-Macaulay rings. Topics include the use of Serre's conditions in proving primeness of ideals, the acyclicity lemma, a proof of the Hilbert-Burch theorem and several applications, and a discussion of Castelnuovo-Mumford regularity. The text proper concludes with a quick glimpse into linkage theory, but the book continues with seven appendices, 190 pages in length. Especially notable are treatments of co-algebra and divided power structures, multi-linear algebra and Koszul-like complexes, and applications of these to determinantal varieties. Exercises explore applications to rational and elliptic normal curves, and scrolls.

This text has "personality''—those familiar with Eisenbud's own research will recognize its traces in his choice of topics and manner of approach. The book conveys infectious enthusiasm and the conviction that research in the field is active and yet accessible.

Consistent with his computer-friendly attitude, Eisenbud maintains a list of corrections (MathSciNet)

Bibliogr. p. [745]-762. Index

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