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**A Singular introduction to commutative algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann.**
*(English)*
Zbl 1023.13001

Berlin: Springer. xvii, 588 p. EUR 39.95/net; sFr. 68.50; £28.00; $ 44.95 (2002).

In recent times the algorithmic and computational aspects of classical algebra became a separate research interest of the subject. For practical computations the most favorite computer algebra systems in commutative algebra with applications in algebraic geometry are CoCoA, Macaulay2, and Singular. The computer algebra system Singular was developed over the last two decades with a strong influence of the authors of the present book. It became a powerful tool for computations in particular for polynomial computations. The version 2.0.4 is freely available under the GNU public license for various platforms under http://www.singular.uni-kl.de. The book under review is completed by a CD-ROM that contains the previous version 2.0.3. The book is motivated by what the authors believe for the most useful way for studying commutative algebra with a view toward algebraic geometry and singularity theory. It grows out of several courses by the authors. The aim of the book is such an introduction to commutative algebra with a view towards to algorithmic aspects and computational practice.

This introduction to commutative algebra is divided into seven chapters: 1. Rings, ideals, and standard bases, 2. Modules, 3. Noether normalization and applications, 4. Primary decomposition and related topics, 5. Hilbert function and dimension, 6. Complete local rings, 7. Homological algebra; and two appendices: A. Geometric background, B. Singular - A short introduction.

The first two chapters introduce the basics about rings, ideals and modules, emphasized for polynomial rings and their factor rings. This is illustrated how to use Singular for computations. For further use (localization and singularity theory) the authors do not restrict to well-orderings for the computation of standard bases. – The following four chapters are devoted to more advanced considerations: Chapter 3 describes Noether normalization as the cornerstone in the theory of affine algebras, theoretically as well as computationally. A highlight is the algorithm for the computation of the non-normal locus and the normalization of an affine ring, based on a criterion of Grauert and Remmert. The chapter ends with a section containing some of the larger procedures written in the Singular programming language. Chapter 4 is devoted to primary decomposition and related topics. For the constructive approach the authors follow the ideas of P. Gianni, B. Trager and G. Zacharias [see J. Symb. Comput. 6, 149-167 (1988; Zbl 0667.13008)]. The chapter 5 is concerned with the Hilbert functions and various related subjects. It culminates with a proof of the Jacobian criterion of affine algebras and its application for the computation of the singular locus. In chapter 6, the authors consider standard bases in power series rings. The basis for local analytic geometry is the fact that standard bases of ideals in power series rings can be computed if the ideal is generated by polynomials. – Chapter 7 gives a short introduction to basic concepts of homological algebra. There are results about Koszul complexes, the proof of the Auslander-Buchsbaum formula and its application to a Cohen-Macaulay criterion.

The first appendix provides an introduction to applications in algebraic geometry, in particular to elimination techniques and singularity theory. More relations might be found in the accompanying libraries of Singular.

The second appendix gives a crash course of the programming language of Singular, data types, functions and control structures of the system, as well as of the procedures appearing in the libraries. Moreover, the authors show how Singular might communicate with other systems (Maple, Mathematica, MuPAD).

In the introduction of the book, the authors describe how to use the book for various courses of different length and difficulties and seminars, using Singular as the tool for explicit computations. Each chapter is completed by exercises that allow the reader to follow the theoretical as well as the computational aspects. The authors’ most important new focus is the presentation of non-well orderings that allow them the computational approach for local commutative algebra. The accompanying CD-ROM also contains all the examples of the book.

Finally one should mention that the book is not at all an introduction to Singular. For that reason one might also consult the manual Singular and the help files. In fact the book provides an introduction to commutative algebra from a computational point of view. So it might be helpful for students and other interested readers (familiar with computers) to explore the beauties and difficulties of commutative algebra by computational experiences. In this respect the book is one of the first samples of a new kind of textbooks in algebra.

This introduction to commutative algebra is divided into seven chapters: 1. Rings, ideals, and standard bases, 2. Modules, 3. Noether normalization and applications, 4. Primary decomposition and related topics, 5. Hilbert function and dimension, 6. Complete local rings, 7. Homological algebra; and two appendices: A. Geometric background, B. Singular - A short introduction.

The first two chapters introduce the basics about rings, ideals and modules, emphasized for polynomial rings and their factor rings. This is illustrated how to use Singular for computations. For further use (localization and singularity theory) the authors do not restrict to well-orderings for the computation of standard bases. – The following four chapters are devoted to more advanced considerations: Chapter 3 describes Noether normalization as the cornerstone in the theory of affine algebras, theoretically as well as computationally. A highlight is the algorithm for the computation of the non-normal locus and the normalization of an affine ring, based on a criterion of Grauert and Remmert. The chapter ends with a section containing some of the larger procedures written in the Singular programming language. Chapter 4 is devoted to primary decomposition and related topics. For the constructive approach the authors follow the ideas of P. Gianni, B. Trager and G. Zacharias [see J. Symb. Comput. 6, 149-167 (1988; Zbl 0667.13008)]. The chapter 5 is concerned with the Hilbert functions and various related subjects. It culminates with a proof of the Jacobian criterion of affine algebras and its application for the computation of the singular locus. In chapter 6, the authors consider standard bases in power series rings. The basis for local analytic geometry is the fact that standard bases of ideals in power series rings can be computed if the ideal is generated by polynomials. – Chapter 7 gives a short introduction to basic concepts of homological algebra. There are results about Koszul complexes, the proof of the Auslander-Buchsbaum formula and its application to a Cohen-Macaulay criterion.

The first appendix provides an introduction to applications in algebraic geometry, in particular to elimination techniques and singularity theory. More relations might be found in the accompanying libraries of Singular.

The second appendix gives a crash course of the programming language of Singular, data types, functions and control structures of the system, as well as of the procedures appearing in the libraries. Moreover, the authors show how Singular might communicate with other systems (Maple, Mathematica, MuPAD).

In the introduction of the book, the authors describe how to use the book for various courses of different length and difficulties and seminars, using Singular as the tool for explicit computations. Each chapter is completed by exercises that allow the reader to follow the theoretical as well as the computational aspects. The authors’ most important new focus is the presentation of non-well orderings that allow them the computational approach for local commutative algebra. The accompanying CD-ROM also contains all the examples of the book.

Finally one should mention that the book is not at all an introduction to Singular. For that reason one might also consult the manual Singular and the help files. In fact the book provides an introduction to commutative algebra from a computational point of view. So it might be helpful for students and other interested readers (familiar with computers) to explore the beauties and difficulties of commutative algebra by computational experiences. In this respect the book is one of the first samples of a new kind of textbooks in algebra.

Reviewer: Peter Schenzel (Halle)

### MSC:

13-04 | Software, source code, etc. for problems pertaining to commutative algebra |

13Pxx | Computational aspects and applications of commutative rings |

14-04 | Software, source code, etc. for problems pertaining to algebraic geometry |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

14Qxx | Computational aspects in algebraic geometry |

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

68W30 | Symbolic computation and algebraic computation |

68-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |