##
**Integral closure of ideals, rings, and modules.**
*(English)*
Zbl 1117.13001

London Mathematical Society Lecture Note Series 336. Cambridge: Cambridge University Press (ISBN 0-521-68860-4/pbk). xiv, 431 p. (2006).

Integral extensions of commutative rings have played an important rôle in number theory and commutative algebra for a long time, starting in the 19th century and leading to the concept for rings and ideals by the work of Krull, Akizuki, Zariski, Rees, Lipman, and others. The study of integral closures culminates by the introduction of a class of rings, nowadays called Krull rings. In more recent time there was a speed up of results in commutative ring theory based on this classical constructions related to Rees rings, reductions of ideals and modules, Briançon-Skoda Theorem and so on. This development emphasizes the importance of the basic concepts of the integral closure for research in recent times.

The main aim of the present book is to collect research material scattered through several papers, and to present it with a unified treatment in order to make it available to a broader public of reseachers and students. With this pretension the book fills up a gap in the textbooks available at the present. The material of the book is presented in a preface, 19 chapters, and two appendices completed by 322 references.

In the first two chapters there is an introduction of integral closures of ideals and rings, with the highlights of Dedekind-Mertens formula and the going-up and going-down theorems. The third chapter is concerned with separability needed in the subsequent for the study of the integral closure under flat homomorphisms and field extensions. The 4th chapter deals with the noetherian properties, covering normalization theorems and the study of Krull rings. In Chapter 5 the Rees rings are introduced and related to integral closures of powers of ideals. The relation of valuations to integral closures, starting with [O. Zariski and P. Samuel, Commutative algebra. Grad. Texts Math. 29 (1976; Zbl 0322.13001)], is the theme of Chapter 6, while their relation to derivations is the theme of Chapter 7. The important notions of reductions, minimal reductions, reduction numbers and related notions are studied in Chapter 8. The final result in this section is Sally’s theorem on extensions.

In Chapter 9 about analytically unramified rings there is the study about the finiteness of integral closures. In Chapter 10 on Rees valuations there is a proof of Rees’ Theorem that for an ideal in a noetherian ring saying that there exist finitely many valuation rings that determine the integral closure of the ideal as well as the integral closures of all powers of it. Chapter 11 “Multiplicity and integral closure” is devoted to the characterization of the equality of two integral closures of two \(0\)-dimensional ideals in a formally equidimensional local ring provided their Samuel multiplicity coincides. Related things are the principle of specialization of integral dependence as introduced by Teissier. Chapter 12 “The conductor” is of prepatory nature to Chapter 13 “The Briançon-Skoda Theorem”. It yields, among others, the J. Lipman–A. Sathaye Theorem [Mich. Math. J. 28, 199–222 (1981; Zbl 0438.13019)] about conductors. As an application this is used in Chapter 13 for the proof of the Briançon-Skoda Theorem, which describes the comparison of powers of ideals with the integral closure of their powers. These kind of results have been in the focus of several research papers, in characteristic \(p\) using the Frobenius as well as in general by differnt approaches. Chapter 12 is concerned with Zariski’s theorems about the structure of integrally closed ideals in a \(2\)-dimensional regular local ring. There are also extensions, e.g. done by Lipman with his reciprocity formula. The computational aspect of the integral closure with modern computer algebra systems is sketched in Chapter 15. There are several approaches for the extension of integral closure to modules. The one of Rees, originated by Zariski and Samuel, is the main subject of Chapter 16. This includes a generalization of the multiplicity criterion by the use of Buchsbaum-Rim multiplicities.

Another subject of recent research, joint reductions, are investigated in Chapter 17. They occur in relation to mixed multiplicities and the Minkowski inequality for them. Adjoint ideals were introduced by J. Lipman [Math. Res. Lett. 1, 739-755 (1994; Zbl 0844.13015)]. The basics about them are subject of Chapter 18. Also the relation to multiplier ideals is discussed. In the final Chapter 19 “Normal homomorphisms” the authors present how integral closures behaves under homomorphisms. The appendices include a few constructions needed in the frame of the book, among others, gradings, complexes, going-down and flatness, dimension formula.

Each chapter is finished with a series of exercises, some of them relatively easy to solve and others that requires much more effort, some with references to the literature. Altogether there are 361 exercises and in addition some instructive examples in the text. Some of the exercises are in fact related to research results that the authors were not able to cover in the text. During their writing the authors, two of the leading experts of the field, give some explanations resp. some discussions of their results and the perspectives for further chapters. The mathematical style of the text is rather compact and uses clever arguments that one might not find in other textbooks or research articles. The book might be used for seminars, extensions to some courses in commutative algebra or as a source for some advanced courses in commutative algebra. Because of the wealth of material of recent research it might be used also as a reference for recent developments around the subject. For an interested student or even an interested researcher it will help to find the relation to research results for the last years and will help him to find the right guide to understand and apply them.

Reviewer’s remark: There is one backdraw of the book: it is a paperback. Even at the time when the reviewer got an impression of it, it was necessary to switch forwards and backwards several times, to references or to other pages, and to see relations to previous subjects. The reviewer guesses an intensive use of any potential reader. This will probably break the spine of the book. In any case it will be, for the reviewer’s point of view, become the standard reference for integral closure in commutative algebra for the next time. For some additional results on the Cohen-Macaulayness of Rees rings, constructive aspects and related material one might consult also W. Vasconcelos’ book [Integral closure. Rees algebras, multiplicities, algorithm. Springer Monographs in Mathematics. (2005; Zbl 1082.13006)].

The main aim of the present book is to collect research material scattered through several papers, and to present it with a unified treatment in order to make it available to a broader public of reseachers and students. With this pretension the book fills up a gap in the textbooks available at the present. The material of the book is presented in a preface, 19 chapters, and two appendices completed by 322 references.

In the first two chapters there is an introduction of integral closures of ideals and rings, with the highlights of Dedekind-Mertens formula and the going-up and going-down theorems. The third chapter is concerned with separability needed in the subsequent for the study of the integral closure under flat homomorphisms and field extensions. The 4th chapter deals with the noetherian properties, covering normalization theorems and the study of Krull rings. In Chapter 5 the Rees rings are introduced and related to integral closures of powers of ideals. The relation of valuations to integral closures, starting with [O. Zariski and P. Samuel, Commutative algebra. Grad. Texts Math. 29 (1976; Zbl 0322.13001)], is the theme of Chapter 6, while their relation to derivations is the theme of Chapter 7. The important notions of reductions, minimal reductions, reduction numbers and related notions are studied in Chapter 8. The final result in this section is Sally’s theorem on extensions.

In Chapter 9 about analytically unramified rings there is the study about the finiteness of integral closures. In Chapter 10 on Rees valuations there is a proof of Rees’ Theorem that for an ideal in a noetherian ring saying that there exist finitely many valuation rings that determine the integral closure of the ideal as well as the integral closures of all powers of it. Chapter 11 “Multiplicity and integral closure” is devoted to the characterization of the equality of two integral closures of two \(0\)-dimensional ideals in a formally equidimensional local ring provided their Samuel multiplicity coincides. Related things are the principle of specialization of integral dependence as introduced by Teissier. Chapter 12 “The conductor” is of prepatory nature to Chapter 13 “The Briançon-Skoda Theorem”. It yields, among others, the J. Lipman–A. Sathaye Theorem [Mich. Math. J. 28, 199–222 (1981; Zbl 0438.13019)] about conductors. As an application this is used in Chapter 13 for the proof of the Briançon-Skoda Theorem, which describes the comparison of powers of ideals with the integral closure of their powers. These kind of results have been in the focus of several research papers, in characteristic \(p\) using the Frobenius as well as in general by differnt approaches. Chapter 12 is concerned with Zariski’s theorems about the structure of integrally closed ideals in a \(2\)-dimensional regular local ring. There are also extensions, e.g. done by Lipman with his reciprocity formula. The computational aspect of the integral closure with modern computer algebra systems is sketched in Chapter 15. There are several approaches for the extension of integral closure to modules. The one of Rees, originated by Zariski and Samuel, is the main subject of Chapter 16. This includes a generalization of the multiplicity criterion by the use of Buchsbaum-Rim multiplicities.

Another subject of recent research, joint reductions, are investigated in Chapter 17. They occur in relation to mixed multiplicities and the Minkowski inequality for them. Adjoint ideals were introduced by J. Lipman [Math. Res. Lett. 1, 739-755 (1994; Zbl 0844.13015)]. The basics about them are subject of Chapter 18. Also the relation to multiplier ideals is discussed. In the final Chapter 19 “Normal homomorphisms” the authors present how integral closures behaves under homomorphisms. The appendices include a few constructions needed in the frame of the book, among others, gradings, complexes, going-down and flatness, dimension formula.

Each chapter is finished with a series of exercises, some of them relatively easy to solve and others that requires much more effort, some with references to the literature. Altogether there are 361 exercises and in addition some instructive examples in the text. Some of the exercises are in fact related to research results that the authors were not able to cover in the text. During their writing the authors, two of the leading experts of the field, give some explanations resp. some discussions of their results and the perspectives for further chapters. The mathematical style of the text is rather compact and uses clever arguments that one might not find in other textbooks or research articles. The book might be used for seminars, extensions to some courses in commutative algebra or as a source for some advanced courses in commutative algebra. Because of the wealth of material of recent research it might be used also as a reference for recent developments around the subject. For an interested student or even an interested researcher it will help to find the relation to research results for the last years and will help him to find the right guide to understand and apply them.

Reviewer’s remark: There is one backdraw of the book: it is a paperback. Even at the time when the reviewer got an impression of it, it was necessary to switch forwards and backwards several times, to references or to other pages, and to see relations to previous subjects. The reviewer guesses an intensive use of any potential reader. This will probably break the spine of the book. In any case it will be, for the reviewer’s point of view, become the standard reference for integral closure in commutative algebra for the next time. For some additional results on the Cohen-Macaulayness of Rees rings, constructive aspects and related material one might consult also W. Vasconcelos’ book [Integral closure. Rees algebras, multiplicities, algorithm. Springer Monographs in Mathematics. (2005; Zbl 1082.13006)].

Reviewer: Peter Schenzel (Halle)

### MSC:

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

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |