##
**Cohen-Macaulay rings.
Rev. ed.**
*(English)*
Zbl 0909.13005

Cambridge Studies in Advanced Mathematics. 39. Cambridge: Cambridge University Press. xiv, 453 p. (1998).

[For the first edition of this book in 1993 see Zbl 0788.13005).]

This book presents basic results in commutative algebra together with their applications in different special fields (algebraic combinatorics, algebraic topology). It can be read immediately after an introductory text, but it brings you quickly to the main problems of the topic. Most of the book studies the Cohen-Macaulay property because after the authors a Cohen-Macaulay ring is a workhorse in commutative algebra.

The first two chapters of the book present the main facts of the theory of Cohen-Macaulay rings, of regular rings and of complete intersection rings. Here the famous theorems of Auslander-Buchsbaum-Serre, Auslander-Buchsbaum-Nagata and beautiful characterizations of the complete intersection property are given. The third chapter presents easy characterizations of the injective dimension and studies the Gorenstein rings, the canonical module, the Matlis duality, the local cohomology and the Grothendieck local duality. All the notions and results are studied also in the graded case, which is very important in algebraic geometry and combinatorics.

The fourth chapter presents the combinatorial theory of commutative rings, mainly describing the Hilbert functions of graded modules over homogeneous rings and some associated numerical invariants. Here the Macaulay theorems, the M. Green theorem, the Gotzmann regularity and persistence theorems (the last ones being newly introduced in this second edition) and the Stanley characterization of homogeneous Gorenstein domains in combinatorial data are given. Chapter 5 contains the theory of Stanley-Reisner rings of simplicial complexes. It studies the Cohen-Macaulay simplicial complexes, the local cohomology and the canonical modules of their Stanley-Reisner rings. Here the Stanley upper bound theorem and the Hochster formula for the Betti numbers of a Stanley-Reisner ring (with proof in this new edition) are given. Chapter 6 studies the normal semigroup rings, which are Cohen-Macaulay by a theorem of Hochster presented here. It describes the canonical modules of normal semigroup rings, which gives the reciprocity laws of Ehrhart and Stanley as a combinatorial application. The chapter ends with a study of rings of invariants of torus actions, where Watanabe’s characterization of Gorenstein invariants, the Shephard-Todd theorem on invariants of reflection groups and the famous Hochster-Roberts theorem concerning the Cohen-Macaulay property of invariants of linearly reductive groups are given. Chapter 7 studies the determinantal rings. It shows that they are Cohen-Macaulay, it describes their canonical modules and it characterizes the Gorenstein determinantal rings. Chapter 8 introduces the characteristic \(p\) methods and shows how they apply to the characteristic zero via the Artin approximation theory. The famous Hochster’s theory on the existence of big Cohen-Macaulay modules over noetherian local rings containing a field ends the chapter. As an application of this theorem, chapter 9 presents Hochster’s direct summand theorem for regular local rings, his canonical element theorem, the Peskine-Szpiro intersection theorem, the theorem of Evans and Griffith on ranks of syzygy modules and bounds for the Bass numbers of modules. Chapter 10 on tight closure (newly introduced in this edition) includes the Smith theorem concerning the pseudo-rationality of \(F\)-rational rings, the Briançon-Skoda theorem and the Hochster-Huneke theorem saying that equicharacteristic direct summands of regular rings are Cohen-Macaulay. A lot of useful informations are packed also in the exercises which follow each section and in the notes which follow each chapter.

This book presents basic results in commutative algebra together with their applications in different special fields (algebraic combinatorics, algebraic topology). It can be read immediately after an introductory text, but it brings you quickly to the main problems of the topic. Most of the book studies the Cohen-Macaulay property because after the authors a Cohen-Macaulay ring is a workhorse in commutative algebra.

The first two chapters of the book present the main facts of the theory of Cohen-Macaulay rings, of regular rings and of complete intersection rings. Here the famous theorems of Auslander-Buchsbaum-Serre, Auslander-Buchsbaum-Nagata and beautiful characterizations of the complete intersection property are given. The third chapter presents easy characterizations of the injective dimension and studies the Gorenstein rings, the canonical module, the Matlis duality, the local cohomology and the Grothendieck local duality. All the notions and results are studied also in the graded case, which is very important in algebraic geometry and combinatorics.

The fourth chapter presents the combinatorial theory of commutative rings, mainly describing the Hilbert functions of graded modules over homogeneous rings and some associated numerical invariants. Here the Macaulay theorems, the M. Green theorem, the Gotzmann regularity and persistence theorems (the last ones being newly introduced in this second edition) and the Stanley characterization of homogeneous Gorenstein domains in combinatorial data are given. Chapter 5 contains the theory of Stanley-Reisner rings of simplicial complexes. It studies the Cohen-Macaulay simplicial complexes, the local cohomology and the canonical modules of their Stanley-Reisner rings. Here the Stanley upper bound theorem and the Hochster formula for the Betti numbers of a Stanley-Reisner ring (with proof in this new edition) are given. Chapter 6 studies the normal semigroup rings, which are Cohen-Macaulay by a theorem of Hochster presented here. It describes the canonical modules of normal semigroup rings, which gives the reciprocity laws of Ehrhart and Stanley as a combinatorial application. The chapter ends with a study of rings of invariants of torus actions, where Watanabe’s characterization of Gorenstein invariants, the Shephard-Todd theorem on invariants of reflection groups and the famous Hochster-Roberts theorem concerning the Cohen-Macaulay property of invariants of linearly reductive groups are given. Chapter 7 studies the determinantal rings. It shows that they are Cohen-Macaulay, it describes their canonical modules and it characterizes the Gorenstein determinantal rings. Chapter 8 introduces the characteristic \(p\) methods and shows how they apply to the characteristic zero via the Artin approximation theory. The famous Hochster’s theory on the existence of big Cohen-Macaulay modules over noetherian local rings containing a field ends the chapter. As an application of this theorem, chapter 9 presents Hochster’s direct summand theorem for regular local rings, his canonical element theorem, the Peskine-Szpiro intersection theorem, the theorem of Evans and Griffith on ranks of syzygy modules and bounds for the Bass numbers of modules. Chapter 10 on tight closure (newly introduced in this edition) includes the Smith theorem concerning the pseudo-rationality of \(F\)-rational rings, the Briançon-Skoda theorem and the Hochster-Huneke theorem saying that equicharacteristic direct summands of regular rings are Cohen-Macaulay. A lot of useful informations are packed also in the exercises which follow each section and in the notes which follow each chapter.

Reviewer: Dorin-Mihail Popescu (Bucureşti)

### MathOverflow Questions:

Does the Peskine–Szpiro intersection theorem imply Krull’s ideal height theorem?### MSC:

13C14 | Cohen-Macaulay modules |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |

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

13D25 | Complexes (MSC2000) |