The main change in the revised edition is the new Chapter 10 on tight closure. This theory was created by Mel Hochster and Craig Huneke about ten years ago and is still strongly expanding. We treat the basic ideas, F-regular rings, and F-rational rings, including Smith's theorem by which F-rationality implies pseudo-rationality. Among the numerous applications of tight closure we have selected the Briançon–Skoda theorem and the theorem of Hochster and Huneke saying that equicharacteristic direct summands of regular rings are Cohen–Macaulay. To cover these applications, Section 8.4, which develops the technique of reduction to characteristic p, had to be rewritten. The title of Part III, no longer appropriate, has been changed.

Another noteworthy addition are the theorems of Gotzmann in the new Section 4.3. We believe that Chapter 4 now treats all the basic theorems on Hilbert functions. Moreover, this chapter has been slightly reorganized.

The new Section 5.5 contains a proof of Hochster's formula for the Betti numbers of a Stanley–Reisner ring since the free resolutions of such rings have recently received much attention. In the first edition the formula was used without proof.

We are grateful to all the readers of the first edition who have suggested corrections and improvements. Our special thanks go to L. Avramov, A. Conca, S. Iyengar, R. Y. Sharp, B. Ulrich, and K.-i. Watanabe.

The concept of a canonical module is of fundamental importance in the study of Cohen–Macaulay local rings. The purpose of this chapter is to introduce the canonical module and derive its basic properties. By definition it is a maximal Cohen–Macaulay module of type 1 and of finite injective dimension.

In the first two sections we investigate the injective dimension of a module, and prove Matlis duality which plays a central role in Grothendieck's local duality theorem. Actually the canonical module has its origin in this theory. Here the canonical module is introduced independently of local cohomology which is an important notion in itself and will be treated later in this chapter.

A ring which is its own canonical module is called a Gorenstein ring. Next to regular rings and complete intersections, Gorenstein rings are in many ways the ‘nicest’ rings. Distinguished by the fact that they are of finite injective dimension, they have various symmetry properties, as reflected in their free resolution, their Koszul homology, and their Hilbert function. The last aspect will be discussed in the next chapter.

Gorenstein rings of embedding dimension at most two are complete intersections. The first non-trivial Gorenstein rings occur in embedding dimension three, and they are classified by the Buchsbaum–Eisenbud structure theorem.

In the final section the canonical module of a graded ring is introduced.

Finite modules of finite injective dimension

In this section we study injective resolutions of finite modules. We shall see that the injective dimension of a finite module M over a Noetherian local ring R either is infinite or equals the depth of R, and is bounded below by the dimension of M.

The notion of a Cohen–Macaulay ring marks the cross–roads of two powerful lines of research in present-day commutative algebra. While its main development belongs to the homological theory of commutative rings, it finds surprising and fruitful applications in the realm of algebraic combinatorics. Consequently this book is an introduction to the homological and combinatorial aspects of commutative algebra.

We have tried to keep the text self-contained. However, it has not proved possible, and would perhaps not have been appropriate, to develop commutative ring theory from scratch. Instead we assume the reader has acquired some fluency in the language of rings, ideals, and modules by working through an introductory text like Atiyah and Macdonald [15] or Sharp [344]. Nevertheless, to ease the access for the non-expert, the essentials of dimension theory have been collected in an appendix.

As exemplified by Matsumura's standard textbook [270], it is natural to have the notions of grade and depth follow dimension theory, and so Chapter 1 opens with the introduction of regular sequences on which their definition is based. From the very beginning we stress their connection with homological and linear algebra, and in particular with the Koszul complex.

Chapter 2 introduces Cohen–Macaulay rings and modules, our main subjects. Next we study regular local rings. They form the most special class of Cohen–Macaulay rings; their theory culminates in the Auslander–Buchsbaum–Serre and Auslander–Buchsbaum–Nagata theorems. Unlike the Cohen–Macaulay property in general, regularity has a very clear geometric interpretation: it is the algebraic counterpart of the notion of a non–singular point. Similarly the third class of rings introduced in Chapter 2, that of complete intersections, is of geometric significance.

The Hilbert function H(M, n) measures the dimension of the n-th homogeneous piece of a graded module M. In the first section of this chapter we study the Hilbert function of modules over homogeneous rings, prove that it is a polynomial for large values of n, and introduce the Hilbert series and multiplicity of a graded module. The next section is devoted to the proof of Macaulay's theorem which describes the possible Hilbert functions. The third section complements these results by Gotzmann's regularity and persistence theorem.

The Hilbert function behaves quite regular, even for graded, non-homogeneous rings. Such rings will be considered in the fourth section, where we will also investigate the Hilbert function of the canonical module.

The passage to the associated graded ring with respect to a filtration allows us to extend some concepts for graded rings like ‘Hilbert function’ or ‘multiplicity’ to non-graded rings, and leads to the Hilbert–Samuel function and the multiplicity of a finite module with respect to an ideal of definition. We shall study basic properties of filtrations and their associated Rees rings and modules, and sketch the theory of reduction ideals. Finally we prove Serre's theorem which interprets multiplicity as the Euler characteristic of a certain Koszul homology.

Hilbert functions over homogeneous rings

We begin by studying numerical properties of finite graded modules over a graded ring R. Our standard assumption in this section will be that R0 is an Artinian local ring, and that R is finitely generated over R0. Notice that for each finite graded R-module M, the homogeneous components Mn of M are finite R0-modules, and hence have finite length.

This chapter opens with the study of affine semigroup rings, i.e. sub-algebras of Laurent polynomial rings generated by a finite number of monomials. We relate the structure of such a ring R to that of the semi-group C formed by the exponent vectors of the monomials in R, and to the cone D spanned by C. From the face lattice of D we then construct a complex for the local cohomology of R.

The connection between R and D is strongest if R is normal: this is the case if and only if R contains all monomials which correspond to the integral points in D. By a theorem of Hochster normal semigroup rings are Cohen–Macaulay. Moreover, we shall determine their canonical modules and, as a combinatorial application, derive the reciprocity laws of Ehrhart and Stanley.

We are led to the second topic of this chapter by the fact that rings of invariants of torus actions are normal semigroup rings. We also treat finite groups, covering Watanabe's characterization of Gorenstein invariants and the famous Shephard–Todd theorem on invariants of reflection groups. The discussion of invariant theory culminates in the Hochster–Roberts theorem which warrants the Cohen–Macaulay property for rings of invariants of all linearly reductive groups.

Affine semigroup rings

An affine semigroup C is a finitely generated semigroup which for some n is isomorphic to a subsemigroup of containing 0. Let k be a field. We write k[C] for the vector space k(C), and denote the basis element of k[C] which corresponds to c ∈ C by Xc.

This chapter is an introduction to ‘combinatorial commutative algebra’, a fascinating new branch of commutative algebra created by Hochster and Stanley in the mid-seventies. The combinatorial objects considered are simplicial complexes to which one assigns algebraic objects, the Stanley–Reisner rings. We study how the face numbers of a simplicial complex are related to the Hilbert series of the corresponding Stanley–Reisner ring. This is the basis of all further investigations which culminate in Stanley's proof of the upper bound theorem for simplicial spheres. It turns out that most of the important algebraic notions introduced in the earlier chapters, such as ‘Cohen–Macaulay’, ‘Gorenstein’, ‘local cohomology’, and ‘Hilbert series’, are the proper concepts in solving purely combinatorial problems. Other applications of commutative algebra to combinatorics will be given in the next chapter.

Simplicial complexes

The present section is devoted to introducing the Stanley–Reisner ring associated with a simplicial complex, and studying its Hilbert series. The most important invariant of a simplicial complex, its f-vector, can be easily transformed into the h-vector, an invariant encoded by the Hilbert function of the associated Stanley–Reisner ring. It is of interest to know when a Stanley–Reisner ring is Cohen–Macaulay, because then the results about Hilbert functions of Chapter 4 may be employed to get information about the f-vector. In concluding this section we show that the Stanley–Reisner ring of a shellable simplicial complex is Cohen–Macaulay, and study systems of parameters of such a ring.

In this chapter we introduce the class of Cohen–Macaulay rings and two subclasses, the regular rings and the complete intersections. The definition of Cohen–Macaulay ring is sufficiently general to allow a wealth of examples in algebraic geometry, invariant theory, and combinatorics. On the other hand it is sufficiently strict to admit a rich theory: in the words of Hochster, ‘life is really worth living’ in a Cohen–Macaulay ring ([183], p. 887). The notion of Cohen–Macaulay ring is a workhorse of commutative algebra.

Regular local rings are abstract versions of polynomial or power series rings over a field. The fascination of their theory stems from a unique interplay of homological algebra and arithmetic. Complete intersections arise as residue class rings of regular rings modulo regular sequences, and, in a sense, are the best singular rings. Their exploration is dominated by methods related to the Koszul complex.

Cohen–Macaulay rings and modules

Let R be a Noetherian local ring, and M a finite module. If the ‘algebraic’ invariant depth M equals the ‘geometric’ invariant dim M, then M is called a Cohen–Macaulay module:

Definition 2.1.1 Let R be a Noetherian local ring. A finite R-module M ≠ 0 is a Cohen–Macaulay module if depth M = dim M. If R itself is a Cohen–Macaulay module, then it is called a Cohen–Macaulay ring. A maximal Cohen–Macaulay module is a Cohen–Macaulay module M such that dim M = dim R.

In general, if R is an arbitrary Noetherian ring, then M is a Cohen–Macaulay module if Mm is a Cohen–Macaulay module for all maximal ideals m ∈ Supp M.