##
**Combinatorics and commutative algebra.**
*(English)*
Zbl 0537.13009

Progress in Mathematics, Vol. 41. Boston-Basel-Stuttgart: BirkhĂ¤user. VIII, 88 p. DM 42.00; sFr. 36.00 (1983).

In this set of lectures the author presents a series of very nice and surprizing applications of commutative algebra techniques to some combinatorial problems in algebra and geometry. Most of these results were obtained in the last decade and the subject seems to be a very promising one. The author is known to be the main contributor and expert in the field; his presentation is clear and concise and the only 85 pages of text contain a large number of results. Some proofs are only sketched or are omitted; however, references abound and the accessibility of the material is ensured by a careful selection of the background information.

The introductory chapter 0 provides most of the fundamental facts from combinatorics (rational power series, diophantine linear homogeneous equations, integer stochastic matrices, convex polytopes), commutative algebra, homological algebra, and topology (homology theory), needed to understand the subsequent two chapters.

Chapter I starts with the presentation of five conjectures on the numbers \(H_ n(r)\), \(n\geq 1\), \(r\geq 0\), of \(n\times n\) integer stochastic matrices with line sum r. Three of these were first formulated by H. Anand, V. C. Dumir, and H. Gupta [Duke Math. J. 33, 757-769 (1966; Zbl 0144.004)]; the proof uses the Elliott-MacMahon algorithm for solving linear homogeneous equations in nonnegative integers and the Hilbert syzygy theorem. Here the author gives a different proof, using other tools from commutative algebra. These are applied to the study of the following objects generated by an \(r\times n {\mathbb{Z}}\)-matrix \(\Phi\) of maximal rank \(r\leq n:\) the monoid \(E_{\Phi}=\{\beta \in {\mathbb{N}}^ n| \Phi \beta =0\},\) the algebra \(R_{\Phi}\) of \(E_{\Phi}\) over a field k, the \(E_{\Phi}\)-module \(E_{\Phi,\alpha}=\{\beta \in {\mathbb{N}}^ n| \Phi \beta =\alpha \}\), for \(\alpha \in {\mathbb{Z}}^ r\), and the \(R_{\Phi}\)-module \(M_{\Phi,\alpha}\) which is the algebra of \(E_{\Phi,\alpha}\) over k. A central role is played by the Hilbert series \(F(R_{\Phi})\) and \(F(M_{\Phi,\alpha})\). Results on the dimension and depth of \(R_{\Phi}\) are used to prove that \(R_{\Phi}\) is Cohen-Macaulay. From considerations on the local cohomology of the modules \(M_{\Phi,\alpha}\), a sufficient condition for Cohen- Macaulayness and a new proof of the reciprocity theorem are derived. Other developments concern the enumeration of rational points in integral convex polytopes and its connection with volume, the proof of a local duality theorem, and the derivation of a necessary and sufficient condition for \(R_{\Phi}\) to be Gorenstein. The proofs of the four conjectures are obtained by applying the general results to the case when \(E_{\Phi}\) is the set of \(n\times n\) integer stochastic matrices with line sum r.

Chapter II is devoted to the study of the face ring (Stanley-Reisner ring) k[\(\Delta]\) of a finite simplicial complex \(\Delta\) over field k. It starts with the basic facts on f-vectors and h-vectors; h-vectors of simplicial complexes are introduced via h-vectors of multicomplexes (semisimplicial complexes) and h-vectors of graded alebras. The clssical characterization theorems (Kruskal-Katona and Macaulay) are presented. There follows a discussion on Cohen-Macaulay complexes, including the proof of an unpublished theorem of M. Hochster on the local cohomology of face rings; the result is used to derive a proof of the upper bound theorem for arbitrary triangulations of spheres. The presentation continues with a review of: some known results on Gorenstein complexes, including the proof of a characterization theorem and a derivation of the Alexander duality theorem; some open problems on Gorenstein complexes; results on the canonical modules of Cohen-Macaulay face rings, with applications to the study of face rings of triangulations of manifolds (M. Hochster) and of doubly Cohen-Macaulay complexes (K. Baclawski); results of P. Schenzel [Math. Z. 178, 125-142 (1981; Zbl 0472.13012)] on Buchsbaum complexes.

The introductory chapter 0 provides most of the fundamental facts from combinatorics (rational power series, diophantine linear homogeneous equations, integer stochastic matrices, convex polytopes), commutative algebra, homological algebra, and topology (homology theory), needed to understand the subsequent two chapters.

Chapter I starts with the presentation of five conjectures on the numbers \(H_ n(r)\), \(n\geq 1\), \(r\geq 0\), of \(n\times n\) integer stochastic matrices with line sum r. Three of these were first formulated by H. Anand, V. C. Dumir, and H. Gupta [Duke Math. J. 33, 757-769 (1966; Zbl 0144.004)]; the proof uses the Elliott-MacMahon algorithm for solving linear homogeneous equations in nonnegative integers and the Hilbert syzygy theorem. Here the author gives a different proof, using other tools from commutative algebra. These are applied to the study of the following objects generated by an \(r\times n {\mathbb{Z}}\)-matrix \(\Phi\) of maximal rank \(r\leq n:\) the monoid \(E_{\Phi}=\{\beta \in {\mathbb{N}}^ n| \Phi \beta =0\},\) the algebra \(R_{\Phi}\) of \(E_{\Phi}\) over a field k, the \(E_{\Phi}\)-module \(E_{\Phi,\alpha}=\{\beta \in {\mathbb{N}}^ n| \Phi \beta =\alpha \}\), for \(\alpha \in {\mathbb{Z}}^ r\), and the \(R_{\Phi}\)-module \(M_{\Phi,\alpha}\) which is the algebra of \(E_{\Phi,\alpha}\) over k. A central role is played by the Hilbert series \(F(R_{\Phi})\) and \(F(M_{\Phi,\alpha})\). Results on the dimension and depth of \(R_{\Phi}\) are used to prove that \(R_{\Phi}\) is Cohen-Macaulay. From considerations on the local cohomology of the modules \(M_{\Phi,\alpha}\), a sufficient condition for Cohen- Macaulayness and a new proof of the reciprocity theorem are derived. Other developments concern the enumeration of rational points in integral convex polytopes and its connection with volume, the proof of a local duality theorem, and the derivation of a necessary and sufficient condition for \(R_{\Phi}\) to be Gorenstein. The proofs of the four conjectures are obtained by applying the general results to the case when \(E_{\Phi}\) is the set of \(n\times n\) integer stochastic matrices with line sum r.

Chapter II is devoted to the study of the face ring (Stanley-Reisner ring) k[\(\Delta]\) of a finite simplicial complex \(\Delta\) over field k. It starts with the basic facts on f-vectors and h-vectors; h-vectors of simplicial complexes are introduced via h-vectors of multicomplexes (semisimplicial complexes) and h-vectors of graded alebras. The clssical characterization theorems (Kruskal-Katona and Macaulay) are presented. There follows a discussion on Cohen-Macaulay complexes, including the proof of an unpublished theorem of M. Hochster on the local cohomology of face rings; the result is used to derive a proof of the upper bound theorem for arbitrary triangulations of spheres. The presentation continues with a review of: some known results on Gorenstein complexes, including the proof of a characterization theorem and a derivation of the Alexander duality theorem; some open problems on Gorenstein complexes; results on the canonical modules of Cohen-Macaulay face rings, with applications to the study of face rings of triangulations of manifolds (M. Hochster) and of doubly Cohen-Macaulay complexes (K. Baclawski); results of P. Schenzel [Math. Z. 178, 125-142 (1981; Zbl 0472.13012)] on Buchsbaum complexes.

Reviewer: J.Weinstein

### MSC:

13C13 | Other special types of modules and ideals in commutative rings |

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

52Bxx | Polytopes and polyhedra |

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

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

15B36 | Matrices of integers |

11C20 | Matrices, determinants in number theory |