Graduate Texts in Mathematics 227. New York, NY: Springer (ISBN 0-387-23707-0/pbk). xiv, 417 p. EUR 36.95/net; sFr. 67.50; £ 28.50; $ 49.95 (2005).
The book under review constitutes a self-contained introduction to the use of combinatorial methods in commutative algebra. The intention is to complement and build on R. Stanley’s classic book “Combinatorics and commutative algebra”, 2nd ed., Prog. Math. 41 (Basel 1996; Zbl 0838.13008)]. Concrete calculations and examples are used to introduce and develop concepts. Numerous exercises provide the opportunity to work through the material and end of chapter notes comment on the history and development of the subject. The authors have provided us with a useful reference and an effective text book.
The book is divided into three parts. The first part in entitled “Monomial Ideals” and contains chapters on squarefree monomial ideals, Borel-fixed monomial ideals, three-dimensional staircases, cellular resolutions, Alexander duality and generic monomial ideals. The second part, on “Toric Algebra”, contains chapters on semigroup rings, multigraded polynomial rings, syzygies of lattice ideals, toric varieties, irreducible and injective modules, Erhart polynomials, and local cohomology. The final section highlights the use of “Determinants” and contains chapters on Plücker coordinates, matrix Schubert varieties, antidiagonal initial ideals, minors in matrix products and Hilbert schemes of points.