Combinatorial commutative algebra. (English) Zbl 1066.13001

Miller, Ezra; Sturmfels, Bernd

Graduate Texts in Mathematics 227. New York, NY: Springer (ISBN 0-387-22356-8/hbk). xiv, 417 p. (2005).

The book is a selfcontained introduction to some of the combinatorical techniques for dealing with multigraded polynomial rings, semi group rings, and determinantal rings. An important role play combinatorically defined ideals and their quotients with the aim to compute numerical invariants and resolutions using gradings more refined than the standard grading.
The book is subdivided in three parts: Monomial Ideals, Toric Algebras, and Determinants. It has altogether 18 chapters containing homological invariants of monomial ideals and their polyhedral resolutions, toric varieties, local cohomology, Hilbert schemes among other subjects, to show how the tools developed can be used for studying algebraic varieties with group actions. Each chapter begins with an overview and ends with notes and references. The book assumes the knowledge of commutative algebra (graded rings, free resolutions, Gröbner bases) and a little simplicial topology and polyhedral geometry. It is interesting for a wide audience of students and researchers.
The book may serve as a basis for a full year course on this topic. It contains a lot of exercises and hints for further studies.

Reviewer: Gerhard Pfister (Kaiserslautern)

MSC:

13-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
13F20	Polynomial rings and ideals; rings of integer-valued polynomials
13P99	Computational aspects and applications of commutative rings
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14M05	Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14C05	Parametrization (Chow and Hilbert schemes)
13C40	Linkage, complete intersections and determinantal ideals

Keywords:

monomial ideal; toric algebra; Hilbert scheme; local cohomology; multigraded polynomial rings