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Monomial ideals. (English) Zbl 1206.13001
Graduate Texts in Mathematics 260. London: Springer (ISBN 978-0-85729-105-9/hbk; 978-1-4471-2594-5/pbk; 978-0-85729-106-6/ebook). xvi, 305 p. (2011).
The present monograph is concerned with combinatorial commutative algebra, centering upon monomials and monomial ideals. The prerequisites are kept to such a minimum as the very basic knowledge of commutative algebra. The monograph is divided into three parts. The first part, consisting of five chapters, is a good introduction to Gröbner bases and initial ideals. The first part culminates in Chapter 2 and Chapter 4, the former is a quick introduction to the theory of Gröbner bases, while the latter gives a detailed description of generic initial ideals. Part II consists of three chapters. Chapter 6 covers the combinatorial characterization of graded ideals due to Macaulay together with its squarefree analogue called the Kruskal-Katona theorem. Chapter 7 discusses minimal free resolutions of monomial ideals. Chapter 8 is concerned with the Eagon-Reiner theorem, which claims that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay. Part III consists of four chapters. Chapter 9 discusses the algebraic aspects of Dirac’s theorem on chordal graphs and the classification problem for Cohen-Macaulay graphs. Chapter 10 is devoted to the study of powers of monomial ideals. Chapter 11 offers a self-contained and systematic presentation of modern shifting theory from the viewpoint of generic initial ideals as well as of graded Betti numbers. Chapter 12 considers discrete polymatroids and polymatroidal ideals.
Binomial ideals, toric rings and convex polytopes are not the main topic of this monograph, and the authors refer the reader to [B. Sturmfels, Gröbner bases and convex polytopes. University Lecture Series. 8. Providece, RI: American Mathematical Society (AMS) (1996; Zbl 0856.13020)], [E. Miller, B. Sturmfels, Combinatorial commutative algebra. Graduate Texts in Mathematics 227. New York, NY: Springe. (2005; Zbl 1066.13001)] and [W. Bruns, J. Gubeladze, Polytopes, rings, and K-theory. Springer Monographs in Mathematics. New York, NY: Springer (2009; Zbl 1168.13001)]. The pioneering work by Richard Stanley on the upper bound conjecture for spheres, for which the reader is referred to [W. Bruns, J. Herzog, Cohen-Macaulay rings. Rev. ed. Cambridge Studies in Advanced Mathematics. 39. Cambridge: Cambridge University Press (1998; Zbl 0909.13005)], [T. Hibi, Algebraic combinatorics on convex polytopes. Glebe: Carslaw Publications (1992; Zbl 0772.52008)] and [R. P. Stanley, Combinatorics and commutative algebra. 2nd ed. Progress in Mathematics 41. Basel: Birkhäuser (2005; Zbl 1157.13302)], is not discussed in this monograph.

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