Combinatorics And Commutative Algebra

by Stanley, Richard P.

• ISBN13: 9780817643690
• ISBN10: 0817643699
• Edition: 2nd
• Format: Paperback
• Copyright: 2004-10-15
• Publisher: Birkhauser

Summary

Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists.New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors.

Table of Contents

Preface to the Second Edition	vii
Preface to the First Edition	viii
Notation	ix
Chapter 0: Background
1. Combinatorics	1	(5)
2. Commutative algebra and homological algebra	6	(13)
3. Topology	19	(6)
Chapter I: Nonnegative Integral Solutions to Linear Equations	25	(28)
1. Integer stochastic matrices (magic squares)	25	(1)
2. Graded algebras and modules	26	(2)
3. Elementary aspects of N-solutions to linear equations	28	(4)
4. Integer stochastic matrices again	32	(1)
5. Dimension, depth, and Cohen-Macaulay modules	33	(3)
6. Local cohomology	36	(2)
7. Local cohomology of the modules ΜΦ,α	38	(4)
8. Reciprocity	42	(2)
9. Reciprocity for integer stochastic matrices	44	(1)
10. Rational points in integral polytopes	45	(1)
11. Free resolutions	46	(2)
12. Duality and canonical modules	48	(4)
13. A final look at linear equations	52	(1)
Chapter II: The Face Ring of a Simplicial Complex	53	(22)
1. Elementary properties of the face ring	53	(1)
2. f-vectors and h-vectors of complexes and multicomplexes	54	(4)
3. Cohen-Macaulay complexes and the Upper Bound Conjecture	58	(2)
4. Homological properties of face rings	60	(4)
5. Gorenstein face rings	64	(2)
6. Gorenstein Hilbert functions	66	(3)
7. Canonical modules of face rings	69	(3)
8. Buchsbaum complexes	72	(3)
Chapter III: Further Aspects of Face Rings	75	(60)
1. Simplicial polytopes, toric varieties, and the g-theorem	75	(3)
2. Shellable simplicial complexes	78	(10)
3. Matroid complexes, level complexes, and doubly Cohen-Macaulay complexes	88	(7)
4. Balanced complexes, order complexes, and flag complexes	95	(11)
5. Splines	106	(4)
6. Algebras with straightening law and simplicial posets	110	(6)
7. Relative simplicial complexes	116	(3)
8. Group actions	119	(7)
9. Subcomplexes	126	(1)
10. Subdivisions	127	(8)
Problems on Simplicial Complexes and their Face Rings	135	(10)
Bibliography	145	(16)
Index	161