Combinatorics And Commutative Algebra

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.

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

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Combinatorics And Commutative Algebra

Summary

Table of Contents

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Combinatorics And Commutative Algebra

Summary

Table of Contents

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