Hilbert Functions of Filtered Modules (eBook)

Hilbert Functions of Filtered Modules 3
Preface 5
Acknowledgements 6
Contents 7
Introduction 9
Chapter 1 Preliminaries 17
1.1 Notation 17
1.2 Superficial Elements 19
1.3 The Hilbert Function and Hilbert Coefficients 24
1.4 Maximal Hilbert Functions 28
Chapter 2 Bounds for e0(M) and e1(M) 31
2.1 The Multiplicity and the First Hilbert Coefficient: Basic Facts 32
2.2 The One-Dimensional Case 36
2.3 The Higher Dimensional Case 46
2.4 The Border Cases 52
Chapter 3 Bounds for e2(M) 63
3.1 The Ratliff--Rush filtration 63
3.2 Bounds for e2(M) 65
Chapter 4 Sally's Conjecture and Applications 76
4.1 A Bound on the Reduction Number 78
4.2 A Generalization of Sally's Conjecture 82
4.3 The Case e1(M)=e0(M)-h0(M)+1 86
4.4 The Case e1(M)=e0(M)-h0(M)+2 88
Chapter 5 Applications to the Fiber Cone 92
5.1 Depth of the Fiber Cone 93
5.2 The Hilbert Function of the Fiber Cone 94
5.3 A Version of Sally's Conjecture for the Fiber Cone 95
5.4 The Hilbert Coefficients of the Fiber Cone 98
5.5 Further Numerical Invariants: The gi 99
Chapter 6 Applications to the Sally Module 102
6.1 Depth of the Sally Module 103
6.2 The Hilbert Function of the Sally Module 103
References 107
Index 112