Table of Contents
Preface vii
1 Introduction 1
2 Rings and Ideals 9
2.1 Rings and Ideals 9
2.2 Localization of a Ring 19
2.3 Ideals in a Polynomial Ring 27
2.4 Gröbner Basis of an Ideal 32
2.5 Elimination and Extension 38
2.6 Implicitization 45
2.7 Schemes 48
2.8 Gröbner Basis Applications 53
2.8.1 Solving Systems of Equations 54
2.8.2 Orthogonal Projection 56
2.8.3 Poncelet's Algebraic Correspondence 58
3.1 Modules 63
3.2 Exact Sequences and Commutative Diagrams 67
3.3 Projective and Injective Modules 71
3.4 Tensor Product of Modules 75
3.5 Flatness 78
3.6 Localization 80
3.7 Local Property 82
3.8 Associated Primes 85
3.9 Primary Decomposition of Modules 89
3.10 Modules of Finite Length 92
3.11 Krull Dimension of a Ring 95
3.12 Dimension of Modules 98
3.13 Rank of Modules 102
3.14 Computational Applications 104
3.14.1 Tensor Products 104
3.14.2 Primary Decomposition 106
3.14.3 Krull Dimension 111
3.14.4 Generators of Syzygy Modules 113
4 Graded and Local Rings and Modules 117
4.1 Graded Rings and Modules 117
4.2 Graded Localization 120
4.3 Graded Associated Primes 122
4.4 Filtration 125
4.5 System of Parameters 129
4.6 Regular and Quasi-regular M-Sequence 131
4.7 Proj of a Graded Ring 134
4.8 Graded Free Resolution 136
4.9 Dimension and Multiplicity 142
4.10 Applications 152
4.10.1 Compute the Free Resolution 152
4.10.2 Implicitization via Syzygies 156
4.10.3 Compute the Rees Algebra 159
4.10.4 Resultants 164
5 Homological Method 171
5.1 Complexes 171
5.2 Complex of Tor 174
5.3 Koszul Complex 178
5.4 Regular Sequences 181
5.5 Regular Rings 186
5.6 Complex of Ext 189
5.7 Exactness Criteria for Complexes 199
5.8 Local Cohomology 203
5.9 Applications 208
5.9.1 Castelnuovo-Mumford Regularity 208
5.9.2 The MacRae's Invariant 211
5.9.3 Approximation Complex 218
Bibliography 225
