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《Complexes and the Cohen-Macaulay Property(复形与Cohen-Macaulay性质）》共分为7章。第一章包含了关于深度、Krull维数以及CM性质等的一些核心结果或者基本事实；其中关于标准代数的CM性与分次CM性的等价性、序列CM性的代数描述两部分内容十《Complexes and the Cohen-Macaulay Property(复形与Cohen-Macaulay性质）》的特色和贡献。第二章是讨论单纯复形的基本事实，特别是描述了两个代数不变量（由复形构造的面环的深度、Krull维数）与复形的拓扑不变量之间的确切关系）。第三章讨论复形的shellable性质，特别是详细推出其用restrictionmap进行的等价刻画、与d-可分性之间的等价关系，是对于shellable性质的深刻描述和讨论。第四章介绍...
《Complexes and the Cohen-Macaulay Property(复形与Cohen-Macaulay性质）》共分为7章。第一章包含了关于深度、Krull维数以及CM性质等的一些核心结果或者基本事实；其中关于标准代数的CM性与分次CM性的等价性、序列CM性的代数描述两部分内容十《Complexes and the Cohen-Macaulay Property(复形与Cohen-Macaulay性质）》的特色和贡献。第二章是讨论单纯复形的基本事实，特别是描述了两个代数不变量（由复形构造的面环的深度、Krull维数）与复形的拓扑不变量之间的确切关系）。第三章讨论复形的shellable性质，特别是详细推出其用restrictionmap进行的等价刻画、与d-可分性之间的等价关系，是对于shellable性质的深刻描述和讨论。第四章介绍了如何由拓扑复形构造代数链复形，介绍相应的导出同调群，并重点介绍了近代文献中有较多应用的Koszul复形以及三种常用复形的详尽构造。第五章是《Complexes and the Cohen-Macaulay Property(复形与Cohen-Macaulay性质）》的核心和重点，全面深刻的介绍CM复形、shellable与CM的关系、线性预解式与线性商，如何从图出发构造好的拓扑与代数复形。第五章包含了作者*新的研究成果，也综述了多个研究专题（包含作者和业界核心专家的成果）。第六章主要介绍Bejorner等人的近期成果，主要是讨论如何从偏序集出发构造系列的shellable复形等。第七章是专门讨论正则度的，既包含中心的传统结果，也包含了作者等人的近期研究成果。
目录 · · · · · ·
Contents
Preface
Notations
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules 1
1.1 Jacobson radical and NAK Lemma 2
1.2 Modules with finite lengths 3
· · · · · · (更多)
Preface
Notations
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules 1
1.1 Jacobson radical and NAK Lemma 2
1.2 Modules with finite lengths 3
· · · · · · (更多)
Contents
Preface
Notations
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules 1
1.1 Jacobson radical and NAK Lemma 2
1.2 Modules with finite lengths 3
1.3 On graded rings and minimal graded free solution 6
1.4 Cohen-Macaulay rings and Cohen-Macaulay Modules 11
1.4.1 Dimension, height and Krull’s PI theorem 11
1.4.2 Depth Lemma and depthI(M) 14
1.4.3 Local rings: Krull dimension and a system of parameters 17
1.4.4 Cohen-Macaulay modules and Cohen-Macaulay rings: local case 22
1.4.5 Cohen-Macaulay rings: non-local case 26
1.4.6 Cohen-Macaulay rings: graded case 28
1.4. 7 Gorenstein rings 34
1.5 Sequentially Cohen-Macaulay modu1es 35
Chapter 2 Abstract Simplicial Complexes 40
2.1 Definitions, fundamiental properties and examples 40
2.1.1 Abstract simplex and abstract simplicial complex＊40
2.1.2 0ther notations and symbols on a simplicial complex ＊ 41
2.1.3 Fundaｍental operations on sub-complexes and geometric realization of an abstract simplicial complex 42
2.2 The facet idea1I(*) and Stanley-Reisner ideal I*of a simplicial complex * 47
2.2.1 Monomial ideals and ideal operations 47
2.2.2 The Stanley-Reisner (nonface） ideal I* and facet ideal I* 47
2.2.3 The Alexander dual simplicial complex * of*and related properties 48
2.2.4 Square-free monomial ideal I: its nonface complex, facet complex; I* and f-ideals 54
2.3 Relative simplicial complexes and relative nonface ideals 55
Chapter 3 Shellable Simplicial Complexes 57
3.1 Destriction and examples 57
3.2 Restriction maps and Rearrangement Lemmas 60
3.3 (r,s)-skeleton * 64
3.4 Shifted， vertex-decomposable and shellable conditions for a simplicial complex 66
3.5 Shellable and k-decomposable 69
Chapter 4 Chain Complex Reduced from a Simplicial Complex and Koszul Complexes 73
4.1 The chain complex reduced from an abstract simplicial complex and reduced homology groups 74
4.2 Koszul complexes of lengths 1 or 2 79
4.3 Koszul complexes of geueral length 80
4.3.1 Exterior algebra constructed from a module 80
4.3.2 Koszul complexes： two commonly used definitions 81
4.4 Koszul complexes: a S田nmary of main results 83
4.5 Other resolutions and complexes of monomial ideals 84
4.5.1 The Taylor resolution 84
4.5.2 The Scarf complex 86
4.5.3 The Lyubeznik resolutions 89
Chapter 5 (Sequentially) Cohen-Macaulay Simplicial Complexes and Graphs 91
5.1 Cohen-Macaulay simplicial complexes 92
5.1.1 Fundamental properties and characterizations 92
5.1.2 Connected in codimension one 97
5.1.3 Minimal Cohen-Macaulay simplicial complexes and shelled over 100
5.2 Matroid complexes 104
5.3 Pure shellable, constructible, and Cohen-Macaulay 107
5.4 A graded ideal with linear quotients and shellable complexes 111
5.4.1 A graded id凶i with linear quotients 111
5.4.2 Shellable complexes and monomial ideals having linear quotients 115
5.4.3 Powers of edge ideals of graphs and regularity 118
5.4.4 A polymatroidal monomial ideal has linear quotients 119
5.4.5 Strongly shellable simplicial complexes 120
5.5 sCM simplicial complexes and sCM graded modules 121
5.6 Clique complex *, edge ideal I(G) and cover ideal Ic(G) 123
5. 7 Vertex-decomposable graphs a且d shellable graphs 124
5.8 Minimal verex covers and standard irredundant primary decomposition of I(G) 127
5.9 Cohen-Macaulay graphs and well-covered graphs 129
5.10 Shellable clutters 131
5.10.1 Clutters with the free vertex property 132
5.10.2 Chordal clutters 133
5.11 Some particular classes of graphs 133
5.11.1 Bipartite graphs 133
5.11.2 Boolean graphs are Cohen-Macaulay 138
5.11.3 Cactus graphs and classes of vertex-decomposable graphs 143
5.11.4 Cameron-Walker graphs 146
5.11.5 Chordal graphs 147
5.11.6 F-simplicial complexes and f-ideals of kind （n，d） 150
5.11.7 Gap-free graphs and related H-free graphs 174
5.11.8 Graphs whose complements are r-partite 176
5.11.9 Graph expansions and graph blow ups 185
5.11.10 Interlacing graphs * and triangular graphs * 188
5.11.11 Vertex clique-whiskered graphs * and their generalizations * 189
5.11.12 1-decomposable graphs 202
Chapter 6 Shellable Simplicial Complexes from Posets 204
6.1 Preliminaries 204
6.2 A bounded, locally upper-semimodular poset is pure shellable 205
6.3 EL-labeling of a poset and EL-shellable graded posets 209
6.4 Admissible lattices and SL-shellable poset 213
6.5 CL-shellable poset and recursive atom orderings 215
6.5.1 Rooted interval and CL-shellable poset 216
6.5.2 Recursive atom orderings 218
Chapter7 Betti Numbers and Castelnuovo-Mumford Regularity 222
7.1 Calculating Betti numbers via the functor Tor 222
7.2 Polarization keeps the Betti numbers and regularity unchanged 224
7.3 Hochster’s Formula and other two reformulations 226
7.4 Graded Betti numbers of graphs: some general reults 230
7.5 Spli也table monomial ideals and Betti splitting 231
7.6 Miscellanies Results on Betti Numbers 232
7.6.1 The join complex* of simplicial complexes 232
7.6.2 Graded S-modules with pure resolutions 233
7.6.3 Stable monomial ideals 233
7.6.4 Monomial ideals with linear quotients 233
7.6.5 Vertex-decomposable simplcial complexes and multiple clique-whiskered graph 234
7.6.6 Chordal graphs 235
7.7 Castelnuovo-Mumford regularity of graphs 235
7.7.1 A brief survey of some general results 235
7.7.2 Graphs whose edge ideals have regularity less than 4 237
References 240
Index 248
· · · · · · (收起)
Preface
Notations
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules 1
1.1 Jacobson radical and NAK Lemma 2
1.2 Modules with finite lengths 3
1.3 On graded rings and minimal graded free solution 6
1.4 Cohen-Macaulay rings and Cohen-Macaulay Modules 11
1.4.1 Dimension, height and Krull’s PI theorem 11
1.4.2 Depth Lemma and depthI(M) 14
1.4.3 Local rings: Krull dimension and a system of parameters 17
1.4.4 Cohen-Macaulay modules and Cohen-Macaulay rings: local case 22
1.4.5 Cohen-Macaulay rings: non-local case 26
1.4.6 Cohen-Macaulay rings: graded case 28
1.4. 7 Gorenstein rings 34
1.5 Sequentially Cohen-Macaulay modu1es 35
Chapter 2 Abstract Simplicial Complexes 40
2.1 Definitions, fundamiental properties and examples 40
2.1.1 Abstract simplex and abstract simplicial complex＊40
2.1.2 0ther notations and symbols on a simplicial complex ＊ 41
2.1.3 Fundaｍental operations on sub-complexes and geometric realization of an abstract simplicial complex 42
2.2 The facet idea1I(*) and Stanley-Reisner ideal I*of a simplicial complex * 47
2.2.1 Monomial ideals and ideal operations 47
2.2.2 The Stanley-Reisner (nonface） ideal I* and facet ideal I* 47
2.2.3 The Alexander dual simplicial complex * of*and related properties 48
2.2.4 Square-free monomial ideal I: its nonface complex, facet complex; I* and f-ideals 54
2.3 Relative simplicial complexes and relative nonface ideals 55
Chapter 3 Shellable Simplicial Complexes 57
3.1 Destriction and examples 57
3.2 Restriction maps and Rearrangement Lemmas 60
3.3 (r,s)-skeleton * 64
3.4 Shifted， vertex-decomposable and shellable conditions for a simplicial complex 66
3.5 Shellable and k-decomposable 69
Chapter 4 Chain Complex Reduced from a Simplicial Complex and Koszul Complexes 73
4.1 The chain complex reduced from an abstract simplicial complex and reduced homology groups 74
4.2 Koszul complexes of lengths 1 or 2 79
4.3 Koszul complexes of geueral length 80
4.3.1 Exterior algebra constructed from a module 80
4.3.2 Koszul complexes： two commonly used definitions 81
4.4 Koszul complexes: a S田nmary of main results 83
4.5 Other resolutions and complexes of monomial ideals 84
4.5.1 The Taylor resolution 84
4.5.2 The Scarf complex 86
4.5.3 The Lyubeznik resolutions 89
Chapter 5 (Sequentially) Cohen-Macaulay Simplicial Complexes and Graphs 91
5.1 Cohen-Macaulay simplicial complexes 92
5.1.1 Fundamental properties and characterizations 92
5.1.2 Connected in codimension one 97
5.1.3 Minimal Cohen-Macaulay simplicial complexes and shelled over 100
5.2 Matroid complexes 104
5.3 Pure shellable, constructible, and Cohen-Macaulay 107
5.4 A graded ideal with linear quotients and shellable complexes 111
5.4.1 A graded id凶i with linear quotients 111
5.4.2 Shellable complexes and monomial ideals having linear quotients 115
5.4.3 Powers of edge ideals of graphs and regularity 118
5.4.4 A polymatroidal monomial ideal has linear quotients 119
5.4.5 Strongly shellable simplicial complexes 120
5.5 sCM simplicial complexes and sCM graded modules 121
5.6 Clique complex *, edge ideal I(G) and cover ideal Ic(G) 123
5. 7 Vertex-decomposable graphs a且d shellable graphs 124
5.8 Minimal verex covers and standard irredundant primary decomposition of I(G) 127
5.9 Cohen-Macaulay graphs and well-covered graphs 129
5.10 Shellable clutters 131
5.10.1 Clutters with the free vertex property 132
5.10.2 Chordal clutters 133
5.11 Some particular classes of graphs 133
5.11.1 Bipartite graphs 133
5.11.2 Boolean graphs are Cohen-Macaulay 138
5.11.3 Cactus graphs and classes of vertex-decomposable graphs 143
5.11.4 Cameron-Walker graphs 146
5.11.5 Chordal graphs 147
5.11.6 F-simplicial complexes and f-ideals of kind （n，d） 150
5.11.7 Gap-free graphs and related H-free graphs 174
5.11.8 Graphs whose complements are r-partite 176
5.11.9 Graph expansions and graph blow ups 185
5.11.10 Interlacing graphs * and triangular graphs * 188
5.11.11 Vertex clique-whiskered graphs * and their generalizations * 189
5.11.12 1-decomposable graphs 202
Chapter 6 Shellable Simplicial Complexes from Posets 204
6.1 Preliminaries 204
6.2 A bounded, locally upper-semimodular poset is pure shellable 205
6.3 EL-labeling of a poset and EL-shellable graded posets 209
6.4 Admissible lattices and SL-shellable poset 213
6.5 CL-shellable poset and recursive atom orderings 215
6.5.1 Rooted interval and CL-shellable poset 216
6.5.2 Recursive atom orderings 218
Chapter7 Betti Numbers and Castelnuovo-Mumford Regularity 222
7.1 Calculating Betti numbers via the functor Tor 222
7.2 Polarization keeps the Betti numbers and regularity unchanged 224
7.3 Hochster’s Formula and other two reformulations 226
7.4 Graded Betti numbers of graphs: some general reults 230
7.5 Spli也table monomial ideals and Betti splitting 231
7.6 Miscellanies Results on Betti Numbers 232
7.6.1 The join complex* of simplicial complexes 232
7.6.2 Graded S-modules with pure resolutions 233
7.6.3 Stable monomial ideals 233
7.6.4 Monomial ideals with linear quotients 233
7.6.5 Vertex-decomposable simplcial complexes and multiple clique-whiskered graph 234
7.6.6 Chordal graphs 235
7.7 Castelnuovo-Mumford regularity of graphs 235
7.7.1 A brief survey of some general results 235
7.7.2 Graphs whose edge ideals have regularity less than 4 237
References 240
Index 248
· · · · · · (收起)
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