作者： 武同锁、郭锦

定价： 128.00

出版社： 科学出版社

版次： 1

出版时间：2021年12月

开本： 16

装帧： 1042.00

页数： 254

ISBN编码：978-7-03-070302-6

购买链接：https://item.jd.com/13592936.html

内容介绍：

全书共分为7章。第一章包含了关于深度、Krull维数以及Cohen-Macaulay性质（以下简称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模的代数描述两部分内容反映了本书的特色。第二章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（维数）之间的确切关系）。第三章讨论复形的shellable性质，特别是详细推演其用restriction map进行的等价刻画、与d-可分性之间的等价关系，这是对于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复形以及来自单项式理想的三种常用代数复形（或正合列）的详尽构造。第五章是本书的核心和重点，全面深刻地介绍CM复形、shellable与CM的关系、线性预解式与线性商以及如何从图出发构造好的拓扑与代数复形。第五章包含了作者最新的研究成果，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：

Preface

Notations

Chapter 1　Preliminaries on Cohen-Macaulay Rings and Modules ..................................1

• 1. Jacobson radical and NAK Lemma..................................2
  2. Modules with finite lengths..................................3
  3. On graded rings and minimal graded free resolution..................................6
  4. Cohen-Macaulay rings and Cohen-Macaulay Modules..................................11
     1. Dimension, height and Krull’s PI theorem..................................11
     2. Depth Lemma and ..................................14
     3. Local rings: Krull dimension and a system of parameters..................................17
     4. Cohen-Macaulay modules and Cohen-Macaulay rings: local case................22
     5. Cohen-Macaulay rings: non-local case..................................26
     6. Cohen-Macaulay rings: graded case..................................28
     7. Gorenstein rings..................................34
  5. Sequentially Cohen-Macaulay modu1es..................................35

Chapter 2　Abstract Simplicial Complexes....................................40

• 1. Definitions, fundamiental properties and examples..................................40
     1. Abstract simplex and abstract simplicial complex..................................40
     2. 0ther notations and symbols on a simplicial complex..................................41
     3. Fundamental operations on sub-complexes and geometric realization of an abstract simplicial complex..................................42
  2. The facet ideal and Stanley-Reisner ideal of a simplicial complex ..................................47
     1. Monomial ideals and ideal operations.................................47
     2. The Stanley-Reisner (nonface) ideal and facet ideal ................................47
     3. The Alexander dual simplicial complexofand related properties.................................48
     4. Square-free monomial ideal : its nonface complex, facet complex; and -ideals.................................54
  3. Relative simplicial complexes and relative nonface ideals.................................55

Chapter 3　Shellable Simplicial Complexes.................................57

• 1. Destriction and examples.................................57
  2. Restriction maps and Rearrangement Lemmas.................................60
  3. -skeleton .................................64
  4. Shifted, vertex-decomposable and shellable conditions for a simplicial complex.................................66
  5. Shellable and -decomposable.................................69

Chapter 4　Chain Complex Reduced from a Simplicial Complex and Koszul Complexes.................................73

• 1. The chain complex reduced from an abstract simplicial complex and reduced homology groups.................................74
  2. Koszul complexes of lengths 1 or 2.................................79
  3. Koszul complexes of geueral length.................................80
     1. Exterior algebra constructed from a module.................................80
     2. Koszul complexes： two commonly used definitions.................................81
  4. Koszul complexes: a summary of main results.................................83
  5. Other resolutions and complexes of monomial ideals.................................84
     1. The Taylor resolution.................................84
     2. The Scarf complex.................................86
     3. The Lyubeznik resolutions.................................89

Chapter 5　(Sequentially) Cohen-Macaulay Simplicial Complexes and Graphs.................................91

• 1. Cohen-Macaulay simplicial complexes.................................92
     1. Fundamental properties and characterizations.................................92
     2. Connected in codimension one.................................97
     3. Minimal Cohen-Macaulay simplicial complexes and shelled over.................................100
  2. Matroid complexes.................................104
  3. Pure shellable, constructible, and Cohen-Macaulay.................................107
  4. A graded ideal with linear quotients and shellable complexes.................................111
     1. A graded ideal with linear quotients.................................111
     2. Shellable complexes and monomial ideals having linear quotients.................................115
     3. Powers of edge ideals of graphs and regularity.................................118
     4. A polymatroidal monomial ideal has linear quotients.................................119
     5. Strongly shellable simplicial complexes.................................120
  5. sCM simplicial complexes and sCM graded modules.................................121
  6. Clique complex , edge ideal and cover ideal ................................123
  7. Vertex-decomposable graphs and shellable graphs.................................124
  8. Minimal verex covers and standard irredundant primary decomposition of I(G) .................................127
  9. Cohen-Macaulay graphs and well-covered graphs.................................129
  10. Shellable clutters.................................131
      1. Clutters with the free vertex property.................................132
      2. Chordal clutters.................................133
  11. Some particular classes of graphs.................................133
      1. Bipartite graphs.................................133
      2. Boolean graphs are Cohen-Macaulay　138
      3. Cactus graphs and classes of vertex-decomposable graphs.................................143
      4. Cameron-Walker graphs.................................146
      5. Chordal graphs.................................147
      6. -simplicial complexes and -ideals of kind .................................150
      7. Gap-free graphs and related -free graphs.................................174
      8. Graphs whose complements are -partite.................................176
      9. Graph expansions and graph blow ups.................................185
      10. Interlacing graphs and triangular graphs .................................188
      11. Vertex clique-whiskered graphs and their generalizations .................................189
      12. 1-decomposable graphs.................................202

Chapter 6　Shellable Simplicial Complexes from Posets.................................204

• 1. Preliminaries.................................204
  2. A bounded, locally upper-semimodular poset is pure shellable.................................205
  3. EL-labeling of a poset and EL-shellable graded posets.................................209
  4. Admissible lattices and SL-shellable poset.................................213
  5. CL-shellable poset and recursive atom orderings.................................215
     1. Rooted interval and CL-shellable poset.................................216
     2. Recursive atom orderings.................................218

Chapter 7　Betti Numbers and Castelnuovo-Mumford Regularity.................................222

• 1. Calculating Betti numbers via the functor Tor.................................222
  2. Polarization keeps the Betti numbers and regularity unchanged.................................224
  3. Hochster’s Formula and other two reformulations.................................226
  4. Graded Betti numbers of graphs: some general reults.................................230
  5. Splittable monomial ideals and Betti splitting.................................231
  6. Miscellanies Results on Betti Numbers.................................232
     1. The join complex of simplicial complexes.................................232
     2. Graded -modules with pure resolutions.................................233
     3. Stable monomial ideals.................................233
     4. Monomial ideals with linear quotients.................................233
     5. Vertex-decomposable simplcial complexes and multiple clique-whiskered graph.................................234
     6. Chordal graphs　235
  7. Castelnuovo-Mumford regularity of graphs.................................235
     1. A brief survey of some general results.................................235
     2. Graphs whose edge ideals have regularity less than 4.................................237

References.................................240

Index.................................248