本书包含组合交换代数近几年来的一些主要研究成果，选题围绕单纯复形、代数复形以及Cohen-Macaulay性质展开，其中的CM性质是交换代数中最为核心的研究课题。全书共分为7章。

Preface
Notations
Chapter 1 Preliminaries on Cohen-Macaulay Rings and Modules
1.1 Jacobson radical and NAK Lemma
1.2 Modules with finite lengths
1.3 On graded rings and minimal graded free resolution
1.4 Cohen-Macaulay rings and Cohen-Macaulay Modules
1.4.1 Dimension, height and Krull's PI theorem
1.4.2 Depth Lemma and depthr(M)
1.4.3 Local rings:Krull dimension and a system of parameters
1.4.4 Cohen-Macaulay modules and Cohen-Macaulay rings:local case
1.4.5 Cohen-Macaulay rings:non-local case
1.4.6 Cohen-Macaulay rings:graded case
1.4.7 Gorenstein rings
1.5 Sequentially Cohen-Macaulay modules
Chapter 2 Abstract Simplicial Complexes
2.1 Definitions, fundamental properties and examples
2.1.1 Abstract simplex and abstract simplicial complex
2.1.2 Other notations and symbols on a simplicial complex A
2.1.3 Fundamental operations on sub-complexes and geometric realization of an abstract simplicial complex
2.2 The facet ideal I(□) and Stanley-Reisner ideal IA of a simplicial complex □
2.2.1 Monomial ideals and ideal operations
2.2.2 The Stanley-Reisner (nonface) ideal I□ and facet ideal I(□)
2.2.3 The Alexander dual simplicial complex Av of A and related properties
2.2.4 Square-free monomial ideal I:its nonface complex, facet complex I□ and f-ideals
2.3 Relative simplicial complexes and relative nonface ideals
Chapter 3 Shellable Simplicial Complexes
3.1 Definitions and examples
3.2 Restriction maps and Rearrangement Lemmas
3.3 (r,s)-skeleton □(r,s)
3.4 Shifted, vertex-decomposable and shellable conditions for a simplicial complex
3.5 Shellable and k-decomposable
Chapter 4 Chain Complex Reduced from a Simplicial Complex and Koszul Complexes
4.1 The chain complex reduced from an abstract simplicial complex and reduced homology groups
4.2 Koszul complexes of lengths 1 or 2
4.3 Koszul complexes of general length
4.3.1 Exterior algebra constructed from a module
4.3.2 Koszul complexes:two commonly used definitions
4.4 Koszul complexes:a summary of main results
4.5 Other resolutions and complexes of monomial ideals
4.5.1 The Taylor resolution
4.5.2 The Scarf complex
4.5.3 The Lyubeznik resolutions
Chapter 5 (Sequentially) Cohen-Macaulay Simplicial Complexes and Graphs
5.1 Cohen-Macaulay simplicial complexes
5.1.1 Fundamental properties and characterizations
5.1.2 Connected in codimension one
5.1.3 Minimal Cohen-Macaulay simplicial complexes and shelled over
5.2 Matroid complexes
5.3 Pure shellable, constructible, and Cohen-Macaulay
5.4 A graded ideal with linear quotients and shellable complexes
5.4.1 A graded ideal with linear quotients
5.4.2 Shellable complexes and monomial ideals having linear quotients
5.4.3 Powers of edge ideals of graphs and regularity
5.4.4 A polymatroidal monomial ideal has linear quotients
5.4.5 Strongly shellable simplicial complexes
5.5 sCM simplicial complexes and sCM graded modules
5.6 Clique complex □G, edge ideal I(G) and cover ideal Ic(G)
5.7 Vertex-decomposable graphs and shellable graphs
5.8 Minimal vertex covers and standard irredundant primary decomposition of I(G)
5.9 Cohen-Macaulay graphs and well-covered graphs
5.10 Shellable clutters
5.10.1 Clutters with the free vertex property
5.10.2 Chordal clutters
5.11 Some particular classes of graphs
5.11.1 Bipartite graphs
5.11.2 Boolean graphs are Cohen-Macaulay
5.11.3 Cactus graphs and classes of vertex-decomposable graphs
5.11.4 Cameron-Walker graphs
5.11.5 Chordal graphs
5.11.6 F-simplicial complexes and f-ideals of kind (n,d)
5.11.7 Gap-free graphs and related H-free graphs
5.11.8 Graphs whose complements are r-partite
5.11.9 Graph expansions and graph blow ups
5.11.10 Interlacing graphs ICn,2