《AN INTRODUCTION TO HOMOLOGICAL ALGEBRA》

1．Generalities concerning modules1

1．1Left modules and right modules1

1．2 Submodules3

1．3 Factor modules3

1．4 A-homomorphisms3

1．5 Some different types of A-homomorphisms4

1．6 Induced mappings5

1．7 Images and kernels6

1．8 Modules generated by subsets7

1．9 Direct products and direct sums9

1．10 Abbreviated notations12

1．11 Sequences of A-homomorphisms13

2．Tensor products and groups of homomorphisms16

2．1The definition of tensor products16

2．2 Tensor products over commutative rings17

2．3 Continuation of the general discussion18

2．4 Tensor products of homomorphisms19

2．5 The principal properties of HomA(B，C)24

3．Categories and functors30

3．1Abstract mappings30

3．2 Categories31

3．3 Additive and A-categories32

3．4 Equivalences32

3．5 The categories ？LΛ and ？RΛ33

3．6 Functors of a single variable33

3．7 Functors of several variables34

3．8 Natural transformations of functors35

3．9 Functors of modules36

3．10 Exact functors38

3．11 Left exact and right exact functors40

3．12 Properties of right exact functors41

3．13 A⊕ΛA C and HomA(B，C)as functors44

4．Homology functors46

4．1Diagrams over a ring46

4．2 Translations of diagrams47

4．3 Images and kernels as functors48

4．4 Homology functors52

4．5 The connecting homomorphism54

4．6 Complexes50

4．7 Homotopic translations62

5．Projective and infective modules63

5．1Projective modules63

5．2 Injective modules67

5．3 An existence theorem for injective modules71

5．4 Complexes over a module75

5．5 Properties of resolutions of modules77

5．6 Properties of resolutions of sequences80

5．7 Further results on resolutions of sequences84

6．Derived functors90

6．1Functors of complexes90

6．2 Functors of two complexes94

6．3 Right-derived functors99

6．4 Left-derived functors109

6．5 Connected sequences of functors113

7．Torsion and extension functors121

7．1Torsion functors121

7．2 Basic properties of torsion functors123

7．3 Extension functors128

7．4 Basic properties of extension functors130

7．5 The homological dimension of a module134

7．6 Global dimension138

7．7 Noetherian rings144

7．8 Commutative Noetherian rings148

7．9 Global dimension of Noetherian rings149

8．Some useful identities155

8．1Bimodules155

8．2 General principles156

8．3 The associative law for tensor products160

8．4 Tensor products over commutative rings161

8．5 Mixed identities164

8．6 Rings and modules of fractions167

9．Commutative Noetherian rings of finite global dimension174

9．1Some special cases174

9．2 Reduction of the general problem184

9．3 Modules over local rings189

9．4 Some auxiliary results202

9．5 Homological codimension204

9．6 Modules of finite homological dimension205

10．Homology and cohomology theories of groups and monoids211

10．1General remarks concerning monoids and groups211

10．2 Modules with respect to monoids and groups214

10．3 Monoid-rings and group-rings215

10．4 The functors AG and AG217

10．5 Axioms for the homology theory of monoids219

10．6 Axioms for the cohomology theory of monoids221

10．7 Standard resolutions of Z223

10．8 The first homology group229

10．9 The first cohomology group230

10．10 The second cohomology group238

10．11 Homology and cohomology in special cases244

10．12 Finite groups249

10．13 The norm of a homomorphism252

10．14 Properties of the complete derived sequence256

10．15 Complete free resolutions of Z259

Notes266

References278

Index281