An introduction to homological algebra

Description

Homological algebra, because of its fundamental nature, is relevant to many branches of pure mathematics, including number theory, geometry, group theory and ring theory. Professor Northcott's aim is to introduce homological ideas and methods and to show some of the results which can be achieved. The early chapters provide the results needed to establish the theory of derived functors and to introduce torsion and extension functors. The new concepts are then applied to the theory of global dimensions, in an elucidation of the structure of commutative Noetherian rings of finite global dimension and in an account of the homology and cohomology theories of monoids and groups. A final section is devoted to comments on the various chapters, supplementary notes and suggestions for further reading. This book is designed with the needs and problems of the beginner in mind, providing a helpful and lucid account for those about to begin research, but will also be a useful work of reference for specialists. It can also be used as a textbook for an advanced course. Title Contents Preface 1. Generalities concerning modules 1.1 Left modules and right modules 1.2 Submodules 1.3 Factor modules 1.4 Lambda-homomorphisms 1.5 Some different types of Lambda-homomorphisms 1.6 Induced mappings 1.7 Images and kernels 1.8 Modules generated by subsets 1.9 Direct products and direct sums 1.10 Abbreviated notations 1.11 Sequences of Lambda-homomorphisms 2. Tensor products and groups of homomorphisms 2.1 The definition of tensor products 2.2 Tensor products over commutative rings 2.3 Continuation of the general discussion 2.4 Tensor products of homomorphisms 2.5 The principal properties of Hom_Lambda(B, C) 3. Categories and functors 3.1 Abstract mappings 3.2 Categories 3.3 Additive and Lambda-categories 3.4 Equivalences 3.5 The categories G^L_Lambda and G^R_Lambda 3.6 Functors of a single variable 3.7 Functors of several variables 3.8 Natural transformations of functors 3.9 Functors of modules 3.10 Exact functors 3.11 Left exact and right exact functors 3.12 Properties of right exact functors 3.13 A (X)_Lambda C and Hom_Lambda (B, C) as functors 4. Homology functors 4.1 Diagrams over a ring 4.2 Translations of diagrams 4.3 Images and kernels as functors 4.4 Homology functors 4.5 The connecting homomorphism 4.6 Complexes 4.7 Homotopic translations 5. Projective and injective modules 5.1 Projective modules 5.2 Injective modules 5.3 An existence theorem for injective modules 5.4 Complexes over a module 5.5 Properties of resolutions of modules 5.6 Properties of resolutions of sequences 5.7 Further results on resolutions of sequences 6. Derived functors 6.1 Functors of complexes 6.2 Functors of two complexes 6.3 Right-derived functors 6.4 Left-derived functors 6.5 Connected sequences of functors 7. Torsion and extension functors 7.1 Torsion functors 7.2 Basic properties of torsion functors 7.3 Extension functors 7.4 Basic properties of extension functors 7.5 The homological dimension of a module 7.6 Global dimension 7.7 Noetherian rings 7.8 Commutative Noetherian rings 7.9 Global dimension of Noetherian rings 8. Some useful identities 8.1 Bimodules 8.2 General principles 8.3 The associative law for tensor products 8.4 Tensor products over commutative rings 8.5 Mixed identities 8.6 Rings and modules of fractions 9. Commutative Noetherian rings of finite global dimension 9.1 Some special cases 9.2 Reduction of the general problem 9.3 Modules over local rings 9.4 Some auxiliary results 9.5 Homological codimension 9.6 Modules of finite homological dimension 10. Homology and cohomology theories of groups and monoids 10.1 General remarks concerning monoids and groups 10.2 Modules with respect to monoids and groups 10.3 Monoid-rings and group-rings 10.4 The functors A^G and A_G 10.5 Axioms for the homology theory of monoids 10.6 Axioms for the cohomology theory of monoids 10.7 Standard resolutions of Z 10.8 The first homology group 10.9 The first cohomology group 10.10 The second cohomology group 10.11 Homology and cohomology in special cases 10.12 Finite groups 10.13 The norm of a homomorphism 10.14 Properties of the complete derived sequence 10.15 Complete free resolutions of Z Notes Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 9 Chapter 10 References Index