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In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL-LINC. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL EILENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. ...

introduction .

chapter i. modules, diagrams, and functors

1. the arrow notation

2. modules

3. diagrams

4. direct sums

5. free and projective modules

6. the functor horn

7. categories

8. functors

chapter if. homology of complexes

1. differential groups

2. complexes

3. cohomology

4. the exact homology sequence

5. some diagram lemmas

6. additive relations

7. singular homology

8. homotopy

9. axioms for homology

.chapter iii. extensions and resolutions

1. extensions of modules

2. addition of extensions

3. obstructions to the extension of homomorphisms

4. the universal coefficient theorem for cohomology

5. composition of extensions

6. resolutions

7. injective modules

8. injective resolutions

9.two exact sequences for extn

10. axiomatic description of ext

11. the injective envelope

chapter iv. cohomol0gy of groups

1. the group ring

2. crossed homomorphisms

3. group extensions

4. factor sets

5. the bar resolution

6. the characteristic class of a group extension

7. cohomology of cyclic and free groups

8. obstructions to extensions

9. realization of obstructions

10. schur's theorem

11. spaces with operators

chapter v. tensor and torsion products

1. tensor products

2. modules over commutative rings

3. bimodules

4. dual modules

5. right exactness of tensor products .

6. torsion products of groups

7. torsion products of modules

8. torsion products by resolutions

9. the tensor product of complexes

10. the konneth formula

11. universal coefficient theorems

chapter vi. types of algebras

1. algebras by diagrams

2. graded modules

3. graded algebras

4. tensor products of algebras

5. modules over algebras

6. cohomology of free abelian groups

7. differential graded algebras

8. identities on horn and

9.coalgebras and hopf algebras

chapter vii. dimension

1. homological dimension

2. dimensions in polynomial rings

3. ext and tor for algebras

4. global dimensions of polynomial rings ..

5. separable algebras

6. graded syzygies

7. local rings

chapter viii. products

1. homology products

2. the torsion product of algebras

3. a diagram lemma

4. external products for ext

5. simplicial objects

6. normalization

7. acyclic models

8. the eilenberg-zilber theorem

9. cup products

chapter ix. relative homological algebra

1. additive categories

2. abelian categories

3. categories of diagrams

4. comparison of allowable resolutions

5. relative abelian categories

6. relative resolutions

7. the categorical bar resolution

8. relative torsion products

9. direct products of rings

chapter x. cohomology of algebraic systems

1. introduction

2. the bar resolution for algebras

3. the cohomology of an algebra

4. the homology of an algebra

5. homology of groups and monoids

6. ground ring extensions and direct products

7. homology of tensor products

8. the case of graded algebras

9. complexes of complexes

10. resolutions and constructions

11. two-stage cohomology of dga-algebras

12. cohomology of commutative dga-algebras

13. homology of algebraic systems

chapter xi. spectral sequences

1. spectral sequences

2. fiber spaces

3. filtered modules

4. transgression

5. exact couples

6. bicomplexes

7. the spectral sequence of a covering

8. cohomology spectral sequences

9. restriction, inflation, and connection

10. the lyndon spectral sequence

11. the comparison theorem

chapter xii. derived functors

1. squares

2. subobjects and quotient objects

3. diagram chasing

4. proper exact sequences

5. ext without projectives

6. the category of short exact sequences

7. connected pairs of additive functors

8. connected sequences of functors

9. derived functors

10. products by universality

11. proper projective complexes

12. the spectral kunneth formula

bibliography

list of standard symbols

index ...

Saunders Mac Lane(1909-2005)已故世界著名数学家。生前曾任美国数学会(MAA)主席，美国数学协会(AMS)主席，美国科学院副院长、院士。麦克莱恩的贡献主要在代数和代数拓扑方面，他是同调代数和范畴论的先驱者之一，因在代数及代数拓扑方面的贡献获1986年美国数学会斯蒂尔奖。1989年曾获美国科学界最高荣誉国家科学奖。

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书籍介绍

In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL-LINC. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL EILENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. ...