同調代數

preface

chapter i. rings and modules

1. preliminaries

2. projective modules

3. injective modules

4. semi-simple rings

5. hereditary rings

6. semi-hereditary rings

7. noetherian tings

exercises

chapter ii. additive functors

1. definitions

2. examples

3. operators

4. preservation of exactness

5. composite functors

6. change of rings

exercises

chapter iii. satellites

1. definition of satellites

2. connecting homomorphisms

3. half exact functors

4. connected sequence of functors

5. axiomatic description of satellites

6. composite functors

7. several variables

exercises

chapter iv. homology

1. modules with differentiation

2. the ring of dual numbers

3. graded modules, complexes

4. double gradings and complexes

5. functors of complexes

6. the homomorphism

7. the homomorphism a (continuation)

8. kiinneth relations

exercises

chapter v. derived functors

1. complexes over modules; resolutions

2. resolutions of sequences

3. definition of derived functors

4. connecting homomorphisms

5. the functors rot and lot

6. comparison with satellites

7. computational devices

8. partial derived functors

9. sums, products, limits

i0. the sequence of a map

exercises

chapter vi. derived functors of ~ and hem

1. the functors tor and ext

2. dimension of modules and rings

3. kiinneth relations

4. change of rings

5. duality homomorphisms

exercises

chapter vli. integral domains

1. generalities

2. the field of quotients

3. inversible ideals

4. priifer rings

5. dedekind rings

6. abelian groups

7. a description of torx (a,c)

exercises

chapter viii. augmented rings

1. homology and cohomology of an augmented ring

2. examples

3. change of rings

4. dimension

5. faithful systems

6. applications to graded and local rings

exercises

chapter ix. associative algebras

1. algebras and their tensor products

2. associativity formulae

3. the enveloping algebra a~

4. homology and cohomology of algebras

5. the hochschild groups as functors of a

6. standard complexes

7. dimension

exercises

chapter x. supplemented algebras

1. homology of supplemented algebras

2. comparison with hochschild groups

3. augmented monoids

4. groups

5. examples of resolutions

6. the inverse process

7. subalgebras and subgroups

8. weakly injective and projective modules

exercises

chapter xi. products

1. external products

2. formal properties of the products

3. lsomorphisms

4. internal products

5. computation of products

6. products in the hochschild theory

7. products for supplemented algebras

8. associativity formulae

9. reduction theorems

exercises

chapter xii. finite groups

1. norms

2. the complete derived sequence

3. complete resolutions

4. products for finite groups

5. the uniqueness theorem

6. duality

7. examples

8. relations with subgroups

9. double cosets

10. p-groups and sylow groups

1. periodicity

exercises

chapter xlli. lie algebras

1. lie algebras and their enveloping algebras

2. homology and cohomology of lie algebras

3. the poincare-witt theorem

4. subaigebras and ideals

5. the diagonal map and its applications

6. a relation in the standard complex

7. the complex v(g)

8. applications of the complex v(g)

exercises

chapter xiv. extensions

1. extensions of modules

2. extensions of associative algebras

3. extensions of supplemented algebras

4. extensions of groups

5. extensions of lie algebras

exercises

chapter xv. spectral sequences

1. filtrations and spectral sequences

2. convergence

3. maps and homotopies

4. the graded case

5. induced homomorphisms and exact sequences

6. application to double complexes

7. a generalization

exercises

chapter xvi. applications of spectral sequences

1. partial derived functors

2. functors of complexes

3. composite functors

4. associativity formulae

5. applications to the change of rings

6. normal subalgebras

7. associativity formulae using diagonal maps

8. complexes over algebras

9. topological applications

10. the almost zero theory

exercises

chapter xvll. hyperhomology

1. resolutions of complexes

2. the invariants

3. regularity conditions

4. mapping theorems

5. kiinneth relations

6. balanced functors

7. composite functors

appendix: exact categories, by david a. buchsbaum

list of symbols

index of terminology

本書是數學發展史上的一個里程碑，在很長一段時間，這是本講述拓撲代數的教程。這本書堪稱是一部同調代數經典，1956年初版，至今已有七次重印出版。這本書曾在純代數領域引起過不小的轟動，作者企圖將這個領域統一起來，並且為這個領域構建一個完整的框架。書中講述的同調理論包含了羣、李代數和結合代數上同調結構，大量的結果都包括在一般框架之中，但每個結果都有不同的講述方式，並且每個理論的特殊性質都給出了具體的講述。本書以環上的模作為出發點，基本計算有二模張量積，以及一個模到其他模的全部同態羣。函子和導出函子也是自然而然的進行了講述。目次：環和模；加性函數；衞星；同調；導出函子；u和hom的導出函子；積分域；增廣環；結合代數；補充代數；乘積；有限羣；李代數；擴張；譜序列；譜序列應用；超拓撲 ^[1]  。