《同調代數》是由世界圖書出版公司在2011年出版的圖書，作者是法國的嘉當。
基本介紹
- 中文名：同調代數
- 作者：嘉當
- 出版社：世界圖書出版公司
- 出版時間：2011年7月
- 定價：59 元
- 開本：24 開
- ISBN：9787510037474
目錄,內容簡介,
目錄
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
內容簡介
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群、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群。函子和導出函子也是自然而然的進行了講述。目次：環和模；加性函式；衛星；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；有限群；李代數；擴張；譜序列；譜序列套用；超拓撲。
