同調(diào)代數(shù)

內(nèi)容概要

　　本書(shū)是數(shù)學(xué)發(fā)展史上的一個(gè)里程碑，在很長(zhǎng)一段時(shí)間，這是唯一本講述拓?fù)浯鷶?shù)的教程。這本書(shū)堪稱是一部同調(diào)代數(shù)經(jīng)典，1956年初版，至今已有七次重印出版。這本書(shū)曾在純代數(shù)領(lǐng)域引起過(guò)不小的轟動(dòng)，作者企圖將這個(gè)領(lǐng)域統(tǒng)一起來(lái)，并且為這個(gè)領(lǐng)域構(gòu)建一個(gè)完整的框架。書(shū)中講述的同調(diào)理論包含了群、李代數(shù)和結(jié)合代數(shù)上同調(diào)結(jié)構(gòu)，大量的結(jié)果都包括在一般框架之中，但每個(gè)結(jié)果都有不同的講述方式，并且每個(gè)理論的特殊性質(zhì)都給出了具體的講述。本書(shū)以環(huán)上的模作為出發(fā)點(diǎn)，基本計(jì)算有二模張量積，以及一個(gè)模到其他模的全部同態(tài)群。函子和導(dǎo)出函子也是自然而然的進(jìn)行了講述。目次：環(huán)和模；加性函數(shù)；衛(wèi)星；同調(diào)；導(dǎo)出函子；u和hom的導(dǎo)出函子；積分域；增廣環(huán)；結(jié)合代數(shù)；補(bǔ)充代數(shù)；乘積；有限群；李代數(shù)；擴(kuò)張；譜序列；譜序列應(yīng)用；超拓?fù)洹?br />　　讀者報(bào)對(duì)象：數(shù)學(xué)、代數(shù)拓?fù)鋵I(yè)的的學(xué)生，教師及相關(guān)專業(yè)的讀者。

作者簡(jiǎn)介

作者：(法國(guó))嘉當(dāng) (Henri Cartan)

書(shū)籍目錄

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

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