An Elementary Approach to Homological Algebra (Chapman & Hall/Crc Monographs and Surveys in Pure and Applied Mathematics.)

homological Algebra Was Developed As An Area Of Study Almost 50 Years Ago, And Many Books On The Subject Exist. However, Few, If Any, Of These Books Are Written At A Level Appropriate For Students Approaching The Subject For The First Time. an Elementary Approach To Homological Algebra Fills That Void. Designed To Meet The Needs Of Beginning Graduate Students, It Presents The Material In A Clear, Easy-to-understand Manner. Complete, Detailed Proofs Make The Material Easy To Follow, Numerous Worked Examples Help Readers Understand The Concepts, And An Abundance Of Exercises Test And Solidify Their Understanding. often Perceived As Dry And Abstract, Homological Algebra Nonetheless Has Important Applications In Many Important Areas. The Author Highlights Some Of These, Particularly Several Related To Group Theoretic Problems, In The Concluding Chapter. Beyond Making Classical Homological Algebra Accessible To Students, The Author's Level Of Detail, While Not Exhaustive, Also Makes The Book Useful For Self-study And As A Reference For Researchers. Title page Preface 1 Modules 1.1 Modules 1.2 Free Modules 1.3 Exact Sequences 1.4 Homomorphisms 1.5 Tensor Product of Modules 1.6 Direct and Inverse Limits 1.7 Pull Back and Push Out 2 Categories and Functors 2.1 Categories 2.2 Functors 2.3 The Functors Hom and Tensor 3 Projective and Injective Modules 3.1 Projective Modules 3.2 Injective Modules 3.3 Baer's Criterion 3.4 An Embedding Theorem 4 Homology of Complexes 4.1 Ker-Coker Sequence 4.2 Connecting Homomorphism - the General Case 4.3 Homotopy 5 Derived Functors 5.1 Projective Resolutions 5.2 Injective Resolutions 5.3 Derived Functors 6 Torsion and Extension Functors 6.1 Derived Functors-Revisited 6.2 Torsion and Extension Functors 6.3 Some Further Properties of Tor^R_n 6.4 Tor and Direct Limits 7 The Functor Ext_R^n 7.1 Ext^l and Extensions 7.2 Baer Sum of Extensions 7.3 Some Further Properties of Ext_R^n 8 Hereditary and Semihereditary Rings 8.1 Hereditary Rings and Dedekind Domains 8.2 Invertible Ideals and Dedekind Rings 8.3 Semihereditary and Prüfer Rings 9 Universal Coefficient Theorem 9.1 Universal Coefficient Theorem for Homology 9.2 Universal Coefficient Theorem for Cohomology 9.3 The Künneth Formula - a Special Case 10 Dimensions of Modules and Rings 10.1 Projectively and Injectively Equivalent Modules 10.2 Dimensions of Modules and Rings 10.3 Global Dimension of Rings 10.4 Global Dimension of Noetherian Rings 10.5 Global Dimension of Artin Rings 11 Cohomology of Groups 11.1 Homology and Cohomology Groups 11.2 Some Examples 11.3 The Groups H0(G,A) and H0(G,A) 11.4 The Groups H^l(G,A) and H_l(G,A) 11.5 Homology and Cohomology of Direct Sums 11.6 The Bar Resolution 11.7 Second Cohomology Group and Extensions 11.8 Some Homomorphisms 11.9 Some Exact Sequences 12 Some Applications 12.1 An Exact Sequence 12.2 Outer Automorphisms of p-groups 12.3 A Theorem of Magnus Bibliography Homological algebra was developed as an area of study almost 50 years ago, and many books on the subject exist. However, few, if any, of these books are written at a level appropriate for students approaching the subject for the first time. An Elementary Approach to Homological Algebra fills that void. Designed to meet the needs of beginning graduate students, it presents the material in a clear, easy-to-understand manner. Complete, detailed proofs make the material easy to follow, numerous worked examples help readers understand the concepts, and an abundance of exercises test and solidify their understanding. Often perceived as dry and abstract, homological algebra nonetheless has important applications in many important areas. The author highlights some of these, particularly several related to group theoretic problems, in the concluding chapter. Beyond making classical homological algebra accessible to students, the author's level of detail, while not exhaustive, also makes the book useful for self-study and as a reference for researchers. Often perceived as dry and abstract, homological algebra nonetheless has important applications in a number of important areas, including ring theory, group theory, representation theory, and algebraic topology and geometry. Although the area of study developed almost 50 years ago, a textbook at this level has never before been available. An Elementary Approach to Homological Algebra fills that void. Designed to meet the needs of beginning graduate students, the author presents the material in a clear, easy-to-understand manner with many examples and exercises. The book's level of detail, while not exhaustive, also makes it useful for self-study and as a reference for researchers. Often perceived as dry and abstract, homological algebra nonetheless has important applications in a number of important areas, including ring theory, group theory, representation theory, and algebraic topology and geometry. Although the area of study developed almost 50 years ago, a textbook at this level has never become available. An Elementary Approach to Homological Algebra fills that void. Designed to meet the needs of beginning graduate students, it presents the material in a clear, easy-to-understand manner with many examples and exercises. The book's level of detail, while not exhaust Front Çover; Contents; Chapter 1: Modules; 1.1 Modules; 1.2 Free Modules; 1.3 Exact Sequences; 1.4 Homomorphisms; 1.5 Tensor Product of Modules; 1.6 Direct and Inverse Limits; 1.7 Pull Back and Push Out; Chapter 2: Categories and Functors; 2.1 Categories; 2.2 Functors; 2.3 The Functors Hom and Tensor; Chapter 3: Projective and Injective Modules; 3.1 Projective Modules; 3.2 Injective Modules; 3.3 Baer's Criterion; 3.4 An Embedding Theorem; Chapter 4: Homology of Complexes; 4.1 Ker - Coker Sequence; 4.2 Connecting Homomorphism - the General Case; 4.3 Homotopy; Chapter 5: Derived Functors 5.1 Projective Resolutions5.2 Injective Resolutions; 5.3 Derived Functors; Chapter 6: Torsion and Extension Functors; 6.1 Derived Functors - Revisited; 6.2 Torsion and Extension Functors; 6.3 Some Further Properties of Tor[sup(R)][sub(n)]; 6.4 Tor and Direct Limits; Chapter 7: The Functor Ext[sup(n)][sub(R)]; 7.1 Ext[sup(1)] and Extensions; 7.2 Baer Sum of Extensions; 7.3 Some Further Properties of Ext[sup(n)][sub(R)]; Chapter 8: Hereditary and Semihereditary Rings; 8.1 Hereditary Rings and Dedekind Domains; 8.2 Invertible Ideals and Dedekind Rings; 8.3 Semihereditary and Prüfer Rings Chapter 9: Universal Coefficient Theorem9.1 Universal Coefficient Theorem for Homology; 9.2 Universal Coefficient Theorem for Cohomology; 9.3 The Künneth Formula - A Special Case; Chapter 10: Dimensions of Modules and Rings; 10.1 Projectively and Injectively Equivalent Modules; 10.2 Dimensions of Modules and Rings; 10.3 Global Dimension of Rings; 10.4 Global Dimension of Noetherian Rings; 10.5 Global Dimension of Artin Rings; Chapter 11: Cohomology of Groups; 11.1 Homology and Cohomology Groups; 11.2 Some Examples; 11.3 The Groups H[sup(0)] (G, A) and H[sub(0)](G, A) 11.4 The Groups H[sup(1)] (G, A) and H[sub(1)](G, A)11.5 Homology and Cohomology of Direct Sums; 11.6 The Bar Resolution; 11.7 Second Cohomology Group and Extensions; 11.8 Some Homomorphisms; 11.9 Some Exact Sequences; Chapter 12: Some Applications; 12.1 An Exact Sequence; 12.2 Outer Automorphisms of p-Groups; 12.3 A Theorem of Magnus; Bibliography; Index; A; B; C; D; E; F; G; H; I; K; L; M; N; O; P; Q; R; S; T; U; Z "Designed to meet the needs of beginning graduate students, it presents the material in a clear, easy-to-understand manner. Complete, detailed proofs make the material easy to follow, numerous worked examples help readers understand the concepts, and an abundance of exercises test and solidify their understanding."--Jacket This chapter is preparatory in nature and we give some results on modules.