http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
elementary approach to homological algebra
1 online resource (315 pages) : illustrations
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.
1 Modules 1 -- 1.2 Free Modules 4 -- 1.3 Exact Sequences 10 -- 1.4 Homomorphisms 18 -- 1.5 Tensor Product of Modules 21 -- 1.6 Direct and Inverse Limits 28 -- 1.7 Pull Back and Push Out 35 -- 2 Categories and Functors 41 -- 2.1 Categories 41 -- 2.2 Functors 54 -- 2.3 The Functors Hom and Tensor 63 -- 3 Projective and Injective Modules 73 -- 3.3 Baer's Criterion 83 -- 3.4 An Embedding Theorem 88 -- 4 Homology of Complexes 97 -- 4.1 Ker -- Coker Sequence 97 -- 4.2 Connecting Homomorphism -- the General Case 105 -- 4.3 Homotopy 112 -- 5 Derived Functors 117 -- 5.1 Projective Resolutions 117 -- 5.2 Injective Resolutions 122 -- 6 Torsion and Extension Functors 145 -- 6.3 Some Further Properties of Tor[superscript R subscript n] 155 -- 6.4 Tor and Direct Limits 161 -- 7 The Functor Ext[superscript n subscript R] 167 -- 7.1 Ext[superscript 1] and Extensions 167 -- 7.2 Baer Sum of Extensions 178 -- 7.3 Some Further Properties of Ext[superscript n subscript R] 184 -- 8 Hereditary and Semihereditary Rings 189 -- 8.1 Hereditary Rings and Dedekind Domains 189 -- 8.2 Invertible Ideals and Dedekind Rings 196 -- 8.3 Semihereditary and Prufer Rings 200 -- 9 Universal Coefficient Theorem 203 -- 9.1 Universal Coefficient Theorem for Homology 203 -- 9.2 Universal Coefficient Theorem for Cohomology 206 -- 9.3 The Kunneth Formula -- A Special Case 209 -- 10 Dimensions of Modules and Rings 215 -- 10.1 Projectively and Injectively Equivalent Modules 215 -- 10.2 Dimensions of Modules and Rings 220 -- 10.3 Global Dimension of Rings 224 -- 10.4 Global Dimension of Noetherian Rings 227 -- 10.5 Global Dimension of Artin Rings 234 -- 11 Cohomology of Groups 239 -- 11.1 Homology and Cohomology Groups 239 -- 11.3 The Groups H[superscript 0] (G, A) and H[subscript 0](G, A) 245 -- 11.4 The Groups H[superscript 1](G, A) and H[subscript 1](G, A) 246 -- 11.5 Homology and Cohomology of Direct Sums 250 -- 11.6 The Bar Resolution 256 -- 11.7 Second Cohomology Group and Extensions 262 -- 11.8 Some Homomorphisms 267 -- 11.9 Some Exact Sequences 278 -- 12.1 An Exact Sequence 293 -- 12.2 Outer Automorphisms of p-Groups 298 -- 12.3 A Theorem of Magnus 303.
Includes bibliographical references (pages 309-312) and index.
Chapman & Hall/CRC monographs and surveys in pure and applied mathematics