An Elementary Approach to Homological Algebra

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.

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.

MODULESModulesFree ModulesExact SequencesHomomorphismsTensor Product of ModulesDirect and Inverse LimitsPull Back and Push OutCATEGORIES AND FUNCTORSCategoriesFunctorsThe Functors Hom and TensorPROJECTIVE AND INJECTIVE MODULESProjective ModulesInjective ModulesBaer's CriterionAn Embedding TheoremHOMOLOGY OF COMPLEXESKer-Coker SequenceConnecting Homomorphism: The General CaseHomotopyDERIVED FUNCTORSProjective ResolutionsInjective ResolutionsDerived FunctorsTORSION AND EXTENSION FUNCTORSDerived Functors--RevisitedTorsion and Extension FunctorsSome Further Properties of Torsion FunctorsTor and Direct LimitsTHE FUNCTOR ExtnRExt1 and ExtensionsBaer Sum of ExtensionsSome Further Properties of Extension FunctorsHEREDITARY AND SEMIHEREDITARY RINGSHereditary Rings and Dedekind DomainsInvertible Ideals and Dedekind RingsSemihereditary and Prufer RingsUNIVERSAL COEFFICIENT THEOREMUniversal Coefficient Theorem for HomologyUniversal Coefficient Theorem for CohomologyThe Kunneth Formula: A Special CaseDIMENSIONS OF MODULES AND RINGSProjectively and Injectively Equivalent ModulesDimensions of Modules and RingsGlobal Dimension of RingsGlobal Dimension of Noetherian RingsGlobal Dimension of Artin RingsCOHOMOLOGY OF GROUPSHomology and Cohomology GroupsSome ExamplesThe Groups H0(G,A) and H0(G,A)The Groups H1(G,A) and H1(G,A)Homology and Cohomology of Direct SumsThe Bar ResolutionSecond Cohomology Group and ExtensionsSome HomomorphismsSome Exact SequencesSOME APPLICATIONSAn Exact SequenceOuter Automorphisms of p-GroupsA Theorem of MagnusBIBLIOGRAPHYINDEX