1st Edition

An Elementary Approach to Homological Algebra

By L.R. Vermani Copyright 2003

    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.

    MODULES
    Modules
    Free Modules
    Exact Sequences
    Homomorphisms
    Tensor Product of Modules
    Direct and Inverse Limits
    Pull Back and Push Out
    CATEGORIES AND FUNCTORS
    Categories
    Functors
    The Functors Hom and Tensor
    PROJECTIVE AND INJECTIVE MODULES
    Projective Modules
    Injective Modules
    Baer's Criterion
    An Embedding Theorem
    HOMOLOGY OF COMPLEXES
    Ker-Coker Sequence
    Connecting Homomorphism: The General Case
    Homotopy
    DERIVED FUNCTORS
    Projective Resolutions
    Injective Resolutions
    Derived Functors
    TORSION AND EXTENSION FUNCTORS
    Derived Functors--Revisited
    Torsion and Extension Functors
    Some Further Properties of Torsion Functors
    Tor and Direct Limits
    THE FUNCTOR ExtnR
    Ext1 and Extensions
    Baer Sum of Extensions
    Some Further Properties of Extension Functors
    HEREDITARY AND SEMIHEREDITARY RINGS
    Hereditary Rings and Dedekind Domains
    Invertible Ideals and Dedekind Rings
    Semihereditary and Prufer Rings
    UNIVERSAL COEFFICIENT THEOREM
    Universal Coefficient Theorem for Homology
    Universal Coefficient Theorem for Cohomology
    The Kunneth Formula: A Special Case
    DIMENSIONS OF MODULES AND RINGS
    Projectively and Injectively Equivalent Modules
    Dimensions of Modules and Rings
    Global Dimension of Rings
    Global Dimension of Noetherian Rings
    Global Dimension of Artin Rings
    COHOMOLOGY OF GROUPS
    Homology and Cohomology Groups
    Some Examples
    The Groups H0(G,A) and H0(G,A)
    The Groups H1(G,A) and H1(G,A)
    Homology and Cohomology of Direct Sums
    The Bar Resolution
    Second Cohomology Group and Extensions
    Some Homomorphisms
    Some Exact Sequences
    SOME APPLICATIONS
    An Exact Sequence
    Outer Automorphisms of p-Groups
    A Theorem of Magnus
    BIBLIOGRAPHY
    INDEX

    Biography

    L.R. Vermani

    "The book is a self-contained one and there are … complete proofs for all the results. In our opinion, it is a highly recommended introductory book in Homological Algebra for everyone interested in this subject."
    - Zentralblatt MATH, 1045

    "… [T]he author on the one hand has included the underlying basics, model and category theory, which are developed from scratch. On the other hand the elementary notions and results of homological algebra are treated in great detail and often their importance within that theory as well as in applications is shown. … All in all the author has managed … admirably to reach the goal he has set [for] himself."
    - Monatshefte fur Mathematik


    "When I next teach an introductory course on homological algebra at my institution, I will certainly consider adopting this book as the textbook for the course.
    -Mathematical Reviews