Introduction to Homological Algebra, 85 book cover

Introduction to Homological Algebra, 85

An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and often, this involves yet another language: spectral sequences. Homological algebra is an accessible subject to those who wish to learn it, and this book is the author’s attempt to make it lovable. This book comprises 11 chapters, with an introductory chapter that focuses on line integrals and independence of path, categories and functors, tensor products, and singular homology. Succeeding chapters discuss Hom and Ⓧ; projectives, injectives, and flats; specific rings; extensions of groups; homology; Ext; Tor; son of specific rings; the return of cohomology of groups; and spectral sequences, such as bicomplexes, Kunneth Theorems, and Grothendieck Spectral Sequences. This book will be of interest to practitioners in the field of pure and applied mathematics.

Hardbound, 400 Pages

Published: June 1979

Imprint: Academic Press

ISBN: 978-0-12-599250-3

Contents


  • Preface

    Contents

    1. Introduction

    Line Integrals and Independence of Path

    Categories and Functors

    Tensor Products

    Singular Homology

    2. Hom and Ⓧ

    Modules

    Sums and Products

    Exactness

    Adjoints

    Direct Limits

    Inverse Limits

    3. Projectives, Injectives, and Flats

    Free Modules

    Projective Modules

    Injective Modules

    Watts’ Theorems

    Flat Modules

    Purity

    Localization

    4. Specific Rings

    Noetherian Rings

    Semisimple Rings

    Von Neumann Regular Rings

    Hereditary and Dedekind Rings

    Semihereditary and Prüfer Rings

    Quasi-Frobenius Rings

    Local Rings and Artinian Rings

    Polynomial Rings

    5. Extensions of Groups

    6. Homology

    Homology Functors

    Derived Functors

    7. Ext

    Elementary Properties

    Ext and Extensions

    Axioms

    8. Tor

    Elementary Properties

    Tor and Torsion

    Universal Coefficient Theorems

    9. Son of Specific Rings

    Dimensions

    Hilbert's Syzygy Theorem

    Serre's Theorem

    Mixed Identities

    Commutative Noetherian Local Rings

    10. The Return of Cohomology of Groups

    Homology Groups

    Cohomology Groups

    Computations and Applications

    11. Spectral Sequences

    Exact Couples and Five-Term Sequences

    Derived Couples and Spectral Sequences

    Filtrations and Convergence

    Bicomplexes

    Künneth Theorems

    Grothendieck Spectral Sequences

    More Groups

    More Modules

    References

    Index


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