Introduction to Homological Algebra, 85

By

  • Joseph Rotman, University of Illinois, Urbana

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.
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Book information

  • Published: June 1979
  • Imprint: ACADEMIC PRESS
  • ISBN: 978-0-12-599250-3


Table of 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