Introduction to Homological Algebra, 85

  • ID: 2986213
  • Book
  • 400 Pages
  • Elsevier Science and Technology
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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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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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Rotman, Joseph J.
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