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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Line Integrals and Independence of Path
Categories and Functors
2. Hom and ?
Sums and Products
3. Projectives, Injectives, and Flats
4. Specific Rings
Von Neumann Regular Rings
Hereditary and Dedekind Rings
Semihereditary and Prüfer Rings
Local Rings and Artinian Rings
5. Extensions of Groups
Ext and Extensions
Tor and Torsion
Universal Coefficient Theorems
9. Son of Specific Rings
Hilbert's Syzygy Theorem
Commutative Noetherian Local Rings
10. The Return of Cohomology of Groups
Computations and Applications
11. Spectral Sequences
Exact Couples and Five-Term Sequences
Derived Couples and Spectral Sequences
Filtrations and Convergence
Grothendieck Spectral Sequences