Introduction to Homological Algebra, 85

Introduction to Homological Algebra, 85, 1st Edition

Introduction to Homological Algebra, 85, 1st Edition,Joseph Rotman,ISBN9780125992503


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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.

Joseph Rotman

Affiliations and Expertise

University of Illinois, Urbana

Introduction to Homological Algebra, 85, 1st Edition

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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