Introduction to Homological Algebra, 85 (eBook)

Joseph J. Rotman (Autor)

eBook Download: PDF | EPUB

1979 | 1. Auflage
400 Seiten
Elsevier Science (Verlag)
978-0-08-087401-2 (ISBN)

Infos

Introduction to Homological Algebra, 85

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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.

Inhalt

Front Cover 1
An Introduction to Homological Algebra 4
Copyright Page 5
Contents 8
Preface 8
Chapter 1. Introduction 16
Line Integrals and Independence of Path 16
Categories and Functors 18
Tensor Products 22
Singular Homology 32
Chapter 2. Hom and × 38
Modules 39
Sums and Products 43
Exactness 49
Adjoints 51
Direct Limits 54
Inverse Limits 64
Chapter 3. Projectives, Injectives, and Flats 72
Free Modules 72
Projective Modules 77
Injective Modules 80
Watts’ Theorems 90
Flat Modules 99
Purity 109
Localization 112
Chapter 4. Specific Rings 123
Noetherian Rings 123
Semisimple Rings 130
Von Neumann Regular Rings 134
Hereditary and Dedekind Rings 135
Semihereditary and Prüfer Rings 142
Quasi-Frobenius Rings 145
Local Rings and Artinian Rings 148
Polynomial Rings 153
Chapter 5. Extensions of Groups 165
Chapter 6. Homology 181
Homology Functors 181
Derived Functors 193
Chapter 7. Ext 209
Elementary Properties 209
Ext and Extensions 217
Axioms 226
Chapter 8. Tor 235
Elementary Properties 235
Tor and Torsion 239
Universal Coefficient Theorems 242
Chapter 9. Son of Specific Rings 247
Dimensions 248
Hilbert's Syzygy Theorem 257
Serre's Theorem 266
Mixed Identities 271
Commutative Noetherian Local Rings 273
Chapter 10. The Return of Cohomology of Groups 280
Homology Groups 281
Cohomology Groups 295
Computations and Applications 307
Chapter 11. Spectral Sequences 314
Exact Couples and Five-Term Sequences 314
Derived Couples and Spectral Sequences 326
Filtrations and Convergence 330
Bicomplexes 334
Künneth Theorems 351
Grothendieck Spectral Sequences 365
More Groups 370
More Modules 373
References 382
Index 386
Pure and Applied Mathematics 392

Produkt-Details

• Erscheint lt. Verlag: 7.9.1979
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-087401-0 / 0080874010
• ISBN-13: 978-0-08-087401-2 / 9780080874012