| Dedication | p. v |
| Preface | p. vii |
| Table of Contents | p. xiii |
| Notes to the Reader | p. xvii |
| Partial List of Notations | p. xix |
| Introduction to Serre's Conjecture: 1955-1976 | p. 1 |
| Foundations | p. 9 |
| Projective Modules | p. 9 |
| Flat Modules, Faithfully Flat Modules and Finitely Presented Modules | p. 12 |
| Local-Global Methods | p. 18 |
| Stably Free Modules and Hermite Rings | p. 24 |
| Appendix to [section]4 | p. 37 |
| Elementary Transformations | p. 43 |
| The Grothendieck Group K[subscript 0] | p. 49 |
| The Whitehead Group K[subscript 1] | p. 52 |
| Examples for SL[subscript n] (R) [not equal] E[subscript n] (R) | p. 53 |
| Suslin's Normality Theorem | p. 59 |
| Notes on Chapter I | p. 67 |
| The "Classical" Results on Serre's Conjecture | p. 71 |
| The Case of Rank 1 Projectives | p. 71 |
| The Case of One Variable | p. 72 |
| The Case of Noncommutative Base Rings | p. 73 |
| The Graded Case | p. 77 |
| The Stable Case | p. 79 |
| The Case of Two Variables | p. 85 |
| The Case of Big Rank | p. 91 |
| Notes on Chapter II | p. 96 |
| The Basic Calculus of Unimodular Rows | p. 99 |
| Suslin's Elementary Proof of Serre's Conjecture | p. 100 |
| Vaserstein's Elementary Proof of Serre's Conjecture | p. 103 |
| Appendix to [section]2 | p. 106 |
| Suslin's Monic Polynomial Theorem | p. 107 |
| Suslin's n! Theorem | p. 111 |
| Sectionable Sequences | p. 116 |
| Self-Duality of Stably Free Modules | p. 125 |
| A Touch of Suslin Matrices | p. 131 |
| Notes on Chapter III | p. 137 |
| Horrocks' Theorem | p. 139 |
| Localization at Monic Polynomials | p. 139 |
| Statement of Horrocks' Theorem | p. 143 |
| Swan's Proof of Horrocks' Theorem | p. 145 |
| Roberts' Proof of Horrocks' Theorem | p. 151 |
| Nashier-Nichols' Proof of Horrocks' Theorem | p. 153 |
| Murthy-Horrocks Theorem | p. 156 |
| Notes on Chapter IV | p. 160 |
| Quaillen's Methods | p. 161 |
| Quillen's Patching Theorem | p. 161 |
| Affine Horrocks' Theorem and Applications | p. 170 |
| Quillen Induction, and the Bass-Quillen Conjecture | p. 177 |
| Appendix to [section]3 | p. 183 |
| Laurent Polynomial Rings | p. 186 |
| Power Series Rings | p. 191 |
| Notes on Chapter V | p. 200 |
| K[subscript 1]-Analogue of Serre's Conjecture | p. 203 |
| Patching Theorems for GL[subscript n] | p. 203 |
| Patching Theorem for Elementary Group Action | p. 210 |
| Mennicke Symbols | p. 215 |
| Suslin's Stability Theorem | p. 218 |
| K[subscript 1]-Analogue of Horrocks' Theorem | p. 220 |
| Structure Theorem on E[subscript n] (R[t, t superscript -1]) | p. 224 |
| Notes on Chapter VI | p. 232 |
| The Quadratic Analogue of Serre's Conjecture | p. 233 |
| Inner Product Spaces | p. 233 |
| Karoubi's Theorem | p. 239 |
| Harder Theorem: Easier Proof | p. 242 |
| Parimala's Counter-Examples | p. 247 |
| Symplectic Spaces and Self-Duality | p. 255 |
| Notes on Chapter VII | p. 268 |
| References for Chapters I-VII | p. 271 |
| Complete Intersections and Serre's Conjecture | p. 277 |
| New Developments (since 1977) | p. 289 |
| R[t subscript 1, ..., t subscript n] for R Noetherian | p. 290 |
| Projective Modules over Affine Algebras | p. 296 |
| Complete Intersections | p. 303 |
| Monomial Algebras and Discrete Hodge Algebras | p. 314 |
| Unimodular Rows | p. 318 |
| The Bass-Quillen Conjecture | p. 328 |
| R[t subscript 1, ..., t subscript n] for R Non-Noetherian | p. 333 |
| Noncommutative Polynomial Rings | p. 338 |
| K[subscript 1] (and Higher K[subscript n]) Analogues | p. 339 |
| Quadratic Analogues of Serre's Conjecture | p. 345 |
| Quantum Versions of Serre's Conjecture | p. 351 |
| Algorithmic Methods | p. 353 |
| Applications of Serre's Conjecture | p. 355 |
| References for Chapter VIII | p. 363 |
| Index | p. 395 |
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