Contents

Preface ix

Acknowledgements xii

Chapter I. Projective modules and vector bundles 1

§1. Free modules, GLn, and stably free modules 1

§2. Projective modules 8

§3. The Picard group of a commutative ring 20

§4. Topological vector bundles and Chern classes 34

§5. Algebraic vector bundles 49

Chapter II. The Grothendieck group K0 69

§1. The group completion of a monoid 69

§2. K0 of a ring 74

§3. K(X), KO(X), and KU(X) of a topological space 89

§4. Lambda and Adams operations 98

§5. K0 of a symmetric monoidal category 114

§6. K0 of an abelian category 124

§7. K0 of an exact category 140

§8. K0 of schemes and varieties 157

§9. K0 of a Waldhausen category 172

Appendix. Localizing by calculus of fractions 189

Chapter III. K1 and K2 of a ring 197

§1. The Whitehead group K1 of a ring 197

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