Table of Contents

Preface vii

1 λ-Rings 1

1.1 Symmetric Functions 1

1.2 λ-Rings 5

1.3 Free λ-Rings 14

1.4 The Verification Principle 17

1.5 The Splitting Principle 18

1.6 Exercises 21

2 Universal λ-Rings 23

2.1 The Universal λ-Ring Λ(R) 23

2.2 Categories and Functors 31

2.3 Comonads and Coalgebras 35

2.4 λ-Rings as Λ-Coalgebras 39

2.5 Exercises 42

3 Adams Operations 43

3.1 Adams Operations 44

3.2 γ-Operations 52

3.3 γ-Filtration 57

3.4 ψ-Rings 65

3.5 ψ-Rings as Ψ-Coalgebras 69

3.6 Wilkerson's Theorem 72

3.7 Localization and Completion of λ-Rings 75

3.8 Exercises 78

4 Witt Vectors 81

4.1 Big Witt Vectors 82

4.2 Artin-Hasse Exponential 90

4.3 Characterization of the Functor Λ 94

4.4 W as a Comonad 97

4.5 Exercises 106

5 Binomial Rings 109

5.1 Binomial Rings as λ-Rings 110

5.2 Integer-Valued Polynomials 116

5.3 A Basis for Integer-Valued Polynomials 119

5.4 Binomial Rings as Integer-Valued Polynomials 124

5.5 Knutson's Theorem 126

5.6 Necklace Rings 128

5.7 Exercises 134

6 Filtered λ-Rings, I 135

6.1 Filtered λ-Rings 136

6.2 Power Series Rings 140

6.3 Classification of λ (Z[x]/(x²)) 142

6.4 Classification of λ (Z[x]/(x³)) 146

6.5 Exercises 158

7 Filtered λ-Rings, II 159

7.1 Power Series Rings 160

7.2 Filtered λ-Rings as Coalgebras 165

7.3 Exercises 167

8 Open Problems 169

8.1 Power Series and Polynomial Rings 169

8.2 Universal Ring Γ 170

8.3 Representation Ring 171

8.4 Topological K-Theory 172

8.5 Mislin Genus 173

Hints to Selected Exercises 175

List of Notations 179

Bibliography 183

Index 187