Introduction | p. 1 |
Preliminaries | p. 1 |
Basic Notions from Algebra | p. 1 |
Basic Notions from Analysis | p. 8 |
The K[subscript 0]-group of a Ring | p. 15 |
Traces and the K[subscript 0]-group | p. 23 |
The Idempotent Conjectures | p. 30 |
The Hattori-Stallings Rank on K[subscript 0](kG) | p. 30 |
Idempotents in CG | p. 34 |
Some First Examples of Groups that Satisfy the Idempotent Conjecture | p. 37 |
Exercises | p. 42 |
Motivating Examples | p. 49 |
The Case of Abelian Groups | p. 49 |
The Geometric Rank Function | p. 50 |
K-theory and the Geometric Rank | p. 52 |
The Connectedness of Spec kG | p. 59 |
The Case of Finite Groups | p. 61 |
The Transfer Homomorphism | p. 61 |
Subgroups of Finite Index | p. 63 |
Swan's Theorem | p. 65 |
Exercises | p. 68 |
Reduction to Positive Characteristic | p. 73 |
The Rationality of the Canonical Trace | p. 73 |
Coefficient Fields of Positive Characteristic | p. 74 |
Lifting to the Field of Algebraic Numbers | p. 77 |
The Kaplansky Positivity Theorem | p. 80 |
Idempotent Matrices with Entries in the Complex Group Algebra | p. 85 |
The Support of the Hattori-Stallings Rank | p. 87 |
Iterates of the Frobenius Operator | p. 87 |
The Main Results | p. 91 |
An Application: the Case of Solvable Groups | p. 100 |
Exercises | p. 106 |
A Homological Approach | p. 111 |
Cyclic Homology of Algebras | p. 111 |
Basic Definitions and Results | p. 112 |
The Relation to K-theory | p. 125 |
The Cyclic Homology of Group Algebras | p. 129 |
The Nilpotency of Connes' Operator | p. 145 |
Idempotent Conjectures and the Nilpotency of S | p. 145 |
Closure Properties | p. 149 |
Exercises | p. 155 |
Completions of CG | p. 159 |
The Integrality of the Trace Conjecture | p. 159 |
Formulation of the Conjecture | p. 160 |
The Case of an Abelian Group | p. 161 |
The Case of a Free Group | p. 170 |
Induced Modules over NG | p. 179 |
The Center-Valued Trace on NG | p. 180 |
Matrices with Entries in NG | p. 195 |
Exercises | p. 202 |
Tools from Commutative Algebra | p. 207 |
Localization and Local Rings | p. 207 |
Integral Dependence | p. 213 |
Noether Normalization | p. 217 |
The Krull Intersection Theorem | p. 222 |
Exercises | p. 225 |
Discrete Ring-Valued Integrals | p. 227 |
Discrete Group-Valued Integrals | p. 227 |
Idempotent-Valued Premeasures | p. 230 |
Exercises | p. 232 |
Frobenius' Density Theorem | p. 235 |
The Density Theorem | p. 235 |
Exercises | p. 238 |
Homological Techniques | p. 239 |
Complexes and Homology | p. 239 |
Chain Complexes | p. 239 |
Double Complexes | p. 240 |
Tor and Ext | p. 242 |
Group Homology and Cohomology | p. 244 |
Basic Definitions | p. 244 |
H[superscript 2] and Extensions | p. 247 |
Products | p. 249 |
Duality | p. 253 |
The (co-)homology of an Extension | p. 255 |
Exercises | p. 257 |
Comparison of Projections | p. 263 |
Equivalence and Weak Ordering | p. 263 |
Exercises | p. 269 |
References | p. 271 |
Index | p. 275 |
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