Algebraic K-Groups as Galois Modules

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This monograph presents the state of the art in the theory of algebraic K-groups. It is aimed at a wide variety of graduate and postgraduate students as well as researchers in related areas such as number theory and algebraic geometry. The techniques presented here are principally algebraic or cohomological. Prerequisites on L-functions and algebraic K-theory are recalled when needed. Throughout number theory and arithmetic-algebraic geometry one encounters objects endowed with a natural action by a Galois group. In particular this applies to algebraic K-groups and etale cohomology groups. This volume is concerned with the construction of algebraic invariants from such Galois actions. Typically these invariants lie in low-dimensional algebraic K-groups of the integral group-ring of the Galois group. A central theme, predictable from the Lichtenbaum conjecture, is the evaluation of these invariants in terms of special values of the associated L-function at a negative integer depending on the algebraic K-theory dimension. In addition, the Wiles unit conjecture is introduced and shown to lead both to an evaluation of the Galois invariants and to explanation of the Brumer-Coates-Sinnott conjectures.