Algebraic K-Groups as Galois Modules: 206 (Progress in Mathematics, 206) - Hardcover

Snaith, Victor P.

Hardcover

ISBN 10: 3764367172 ISBN 13: 9783764367176

Publisher: Birkhäuser, 2002

Synopsis

This volume began as the last part of a one-term graduate course given at the Fields Institute for Research in the Mathematical Sciences in the Autumn of 1993. The course was one of four associated with the 1993-94 Fields Institute programme, which I helped to organise, entitled "Artin L-functions". Published as [132]' the final chapter of the course introduced a manner in which to construct class-group­ valued invariants from Galois actions on the algebraic K-groups, in dimensions two and three, of number rings. These invariants were inspired by the analogous Chin­ burg invariants of [34], which correspond to dimensions zero and one. The classical Chinburg invariants measure the Galois structure of classical objects such as units in rings of algebraic integers. However, at the "Galois Module Structure" workshop in February 1994, discussions about my invariant (0,1 (L/ K, 3) in the notation of Chapter 5) after my lecture revealed that a number of other higher-dimensional co­ homological and motivic invariants of a similar nature were beginning to surface in the work of several authors. Encouraged by this trend and convinced that K-theory is the archetypical motivic cohomology theory, I gratefully took the opportunity of collaboration on computing and generalizing these K-theoretic invariants. These generalizations took several forms - local and global, for example - as I followed part of number theory and the prevalent trends in the "Galois Module Structure" arithmetic geometry.

About this title

Synopsis

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.

Edition details

• Publisher: Birkhäuser
• Publication date: 2002
• Language: English
• ISBN 10: 3764367172
• ISBN 13: 9783764367176
• Binding: Hardcover
• Number of pages: 319