Overview

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.

Full Product Details

Author:   Victor P. Snaith
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2002 ed.
Volume:   206
Dimensions:   Width: 15.50cm , Height: 2.00cm , Length: 23.50cm
Weight:   0.680kg
ISBN:  

9783764367176

ISBN 10:   3764367172
Pages:   309
Publication Date:   01 March 2002
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

1 Galois Actions and L-values.- 1.1 Analytic prerequisites.- 1.2 The Lichtenbaum conjecture.- 1.3 Examples of Galois structure invariants.- 2 K-groups and Class-groups.- 2.1 Low-dimensional algebraic K-theory.- 2.2 Perfect complexes.- 2.3 Nearly perfect complexes.- 2.4 Higher-dimensional algebraic K-theory.- 2.5 Describing the class-group by representations.- 3 Higher K-theory of Local Fields.- 3.1 Local fundamental classes and K-groups.- 3.2 The higher K-theory invariants ?s(L/K,2).- 3.3 Two-dimensional thoughts.- 4 Positive Characteristic.- 4.1 ?1(L/K,2) in the tame case.- 4.2 $$ Ext_{Z[G(L/K)]}^2(F_{{v^d}}^*,F_{{v^{2d}}}^*) $$.- 4.3 Connections with motivic complexes.- 5 Higher K-theory of Algebraic Integers.- 5.1 Positive étale cohomology.- 5.2 The invariant ?n(N/K,3).- 5.3 A closer look at ?1(L/K,3).- 5.4 Comparing the two definitions.- 5.5 Some calculations.- 5.6 Lifted Galois invariants.- 6 The Wiles unit.- 6.1 The Iwasawa polynomial.- 6.2 p-adic L-functions.- 6.3 Determinants and the Wiles unit.- 6.4 Modular forms with coefficients in ?[G].- 7 Annihilators.- 7.1K0(Z[G], Q) and annihilator relations.- 7.2 Conjectures of Brumer, Coates and Sinnott.- 7.3 The radical of the Stickelberger ideal.

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