Algebraic K-Groups as Galois Modules

ISBN-13: 9783764367176 / Angielski / Twarda / 2002 / 309 str.

Snaith; Victor P. Snaith; V. P. Snaith

Opis

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.

Spis treści

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 $$ Ext_{Z[G(L/K)]}^2(F_{{v^d}}^*,F_{{v^}}^*) $$.- 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.