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By Victor P. Snaith

This monograph provides the state-of-the-art within the thought of algebraic K-groups. it's of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in similar parts equivalent to quantity idea and algebraic geometry. The ideas provided listed below are largely algebraic or cohomological. all through quantity thought and arithmetic-algebraic geometry one encounters gadgets endowed with a average motion by way of a Galois staff. particularly this is applicable to algebraic K-groups and ?tale cohomology teams. This quantity is worried with the development of algebraic invariants from such Galois activities. mostly those invariants lie in low-dimensional algebraic K-groups of the necessary group-ring of the Galois crew. A crucial subject, predictable from the Lichtenbaum conjecture, is the review of those invariants when it comes to designated values of the linked L-function at a detrimental integer counting on the algebraic K-theory measurement. furthermore, the "Wiles unit conjecture" is brought and proven to guide either to an overview of the Galois invariants and to clarification of the Brumer-Coates-Sinnott conjectures. This publication is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in parts concerning algebraic K-theory akin to quantity concept and algebraic geometry. The recommendations offered listed here are mostly algebraic or cohomological. necessities on L-functions and algebraic K-theory are recalled whilst wanted.

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Algebraic K-groups as Galois modules

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Example text

X f : ejF2j8 Q 2 @ F2j respect to a Q[G]-basis. It will suffice to show for each representation, X, that d e t ( x ( ~ )= ) 1. Now the change in q2k-1 affects only the part of the composition for X, X f written out above and then only through the change in $2k. In other words, the only part of the composition to change is F 2 k €3 Q (B2k-' @ Z2k) @ Q and the difference in this part is the map, W2k H (0, (q2k-' - ~ & ~ - ~ ) d 2 k ( wNOW ~~)). qik-l)d2k(~2k)) to X-' and (XI)-', each map (0, ( ~ 2 k ~ the inverses, both (r/2k-1 - T&k-l)d2k(~2k) E Z2k Q C F 2 k @ Q.

Serre (see also [I311 Chapter 7). It is a 2extension of Galois modules and (see [132]) is filtered by the usual filtration of the multiplicative group of a local field so as to remain exact at each level. In particular, in the tame case, we may truncate Serre's fundamental class at level one to give a 2-extension of Z[G(L/K)]-modules of the form where ~ ( ga' (7, m)) = g'-l @ (av,m). 22. In this case, recalling the isomorphism of Galois modules G(L/K) = (a, g I gd = ac, a' = 1, gag-1 = a") where v = IKI, the order of the residue field, K,of K .

Now define q([M]) by the formula Chapter 3 Higher K-theory of Local Fields In this chapter we shall examine invariants of the Galois module structure on the higher algebraic K-groups of local fields. 8 (i) as the Euler characteristic of suit able 2-extensions, ) ~ ( K ~for~ ( L ) called the local fundamental classes lying in E X ~ ~ ~ ~ ( ~ / ~K2rcl(L)) r 2 1, where L/K is a Galois extension with group G(L/K). 8(i) to the canonical corresponding element in E X ~ ~ [ ~ ( ~ ~ ~ )K2r+1 ~ ( K(L)/A) ~ , - (where L ) , A is a cohomologically trivial Z [G(L/K)]-submodule chosen so that K2T+1(L)/A is finitely generated.

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