By Victor P. Snaith
This monograph offers the cutting-edge within the thought of algebraic K-groups. it's of curiosity to a large choice of graduate and postgraduate scholars in addition to researchers in similar components similar to quantity idea and algebraic geometry. The ideas awarded listed here are mostly algebraic or cohomological. all through quantity thought and arithmetic-algebraic geometry one encounters gadgets endowed with a ordinary motion through a Galois workforce. specifically this is applicable to algebraic K-groups and ?tale cohomology teams. This quantity is anxious with the development of algebraic invariants from such Galois activities. in general those invariants lie in low-dimensional algebraic K-groups of the vital group-ring of the Galois workforce. A vital subject, predictable from the Lichtenbaum conjecture, is the review of those invariants by way of specific values of the linked L-function at a damaging integer counting on the algebraic K-theory size. furthermore, the "Wiles unit conjecture" is brought and proven to steer either to an assessment of the Galois invariants and to rationalization of the Brumer-Coates-Sinnott conjectures. This publication is of curiosity to a large choice of graduate and postgraduate scholars in addition to researchers in parts with regards to algebraic K-theory similar to quantity conception and algebraic geometry. The concepts offered listed below are largely algebraic or cohomological. necessities on L-functions and algebraic K-theory are recalled while wanted.
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If C,=,,,, the Pl 8 Qo-component and recalling that y = (1 a . . a'-')z2, + + + If i = d we obtain in Pl 8 Qo. Since is an injective Z[G(L/K)]-module homomorphism, we see that al,j = 0 for all 1 5 j 5 d. Also, if ul@u2 E Pl@Qo then (a-l)(ul@u2) = ((a-l)ul)@u2 so that (a - 1)z2,j= 0 for all 1 5 j 5 d. Therefore, for each j, a 2 j = (1 + a . a'-')~;,~ for some a;, E Z[g]/(gd - 1). Hence we have + = tr Now we have PZd-' SO that Notice that, in Z[g]/(gd - 1) In Po 8 Q1 @ PI €3 Qo we define elements This means that ) K3(Fvd)).
22, since a E G(L/W) acts triviallyon the roots of unity. The reason for distinguishing Ind(1) and Ind(2) is that they sit in different 2-extensions, which we shall now describe. -P. Serre (see also [I311 Chapter 7). It is a 2extension of Galois modules and (see ) 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).
Q-lad-1, ad) = C gi 8 q-'ad-i. i=O 68 Chapter 3. 1. Local fundamental classes and K-groups 69 Proof. 16. However, as abelian groups there are isomorphisms of the form and w w @ (@f=lz[l/p]). Furthermore the resulting map on the quotients of these direct sums by the submodules given by the first summands has the form by the formula (b E (Q/Z)(r) [llp], m E Z[l/p]) and is given by If u E Z satisfies uv 1 (modulo t) then 9-'agi(O, 1) = aui( 0 , l ) = (ui&, 1) E (QlZ) (r) [llp]6 Z [l/p], writing the first coordinate additively, as usual.
Algebraic K-groups as Galois modules by Victor P. Snaith