By Victor P. Snaith
This monograph offers the state-of-the-art within the thought of algebraic K-groups. it truly is of curiosity to a wide selection of graduate and postgraduate scholars in addition to researchers in similar components resembling quantity concept and algebraic geometry. The options offered listed here are largely algebraic or cohomological. all through quantity thought and arithmetic-algebraic geometry one encounters items endowed with a common motion by means of a Galois workforce. particularly 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 necessary group-ring of the Galois staff. A imperative topic, predictable from the Lichtenbaum conjecture, is the review of those invariants when it comes to exact 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 review 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 components relating to algebraic K-theory resembling quantity conception and algebraic geometry. The options offered listed here are mostly algebraic or cohomological. necessities on L-functions and algebraic K-theory are recalled while wanted.
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Extra resources for Algebraic K-groups as Galois modules
Therefore we may choose g E G(L/K) of order d which maps to the Frobenius automorphism in G(EIK). For each positive integer, m, set L(m) = FvdmL so that L(m) = F V d m and there is an extension of the form in which the g's may be chosen to satisfy rm(g) = g. The kernel of r,, G(L(m)/L), is isomorphic to G(FVdm/Fvd)which is cyclic of order m generated by gd. 2, then Lo/K is equal to the limit of the extensions, L(m)/ K. 1. I 7 defined using tt while the lower sequence is that defined using [,b. 5.
However + + + + Det (1 = det + + + since a 2 b2 - c2 - d2 is a unit in the 2-adic integers. Therefore the homomorphism, A, is well defined. 37, that the domain of X is generated by the homomorphism, h, which sends 21 to minus one and the other one-dimensional repreI7 sentations to one. Thus X[h] = -1, which completes the proof. 40 A generator for ~ o r s K (z[Qgb]; 0 Q) Therefore multiplying by determinantal functions of this kind can only change f (v) modulo four. Since the function which is minus one on v and trivial on all the onedimensional representations is a generator for the class-group of Z[Q8]the previous argument shows that TorsKO(Z [Q8];Q) maps onto CC(z[Q8])CB Tors~O (Z [Qgb]; Q) and that the order of TorsKO(Z[Q8]; Q) is at most four, which completes the proof.
We have an isomorphism of Z[G(L/K)]-modules is also an isomorphism. Define w by the pushout diagram ( ( a. 13. where G(L/ W) acts trivially on (QIZ) (r) [lip]. Hence we may form the following push-out diagram, which defines the module 62,~ 2 , - 1(LO). I4 is cohomologically trivial. 16 On (Q/Z)(r)[l/p] Fo acts by multiplication by qd where q = vr, v = ~ O K / T KHence, ~. 13 The Z[G(L/K)]-module, ial. 12 is cohomologically trivwhere, on the right, G(L/W) acts trivially on (Q/Z)(r) [llp] and Proof.