Approximation in complex and real Lipschitz algebras by Honary T.G.

By Honary T.G.

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Kp[G(L/K)]ap 2 We shall abbreviate OK, [G(LQ/Kp)]ap to XQ and set X = OK [G(L/K)]a. This is to be interpreted as meaning that X is the intersection of L with the product of its P-completions, Xp, where both are considered as subgroups of the adkles. Hence X is a locally free OKIG(L/K)]-module whose P-completion is where p is the residue characteristic. There are many to choose from and several are described in ([I341 55), which may be derived (sometimes a little work is needed) from ([ill, [I317 [I617 [461, [511, [551, [661, 1671, [731, [go], [991, [loo], [1411, [143], [162]).

For p-adic fields when p is odd the conjecture was proved recently [74]. When L I K is a Galois extension of local fields of characteristic p the Lichtenbaum-Quillen Conjecture is now known to be true by [60], which shows that the K-theory of L has no p-torsion, combined with the results of [148]. These advances made it possible to construct the local fundamental classes associated to the higher K-groups of local fields without any assumptions (for further details, see [134], [135], [136] and [137]).

H(ad)). 66 Chapter 3. Higher K-theory of Local Fields Consider the Z [G(L/K)]-submodule, (Q/z) (r) [lM contained in @f=l K2r- (LO). Since L/W is totally ramified, G(L/W) acts trivially on this submodule. 1. 7. Since (Q/Z)(r)[l/p] c U we have ( Q / ~ ) ( r ) [ l / p ]C W. Also G(L/W) acts component-by-component on W so that W (@:=;'K~~-I (Lo)) @U as a Z[G(L/W)]-module. 4, the inclusion induces a cohomology isomorphism of the following form " ) )(qa2, gas,. . ,qad, ql-dal) g(a1,. . ,ad) = (F(a2), .

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