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By Chambadal L., Ovaert J.-L.

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To define f2, f3, f4 we need another element, y E Q/Z(l) [llp], which we shall choose in the same manner as 1341. We begin by considering Q/Z(l)[l/p] as a subgroup of the multiplicative group of the maximal unramified extension of L. If nL is a prime of L such that then we choose y to satisfy yvd-' = &- SO that y is a primitive (vd - 1)r-th root of unity. Chapter 4. 1. R1(L/K,2) in the tame case Next set Then f4(x) E F;, x Z has an image in 1;d(l) of the form (. :,::c( which has second coordinate equal to 4(1) so that g-i)gd-'@l), and so f 4 is surjective.

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. 5. Then F is a Z[G(L/K)]-module homomorphism since If (ml,. .

For r 2 1, K2r(L(m)) and K2,+1(L(m)) have no ptorsion [60] and so the results of [148] imply that the tame symbol 72 Chapter 3. 2. The higher K-theory invariants 0, (LIK, 2) and the map induced by the inclusion of the field of constants where each have uniquely divisible kernel and cokernel. 17. Proof. 11. 21, the module ~ n d z [ $ $ { ~ )(pm[ l l p ] 6 z ) is a submod- for all i > 0 and Incidentally, for any one-dimensional local field, L, the indecomposable Kgroup K ~ ( L [80] ) differs fiom Kj(L) only by uniquely divisible groups when j 3 by [15].

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Algebra lineaire et tensorielle by Chambadal L., Ovaert J.-L.

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