Codescente en K-théorie étale et corps de nombres

Galois co-descent for étale wild kernels and capitulation

“…-The possibility of the failure of Galois co-descent in Proposition 2.3 was already observed in [2]. The situations where this happens are easily described: First of all we must have i = 1 mod d and L f. We recall that by Lemma 2.1, the natural map…”

“…Results for arbitrary finite Galois p-extensions have been obtained by Assim (cp. [I], [2]) in terms of primitive ramification, however under the assumption that Leopoldt's Conjecture holds for the fields L(apn) for all n. Let S be the finite set of primes of F, consisting of the set Sp and the tamely ramified primes in L/F. We have the following exact sequence: is always surjective and that the kernel of this map is isomorphic to the cokernel of the map where S^ consists of the primes in L above S \ Sp.…”