Codescent in étale \(K\)-theory and number fields
Let \(F\) be a number field with ring of integers \(o_F\) and \(p\) a fixed odd prime number. The author studies codescent properties for the étale \(K\)-theory groups \(K^{\text{ét}}_{2i- 2} (o_F)\) for \(i\geq 2\). In ``Algebraic and étale \(K\)-theory'', Trans. Am. Math. Soc. 292, 247--280 (1985; Zbl 0581.14012), \textit{W. G. Dwyer} and \textit{E. M. Friedlander} introduced étale \(K\)-theory as a link between algebraic \(K\)-theory and étale cohomology. They showed that there is a surjective map from the \(p\)-primary part \(K_{2i- 2} (o_F) (p)\) of \(K_{2i- 2} (o_F)\) to \(K^{\text{ét}}_{2i- 2} (o_F)\), which is expected to be an isomorphism -- this being the case for \(i=2\). Furthermore they showed that \(K^{\text{ét}}_{2i- 2} (o_F)\) is canonically isomorphic to the étale cohomology group \(H^2_{\text{ét}} (\text{spec } o_F [{1\over p} ], \mathbb{Z}_p (i))\). The main result of the paper is the following: Assume that \(E/F\) is a Galois \(p\)-extension of number fields with Galois group \(G\) and that Leopoldt's conjecture holds in the cyclotomic tower of \(E\). Then \(K^{\text{ét}}_{2i- 2} (o_E)_G\) is isomorphic to \(K^{\text{ét}}_{2i- 2} (o_F)\) if and only if the ramification in \(E/F\) satisfies a certain primitivity condition.
scientific article; zbMATH DE number 784055
scientific article