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Galois descent and \(K_ 2\) of number fields. (Descente galoisienne et \(K_ 2\) des corps de nombres.) (French) Zbl 0780.12007
Let \(E/F\) be a finite Galois extension with group \(G\). Let \(f_{E/P}: K_ 2(F)\to K_ 2(E)^ G\) denote the canonical map. The main results are natural isomorphisms of the form \[ \text{Ker} (f_{E/P}) \cong H^ 1(G; K_ 3^{\text{ind}}(E_ 0)) \qquad \text{and} \qquad \text{Coker} (f_{E/P}) \cong H^ 2(G; K_ 3^{\text{ind}}(E_ 0)) \] where \(E_ 0\) denotes the field of constants and \(K_ 3^{\text{ind}}(E)\) denotes the indecomposable \(K_ 3\)-group of \(E\). The proof uses the hypercohomology of the \(\Gamma(2)\)-complex of [S. Lichtenbaum, Invent. Math. 88, 183-215 (1987; Zbl 0615.14004)]. Several applications are given to function fields of varieties and to number fields. In the latter situation several of the author’s results were obtained independently by J. Brinkhuis [Lect. Notes Math. 1046, 13-28 (1984; Zbl 0525.18009)] and M. Kolster [J. Pure Appl. Algebra 74, 257-273 (1991; Zbl 0752.19003)].

MSC:
12G05 Galois cohomology
11R70 \(K\)-theory of global fields
19C99 Steinberg groups and \(K_2\)
13D25 Complexes (MSC2000)
19C30 \(K_2\) and the Brauer group
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