## 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

### Citations:

Zbl 0615.14004; Zbl 0525.18009; Zbl 0752.19003
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