For a number field \(F,\) Moore’s reciprocity uniqueness theorem is summarized in an exact sequence \(K_2 (F)\rightarrow \bigoplus_{v\text{ non complex}} \mu(F_v) \rightarrow \mu (F) \rightarrow 0.\) This has been generalized by several authors to the \(\ell\)-part of higher even \(K\)-groups as follows: if \(\ell\) is an odd prime, or \(\ell = 2\) and \(F\) is totally imaginary, there is an exact sequence \((i \geq 1)\):
\[ K_{2i} (F) \{ \ell \} \rightarrow \bigoplus_{v \text{ non complex}} H^0(F_v, {\mathbb Q}_\ell/{\mathbb Z}_\ell (i)) \rightarrow H^0 (F, {\mathbb Q}_\ell/{\mathbb Z}_\ell (i)) \rightarrow 0 \]
[see, e.g., B. Kahn, \(K\)-Theory 30, 211–241 (2003; Zbl 1053.19001)]. Here, for \(\ell = 2,\) the authors show the existence of a complex
\[ K_{2i} (F) \{ 2 \} \rightarrow \bigoplus_{v\text{ non complex}} \widehat H^0(F_v, {\mathbb Q}_2/{\mathbb Z}_2 (i)) \rightarrow H^0 (F, {\mathbb Q}_2/{\mathbb Z}_2 (i)) \rightarrow 0, \]
which is exact for \(i \equiv 0, 1, 2\) (mod 4) and, for \(i \equiv 3\pmod 4\), then the homology group at the second term is isomorphic to \(({\mathbb Z}/2 {\mathbb Z})^{r_1},\) \(r_1\) being the number of real embeddings of \(F.\) The main step is the construction of the leftmost map. This is done by successively applying:

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a split inverse \(K_{2i} (F) \{ 2 \} \to K_{2i+1} (F, {\mathbb Q}_2 /{\mathbb Z}_2)\) of the natural isomorphism
\[ K_{2i+1} (F, {\mathbb Q}_2/{\mathbb Z}_2)/\operatorname{Div}{\buildrel\sim\over\to}\;K_{2i} (F) \{ 2 \}, \]

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the edge map in the Bloch-Lichtenbaum spectral sequence [for details, see J. Rognes and C. Weibel, J. Am. Math. Soc. 13, 1–54 (2000; Zbl 0934.19001)]
\[ K_{2i+1} (F, {\mathbb Q}_2 /{\mathbb Z}_2) \to H^1 (F, {\mathbb Q}_2 /{\mathbb Z}_2 (i+1)), \]

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the natural coboundary map \(H^1 (F, {\mathbb Q}_2/{\mathbb Z}_2 (i+1)) \to H^2 (F, {\mathbb Z}_2 (i+1)) \{ 2 \}\)

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the localization map \(H^2 (F, {\mathbb Z}_2 (i+1)) \to H^2 (F_v, {\mathbb Z}_2 (i+1)) \displaystyle{\buildrel\sim\over\to}\;\widehat H^0 (F_v, {\mathbb Q}_2/{\mathbb Z}_2(i)).\)

It is shown that the composite map is independent of the choice of the initial split inverse.