\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \prod_{\sigma/\mathbb{Z}_p} \mathbb{G}_m \arrow[r] & T_{\sigma_K} \arrow[r] & \mathbb{Z} \arrow[r] & 0 \\
0 \arrow[r] & g_\sigma \arrow[r] & N \arrow[r] & \mathbb{Z} \arrow[r] & 0 \\
0 \arrow[r] & g_\sigma \arrow[u, Rightarrow] \arrow[r] & \tilde{N} \arrow[u] \arrow[r] & \mathbb{Z}_\sigma \arrow[u] \arrow[r] & 0 \\
0 \arrow[r] & g_K \arrow[r] & \tilde{N}_K \arrow[r] & \mathbb{Z}_K \arrow[r] & 0 \\
 & & g_K \times \mathbb{Z}_K \arrow[u, no head, "\simeq" description] & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 0 & \\
 & \mathbb{Z} \arrow[u] & \\
 & K^{*} \arrow[u] & \\
 & \sigma^{*}_K \arrow[u] & \\
 & 0 \arrow[u] & \\
\Gamma(N) \arrow[rr, "\simeq"] & & \Gamma(g) = K^{*} \arrow[ddll, "\pi"'] \\
\Gamma(\tilde{N}) \arrow[u, "2"'] \arrow[d, hook] & & \\
\Gamma(\tilde{N}_K) \simeq K^{*} \times \mathbb{Z} & &
\end{tikzcd}