\begin{tikzcd}
  & G_{\alpha}/G_{\alpha}^{0} \arrow[dr, dashed] & \\
  \pi_{1}(S, \xi) \arrow[r, "\rho_{\ell}"] \arrow[rr, bend right=40, "\text{hom can.}"'] & G_{\alpha}(\mathbb{Q}_{\ell}) \arrow[r] \arrow[u, dashed] & \pi_{\alpha}
\end{tikzcd}

\begin{tikzcd}
  M^{+}(X) \arrow[d, hook] & = & \mathbb{Z}^{(\Sigma^{+}(X))} \arrow[d, hook] \\
  M(X) & = & \mathbb{Z}^{(\Sigma(X))}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & M^{\pi} \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell} \arrow[d, no head, "\wr" description] & & & \\
0 \arrow[r] & H^{0}(S, A) \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell} \arrow[r] \arrow[d] & H^{1}(S, {}_{\ell^{\infty}}A) \arrow[r] \arrow[d] & H^{1}(S, A)(\ell) \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & (H^{0}(\overline{S}, \overline{A}) \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})^{\pi} \arrow[r] \arrow[d, no head, "\wr" description] & H^{1}(\overline{S}, {}_{\ell^{\infty}}\overline{A})^{\pi} \arrow[r] & H^{1}(\overline{S}, \overline{A})(\ell)^{\pi} \arrow[r] & 0 \\
 & (M \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})^{\pi} & & &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
  \cdots \arrow[r] & H^{1}(C, A) \arrow[r] \arrow[d, no head, "\Vert" description] & H^{1}(U, A) \arrow[r] & H^{2}_{x}(C, A) \arrow[r] & H^{2}(C, A) \arrow[d, no head, "\Vert" description] \\
  & 0 & & & 0
\end{tikzcd}

\begin{tikzcd}
  X \arrow[r, no head] \arrow[d] & X_{K} \arrow[d, no head] \\
  Y \arrow[r, no head] & K
\end{tikzcd}

\begin{tikzcd}[column sep=small]
  H^{1}(\widetilde{Y}) \arrow[dr] & & \\
  H^{3}_{Y}(X) \arrow[r] & H^{3}(X) \arrow[r, no head] & H^{3}(U)
\end{tikzcd}

\begin{tikzcd}[column sep=small]
  H^{1}_{!}(Y) & H^{1}_{!}(X) \arrow[l] \arrow[dl] & H^{1}_{!}(U) \arrow[l] \\
  H^{1}_{!}(\widetilde{Y}) & &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
  H^{0}(\widetilde{T}') \arrow[r] & H^{0}(\widetilde{T}) \arrow[r] & H^{1}_{!}(\widetilde{Y}) \arrow[r] & H^{1}(\widetilde{T}') \\
  H^{0}(\mathbb{P}^{2}) \arrow[r] \arrow[u] & H^{0}(T') \arrow[r] \arrow[u] & H^{1}_{!}(X) \arrow[r] \arrow[u] & H^{1}(\mathbb{P}^{2}) \arrow[u] \\
  H^{0}(\mathbb{P}^{2}) \arrow[r] & H^{0}(T) \arrow[r] \arrow[u] & H^{1}_{!}(U) \arrow[r] \arrow[u] & H^{1}(\mathbb{P}^{2}) \arrow[u, no head, "\Vert" description]
\end{tikzcd}

\begin{tikzcd}[column sep=small]
  & H^{0}(T')/\operatorname{Im} H^{0}(\mathbb{P}^{2}) \arrow[dl, "\text{inj ?}"'] & H^{0}(T)/\operatorname{Im} H^{0}(\mathbb{P}^{2}) \arrow[l] \\
  H^{0}(\widetilde{T})/\operatorname{Im} H^{0}(\widetilde{T}') & &
\end{tikzcd}

\begin{tikzcd}
H^{2i}_{\ell}(Z) \arrow[r, "f^{*}"] & H^{2i}_{\ell}(X) \\
C^{i}(Z) \arrow[u, "\rho_{Z}"] \arrow[r, "f^{*}"] & C^{i}(X) \arrow[u, "\rho_{X}"']
\end{tikzcd}

\begin{tikzcd}
H^{2i}_{\ell}(X) \arrow[r, "f_{*}"] & H^{2i-2r}_{\ell}(Z) \\
C^{i}(X) \arrow[u, "\rho_{X}"] \arrow[r, "f_{*}"] & C^{i-r}(Z) \arrow[u, "\rho_{Z}"']
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, hook] \arrow[d, "f"'] & X' \arrow[dl, "f^{\prime}"] \\
\operatorname{Spec} k &
\end{tikzcd}

\begin{tikzcd}
X \arrow[r, "g"] \arrow[dr, "f"'] & Y' \arrow[d, "h"] \\
& Y
\end{tikzcd}

\begin{tikzcd}
X \arrow[d, "f"] \\
Y
\end{tikzcd}

\begin{tikzcd}
X' \arrow[r, "f'"] \arrow[d, "\varphi"'] & Y' \arrow[d, "g"] \\
X \arrow[r, "f"'] & \uncertain{Y}
\end{tikzcd}

\begin{tikzcd}
& X' \arrow[d, "f"] \\
U \arrow[ur, hook, "i'"] \arrow[r, hook, "i"'] & X
\end{tikzcd}

\begin{tikzcd}
V \arrow[r, hook, "j"] \arrow[d, "t"'] & \hat{Y} \arrow[d, "g"] \\
U \arrow[r, hook, "i"'] & X
\end{tikzcd}

\begin{tikzcd}
E_{\ell} \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell} \arrow[r, "\alpha_{\ell}"] \arrow[d, "u_{\ell} \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell}"'] & T_{\ell}(M) \arrow[d, "T_{\ell}(u)"] \\
E'_{\ell} \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell} \arrow[r, "\alpha'_{\ell}"'] & T_{\ell}(M')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
  X \arrow[r, "\pi"] \arrow[d, "f"'] & X \arrow[d, "f"] \\
  X' \arrow[r, "\pi'"] \arrow[d, "g"'] & X' \arrow[d, "g"] \\
  X'' \arrow[r, "\pi''"] & X''
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
  P \simeq \pi P + \pi' P \arrow[r] & \pi P \arrow[d, "\wr"] \\
  Q \simeq \varpi Q + \varpi' Q & \varpi Q \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
  P \arrow[rr, "u"] \arrow[d] & & Q \\
  P' \arrow[rr, "\sim"] & & Q' \arrow[u]
\end{tikzcd}

\begin{tikzcd}
  \mathcal{V} \arrow[rr, "H"] \arrow[dr, "\underline{h}"'] & & \mathcal{C} \\
  & \mathcal{M} \arrow[ur, "T"'] &
\end{tikzcd}

\begin{tikzcd}
  M \arrow[r, leftrightarrow, "u^{\sharp}"] \arrow[d, "\widetilde{\alpha}"'] & N \arrow[d, "\widetilde{\beta}"] \\
  M^{*} & N^{*} \arrow[l, "{}^{t}u^{*}"']
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
  \underline{h}^{i}(X) \arrow[dr, bend left=20, "\lambda'"] & & & \underline{h}^{j}(Y) \arrow[dl, bend right=20, "\mu'"'] \\
  & M \arrow[ul, hook, bend left=20, "\lambda"] \arrow[r, "u"] & M \arrow[ur, hook', bend right=20, "\mu"'] & \\
  & \varphi_{0} & \psi_{0} &
\end{tikzcd}