\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & V(K) \arrow[dr, "\text{diviseurs abs. principaux}"'] \arrow[drr, "{\mathcal{D}_{\mathrm{abs},0}(K)}"] \arrow[drrr, bend left=20, "{\mathcal{D}_{\mathrm{abs}}(K)}"] & & & \\
e \arrow[r] & U(K) \arrow[ur] \arrow[dr] & & K^{*}V(K) \arrow[r, "{\mathcal{P}_{\mathrm{abs},0}(K)}"'] & J_0(K) \arrow[r] & J(K) \\
 & & K^{*} \arrow[ur, "\text{\uncertain{multiplication}}"] \arrow[urr, "C_0(K)"'] \arrow[urrr, bend right=20, "C(K)"'] & & & \mathbb{R}^{*+}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 0 \arrow[d] & & & \\
G(C) \arrow[r] \arrow[dr] & U_G(k) \arrow[dr] & & & \\
0 \arrow[r] & G(K) \arrow[r] & \mathbf{V}_G(k) \arrow[r] \arrow[dr] & J_G(k) \arrow[r] \arrow[d] & 0 \\
 & & & \mathcal{D}_G(k) \arrow[r] \arrow[d] & J_{G,C} \arrow[r, dashed] & \mathrm{Hom}(\mathbb{G}_m, G) \\
 & & & 0 & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 0 \arrow[d] & & & \\
k^{*} \arrow[r] \arrow[dr] & U(k) \arrow[dr] & & & \\
0 \arrow[r] & K^{*} \arrow[r] & \mathbf{V}(k) \arrow[r] \arrow[dr] & J(k) \arrow[r] \arrow[d] & 0 \\
 & & & \mathcal{D}(k) \arrow[r] \arrow[d] & J_C(k) \arrow[r] & \mathrm{Hom}(\mathbb{G}_m, \mathbb{G}_m) = \mathbb{Z} \\
 & & & 0 & &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\text{pts adéliques \uncertain{entiers}}\ \prod_{\text{\uncertain{locaux}}} G(\hat{\mathcal{O}}_x) \arrow[d] & 0 \arrow[l] \arrow[d] \\
\text{pts adéliques \uncertain{ordinaires}}\ \prod_{\text{local}} G(\hat{K}_x) & G(K_\xi) \arrow[l] \\
\text{« \uncertain{torsion} »}\ \coprod_x H^{0}_x(G) \arrow[d] & 0 \arrow[l] \arrow[d] \\
\text{diviseurs}\ \coprod_x H^{1}_x(G) & H^{0}_{\xi}(G) \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\prod_x G(\hat{\mathcal{O}}_x) \arrow[d] & 0 \arrow[l] \arrow[d] & & 0 & 0 \arrow[l] \\
\prod_{\text{local}} G(\hat{K}_x) & G(K) \arrow[l] & & G\text{-diviseurs sur } X & G(K) \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\prod_{\text{local}} J^{*}_{\mathcal{O}_D/k} \arrow[r] & J^{*}_{K/k} \\
\prod_{\text{local}} J^{0}_{\mathcal{O}_D/k} \arrow[u] \arrow[r, no head] & \prod J^{0}_{\mathcal{O}_D/k} \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
? \arrow[r] & \Bigl(\prod_{D\ \text{local}} J^{*}_{\mathcal{O}_D/k}\Bigr)^{0} \arrow[r] \arrow[dr] & J^{0}_{K/k} \arrow[dr] & \\
0 \arrow[r] & ? \arrow[u] & \Bigl(\coprod_D J^{*}_{k(D)/k}\Bigr)^{0} \arrow[r] & J^{0}_{X/k}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A & A' \\
A_1 \arrow[u] & A'_1 \arrow[u] \\
A_2 \arrow[u] & A'_2 \arrow[u] \\
\cdots \arrow[u] & 0 \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\scriptsize}]
0 \arrow[r] & \text{Noyau d'Albanese} \arrow[r] \arrow[d] & Z^{0}_{X} \arrow[r] \arrow[d] & J^{0}_{X/k} \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & G(K)/G(Y) \arrow[r] & \mathcal{D}iv^{0}(Y, G) \arrow[r] & \mathrm{Pic}^{0}(Y, G) \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
S_n \arrow[r, "i_n"] \arrow[d, "f_n"'] & X \arrow[d, "p_n"] \\
Y \arrow[r, "\varepsilon_n"] & X_n
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
Q & Q'' \\
X - S \arrow[r] & \mathrm{Pic}_{S, X/S}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\scriptsize}]
\underline{\mathrm{Ext}}^1(\mathrm{Pic}^{0}_{X/Y}, \mathbb{G}_m) \arrow[d, "2"] & \text{\emph{autodualité des jacobiennes}} \\
\underline{H}^{0}(Y, \mathrm{Pic}^{0}_{X/Y}) &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
0 \arrow[r] & \mathrm{Ext}^{1}(\mathbb{Z}, G) \arrow[r] \arrow[d, "\wr"] & \mathrm{Ext}^{1}(J, G) \arrow[r] \arrow[d] & \mathrm{Ext}^{1}(J^{0}, G) \arrow[r] \arrow[d, "\wr"] & 0 \\
0 \arrow[r] & \mathrm{Hom}(\mathbb{Z}, H^{1}(G)) \arrow[r] & \mathrm{Hom}(J(k), H^{1}(G)) \arrow[r] & \mathrm{Hom}(J^{0}(k), H^{1}(G)) \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\prod_{x_0} \mathcal{O}_{[x_0]} \arrow[d, "\mathrm{localisant}"] & & \\
\prod_{x_1} \prod^{\mathrm{local}}_{x_0} \mathcal{O}_{[x_0, x_1]} \arrow[d, "\mathrm{localisant}"] & \prod_{x_1} \mathcal{O}_{[x_1]} \arrow[l] \arrow[d, "\mathrm{localisant}"] & \\
\prod^{\mathrm{vert.}}_{x_0} \prod^{\mathrm{vert.}}_{x_1} \mathcal{O}_{[x_0, x_1, x_2]} & \prod^{\mathrm{local}}_{(x_1, x_2)} \mathcal{O}_{[x_1 x_2]} \arrow[l] & \prod_{x_2} \mathcal{O}_{[x_2]} = K \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\prod_{x} \hat{\underline{\mathcal{O}}}_{x}^{*} \arrow[d] & & \\
\prod_{D} \prod^{\mathrm{local}}_{x \in |D|} \hat{\mathcal{O}}_{D,x}^{*} \arrow[d] & \prod_{D} \hat{\mathcal{O}}_{D}^{*} \arrow[l] \arrow[d] & 0 \arrow[l, dashed] \arrow[d, dashed] \\
\prod_{x} \prod_{|D| \ni x} (\hat{\mathcal{O}}_{D,x} \otimes_{\underline{\mathcal{O}}_{D}} K)^{*} & \prod_{\mathrm{local}} \hat{K}_{D}^{*} \arrow[l] & K^{*} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
\hat{\underline{\mathcal{O}}}_{D_x} & \hat{\mathcal{O}}_{D} \arrow[l] \arrow[d] & \underline{\mathcal{O}}_{D} \arrow[l] \arrow[d] \\
 & \hat{K}_{D} & K^{*} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
0 & \hat{\mathcal{O}}_{\mathfrak{q}_1} \arrow[l] \arrow[d] & \mathcal{O}_{\mathfrak{q}_1} \arrow[l] \arrow[d] \\
 & \hat{\mathcal{O}}_{\mathfrak{q}_1 \mathfrak{q}_0} & \mathcal{O}_{\mathfrak{q}_0} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
A_{\mathfrak{q},\mathfrak{p}''} \arrow[d] & & \\
A_{\mathfrak{q},\mathfrak{p}'} \arrow[d] & A_{\mathfrak{q}',\mathfrak{p}'} \arrow[l] \arrow[d] & \\
A_{\mathfrak{q},\mathfrak{p}} & A_{\mathfrak{q}',\mathfrak{p}} \arrow[l] & A_{\mathfrak{q}'',\mathfrak{p}} \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & A_{[x_0 x_1 x_2]} & \\
 & & A_{[x_0, x_2]} \arrow[u] & \\
\Sigma A_{[x_0 x_1]} & \Sigma A_{[x_1, x_2]} & \Sigma A_{[x_0, x_1, x_2]} & A_{[x_0, x_2]} \arrow[l] \\
A_{[x_0]} \arrow[u] \arrow[ru] & A_{[x_1]} \arrow[u] & A_{[x_2]} \arrow[ru] &
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\hat{A} \arrow[d] & & \\
\varprojlim \hat{A}/\mathfrak{r}^{n}\hat{A} \otimes_{A} A_{\mathfrak{r}}/\mathfrak{r}^{n} A_{\mathfrak{r}} \arrow[d] & \widehat{A_{\mathfrak{r}}} \arrow[l] \arrow[d] & \\
\prod_{\mathfrak{r}} A_{[\mathfrak{m}, \mathfrak{r}, 0]} & \hat{A}_{\mathfrak{r}} \otimes_{A} K = \widehat{K_{\mathfrak{r}}} \arrow[l] & K \arrow[l]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\hat{A}^{*} \arrow[d] & & \\
\varprojlim (\hat{A} \otimes_{A} A_{\mathfrak{r}}/\mathfrak{r}^{n} A_{\mathfrak{r}})^{*} \arrow[d] & \widehat{A_{\mathfrak{r}}}^{*} \arrow[l] \arrow[d] & \\
\cdot & \widehat{K_{\mathfrak{r}}}^{*} \arrow[l] & K^{*} \arrow[l]
\end{tikzcd}

\begin{tikzcd}
A_{[x_1, \ldots, x_{n-1}]} \arrow[r, "u_2"] \arrow[d, "u_1"'] & A_{[x_0, x_1, \ldots, x_{n-1}]} \arrow[d, "u_1"] \\
A_{[x_1, \ldots, x_n]} \arrow[r] & A_{[x_0, x_1, \ldots, x_{n-1}, x_n]}
\end{tikzcd}

\begin{tikzcd}
\hat{A}_{\mathfrak{p}} \arrow[r, "{\mathfrak{q}\hat{A}_{\mathfrak{p}}}"] & A_{[\mathfrak{p}, \mathfrak{q}]} \\
A \arrow[u] \arrow[r] & \hat{A}_{\mathfrak{q}} \arrow[u]
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & A_{[\mathfrak{p}, \mathfrak{q}, \mathfrak{r}]} & & \\
A_{[\mathfrak{p}, \mathfrak{q}]} \arrow[ur] & & A_{[\mathfrak{q}, \mathfrak{r}]} \arrow[ul] & \\
A_{[\mathfrak{p}]} \arrow[u] & A_{[\mathfrak{q}]} \arrow[ul] \arrow[ur] & & A_{[\mathfrak{r}]} \arrow[ul] \\
 & A_{\mathfrak{p}} \arrow[ul] \arrow[u] \arrow[urr] & &
\end{tikzcd}

\begin{tikzcd}[column sep=small]
A_{[\mathfrak{p}_1 \cdots \mathfrak{p}_{n-1}]} \arrow[r] & A_{[\mathfrak{p}_1, \ldots, \mathfrak{p}_n]} \\
\hat{A}_{\mathfrak{p}_1} \cdots \hat{A}_{\mathfrak{p}_i} \cdots \arrow[ur] & \hat{A}_{\mathfrak{p}_n} \arrow[u]
\end{tikzcd}

\begin{tikzcd}
A_{c''} \times A_{c'} \times A_{c} \arrow[r] \arrow[d] & A_{c''c'} \times A_{c} \arrow[d] \\
A_{c''} \times A_{c'c} \arrow[r] & A_{c''c'c}
\end{tikzcd}

\begin{tikzcd}[column sep=small]
 & K(x) \arrow[r] & K(y) \arrow[dr] & \\
K(x') \arrow[ur] \arrow[rrr] & & & K(y')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & \text{faisceaux fpqc} & \\
\text{$k$-schémas en groupes} \arrow[ur] & & \text{faisceaux fppf} \arrow[ul] \\
\text{$k$-gr.\ qu.-cpts} \arrow[u] \arrow[drr, dashed] & \text{$k$-préschémas loc.\ alg.} \arrow[ul] \arrow[ur] & \text{gr.\ pro-alg.} \arrow[u] \\
 & \text{$k$-gr.\ alg.} \arrow[ul, "F"'] \arrow[u] \arrow[r] & \text{$k$-gr.\ qu.-cpts $F_s$} \arrow[u]
\end{tikzcd}

\begin{tikzcd}
\text{site fpqc} \arrow[d, "u"] \\
\text{site fppf}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & & (1) & \\
(2) \arrow[urr] \arrow[d, no head] & & & \\
(3) \arrow[dr, no head] & (4) \arrow[uur] \arrow[ul, no head] \arrow[d, no head] \arrow[dr, no head] & & (5) \arrow[d, no head] \\
 & (6) \arrow[dr, no head] & (7) \arrow[ur, no head] \arrow[d, no head] & (8) \\
 & & (9) \arrow[r, no head] & (10) \arrow[u, no head]
\end{tikzcd}

\begin{tikzcd}
K\,\mathrm{Gr}_{K} \arrow[d, "T"'] \\
K\,\mathrm{QuGr}_{k}
\end{tikzcd}

\begin{tikzcd}[column sep=large]
K^{-}_{\mathrm{fini}}(K)^{\circ} \arrow[r, "D"] \arrow[d, "T"'] & K^{+}_{\mathrm{fini}}(K) \arrow[d, "T(-1)"] \\
K^{-}(k)'^{\circ} \arrow[r, "\Delta"] \arrow[d, "S"'] & K^{+}(k)' \arrow[d, "S(-1)"] \\
K^{-}\mathcal{C}^{\circ} \arrow[r, "\Theta"] & K^{+}\mathcal{C}
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
0 \arrow[r] & H^{0}(K, {}_{\infty}B) \arrow[r] & DH^{1}(K, A) \arrow[d, no head, "\parallel"'] \arrow[dr, bend left=20] & & \\
& & H^{0}(K, B) & H^{0}(K, B) \otimes \mathbb{Q}/\mathbb{Z} \arrow[d, bend left=20] & \\
& & & H^{1}(K, {}_{\infty}B) \arrow[r] & DH^{0}(K, A) \arrow[r] & 0
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\underline{H}^{i-1}_{S/k}(G) \arrow[d, no head] & \underline{H}^{2-(i-1)}_{Y/k}(S, \hat{G}) \arrow[d] \\
\underline{H}^{i-1}_{U/k}(G) \arrow[d] & \underline{H}^{1-(i-1)}_{U/k}(\hat{G}) \\
\underline{H}^{i}_{Y/k}(S, G) \arrow[d] & \underline{H}^{2-i}_{S/k}(\hat{G}) \arrow[u, Rightarrow] \\
\underline{H}^{i}_{S/k}(G) \arrow[d] & \underline{H}^{2-i}_{Y/k}(\hat{G}) \arrow[u] \\
\underline{H}^{i}_{U/k}(G) & \underline{H}^{1-i}_{U/k}(\hat{G}) \arrow[u]
\end{tikzcd}

\begin{tikzcd}
\mathcal{I}_{\ell}(S)^{\circ} \arrow[r, "\approx"] \arrow[d] & \mathcal{M}_{\ell}(S) \arrow[r] \arrow[d] & \mathcal{V}_{\ell}(S) \arrow[d] \\
\mathcal{I}_{\ell}(S')^{\circ} \arrow[r, "\approx"] & \mathcal{M}_{\ell}(S') \arrow[r] & \mathcal{V}_{\ell}(S')
\end{tikzcd}

\begin{tikzcd}
 & \bar{K}_{2} \arrow[d, no head] \\
\bar{K}_{1} \arrow[r, no head] \arrow[d, no head] & (\bar{K}_{1}, K_{2}) \arrow[d, no head] \\
K_{1} \arrow[r, no head] \arrow[d, no head] & K_{2} \\
k_{1} &
\end{tikzcd}

\begin{tikzcd}
G \times G \arrow[r, "\varphi \times \varphi"] \arrow[d, "\pi"'] & E \times E \arrow[d, "\pi'"] \\
G \arrow[r, "\varphi"] & E
\end{tikzcd}

\begin{tikzcd}[column sep=small]
\Gamma \times \Gamma & E \times E \arrow[r] \arrow[d] & G \times G \arrow[d] \\
\Gamma & E \arrow[r] & G
\end{tikzcd}

\begin{tikzcd}
E \times E \arrow[r] \arrow[d] & G \times G \arrow[d] \\
E \arrow[r] & G
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & & \mathbb{Z}/\ell\mathbb{Z} \\
\mathbb{G}_m \arrow[r] & G \arrow[ur] \arrow[dr] & \\
 & & \mu_p
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & & \mathbb{Z}/\ell\mathbb{Z} \arrow[dl] \\
\mathbb{Z} & G \arrow[l] & \\
 & & \mathbb{Z}/p\mathbb{Z} \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & & \mathbb{Z}/p\mathbb{Z} \\
\mathbb{G}_a \arrow[r] & G \arrow[ur] \arrow[dr] & \\
 & & \alpha_p
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
 & & \mu_p \arrow[dl] \\
\mathbb{V} & G \arrow[l] & \\
 & & \alpha_p \arrow[ul]
\end{tikzcd}

\begin{tikzcd}[row sep=small]
C_1 \arrow[dr, "\alpha"] & \\
 & C_0 \\
C_2 \arrow[ur, "\beta"'] &
\end{tikzcd}

\begin{tikzcd}
H^0(X) \arrow[r] \arrow[d, "\wr"] & H^0(X') \arrow[d, "\wr"] \\
D\,H^0(Y) \arrow[r] & D\,H^0(Y')
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=large, nodes={font=\scriptsize}]
\text{\emph{groupes formels raisonnables}} \arrow[r, leftrightarrow, "\text{contrav.}"] \arrow[d, leftrightarrow, "\text{contrav.}"] & \text{Modules \ldots\ sur } A \\
\text{\emph{pro-(groupes alg.\ finis) raisonnables}} \arrow[ur, leftrightarrow, "\text{cov.}"'] &
\end{tikzcd}

\begin{tikzcd}
W_{n,k} \arrow[r, "i"] \arrow[d, "V"'] & W_{n,k'} \arrow[d] \\
W_{n+1,k} \arrow[r, "i"] & W
\end{tikzcd}

\begin{tikzcd}
W_4 \arrow[r, no head, "V_{4,5}"] & W_5 \arrow[r, no head] & W_{?} \\
W \arrow[u] \arrow[r, "V"] & W \arrow[u] \arrow[r, no head] & W_1
\end{tikzcd}

\begin{tikzcd}[column sep=small, row sep=small]
  & \mathcal{C} & \\
  & C_1 \arrow[u] & \\
  C_F \arrow[ur] & & C'_F \arrow[ul] \\
  & C_0 \arrow[ul] \arrow[ur] \arrow[d, no head] & \\
  & 0 &
\end{tikzcd}