\[
K^{*}/K^{*n} \simeq \mathbb{Z}/n\mathbb{Z} \qquad \text{(par la valuation)}
\]

\[
\Gamma(\overline{K}, K) \overset{\text{déf}}{=} T_{\infty}(k)
= \varprojlim_{n} \underbrace{\mu_n(k)}_{\simeq\, \mathbb{Z}/n\mathbb{Z}}
\qquad (\simeq \widehat{\mathbb{Z}})
\]

\[
\pi_1(K)^{\text{premier à } p} \longrightarrow
\varprojlim_{n \text{ premier à } p} \mu_n(k)
\]

\[
\pi_1(S, \bar{s}) \longrightarrow \varprojlim_{n \text{ \uncertain{premier à} } s} \mu_n(S)
\]

\[
0 \to H^0(S, \mu_n) \to H^0(S, \mathbb{G}_m) \to H^1(S, \mu_n) \to H^1(S, \mathbb{G}_m)
\]

\[
0 \to \mathbb{G}_m(S)_n \longrightarrow \mathrm{Rev}(S, \mu_n) \to
\uncertain{{}_{n}\mathrm{Pic}}(S) \to 0
\]

\[
a \longmapsto \mathrm{Spec}\bigl(\mathcal{O}_S[T]/(T^n - a)\bigr),
\qquad A[T]/(T^n - a)
\]

\[
\mathrm{hot}(X, B) \overset{h_X}{\longrightarrow} H^1(X, \pi)
\qquad \Bigl(\overset{\text{déf}}{=} \text{classes d'isom. de $\pi$-rev. principaux sur $X$}\Bigr)
\]

\[
E_{\pi} \times_{\pi} X = (E_{\pi} \times X)/\pi
\]

\[
\underset{\substack{\| \\ 0}}{\pi_2(B_{\pi})} \longrightarrow \pi_1(X) \to
\pi_1(X, \pi) \longrightarrow \pi \longrightarrow
\underset{\substack{\| \\ 1}}{\pi_0(X)}
\]

\[
\tilde{x} \in (\widetilde{X} \supset \widetilde{K} \supset \widetilde{S}) \simeq C_{p,q}
\qquad \mathbf{\Gamma_{p,q}} \supset \pi
\qquad \mathrm{Norm}(\pi)/\pi
\]

\[
\Gamma_{\infty,\infty} \supset \pi,
\qquad \Gamma_{\infty,\infty} \to \Gamma_{\infty,3},
\qquad \Gamma_{\infty,\infty} \to \Gamma_{p,q} \supset \pi
\qquad\qquad \Gamma_{2,2}
\]

\[
1 \to \uncertain{\Gamma_1} \to \Gamma^{0}_{\infty,3} = SL(2, \mathbb{Z})/(\pm 1)
\to \mathfrak{S}_3 \to 1
\]

\[
V \ni e, \qquad \textbf{$\omega_1, \omega_2$}, \qquad I(\omega_2/\omega_1) > 0
\qquad\qquad \omega_1, \omega_2, \quad \omega_2/\omega_1 = \tau
\]

\[
\{\sigma_0, \sigma_1, \sigma_2 \mid \rho_s^{3} = 1\} = \Gamma_{\infty,3} \supset \pi
\ni \rho_s, \qquad \textbf{$\rho_s^{\,p}$}
\]

\[
C_{\infty,3} \longrightarrow C_{p,3}
\]

\[
R_{(p,p',q,q')}, \qquad
\Gamma_{p,q} \longrightarrow \Gamma_{p',q'}, \qquad
C_{p,q} \longrightarrow C_{p',q'} = C_{p,q}/R, \qquad
p' \mid p, \quad q' \mid q.
\]

\[
C_{2,1} \qquad C_{1,2} \qquad C_{1,1}
\]

\[
X \text{ surface top.}, \qquad K \text{ sous-\uncertain{variété} fermée de dim } 1
\]

\[
\underbrace{|\Pi_n| \times [0,1]} \supset |\Pi_n| \supset S_n
\]

\[
D_n^{*} = |\Pi_n| \times \left]0,1\right] \smallsetminus S_n
\]

\[
\underbrace{\sigma_0\, \sigma_1}\, \sigma_2 \qquad \mathbf{E} \qquad
\boxed{\text{Déc}(\widehat{X}, \uncertain{W})} \qquad
\widetilde{X} \smallsetminus F
\]

\[
n \left\{ \begin{array}{ll} L & d \text{ gén.} \\ \cup & \\ L' & d' \text{ gén.} \end{array} \right.
\qquad (d' - 1) = n(d - 1), \qquad 2 = 2 \cdot 1
\]

\[
\begin{array}{lcl}
L \simeq \pi_1(X) & & 1 - d \\
\cup & & \\
L' \simeq \pi_1(X') & & 1 - d'
\end{array}
\]

\[
\pi^{+} \qquad \rho_0 \quad \rho_1 \quad \rho_\infty
\]

\[
\rho_0 \mapsto \rho_0^{-1}, \qquad \rho_\infty \mapsto \rho_\infty^{-1}, \qquad
\rho_1 \mapsto \rho_\infty \rho_0
\]

\[
\pi \qquad \mathrm{Ens}(\pi) \xrightarrow[\text{aux } \varinjlim]{\;\text{commutant}\;} \mathcal{C}
\]

\[
\updownarrow
\]

\[
F \longmapsto F(\pi_s)
\]

\[
X \longmapsto F_X : E \longmapsto \underbrace{X \times_{\pi} E}_{\in\, \mathrm{Ob}\,\mathcal{C}}
\]

\[
\mathrm{Hom}_{\mathcal{C}}(X \times_{\pi} E, Y) \simeq \mathrm{Hom}_{\pi}(E, \mathrm{Hom}(X, Y)),
\qquad Y \in \mathrm{Ob}\,\mathcal{C}
\]

\[
E \simeq \coprod_{i \in I} \pi/H_i
\]

\[
X \times_{\pi} E = \coprod_i \underbrace{X \times_{\pi} (\pi/H_i)}_{X/H_i}
\]

\[
\mathbf{Hom}_{!}\bigl(\mathrm{Ens}(\pi), \mathrm{Ens}(\pi')\bigr) \simeq {}_{\pi'}\mathrm{Ens}_{\pi}
\ni {}_{\pi'}X_{\pi}
\]

\[
\begin{array}{ccl}
X & & E \in \mathrm{Ens}(\pi) \\
\updownarrow & & \\
F : & & E \longmapsto X \times_{\pi} E
\end{array}
\]

\[
{}^{X}\pi \overset{\text{déf}}{=} X \times_{\pi} \mathrm{int}(\pi) = \mathrm{Aut}_{\pi}(X)
\]

\[
\pi' \longrightarrow \mathrm{Aut}_{\pi}(X)
\]

\[
\lambda e = e \lambda \quad \forall \lambda \in \pi
\]

\[
\mathbf{Bit.invol}(\pi) \xrightarrow{\approx} \mathbf{Invol}\bigl(\mathrm{Ens}(\pi)\bigr)
\xrightarrow{\approx} \mathbf{Ext}(\mathbb{Z}/2, \pi)
\]

\[
1 \to \pi \longrightarrow \widetilde{\pi} \overset{p}{\longrightarrow} \mathbb{Z}/2 \to 1
\]

\[
\pi_I = (\mathbb{Z}/2)^{*I}
\]

\[
1 \to \pi_I^{+} \longrightarrow \pi_I \to \mathbb{Z}/2 \to 1,
\qquad \pi_I^{+} \simeq \underbrace{L_I/\uncertain{-}}
\]

\[
\pi \qquad \lambda_0 \; \lambda_1 \; \lambda_2 \; \lambda_3 \qquad \varepsilon
\]

\[
\struck{\lambda_0 \lambda_1 \lambda_2 \lambda_3 = 1} \qquad
\lambda_3 \lambda_2 \lambda_1 \lambda_0 = 1 \qquad \varepsilon^{2} = 1
\]

\[
\begin{cases}
\varepsilon \lambda_0 \varepsilon^{-1} = \lambda_0^{-1} \\
\varepsilon \lambda_1 \varepsilon^{-1} = \lambda_1^{-1}
\end{cases}
\qquad \varepsilon \lambda_i \varepsilon^{-1}
\]

\[
\mathrm{Hom}(i, -i) \qquad
\underset{\substack{\wr \\ \pi}}{\mathrm{Hom}(-i, i)} \qquad
\underset{\substack{\wr \\ \pi}}{\mathrm{Hom}(i, i)}
\qquad \lambda_1^{-1} \lambda_2^{-1} \lambda_1
\qquad \ell \circ \bar{\ell}^{-1}
\]

\[
i \xrightarrow{\;\varepsilon^{-1}\;} \bar{\imath} \qquad
i \xrightarrow{\;\lambda_i\;} \bar{\imath} \qquad
\bar{\imath} \xrightarrow{\;\varepsilon\;} i
\qquad\qquad \varepsilon \circ \bar{\lambda}_i \circ \varepsilon^{-1}
\]

\[
\underset{\substack{\in \\ a}}{X} \qquad \Gamma
\]

\[
\bigl(\gamma \in \Gamma, \ \ell \in \mathrm{Hom}(\gamma a, \struck{\ill{}}\, b)\bigr)
\qquad
\bigl(\gamma' \in \Gamma, \ \ell' \in \mathrm{Hom}(\gamma' b, \struck{\ill{}}\, c)\bigr)
\]

\[
\gamma a \xrightarrow{\;\ell\;} b \qquad
\gamma' b \xrightarrow{\;\ell'\;} c
\]

\[
a \xrightarrow{\;(\gamma\ell') \circ \ell\;} \gamma\gamma' c \qquad
\bigl(\gamma\gamma', (\gamma\ell') \circ \ell\bigr)
\]

\[
\pi_1(X, \Gamma, a) \longrightarrow \Gamma \qquad
\gamma'\gamma a \xrightarrow{\;\gamma'(\ell)\;} \gamma' b \xrightarrow{\;\ell'\;} c
\]

\[
(\gamma', \ell')(\gamma, \ell) = \bigl(\gamma'\gamma, \ \ell' \circ \gamma'(\ell)\bigr)
\]

\[
\mathrm{Ens}(\pi) \xrightarrow{\;F_X\;} \mathrm{Ens}(\pi') \xrightarrow{\;F_Y\;} \mathrm{Ens}(\pi'')
\qquad {}_{\pi'}X_{\pi} \qquad {}_{\pi''}Y_{\pi'}
\]

\[
F_Y \circ F_X = F_{Y \wedge_{\pi'} X}
\]

\[
\pi' \xrightarrow{\sim} {}^{X}\pi \overset{\text{non can.}}{\simeq} \pi
\]

\[
\pi_0\bigl(\mathbf{Bit}(\pi', \pi)\bigr) \xrightarrow{\;\sim\;} \mathbf{Isomext}(\pi', \pi)
\]

\[
F^{2} \overset{\mu}{\xrightarrow{\sim}} \mathrm{id} \qquad
F(F(E)) \xrightarrow{\;\mu_E\;} E
\]

\[
F(F(F(E))) \underset{\mu_{F(E)}}{\overset{F(\mu_E)}{\rightrightarrows}} F(E)
\]

\[
F\mu = \mu F : F^{3} \to F
\]