\[
  \pi = \operatorname{Gal}(\overline{K}/K) .
\]

\[
  (*) \qquad \pi \longrightarrow \operatorname{Aut}_{\mathbb{Z}_\ell}(E) ,
\]

\[
  \pi \longrightarrow \operatorname{Aut}_{\mathbb{Q}_\ell}(E) ,
\]

\[
  E = H^i(X_{\overline{K}}, \mathbb{Z}_\ell) \overset{\mathrm{df}}{=} \varprojlim_\nu H^i(X_{\overline{K}}, \mathbb{Z}/\ell^\nu\mathbb{Z})
\]

\[
  E = T_\ell(A(\overline{K})) \overset{\mathrm{df}}{=} \varprojlim_\nu {}_{\ell^\nu}A(\overline{K}) .
\]

\[
  H^1(A_{\overline{K}}, \mathbb{Z}_\ell) \simeq \text{\struck{\ill{}}}\ \operatorname{Hom}_{\mathbb{Z}_\ell}(T_\ell(A(\overline{K})), \mathbb{Z}_\ell) ,
\]

\[
  T_\ell(A(\overline{K})) = \operatorname{Hom}_{\mathbb{Z}_\ell}(H^1(A_{\overline{K}}, \mathbb{Z}_\ell), \mathbb{Z}_\ell) .
\]

\[
  H^1(X_{\overline{K}}, \mathbb{Z}_\ell(1)) \simeq \varprojlim_\nu H^1(X_{\overline{K}}, \mu_{\ell^\nu}) \simeq T_\ell(\operatorname{Pic}(X_{\overline{K}}))
  \simeq T_\ell(\underline{\operatorname{Pic}}_{X/K}(\overline{K}))
\]

\[
  H^1(X_{\overline{K}}, \mathbb{Z}_\ell) \simeq T_\ell(\underline{\operatorname{Pic}}_{X/K}(\overline{K})) \otimes_{\mathbb{Z}_\ell} \mathbb{Z}_\ell(-1) .
\]

\[
  f : X \longrightarrow S
\]

\[
  \mathbb{Z} \simeq \operatorname{Pic}(\mathbb{P}^3) \xrightarrow{\ \sim\ } \operatorname{Pic}(X_0) ,
\]

\[
  \widetilde{V} \otimes_V k = \bar{k}
\]

\[
  \begin{cases} \widetilde{S} = \operatorname{Spec}\widetilde{V} = \text{hensélisé strict de } S \\ \bar{s} = \operatorname{Spec}\bar{k} . \end{cases}
\]

\[
  \operatorname{Hom}_S(\bar{s}, S') \xleftarrow{\ \sim\ } \operatorname{Hom}_S(\text{\struck{Spec}}\ \widetilde{S}, S') \xrightarrow{\ \sim\ } \operatorname{Hom}_S(\text{\struck{Spec}}\ \bar\eta, S') ,
\]

\[
  (2.3.1) \qquad \operatorname{Hom}_S(\bar{s}, S') \simeq \operatorname{Hom}_S(\bar\eta, S') ,
\]

\[
  (2.3.2) \qquad \mu_n(\bar{s}) = {}_n\bar{k}^{*} \simeq \mu_n(\overline{K}) = {}_n\overline{K}^{*} ,
\]

\[
  (2.3.3) \qquad T_\ell(\bar{k}^{*}) \simeq T_\ell(\overline{K}^{*}) .
\]

\[
  (2.3.4) \qquad \mathbb{Z}_\ell(n) \simeq \varprojlim_\nu \mu_{\ell^\nu}(\bar{k}^{*})^{\otimes n} \simeq \varprojlim_\nu \mu_{\ell^\nu}(\overline{K}^{*})^{\otimes n} .
\]

\[
  (2.2.1) \qquad \pi_0 = \operatorname{Gal}(\bar{k}/k) .
\]

\[
  (2.2.2) \qquad 1 \longrightarrow I \longrightarrow \pi \longrightarrow \pi_0 \longrightarrow 1 ,
\]

\[
  I = \operatorname{Gal}(\overline{K}/\widetilde{K})
\]

\[
  (2.4.1) \qquad 1 \longrightarrow P \longrightarrow I \longrightarrow I_t \longrightarrow 1 ,
\]

\[
  (2.4.2) \qquad I_t \simeq \prod_{\ell \neq p} \mathbb{Z}_\ell(1) .
\]

\[
  (2.4.3) \qquad P = \operatorname{Gal}(\overline{K}/\overline{K}_t) ,
\]

\[
  (2.4.4) \qquad K_n = \widetilde{K}(\sqrt[n]{t}) \subset \overline{K} ,
\]

\[
  \mu_n(\overline{K}) = {}_n\overline{K}^{*}
\]

\[
  I_t = \operatorname{Gal}\bigl((\varinjlim_n K_n)/\widetilde{K}\bigr) \simeq \varprojlim_n \operatorname{Gal}(K_n/\widetilde{K}) ,
\]

\[
  \mathbb{Z}_\ell(1) \simeq \mathbb{Z}_\ell ,
\]

\[
  (3.1) \qquad f : S' = \operatorname{Spec}(V') \longrightarrow S = \operatorname{Spec} V ,
\]

\[
  (3.3) \qquad (3.4)
\]

\[
  (3.6) \qquad \pi'_0 = \operatorname{Gal}(\bar{k}'/k') \longrightarrow \operatorname{Gal}(\bar{k}/k) = \pi_0
\]

\[
  (3.8) \qquad I'_t \simeq \prod_{\ell \neq p} \mathbb{Z}_\ell(1) \longrightarrow \prod_{\ell \neq p} \mathbb{Z}_\ell(1) \simeq I_t
\]

\[
  (3.9) \qquad
  \begin{cases}
    K_n = \widetilde{K}(\theta_n) \subset \overline{K}_t & \text{où } \theta_n^{\,n} = t , \\
    K'_n = \widetilde{K'}(\theta'_n) \subset \overline{K'}_t & \text{où } \theta'^{\,n}_n = t'
  \end{cases}
\]

\[
  \operatorname{Gal}(K'_n/\widetilde{K'}) \simeq \mu_n(\widetilde{K'}) \xleftarrow{\ \sim\ } \mu_n(\widetilde{K})
\]

\[
  \varepsilon_{K'_n}(\theta'_n) = \varepsilon\,\theta'_n ,
\]

\[
  (3.10) \qquad t = u'\,t'^{\,e} \qquad (e \in \mathbb{N},\ u' \in \widetilde{V}'^{*})
\]

\[
  \theta_n^{\,n} = u' (\theta'^{\,e}_n)^{n} ,
\]

\[
  (3.11) \qquad \theta_n = \lambda\,\theta'^{\,e}_n , \qquad (\lambda \in K'_n) ,
\]

\[
  \lambda^n = u' ,
\]

\[
  \varepsilon_{K'_n}(\theta_n) = \underbrace{\varepsilon_{K'_n}(\lambda)}_{=\lambda}\, \varepsilon_{K'_n}(\theta'^{\,e}_n) = \lambda (\varepsilon\theta'_n)^{e} = \varepsilon^{e}\,\theta_n
\]

\[
  \varepsilon_{K'_n}(\theta_n) = (\varepsilon^{e})_{K_n}(\theta_n)
\]

\[
  \varepsilon_{K'_n} | K_n = (\varepsilon^{e})_{K_n} .
\]

\[
  (3.14.1) \qquad I' = I \cap \pi' .
\]

\[
  (3.14.2) \qquad \bigl[ I'/(I',I') \bigr]^{(\ell)} \simeq \mathbb{Z}_\ell(1) \qquad (\ell \neq p)
\]

\[
  e = [I : I'] = \operatorname{card} I/I' \quad \text{si } k'/k \text{ est \uncertain{séparable}} .
\]

\[
  I' = I \cap \pi'
\]

\[
  (3.15.1) \qquad I'_t = \prod_{\ell \neq p} \bigl( I'/[I',I'] \bigr)(\ell) \simeq \prod_{\ell \neq p} \mathbb{Z}_\ell(1) .
\]

\[
  (4.1) \qquad E = \varprojlim_\nu E_\nu , \qquad E_\nu = E \otimes_{\mathbb{Z}_\ell} \mathbb{Z}/\ell^{\nu+1}\mathbb{Z} ,
\]

\[
  (4.2) \qquad G = \operatorname{Aut}_{\mathbb{Z}_\ell}(E) = \varprojlim_\nu \operatorname{Aut}_{\mathbb{Z}/\ell^{\nu+1}\mathbb{Z}}(E_\nu) .
\]

\[
  G^{1} = \operatorname{Ker}\bigl( \operatorname{Aut}_{\mathbb{Z}_\ell}(E) \to \operatorname{Aut}_{\mathbb{Z}/\ell\mathbb{Z}}(E_0) \bigr)
\]

\[
  \operatorname{Ker}\bigl( \operatorname{Aut}(E_\nu) \to \operatorname{Aut}(E_0) \bigr)
\]

\[
  u \equiv \mathrm{id} \ (\ell) ,
\]

\[
  Q \longrightarrow \operatorname{Aut}_{\mathbb{Z}/\ell\mathbb{Z}}(E_0)
\]

\[
  \prod_{\ell' \neq \ell, p} \mathbb{Z}_{\ell'}(1) \quad \text{de} \quad I/P \simeq \prod_{\ell'} \mathbb{Z}_{\ell'}(1) ,
\]

\[
  I'/I' \cap N \longrightarrow \operatorname{Aut}_{\mathbb{Z}/\ell\mathbb{Z}}(E_0)
\]

\[
  U \longrightarrow G
\]

\[
  U/U \cap I' = \operatorname{Im}\bigl(U \to \mathbb{Z}_\ell(1)\bigr) \longrightarrow \text{\struck{\ill{}}}\ G .
\]

\[
  U \longrightarrow \operatorname{Im}\bigl(U \to \mathbb{Z}_\ell(1)\bigr) \longrightarrow G
\]

\[
  (4.7) \qquad \mathbb{Z}_\ell(1) \xrightarrow{\ \sim\ } U' ,
\]

\[
  P \cap U \subset P \cap N ,
\]

\[
  U' = \text{\struck{$\operatorname{Im}$}}\ \operatorname{Im}(U \to I_t) \longrightarrow G .
\]

\[
  (4.8) \qquad U' \simeq I_t \simeq \prod_{\ell' \neq p} \mathbb{Z}_{\ell'}(1)
\]

\[
  (4.10.1) \qquad u : \pi \longrightarrow \operatorname{Aut}_{\mathbb{Q}_\ell}(E) .
\]

\[
  (4.10.2) \qquad u_{\mathbb{Z}_\ell} : \pi \longrightarrow \operatorname{Aut}_{\mathbb{Z}_\ell}(E_{\mathbb{Z}_\ell}) .
\]

\[ \pi \longrightarrow (G/G^{0})(R) \simeq G(R)/G^{0}(R) \]

\[ \text{\struck{$E^{(0)} = 0$, \quad $E^{(i+1)} = $ image inverse de $(E/E^{i})^{U} = (E/E^{i})^{G^{0}}$ dans $E$.}} \]

\[ \text{(5.4.1)} \qquad E^{i+1} = \text{image inverse dans $E$ de } (E/E^{i})^{G^{0}} . \]

\[ H^{i}(\Gamma, V) = 0 \quad \text{pour } i \geq 0 \]

\[ \text{(5.5.1)} \qquad G \simeq \Gamma \cdot G^{0} , \]

\[ \text{(5.5.1)} \qquad \pi/U \simeq (G/G^{0})(R) = G(R)/G^{0}(R) . \]

\[ E^{U'} = E^{U} = E^{G^{0}} . \]

\[ U \longrightarrow G^{0}(\mathbf{Q}_{\ell}) \simeq \mathbf{Q}_{\ell}^{d} , \]

\[ \text{(5.6.1)} \qquad N = \operatorname{Ker}\bigl(u : \pi \longrightarrow G(\mathbf{Q}_{\ell}) \subset \mathrm{Aut}(E)\bigr), \quad \text{(donc $N \subset U$)}, \]

\[ U/U \cap N \hookrightarrow G^{0}(R) \]

\[ U/N \longrightarrow U/N \otimes_{\mathbf{Z}_{\ell}} \mathbf{Q}_{\ell} . \]

\[ U/N \otimes_{\mathbf{Z}_{\ell}} \mathbf{Q}_{\ell} , \]

\[ \underbrace{U/[U,U]^{(\ell)}}_{\text{complété $\ell$-primaire}} \otimes_{\mathbf{Z}_{\ell}} \mathbf{Q}_{\ell} . \]

\[ \text{(5.7.1)} \qquad U/[U,U]^{(\ell)} \simeq \mathbf{Z}_{\ell}(1) \]

\[ \pi/U = \Gamma \longrightarrow \mathbf{Z}_{\ell}^{*} \]

\[ \text{(5.7.2)} \qquad G \simeq G^{0} \times \Gamma \quad \text{\struck{\ill{}}} \]

\[ \text{(5.7.3)} \qquad \begin{cases} G(\mathbf{Q}_{\ell}) \simeq \Gamma = \pi/U & \text{si $G$ fini, i.e.\ $U$ opère trivialement,} \\ G(\mathbf{Q}_{\ell}) \simeq \mathbf{Q}_{\ell}(1) \times \Gamma & \text{sinon.} \end{cases} \]

\[ \text{\struck{(5.10.1) \quad $E^{P}$}} \]

\[ \text{\struck{(6.1) \quad card $k = q = p^{f}$}} \]

\[ \text{\struck{$\pi_{0} \simeq \widehat{\mathbf{Z}}$ ,}} \]

\[ \text{\struck{(6.3) \quad $\mathrm{frob}_{q}(\lambda) = \lambda^{q}$ \quad ($\lambda \in \overline{k}$) .}} \]

\[ \text{\struck{$u : \pi \longrightarrow \mathrm{Aut}_{R}(E)$}} \]

\[ \text{\struck{$\widehat{\mathbf{Z}} \simeq \prod_{\ell} \mathbf{Z}_{\ell}$}} \]

\[ [I', \pi'] \quad \text{\emph{est d'indice fini dans} } I' , \quad \text{\emph{pour tout $\pi'$ ouvert}} \]

\[ H \subset G , \quad H \text{ invariant dans } G , \]

\[ [G, H] = H . \]

\[ [G, H] = H \quad \text{implique} \quad \text{\struck{$H_{r} = 1$,}} \]

\[ 1 \longrightarrow I \longrightarrow \pi \longrightarrow \pi_{0} \longrightarrow 1 \]

\[ u : \pi \longrightarrow \mathrm{Aut}_{\mathbf{Q}_{\ell}}(E_{\ell}) \]

\[ A_{(G^{0})} = 0 , \quad \text{\emph{i.e.\ $A$ est} \struck{\emph{engendré}} \emph{engendré par les} } (1-g)(A), \ \text{\emph{où} } g \in G^{0}(\mathbf{Q}_{\ell}) . \]

\[ I \longrightarrow \mathrm{Aut}_{\mathbf{Z}_{\ell}}\bigl(H^{i}(X_{\overline{K}}, \mathbf{Z}_{\ell})\bigr) \]

\[ \text{(7.3.1)} \qquad \mathrm{Tr}\, g_{H^{i}(\overline{X}, \mathbf{Q}_{\ell})} = \mathrm{Tr}\, g_{H^{i}(\overline{X}, \mathbf{Q}_{\ell'})} \]

\[ \mathrm{Gal}(\overline{k}/k) \simeq \widehat{\mathbf{Z}} \]

\[ \mathbf{Z}_{\mathbf{Q}}(G_{\mathbf{Q}} \times \Gamma_{\mathbf{Q}}) = G \]

\[ \dim \mathcal{O}_{S_{\alpha}, x_{\alpha}} \leq \dim \mathcal{O}_{S, x} = 1 , \]

\[ K = \varinjlim \underbrace{(V_{\alpha})_{t}}_{K_{\alpha}} , \]

\[ E_{U_{\alpha}} = R^{i} f_{\alpha *}(\mathbf{Z}_{\ell}) \text{ sur } U_{\alpha} \]

\[ R f_{K *}\bigl((\mathbf{Z}_{\ell})_{X_{K}}\bigr) = E_{K} \]

\[ \pi \longrightarrow \pi_{\alpha} \qquad \text{\struck{$\ldots \to \pi_{\alpha}$}} \]

\[ \pi \longrightarrow \pi' . \]

\[ \pi'_{0} = \mathrm{Gal}(\overline{k}'/k') \simeq \pi_{1}(S'_{\alpha}) \]

\[ I \longrightarrow I' \subset \pi' . \]

\[
(8.5.1) \qquad \pi_1(S,\xi) \simeq \pi_1(\operatorname{Spec} k, \bar k) = \operatorname{Gal}(\bar k/k).
\]

\[
(8.5.2) \qquad \pi_1(U,\xi) \longrightarrow \pi_1(S,\xi) \simeq \operatorname{Gal}(\bar k/k)
\]

\[
(8.5.3) \qquad 1 \longrightarrow I \longrightarrow \pi_1(U,\xi) \longrightarrow \operatorname{Gal}(\bar k/k) \longrightarrow 1 ,
\]

\[
(8.5.4) \qquad I \simeq \pi_1(\tilde U, \tilde\xi) ,
\]

\[
(8.5.5) \qquad I_t \simeq \Bigl(\prod_{\ell \neq p} \mathbf{Z}_\ell(1)\Bigr)^n ,
\]

\[
\mathbf{Z}_\ell(1) \simeq T_\ell(k(\xi)^*) \simeq T_\ell(\bar k^*) .
\]

\[
I_t \xrightarrow{\ \sim\ } I'_t .
\]

\[
\dim X_K \leq n .
\]

\[
H^i_!(X_{\bar K}, \mathbf{Q}_\ell) \qquad (\text{pour } X_K \text{ \emph{régulier}}).
\]

\[
H^i(X_{\bar K}, \mathbf{Q}_\ell) \simeq H^{2n-i}_!(X_{\bar K}, \mathbf{Q}_\ell)'(-n) ,
\]

\[
\cdots \to H^i_!(Y_{\bar K}, \mathbf{Q}_\ell) \to H^i_!(X_{\bar K}, \mathbf{Q}_\ell) \to H^i_!(U_{\bar K}, \mathbf{Q}_\ell) \to H^{i+1}_!(U_{\bar K}, \ldots
\]

\[
X' \longrightarrow X ,
\]

\[
H^*(X_{\bar K}, \mathbf{Q}_\ell) \Longleftarrow E_2^{pq} = H^p\bigl(\alpha \longmapsto H^q((X'/X)^{\alpha+1}_{\bar K}, \mathbf{Q}_\ell)\bigr) .
\]

\[
0 \to H^1\bigl(\alpha \mapsto H^0((X'/X)^{\alpha+1}_{\bar K}, \mathbf{Q}_\ell)\bigr) \to H^1(X_{\bar K}, \mathbf{Q}_\ell) \to \operatorname{Ker}\bigl(H^1(X'_{\bar K}, \mathbf{Q}_\ell) \overset{\partial}{\rightrightarrows} H^1(X''_{\bar K}, \mathbf{Q}_\ell)\bigr) .
\]

\[
(1.1.1) \qquad 1 \to \pi_1(\bar X, \xi) \longrightarrow \pi_1(X, \xi) \longrightarrow \pi_1(S, \xi) \longrightarrow 1 ,
\]

\[
(1.1.2) \qquad 1 \longrightarrow I \longrightarrow \pi \longrightarrow \pi_0 \longrightarrow 1 .
\]

\[
(1.1.3) \qquad 1 \longrightarrow I(\ell) \longrightarrow \pi' \longrightarrow \pi_0 \longrightarrow 1 .
\]

\[
(I^{\mathrm{ab}})_{\pi_0} \longrightarrow \pi^{\mathrm{ab}} \longrightarrow \pi_0^{\mathrm{ab}} \longrightarrow 1
\]

\[
I(\ell)_{\pi_0} \longrightarrow \pi^{\mathrm{ab}}(\ell) \longrightarrow \pi_0^{\mathrm{ab}}(\ell) \longrightarrow 1 ,
\]

\[
\pi^{\mathrm{ab}}(\ell) \longrightarrow \pi_0^{\mathrm{ab}}(\ell)
\]

\[
u : \pi_1(X,\xi) \longrightarrow \operatorname{Aut}_{\mathbf{Q}_\ell}(E)
\]

\[
\pi_0 = \operatorname{Im}\bigl(\pi \to \pi_1(S,\xi)\bigr) .
\]

\[
1 \to \pi_1(\bar X_0, \xi) \longrightarrow \pi_1(X_0, \xi) \longrightarrow \pi_1(S_0, \xi) \to 1 .
\]

\[
(1.5.1) \qquad \pi_1(X,\xi) \longrightarrow \pi_1(\bar X_0, \xi)
\]

\[
\Gamma \longrightarrow \operatorname{Aut}(B)
\]

\[
B \otimes_{k'} K' \simeq A \otimes_K K' ,
\]

\[
B_{X'} \longrightarrow A_{X'} ,
\]

\[
  \text{\struck{$1 \to H \to \pi' \to \pi_0 \to 1$}} .
\]

\[
  (\check{V})^{\pi_0} = 0
\]

\begin{equation}
  H^1(\overline{X}, \mathbb{Z}_\ell)^{\pi_0} = 0 . \tag{2.4.1}
\end{equation}

\[
  0 \to H^1(\overline{X'}, \mathbb{Z}_\ell) \to H^1(\overline{X}, \mathbb{Z}_\ell) \to \prod_{1 \leq i \leq n} H^2_{x_i}(\overline{X}, \mathbb{Z}_\ell)
\]

\[
  \to H^2(\overline{X'}, \mathbb{Z}_\ell) \to 0 \tag{2.5.1}
\]

\[
  H^2_{x_i}(\overline{X}, \mathbb{Z}_\ell) \simeq \mathbb{Z}_\ell(-1) \tag{2.5.2}
\]

\[
  \mathbb{Z}_\ell(-1) = \mathbb{Z}_\ell(1)^{\otimes(-1)}, \qquad \mathbb{Z}_\ell(1) = \varprojlim_\nu \left( {}_{\ell^\nu}\overline{k}^* \right) . \tag{2.5.3}
\]

\[
  H^1(\overline{X}, \mathbb{Z}_\ell)^{\pi_0} = H^1(\overline{X'}, \mathbb{Z}_\ell)^{\pi_0} . \tag{2.5.4}
\]

\[
  H^1(\overline{X'}, \mathbb{Z}_\ell)^{\pi_0} = 0 . \tag{2.5.5}
\]

\[
  H^0(S, R^1 f_*(\mathbb{Z}_\ell)) = 0
\]

\[
  H^1(C_{\overline{s}}, \mathbb{Z}_\ell)^{\pi_1(s,\overline{s})} = 0 \tag{2.5.6}
\]

\[
  1 \to \pi_1(\overline{X},\xi) \to \pi_1(X,\xi) \to \pi_1(S,\xi) \to 1 . \tag{3.1.1}
\]

\[
  C' = L \cap X'_K .
\]

\[
  C \to X
\]

\[
  \begin{array}{ccc}
    \xi_1 & \longrightarrow & \xi \\
    \downarrow & & \downarrow \\
    C & \longrightarrow & X \\
    \downarrow & & \downarrow \\
    \mathrm{Spec}\,K & \longrightarrow & S
  \end{array} \tag{3.1.2}
\]

\[
  \begin{array}{ccc}
    k & \longrightarrow & K = k(t) \\
    \downarrow & & \downarrow \\
    \overline{k} & \longrightarrow & \overline{K}
  \end{array}
\]

\[
  C \to X .
\]

\[
  \overline{[\pi', I(\ell)]} = I(\ell),
\]

\[
  H^1(\overline{X}, \mathbb{Z}/\ell\mathbb{Z})^{\pi_0} = 0 .
\]

\[
  \to H^1(\overline{X'}, \mathbb{Z}/\ell\mathbb{Z}) \to H^1(\overline{X}, \mathbb{Z}/\ell\mathbb{Z}) \to \prod_i H^2_{x_i}(\overline{X}, \mathbb{Z}/\ell\mathbb{Z})
\]

\[
  \to H^2(\overline{X'}, \mathbb{Z}/\ell\mathbb{Z}) \tag{4.2.1}
\]

\[
  N = \text{card du groupe des racines de 1 dans } k^* \tag{4.2.2}
\]

\[
  H^1(\overline{X}, \mathbb{Z}/\ell\mathbb{Z})^{\pi} \subset H^1(\overline{X'}, \mathbb{Z}/\ell\mathbb{Z})^{\pi} \quad \text{si } \ell \nmid N . \tag{4.2.3}
\]

\[
  \text{\struck{$H^1(\overline{X'}, \mathbb{Z}/\ell\mathbb{Z}) \simeq {}_\ell A(\overline{k})^\vee$}} ,
\]

\[
  ({}_\ell A(\overline{k}))_\pi = 0 \iff ({}_\ell A(\overline{k}))^\pi = 0 \quad \text{i.e.} \quad {}_\ell A(k) = 0 .
\]

\[
  {}_\ell A(k) = 0 \quad \text{si } \ell \nmid N' .
\]

\[
  H^1(\overline{X'}, \mathbb{Z}/\ell\mathbb{Z})^\pi = 0, \quad \text{cqfd.}
\]

\[
  H^1(\overline{X}, \mathbb{Z}_\ell) \to H^1(\overline{C}, \mathbb{Z}_\ell)
\]

\[
(1.1) \qquad \bar K = \varinjlim_{\alpha} K_\alpha ,
\]

\[
(1.2) \qquad X_{\bar\eta} = \varprojlim_{\alpha} X_\alpha , \qquad \text{où } X_\alpha = X_{K_\alpha},
\]

\[
(1.3) \qquad H^*(X_{\bar\eta}, \Lambda_\nu) = \varinjlim_{\alpha} H^*(X_{\bar K_\alpha}, \Lambda_\nu) .
\]

\[
(1.4) \qquad X_\alpha = X \times_S S_\alpha , \qquad X_{\alpha 0} = X_{\alpha s_\alpha} = X_0 \otimes_k k_\alpha .
\]

\[
\text{\struck{$X_{\alpha s_\alpha} = X_0 \otimes_k k_\alpha$}}
\]

\[
(1.5) \qquad H^*(X_s) \xrightarrow{\ \sim\ } H^*(X_{\alpha s_\alpha}) .
\]

\[
(1.7) \qquad H^*(X) \xrightarrow{\ \sim\ } H^*(X_\alpha) .
\]

\[
\begin{aligned}
& (1.8) \qquad \cdots \to H^i_{X_{\alpha 0}}(X_\alpha, \Lambda_\nu) \to H^i(X_\alpha, \Lambda_\nu) \to H^i(X_{\eta_\alpha}, \Lambda_\nu) \\
& \qquad \to H^{i+1}_{X_{\alpha, 0}}(X_{\eta_\alpha}, \Lambda_\nu) \to \cdots
\end{aligned}
\]

\[
(1.9) \qquad \cdots \to \Phi^i_\nu \to H^i(X_s, \Lambda_\nu) \to H^i(X_{\bar\eta}, \Lambda_\nu) \to \Phi^{i+1}_\nu \to \cdots
\]

\[
(1.10) \qquad \Phi^i_\nu = \varinjlim_\alpha H^i_{X_{\alpha 0}}(X_\alpha, \Lambda_\nu) .
\]

\[
(1.9\ \text{bis}) \qquad \cdots \to \Phi^i \to H^i(X_s, \mathbf{Z}_\ell) \to H^i(X_{\bar\eta}, \mathbf{Z}_\ell) \to \Phi^{i+1} \to \cdots ,
\]

\[
(1.10\ \text{bis}) \qquad \Phi^i = \varprojlim_\nu \Phi^i_\nu = \varprojlim_\nu \varinjlim_\alpha H^i_{X_{\alpha 0}}(X_\alpha, \Lambda_\nu) .
\]

\[
(1.13) \qquad H^*_{X_{\alpha 0}}(X_\alpha, \Lambda_\nu) \Longleftarrow E_2^{pq} = H^p(X_{\alpha 0}, \underline{H}^q_{X_{\alpha 0}}(\Lambda_{\nu X_\alpha})) .
\]

\[
(1.14) \qquad \Phi^*_\nu \Longleftarrow E_2^{pq} = H^p(X_0, \underline{\Phi}^q_\nu)
\]

\[
(1.15) \qquad \underline{\Phi}^q_\nu = \varinjlim_\alpha \underline{H}^*_{X_{\alpha 0}}(\Lambda_{\nu X_\alpha}) .
\]

\[
(1.14\ \text{bis}) \qquad \Phi^* \Longleftarrow E_2^{pq} = H^p(X_0, \underline{\Phi}^q)
\]

\[
(1.15\ \text{bis}) \qquad \underline{\Phi}^q = \varprojlim_\nu \underline{\Phi}^q_\nu ,
\]

\[
\underline{H}^2_{X_{\alpha 0}}(\Lambda_{\nu X_\alpha}) \simeq \bigl(\mu_{\ell^\nu}^{\otimes(-1)}\bigr)_{X_{\alpha 0}}
\]

\[
\underline{\Phi}^2_\nu = \text{\struck{$\ill{}$}}\ \varinjlim_\alpha (\mu_{\ell^\nu})_{X_{\alpha 0}} .
\]

\[
(1.16.1) \qquad H^i(X_s, \Lambda_\nu) \xrightarrow{\ \sim\ } H^i(X_{\bar\eta}, \Lambda_\nu)
\]

\[
(1.18) \qquad \Phi^i_\nu = \sum_{t \in T} (\underline{\Phi}^i_\nu)_t ,
\]

\[
(1.19) \qquad (\underline{\Phi}^i_\nu)_t = \varinjlim_\alpha \underline{H}^i_{X_{\alpha 0}}(\Lambda_{\nu X_\alpha})_{t_\alpha}
\]

\[
(1.22) \qquad \cdots \to \Phi^i_\nu \to H^i(X_s, \Lambda_\nu) \to H^i(U, \Lambda_\nu) \to \Phi^{i+1}_\nu \to \cdots
\]

\[
(1.23) \qquad \Phi^i_\nu = \varinjlim_\alpha H^i_{Y_\alpha}(X_\alpha, \Lambda_\nu) ,
\]

\[
(1.14) \qquad \Phi^*_\nu \Longleftarrow E_2^{pq} = H^p(X_0, \underline{\Phi}^q_\nu) ,
\]

\[
(1.15) \qquad \underline{\Phi}^q_\nu = \varinjlim_\alpha \bigl(\underline{H}^q_{Y_\alpha}((\Lambda_\nu)_{X_\alpha}) \,|\, X_{\alpha 0}\bigr) .
\]

\[
U_{\bar K} \to U_{\bar K_t}
\]

\[
H^*(U_{\bar K_t}, \Lambda) \Longleftarrow E_2^{p,q} = H^p(P, H^q(U_{\bar K}, \Lambda_\nu)) ,
\]

\[
(1.27) \qquad H^*(U_{\bar K_t}, \Lambda_\nu) \xrightarrow{\ \sim\ } H^*(U_{\bar K}, \Lambda_\nu)^P ,
\]

\[
\varinjlim_\alpha H^*(U_{K_\alpha}, \Lambda_\nu) ,
\]

\[
(1.28) \qquad K(n) = K(\pi^{1/n}) \qquad \text{pour } (p, n) = 1 .
\]

\[
\begin{aligned}
& (1.29) \qquad \cdots \to (\Phi^i_\nu)_{\mathrm{mod}} \to H^i(X_s, \Lambda_\nu) \\
& \qquad \to H^i(U_{\bar\eta}, \Lambda_\nu)^P \to (\Phi^{i+1}_\nu)_{\mathrm{mod}} \to \cdots
\end{aligned}
\]

\[
(1.30) \qquad (\Phi^i_\nu)_{\mathrm{mod}} = \varinjlim_{(n,p)=1} H^i_{Y_n}(X_n, \Lambda_\nu) ,
\]

\[
(1.31) \qquad (\Phi^*_\nu)_{\mathrm{mod}} \Longleftarrow E_2^{pq} = H^p(X_0, (\underline{\Phi}^q_\nu)_{\mathrm{mod}})
\]

\[
(1.32) \qquad (\underline{\Phi}^q_\nu)_{\mathrm{mod}} = \varinjlim_{(n,p)=1} \bigl(\underline{H}^q_{Y(n)}((\Lambda_\nu)_{X(n)}) \,|\, X(n)_0\bigr) ,
\]

\[
(2.3) \qquad \tilde X \xleftarrow{\ g\ } \bar U ,
\]

\[
(2.4) \qquad H^*(\bar U, \Lambda_\nu) \Longleftarrow E_2^{pq} = H^p(\tilde X, R^q g_*((\Lambda_\nu)_{\bar U})) .
\]

\[
(2.5) \qquad H^*(\bar U, \Lambda_\nu) \Longleftarrow E_2^{pq} = H^p(\tilde X_0, \underline{\Psi}^q_\nu)
\]

\[
(2.6) \qquad \underline{\Psi}^q_\nu = R^q g_*((\Lambda_\nu)_{\bar U} \,|\, \tilde X_0) = \varinjlim_\alpha R^q g_{\alpha *}((\Lambda_\nu)_{U_\alpha}) \,|\, \tilde X_0 ,
\]

\[
(2.8) \qquad
\begin{cases}
\underline{\Phi}^q_\nu = \underline{\Psi}^{q-1}_\nu & \text{si } q \geq 2 \\
0 \to \underline{\Phi}^0_\nu \to (\Lambda_\nu)_{X_0} \to \underline{\Psi}^0_\nu \to \underline{\Phi}^1_\nu \to 0 .
\end{cases}
\]

\[
(2.5\ \text{bis}) \qquad H^*(\bar U, \Lambda_\nu)^P \Longleftarrow E_2^{pq} = H^p(\tilde X_0, (\underline{\Psi}^q_\nu)_{\mathrm{mod}}) ,
\]

\[
\begin{aligned}
& (2.6\ \text{bis}) \qquad (\underline{\Psi}^q_\nu)_{\mathrm{mod}} = R^q g_{t *}((\Lambda_\nu)_{U_{\bar K_t}}) \,|\, \tilde X_0 \\
& \qquad = \varinjlim_{(n,\ell)=1} R^q g_{n *}((\Lambda_\nu)_{U_{K(n)}}) \,|\, \tilde X_0 ,
\end{aligned}
\]

\[
K(n) = \tilde K(\pi^{1/n}) \subset \bar K_t \subset \bar K
\]

\[
g_n : U_{K(n)} \to \tilde X
\]

\[
\text{\struck{$X_0$}}\ \text{\struck{$Y_0$}} \qquad X_0 = \sum m_\lambda X_0^{(\lambda)}
\]

\[
(3.3) \qquad (\mathrm{id} - g_E^{m'})^{i+1} = 0 .
\]

\[
(4.1.1) \qquad \underline{H}^i_Z(\Lambda_Y) = 0 \quad \text{pour } i \neq 2
\]

\[
(4.1.2) \qquad R^i f_*(\Lambda_U) = 0 \quad \text{pour } i \neq 0, 1 .
\]

\[
H^i(Z - Y, \mathbf{Z}/n\mathbf{Z}) = 0 \quad \text{si } i \neq 0, 1
\]

\[
(4.2.1) \qquad \underline{H}^i_Y(\Lambda_X) = 0 \quad \text{si } i \neq 2d ,
\]

\[
(4.2.2) \qquad \underline{H}^{2d}_Y(\Lambda_X) \simeq \bigl(\mu_n^{\otimes(-d)}\bigr)_Y
\]

\[
(4.2.2 \text{ bis}) \qquad \underline{H}^{2d}_Y(\mu_n^{\otimes d}) \simeq \Lambda_Y\ ).
\]

\[
\underline{H}^*_Y(\Lambda_X) \Longleftarrow E_2^{pq} = \underline{H}^p_Y\bigl(\underline{H}^q_Z(\Lambda_X)\bigr) ,
\]

\[
\begin{aligned}
(4.3.1) \qquad \cdots \to H^{i-2d}\bigl(Y, F(-d)\bigr) &\xrightarrow{\alpha} H^i(X, F) \xrightarrow{\beta} H^i(X - Y, F) \\
&\xrightarrow{\gamma} H^{i-2d+1}\bigl(Y, F(-d)\bigr) \to \cdots ,
\end{aligned}
\]

\[
H^{i-2}\bigl(X, F(-d)\bigr) \xrightarrow{\text{restr.}} H^i\bigl(Y, F(-d)\bigr) \xrightarrow{\alpha} H^i(X, F)
\]

\[
(4.3.2) \qquad \gamma(Y) \in H^{2d}\bigl(X, \Lambda_X(d)\bigr) .
\]

\[
0 \to \mu_{n\,X} \to \mathbf{G}_{m\,X} \xrightarrow{\lambda \mapsto \lambda^n} \mathbf{G}_{m\,X} \to 0
\]

\[
\cdots \to H^i_Y(X, F) \to H^i(X, F) \to H^i(X - Y, F) \to \cdots ,
\]

\[
H^i_Y(X, F) \simeq H^{i-2d}\bigl(Y, F(-d)\bigr)
\]

\[
H^*_Y(X, F) \Longleftarrow E_2^{pq} = H^p\bigl(Y, \underline{H}^q_Y(F)\bigr) ,
\]

\[
H^{2d}_Y\bigl(X, \Lambda(d)\bigr) \simeq H^0(Y, \Lambda) \longrightarrow H^{2d}\bigl(X, \Lambda(d)\bigr)
\]

\[
(4.5.1) \qquad R^1 f_*(\Lambda_U) \simeq \coprod_{1 \leq i \leq r} \bigl(\mu_n^{\otimes -1}\bigr)_{D_i} \simeq \coprod_{1 \leq i \leq r} \Lambda_{D_i}(-1) .
\]

\[
(4.5.2) \qquad \textstyle\bigwedge^p R^1 f_*(\Lambda_U) \longrightarrow R^p f_*(\Lambda_U)
\]

\[
\begin{aligned}
(4.5.3) \qquad R^p f_*(\Lambda_U) &\simeq \coprod_{1 \leq i_1 < i_2 < \cdots < i_p \leq r} \bigl(\Lambda_X(-p)\bigr)_{D_{i_1 \ldots i_p}} \\
&\simeq \Lambda_X(-p) \otimes \coprod_{1 \leq i_1 < \cdots < i_p \leq r} \Lambda_{D_{i_1 \ldots i_p}} ,
\end{aligned}
\]

\[
U \xrightarrow{\;g_i\;} U_i \xrightarrow{\;f_i\;} X , \qquad f = f_i g_i .
\]

\[
(*{*}) \qquad R^1 f_{i*}(\Lambda_{U_i}) \longrightarrow R^1 f_*(\Lambda_U) .
\]

\[
R^1 f_{i*}(\Lambda_{U_i}) \simeq \underline{H}^2_{D_i}(\Lambda_X) \simeq \Lambda_{D_i}(-1) ,
\]

\[
\varphi_i : \Lambda_{D_i}(-1) \longrightarrow R^1 f_*(\Lambda_U) ,
\]

\[
\varphi : \coprod_{1 \leq i \leq r} \Lambda_{D_i}(-1) \longrightarrow R^1 f_*(\Lambda_U) .
\]

\[
\text{\struck{$R^* f_*(\Lambda_U) \Longleftarrow E_2^{pq} = H^p\bigl((i_0 \ldots i_p) \mapsto R^q f_{i_0 \ldots i_p *}(\Lambda_{U_{i_0 \ldots i_p}})\bigr)$}}
\]

\[
\text{\struck{$R^0 f_{i_0 \ldots i_p *}(\Lambda_{U_{i_0 \ldots i_p}}) = \Lambda_X$ ,}}
\]

\[
\text{\struck{$(*{*}*) \quad R^1 f_*(\Lambda_U) \xrightarrow{\;\sim\;} E_2^{01} \subset \coprod_{1 \leq i \leq r} R^1 f_{i*}(\Lambda_{U_i})$}}
\]

\[
\text{\struck{$\textstyle\bigwedge^p H^1(U, \Lambda) \longrightarrow H^p(U, \Lambda)$}}
\]

\[
(4.5.4) \qquad \coprod_{1 \leq i \leq r} (\Lambda)_{D_i} \longrightarrow R^1 f_*\bigl(\mu_{n\,X}\bigr)
\]

\[
\text{\struck{$\coprod$}} \quad \Lambda^r \longrightarrow H^1\bigl(U, \mu_n\bigr)
\]

\[
\textstyle\bigwedge^i H^1(U, \Lambda) \longrightarrow H^i(U, \Lambda) ,
\]

\[
U' = X - \bigcup_{1 \leq i \leq r-1} D_i , \qquad D'_r = U' \cap D_r ,
\]

\[
U = U' - D'_r , \qquad D'_r = D_r - \operatorname{supp} \Delta_r , \qquad \Delta_r = \sum_{1 \leq i \leq r-1} D_i \cap D_r .
\]

\[
H^*_{D'_r}(U', \Lambda) \Longleftarrow E_2^{pq} = H^p\bigl(D'_r, \underline{H}^q_{D'_r}(\Lambda_{U'})\bigr)
\]

\[
H^i_{D'_r}(U', \Lambda_{U'}) \simeq H^{i-2}\bigl(D'_r, \mu_n^{\otimes -1}{}_{D'_r}\bigr) \simeq H^{i-2}(D'_r, \Lambda) \otimes \mu_n^{\otimes -1}(k) ,
\]

\[
H^i(D'_r, \Lambda) \simeq \textstyle\bigwedge^i H^1(D'_r, \Lambda) \simeq \bigl(\bigwedge^i(\Lambda^{r-1})\bigr) \otimes \mu_n^{\otimes -i}(k)
\]

\[
H^i(U', \Lambda) \simeq \textstyle\bigwedge^i H^1(U', \Lambda) \simeq \bigl(\bigwedge^i(\Lambda^{r-1})\bigr) \otimes \mu_n^{\otimes -i}(k) .
\]

\[
\text{\struck{$H^i_{D'_r}(U', \Lambda) \simeq H^{i-2}(D'_r, \Lambda) \otimes \mu_n^{\otimes -1}(k) \simeq \bigwedge^{i-2}(\Lambda^{r-1}) \otimes \mu_n^{\otimes((i-2)-1)}(k)$}}
\]

\[
H^i(U', \Lambda) \simeq \textstyle\bigwedge^i(\Lambda^{r-1}) \otimes \mu^{\otimes -i}(k)
\]

\[
\begin{aligned}
(4.5.5) \qquad \cdots \to H^{i-2}\bigl(D'_r, \Lambda(-1)\bigr) &\xrightarrow{\alpha_i} H^i(U', \Lambda) \to H^i(U, \Lambda) \\
&\to H^{i-1}\bigl(D'_r, \Lambda(-1)\bigr) \to \cdots
\end{aligned}
\]

\[
\textstyle\bigwedge^* H^1(U', \Lambda) \quad \text{et} \quad \bigl(\bigwedge^* H^1(D'_r, \Lambda)\bigr)(-1) .
\]

\[
H^*(U', F) \longrightarrow H^*(D'_r, F)
\]

\[
H^{i-2}\bigl(U', \Lambda(-1)\bigr) \longrightarrow H^{i-2}\bigl(D'_r, \Lambda(-1)\bigr) \xrightarrow{\alpha_i} H^i(U', \Lambda)
\]

\[
(4.5.6) \qquad 0 \to H^i(U', \Lambda) \to H^i(U, \Lambda) \to H^{i-1}(D'_r, \Lambda)(-1) \to 0 .
\]

\[
H^1(U, \Lambda) \simeq \coprod_{1 \leq i \leq r} H^1(U_i, \Lambda) \simeq \coprod_{1 \leq i \leq r-1} H^1(U_i, \Lambda) \times H^1(U_r, \Lambda)
\]

\[
\simeq H^1(U', \Lambda) \times \mu_n(X)^{-1}
\]

\[
\textstyle\bigwedge^i H^1(U, \Lambda) \simeq \bigwedge^i H^1(U', \Lambda) \times \Bigl(\bigwedge^{i-1} H^1(U', \Lambda) \otimes \mu_n(X)^{-1}\Bigr)
\]

\[
(4.5.7) \qquad 0 \to \textstyle\bigwedge^i H^1(U', \Lambda) \longrightarrow \bigwedge^i H^1(U, \Lambda) \longrightarrow \Bigl(\bigwedge^{i-1} H^1(U', \Lambda)\Bigr)(-1) \to 0 .
\]

\[
\text{\struck{$U = U' - D'_r$}}
\]

\[
D_i = V(a_i) , \quad \text{où} \quad a_i \in \Gamma(\underline{O}_X) .
\]

\[
X' = \operatorname{Spec} \underline{O}_X[T_1, \ldots, T_r]/(T_1^m - a_1, \ldots, T_r^m - a_r) .
\]

\[
p^*(D_i) = m \Delta_i .
\]

\[
U' = p^{-1}(U) .
\]

\[
(4.6.1) \qquad p^*\bigl(R^* f_*(\Lambda_U)\bigr) \longrightarrow R^* f'_*(\Lambda_{U'}) ,
\]

\[
X'_m = \operatorname{Spec} A[T_1, \ldots, T_r]/(T_1^m - a_1, \ldots, T_r^m - a_r) ,
\]

\[
H^*(U, \Lambda) \longrightarrow H^*(U', \Lambda)
\]

\[
D = \operatorname{div}\Bigl(\prod a_i\Bigr) = \sum D_i \qquad (\text{où } D_i = \operatorname{div} a_i)
\]

\[
U = X - \operatorname{supp} D = X - \bigcup D_i ,
\]

\[
X(m) = \operatorname{Spec} A[T_1, \ldots, T_r]/(T_1^m - a_1, \ldots, T_r^m - a_r) ,
\]

\[
\begin{array}{ccc}
\mu_{m'}^r(X) & \longrightarrow & \mu_m^r(X) \\
\| & & \| \\
G(m') & & G(m)
\end{array}
\qquad (m \mid m') .
\]

\[
U(\infty) = \varprojlim U(m)
\]

\[
(4.8.1) \qquad G = \varprojlim_m G(m) = \prod_{\ell \neq p} \mathbf{Z}_\ell(1)^r ,
\]

\[
(4.8.2) \qquad \mathbf{Z}_\ell(1) = \varprojlim_\nu \mu_{\ell^\nu}(X) .
\]

\[
H^*\bigl(U(\infty), \Lambda\bigr) \simeq \varinjlim_m H^*\bigl(U(m), \Lambda\bigr) ,
\]

\[
H^*\bigl(U(m), \Lambda\bigr) \longrightarrow H^*\bigl(U(m'), \Lambda\bigr) , \quad \text{pour } m \mid m'
\]

\[
(4.9.1) \qquad
\begin{cases}
H^i\bigl(U(\infty), \Lambda\bigr) = 0 & \text{si } i \neq 0 \\
H^0\bigl(U(\infty), \Lambda\bigr) \simeq \Lambda .
\end{cases}
\]

\[
(4.10.1) \qquad \varphi : G \longrightarrow \Gamma ,
\]

\[
H^*(H, \Lambda) \overset{G}{\times} \Gamma .
\]

\[
\widetilde{U}' = U(\infty) \overset{G}{\times} \Gamma
\]

\[
(4.10.1) \qquad H^*(\widetilde{U}', \Lambda) \simeq H^*(H, \Lambda) \overset{G}{\times} \Gamma .
\]

\[
U' = (U(\infty)/H) \overset{K}{\times} \Gamma ,
\]

\[
H^*(U', \Lambda) \simeq H^*(U(\infty)/H, \Lambda) \overset{K}{\times} \Gamma
\]

\[
H^*(U(\infty)/H, \Lambda) \simeq H^*(H, \Lambda) ,
\]

\[
H^*(U(\infty)/H, \Lambda) \Longleftarrow E_2^{pq} = H^p\bigl(H, H^q(U(\infty), \Lambda)\bigr) ,
\]

\[
H^*(U(\infty)/H, \Lambda) \simeq H^*(H, \Lambda) .
\]

\[
(4.13.1) \qquad \Lambda = \mathbf{Z}/\ell^{\nu+1}\mathbf{Z} ,
\]

\[
H = \prod_{\ell \neq p} H(\ell) , \qquad H(\ell) \subset G(\infty)(\ell) \simeq \mathbf{Z}_\ell(1)^r .
\]

\[
(4.13.2) \qquad \Lambda = \mathbf{Z}/n\mathbf{Z} , \quad \text{avec } n = \ell^{\nu+1} ,
\]

\[
(4.13.1) \qquad
\begin{cases}
H^*(H, \Lambda) \simeq \bigwedge^* H^1(H, \Lambda) \\
H^1(H, \Lambda) \simeq \operatorname{Hom}(H, \Lambda) \simeq \Lambda^s , \quad \text{où } H(\ell) \simeq \mathbf{Z}_\ell^s .
\end{cases}
\]

\[
(4.14.1) \qquad t = a_1^{m_1} \cdots a_r^{m_r}
\]

\[
(4.14.2) \qquad X'(m) = \operatorname{Spec} A[T]/(T^m - t)
\]

\[
(4.14.3) \qquad \text{\struck{$X(m) =$}} \quad U'(m) = X'(m) \times_X U ,
\]

\[
(4.14.4) \qquad U' = \varprojlim_m U'(m) ,
\]

\[
(4.14.5) \qquad \Gamma = \varprojlim \mu_m(X) = \prod_{\ell \neq p} \mathbf{Z}_\ell(1) .
\]

\[
U' = U(n) \times^{G} \Gamma ,
\]

\[
\text{(4.14.6)} \qquad \varphi(\lambda_1, \ldots, \lambda_r) = \sum_{1}^{r} m_i \lambda_i .
\]

\[
t \in \Gamma(X, \mathcal{O}_X) .
\]

\[
U = X - \operatorname{supp} D ,
\]

\[
t|U \in \Gamma(U, \mathcal{O}_U^*) , \quad \text{i.e.} \quad \operatorname{supp} \operatorname{div} t \subset \operatorname{supp} D .
\]

\[
\begin{cases}
X'(n) = \operatorname{Spec} \mathcal{O}_X[T]/(T^n - t) , \\
U'(n) = X'(n) \times_X U ,
\end{cases}
\]

\[
U' = \varprojlim U'(n) ,
\]

\[
G = \varprojlim \mu_n ,
\]

\[
\Lambda = \mathbf{Z}/n\mathbf{Z} ,
\]

\[
R^i f_*(\Lambda_{U'})|X_0 , \qquad X_0 = V(t)
\]

\[
G' = m'G \;(= mG) \subset G
\]

\[
\text{\struck{(5.1)}} \quad \text{\struck{$H^*(\overline{U}, \mathbf{Z}_\ell)^P \Longleftarrow H^p(\widetilde{X}_0, (\underline{\Psi}^q)_{\mathrm{tame}})$}}
\]

\[
\text{\struck{(5.2)}} \quad \text{\struck{$(\underline{\Psi}^q)_{\mathrm{tame}} = \varprojlim_\nu (\underline{\Psi}^q_\nu)_{\mathrm{tame}}$}}
\]

\[
\text{(5.1)} \qquad H^*(\overline{U}, \mathbf{Z}_\ell)^P \Longleftarrow E_2^{pq} = \varprojlim_\nu H^p(\widetilde{X}_0, (\underline{\Psi}^q_\nu)_{\mathrm{tame}}) .
\]

\[
\text{(6.1.1)} \qquad (\underline{\Psi}^q_\nu)_{\mathrm{tame}} = (\underline{\Psi}^q_\nu)^P
\]

\[
\Lambda_\nu = \mathbf{Z}/\ell^{\nu+1}\mathbf{Z} \qquad (\nu \geqslant 0) .
\]

\[
(\underline{\Psi}^q)_{\mathrm{tame}} = \varprojlim_\nu (\underline{\Psi}^q_\nu)_{\mathrm{tame}} .
\]

\[
\text{(6.2.1)} \qquad (\underline{\Psi}^q_\nu)_{\mathrm{tame}} = R^q g_*(\Lambda_\nu)|X_0
\]

\[
\text{(6.2.2)} \qquad R^* g_*(\Lambda_\nu) = (\underline{\Psi}^*_\nu)_{\mathrm{tame}} \Longleftarrow E_2^{pq}(\nu) = R^p p_*(R^q g'_*(\Lambda^q_\nu)) .
\]

\[
\text{\struck{(1.1.1)}} \quad \text{(1.1.1)} \qquad f' : X' \longrightarrow \widetilde{S}
\]

\[
\text{(1.1.2)} \qquad U' \subset X'_{\widetilde{\eta}} .
\]

\[
(\underline{\Psi}^{(q)}_\nu)_{\overline{x}} \simeq \text{\struck{$H^q(\overline{U}' \times_{\widetilde{X}} \overline{U})$}} \; H^q(\overline{U}', \Lambda_\nu) , \quad \text{avec} \quad \overline{U}' \simeq X' \times_{\widetilde{X}} \overline{U} ,
\]

\[
X' \times_{\widetilde{X}} \overline{U} \simeq U' \times_{\widetilde{U}} \overline{U} \simeq U' \times_{\widetilde{\eta}} \overline{\eta} ,
\]

\[
\text{(1.1.3)} \qquad (\underline{\Psi}^{(q)}_\nu)_{\overline{x}} \simeq H^q(\overline{U}', \Lambda_\nu) , \qquad \overline{U}' = U' \otimes_{\widetilde{\eta}} \overline{\eta} = U' \otimes_{\widetilde{K}} \overline{K} .
\]

\[
\text{(1.2)} \qquad (\underline{\Psi}^{(q)}_\nu)_{\overline{x}} = H^q(X' \times_{\widetilde{S}} \overline{\eta}, \Lambda_\nu) ,
\]

\[
\text{\struck{(1.1.3)}} \qquad ((\underline{\Psi}^{(q)}_\nu)_{\mathrm{tame}})_{\overline{x}} \simeq H^q(U' \otimes_{\widetilde{K}} \overline{K}_t) ,
\]

\[
\text{(1.3)} \qquad ((\underline{\Psi}^q_\nu)_{\mathrm{tame}})_{\overline{x}} = H^q(X \otimes_{\widetilde{S}} \operatorname{Spec} \overline{K}_t) \; ;
\]

\[
X' = \varprojlim_\alpha X_\alpha ,
\]

\[
U' = \varprojlim_\alpha X_\alpha \times_X U , \qquad U'_{\overline{\eta}} = \varprojlim_\alpha (X_\alpha \times_X U)_{\overline{\eta}} ,
\]

\[
\text{(1.4.1)} \qquad (\underline{\Psi}^{(i)}_\nu)_{\overline{x}} \simeq H^i(U'_{\overline{\eta}}, \Lambda) \simeq \varinjlim_\alpha H^i((X_\alpha \times_X U)_{\overline{\eta}}, \Lambda) ,
\]

\[
\text{(1.4.2)} \qquad (\underline{\Psi}^{(i)}_\nu)_{\overline{x}} = H^i(X'_{\overline{\eta}}, \Lambda) \simeq \varinjlim_\alpha H^i(X_{\alpha\overline{\eta}}, \Lambda) ,
\]

\[
\text{(2.1.0)} \qquad n_x = \text{\struck{$\dim X_\eta$}} \; \sup_i \dim X_{i\eta} ,
\]

\[
\text{(2.1.1)} \qquad (\underline{\Psi}^i_\nu)_{\overline{x}} = 0 \quad \text{pour } i > n_x .
\]

\[
\text{(2.1.2)} \qquad (\underline{\Psi}^i_\nu)_{\overline{x}} = 0 \quad \text{pour } i > n_x - \dim \overline{\{x\}} ,
\]

\[
\text{(2.1.3)} \qquad (\underline{\Psi}^i_\nu)_{\overline{x}} = 0 \quad \text{pour } i > \dim \mathcal{O}_{X_0, x} = \dim_x(X_0) - \dim \overline{\{x\}}
\]

\[
n = \dim X_\eta = n_x \; ;
\]

\[
t_1, \ldots, t_r \in \Gamma(X, \mathcal{O}_X)
\]

\[
h : X \longrightarrow Y = S[T_1, \ldots, T_r] ,
\]

\[
\dim_x h^{-1}(y) = \dim \mathcal{O}_{X_0, x} = n - r ,
\]

\[
\text{(2.2.1)} \qquad H^i(U' \times_{S_1} \overline{\lambda}, \Lambda) = 0 \quad \text{si } i > n - r , \quad \text{i.e.} \quad R^i \varphi_*(\Lambda_{U'_{\overline{\eta}}}) = 0 \text{ si } i > n - r .
\]

\[
H^i(U'_{\overline{\eta}}, \Lambda) \Longleftarrow E_2^{p,q} = H^p(\lambda, R^q \varphi_*(\Lambda_{U'_{\overline{\eta}}}))
\]

\[
H^i(U'_{\overline{\eta}}, \Lambda) \simeq H^0(\lambda, R^i \varphi_*(\Lambda_{U'_{\overline{\eta}}})) \subset H^i(U' \times_{S_1} \overline{\lambda}, \Lambda)
\]

\[
H^i(U'_{\overline{\eta}}, \Lambda) = 0 \quad \text{pour } i > n - r ,
\]

\[
(\underline{\Psi}^q_\nu)_{\overline{x}} = 0 \quad \text{pour } q > n_x - \dim \overline{\{x\}}
\]

\[
(\underline{\Phi}^q_\nu)_{\overline{x}} = 0 \quad \text{si } q > n_x + 1 - \dim \overline{\{x\}} , \quad q \geqslant 1 .
\]

\[
H^p(X_0, (\underline{\Phi}^q_\nu)|X_0) = 0 \quad \text{si } p \geqslant 2(n + 1 - q) ,
\]

\[
\text{i.e.} \quad p + q \geqslant 2n + 2 - q .
\]

\[
\underline{\Psi}^q_\nu = 0 \quad \text{si } q > \dim X_\eta , \qquad \underline{\Phi}^q_\nu = 0 \quad \text{si } q > \dim X_\eta + 1 .
\]

\[
(*) \qquad H^p(X_0, \underline{\Phi}^q_\nu) \neq 0 \;\; \text{\struck{\ill{}}} \;\; \text{implique} \;\; p \leqslant 2d .
\]

\[
\delta(\underline{\Phi}^q_\nu) \leqslant n + 1 - q \quad \text{si } q \geqslant 1
\]

\[
q > n + 1 - d \; ; \quad \text{\struck{$2\delta(\underline{\Phi}^q_\nu) =$}}
\]

\[
(**) \qquad H^p(X_0, \underline{\Phi}^q_\nu) \neq 0 \;\; \text{implique} \;\; p \leqslant 2(n + 1 - q) ,
\]

\[
\text{i.e.} \quad p + q \leqslant 2n + 2 - q ,
\]

\[
H^p(X_0, \underline{\Phi}^q_\nu) \neq 0 \Longrightarrow p + q \leqslant n + d + 1 .
\]

\[
\underline{\Phi}_\nu = 0 \quad \text{pour } i > n + d + 1 .
\]

\[
H^i(X_0, \Lambda_\nu) \longrightarrow H^i(X_{\overline{\eta}}, \Lambda_\nu)
\]

\[
H^i(Y_{\overline{\eta}}, \Lambda_\nu) \longrightarrow H^{i + 2(d+1)}(X_{\overline{\eta}}, \Lambda_\nu(d + 1))
\]

\[
\begin{array}{ccc}
H^i(Y_s, \Lambda) & \longrightarrow & H^{i + 2(d+1)}(X_s) \\
\downarrow{\scriptstyle \varphi_Y} & & \downarrow{\scriptstyle \varphi_X} \\
H^i(Y_{\overline{\eta}}, \Lambda) & \longrightarrow & H^{i + 2(d+1)}(X_{\overline{\eta}})
\end{array}
\]

\[
X_{(m)} = \operatorname{Spec} A[T]/(T^m - t) , \qquad U_{(m)} = X_{(m)} \times_X U ,
\]

\[
U(\infty) = \varprojlim U(m) .
\]

\[
\text{(3.1.1)} \qquad \underbrace{\operatorname{prof\,ét}_x X_s}_{\operatorname{prof\,ét}^*_x(X_s)} + \dim \overline{\{x\}} \geqslant m
\]

\[
\text{(3.1.2)} \qquad \underbrace{\operatorname{prof\,ét}_x (X_{\overline{\eta}})}_{\operatorname{prof\,ét}^*_x(X_{\overline{\eta}})} + \dim \overline{\{x\}} \geqslant m ,
\]

\[
H^i(X_s, \Lambda) \longrightarrow H^i(Y_s, \Lambda) \qquad (\text{resp. } H^i(X_{\overline{\eta}}, \Lambda) \to H^i(Y_{\overline{\eta}}, \Lambda))
\]

\[
\text{(3.2)} \qquad \varphi^i_X : H^i(X_s, \Lambda) \longrightarrow H^i(X_{\overline{\eta}}, \Lambda)
\]

\[
\text{(3.1.3)} \qquad \underline{\Phi}^i_\nu = 0 \quad \text{pour } i \leqslant m - d - 1 .
\]

\[
\text{\struck{(3.2.1)}} \quad \text{\struck{$\underline{\Phi}^i_\nu = 0$ pour $i \leqslant n - d - 1$,}}
\]

\[
\text{\struck{(3.2.2)}} \quad \text{\struck{$\underline{\Phi}^i_\nu = 0$ si $i \notin [n - d, n + d + 1]$.}}
\]

\[
\text{\struck{(3.1.6)}} \quad \text{\struck{$\underline{\Phi}^i_\nu = 0$ si $i \neq n, n+1$.}}
\]

\[
(4.1.1)\qquad V = X' - \{\bar x\}, \quad V_s = V \times_S s, \quad
V_\eta = V \times_S \eta = V - V_s,
\]

\[
V_{\bar\eta} = \text{\struck{$V \times_S \bar\eta$}} = V_\eta \times_\eta \bar\eta .
\]

\[
(4.1.2)\qquad \bar S \longrightarrow S, \qquad \bar S = \varprojlim S_\alpha ,
\]

\[
(4.1.3)\qquad \bar V = V \times_S \bar S = \varprojlim V_\alpha, \qquad
V_\alpha = V \times_S S_\alpha .
\]

\[
(4.1.4)\qquad V_{\bar\eta} \hookrightarrow \bar V ,
\]

\[
(4.1.5)\qquad V_{\bar s} = V_s \otimes_{k(s)} k(\bar s),
\]

\[
(4.1.6)\qquad V_{\bar s} \longrightarrow V_s
\]

\[
(4.4.1)\qquad (\underline{\Phi}^i_\nu)_{\bar x} = 0 \quad \text{pour } i < m ,
\]

\[
(4.4.2)\qquad H^i(V_s, \underline{\Phi}^\bullet_\nu) = 0 \quad \text{pour } i < m ,
\]

\[
(4.4.3)\qquad H^i(\bar V, \Lambda_\nu) \xrightarrow{\ \bar j^*\ } H^i(V_{\bar\eta}, \Lambda_\nu)
\quad \begin{cases} \text{injectif pour } i \leq m-1 \\ \text{bijectif pour } i \leq m-2 . \end{cases}
\]

\[
\mathbb{R}(\Gamma, I)(V_s, \underline{\Phi}^\bullet_\nu) = 0 ,
\]

\[
\mathbb{R}(\Gamma, I)(\bar V, \Lambda_\nu) \xrightarrow{\ \bar j^*\ }
\mathbb{R}(\Gamma, I)(V_{\bar\eta}, \Lambda_\nu)
\]

\[
H^i(\bar V, \Lambda_\nu) \xrightarrow[\sim]{\ \bar j^*\ } H^i(V_{\bar\eta}, \Lambda_\nu)
\quad \text{pour tout } i ,
\]

\[
(4.5.1)\qquad \mathbb{R}(\Gamma, I)(\bar V, \Lambda_\nu) \longrightarrow
\mathbb{R}(\Gamma, I)(V_s, \Lambda_\nu)
\]

\[
(5.1.1)\qquad (\underline{\Phi}^i_\nu)_{\bar x'} = 0
\]

\[
(*)\qquad (\underline{\Phi}^i_\nu)_{\bar x} = 0 \quad \text{si } i \leq m .
\]

\[
\text{\struck{$(5.1.1)\quad H^i(\bar V, \Lambda_\nu) \xrightarrow{\sim} H^i(X_{\bar\eta}, \Lambda_\nu)$ si $i < m'$}}
\]

\[
(5.1.2)\qquad (\underline{\Phi}^i_\nu)_{\bar x} = H^i_x(\bar X', \Lambda_\nu)
= \varinjlim_h H^i_x(X(h), \Lambda_\nu) \quad \text{pour } i \leq m' .
\]

\[
(5.1.3)\qquad \operatorname{prof}_x(X(h)) \geq m'+1 \quad \text{pour tout } h .
\]

\[
\operatorname{prof}_{\bar x'}(X(h)) \geq m'+1 ,
\]

\[
(5.2.1)\qquad (\underline{\Phi}^i_\nu)_x = 0 \quad \text{si } i \leq m' ,
\]

\[
(5.2.2)\qquad H^i(X_0, \Lambda_\nu) \longrightarrow H^i(X_{\bar\eta}, \Lambda_\nu)
\]

\[
(5.3.1)\qquad
\begin{cases}
\operatorname{prof\,et}^*_{x'}(X_0) \geq m'+1 & \text{si } x' \in Z \\
\operatorname{prof\,et}^*_{x'}(X_{\bar\eta}) \geq m' + \dim \overline{\{x\}} & \text{si } x' \in X_{\bar\eta} ,
\end{cases}
\]

\[
(5.3.2)\qquad
\begin{cases}
\operatorname{prof\,et}^x_{x'}(X_0) \geq m & \text{pour toute \add{générisation} } x' \in Z \text{ de } x \\
\operatorname{prof\,et}^x_{x'}(X_{\bar\eta}) \geq m-1 & \text{pour toute gén. $x'$ de $x$ dans $X$, avec } x' \in X_{\bar\eta} .
\end{cases}
\]

\[
\text{\struck{$\ill{}$}}\quad m' = \operatorname{prof\,géom}^*_x(X_0) - d ,
\]

\[
(*)\qquad \operatorname{prof\,et}_x(X) \geq m'+1 .
\]

\[
\operatorname{prof\,et}_{x'}(X') + \dim \overline{\{x'\}}^{(X_{\bar\eta})}
\geq m' + \dim \overline{\{x\}} ,
\]

\[
\dim \overline{\{x'\}}^{(X_{\bar\eta})} = \dim \overline{\{x'\}}^{X} - 1
= \dim \overline{\{x\}} + \dim \overline{\{\bar x'\}}^{(X')} - 1
\]

\[
\operatorname{prof\,et}_{x'}(V) + \dim \overline{\{x'\}}^{(V)} \geq m'
\ \text{\struck{$\ill{}$}} ,
\]

\[
H^i(V) \longrightarrow H^i(V_s) \quad \text{est}\quad
\begin{cases} \text{bijectif pour } i < m'-1 \\ \text{injectif pour } i = m'-1 . \end{cases}
\]

\[
(**)\qquad \text{\struck{$\operatorname{prof\,et}_{x'}(X) \geq m'+1$}}
\]

\[
\text{\struck{$\operatorname{prof\,et}_{x'}(X) \geq \operatorname{prof\,et}'_x$}}
\]

\[
\operatorname{prof\,géom}_{x'}(X_0) \geq \operatorname{prof\,géom}_x(X)
\ \text{\struck{\ill{}}}\ \operatorname{codim}(\overline{\{x\}}, \overline{\{x'\}})
\]

\[
\operatorname{prof\,géom}_{x'}(X_0) \geq \operatorname{prof\,géom}^*_x(X_0)
\ \text{\struck{\ill{}}}\ - d = m' .
\]

\[
\operatorname{prof\,géom}_{x'}(X) \geq \operatorname{prof\,géom}_{x'}(X_0) + 1 ,
\]

\[
\operatorname{prof\,géom}_{x'}(X) \geq m'+1 .
\]

\[
\operatorname{prof\,et}_{x'}(X) \geq \operatorname{prof\,géom}_{x'}(X) ,
\]

\[
\operatorname{prof\,géom} A_{\mathfrak{p}} \geq \operatorname{prof\,géom} A - \dim A/\mathfrak{p} .
\]

\[
\operatorname{prof\,géom} B \geq \operatorname{prof\,géom} A + \operatorname{prof\,géom} B \otimes_A k .
\]

\[
\underline{\Phi}^i_\nu = 0 \quad \text{pour}\quad
i \notin [n-d+1, n+d+1] ,
\]

\[
\underline{\Phi}^i_\nu = 0 \quad \text{si } i \neq n+1 .
\]

\[
\underline{\Phi}^i_\nu = 0 \quad \text{si } i \neq n+1 ,
\]

\[
H^i(X_0, \Lambda_\nu) \longrightarrow H^i(X_{\bar\eta}, \Lambda_\nu)
\]

\[
(5.5.1)\qquad H^i(\bar V, G) \xrightarrow{\ \sim\ } H^i(V_{\bar\eta}, G)
\quad \text{si } i \leq 1 .
\]

\[
(5.5.2)\qquad H^i(V(h), G) = 0 \quad \text{si } i \leq 1 .
\]

\[
H^1(X_0, G) \longrightarrow H^1(X_{\bar\eta}, G)
\]

\[
(6.1.1)\qquad \underline{\Phi}^i_\nu | V = 0, \ \text{i.e.}\ 
(\underline{\Phi}^i_\nu)_{\bar x'} = 0 \ \text{pour toute gén. } x' \neq x \text{ dans } X_0 .
\]

\[
V_{\bar\eta} \longleftarrow \bar V
\]

\[
(6.1.2)\qquad \mathbb{R}\Gamma(\bar V, \Lambda_\nu) \xrightarrow{\ \sim\ }
\mathbb{R}\Gamma(V_{\bar\eta}, \Lambda_\nu)
\]

\[
(6.1.3)\qquad H^i(\bar V, \Lambda_\nu) \simeq H^i(V_{\bar\eta}, \Lambda_\nu) .
\]

\[
(6.1.4)\qquad \mathbb{R}(\Gamma, \pi)(\bar V, \Lambda_\nu) \xrightarrow{\ \sim\ }
\mathbb{R}(\Gamma, \pi)(V_{\bar\eta}, \Lambda_\nu) .
\]

\[
(6.2.1)\qquad u = (k^*_\pi)^* : \mathbb{R}(\Gamma, \pi)(V_{\bar\eta}, \Lambda_\nu)
\longrightarrow \mathbb{R}(\Gamma, \pi)(V_{\bar s}, \Lambda_\nu) ,
\]

\[
V_{\bar s} \xrightarrow{\ \bar i\ } \bar V \overset{\mathrm{coh.}}{\simeq} V_{\bar\eta} ,
\]

\[
(6.2.5)\qquad \mathbb{R}\Gamma^\pi(\Lambda_\nu) \simeq \Lambda_\nu \oplus \Lambda_\nu(-1)[-1] .
\]

\[
(6.2.6)\qquad \mathbb{R}\Gamma^\pi(k^*) = (\alpha, \beta) :
\mathbb{R}\Gamma(V_\eta, \Lambda_\nu) = \mathbb{R}\Gamma^\pi(K^\bullet)
\]

\[
\longrightarrow \mathbb{R}\Gamma(V_s, \Lambda_\nu) + \mathbb{R}\Gamma(V_s, \Lambda_\nu(-1))
\]

\[
(6.2.7)\qquad \alpha = k^* h^* ,
\]

\[
(6.2.8)\qquad \beta : \mathbb{R}\Gamma(V_\eta, \Lambda_\nu) \longrightarrow
\mathbb{R}\Gamma(V_s, \Lambda_\nu)(-1)[-1] .
\]

\[
(6.3.1)\qquad \underline{H}^i_{V_s}(\Lambda_{\nu\,V}) \simeq
\begin{cases} 0 & \text{si } i \neq 2 \\ \Lambda_{\nu\,V_s}(-1) & \text{si } i = 2 , \end{cases}
\]

\[
(6.3.2)\qquad \mathbb{R}\underline{\Gamma}_{V_s}(\Lambda_{\nu\,V}) \simeq \Lambda_{\nu\,V_s}(-1)[-2] ,
\]

\[
(6.3.3)\qquad \mathbb{R}\Gamma_{V_s}(V, \Lambda_\nu) \simeq \mathbb{R}\Gamma(V_s, \Lambda_\nu)(-1)[-2] .
\]

\[
\begin{aligned}
\mathbb{R}j_*(\Lambda_{V_\eta}) &\simeq \mathbb{R}j_*\,\mathbb{R}\Gamma^\pi\,\mathbb{R}h_*(\Lambda_{V_{\bar\eta}}) \\
&\simeq \mathbb{R}\Gamma^\pi\,\mathbb{R}j_*\,\mathbb{R}h_*(\Lambda_{V_{\bar\eta}}) \\
&\simeq \mathbb{R}\Gamma^\pi\,\mathbb{R}g_*\,\mathbb{R}\bar j_*(\Lambda_{V_{\bar\eta}}) \\
&\xrightarrow{\ \sim\ } \mathbb{R}\Gamma^\pi\,\mathbb{R}g_*(\Lambda_{\bar V})
\end{aligned}
\]

\[
\begin{aligned}
i^*\mathbb{R}j_*(\Lambda_{V_\eta}) &\simeq \mathbb{R}\underline{\Gamma}^\pi\, i^*\mathbb{R}g_*(\Lambda_{\bar V}) \\
&\simeq \mathbb{R}\underline{\Gamma}^\pi\,\mathbb{R}g_{0*}(\Lambda_{V_{\bar s}}) \\
&\simeq \mathbb{R}\underline{\Gamma}^\pi(\Lambda_{V_s})^{\mathrm{triv}} \\
&\simeq \Lambda_{V_s} \oplus \Lambda_{V_s}(-1)[-1]
\end{aligned}
\]

\[
(6.5.1)\qquad n = \dim \mathcal{O}_{X_0,x}
\]

\[
(6.5.2)\qquad H^i(V_{\bar\eta}) = 0 \quad \text{si } i > n .
\]

\[
H^*(V_\eta) \Longleftarrow E_2^{pq} = H^p(\pi, H^q(V_{\bar\eta}))
\]

\[
(6.5.3)\qquad \cdots \longrightarrow H^{i-1}(V_{\bar\eta})_\pi \longrightarrow H^i(V_\eta)
\longrightarrow H^i(V_{\bar\eta})^\pi \longrightarrow 0 ,
\]

\[
(6.5.4)\qquad H^i(V_\eta) = 0 \quad \text{si } i > n+1 .
\]

\[
(6.5.5)\qquad \longrightarrow H^{i-2}(V_s)(-1) \longrightarrow H^i(V) \longrightarrow H^i(V_\eta)
\longrightarrow H^{i-1}(V_s) \longrightarrow H^{i-1}(V) \longrightarrow \cdots ,
\]

\[
(6.5.6)\qquad i_* : H^i(V, \Lambda) \longrightarrow H^{i-2}(V_s)
\]

\[
  H^{i}(X) \longrightarrow \prod H^{i}(Y_{\alpha}) \longrightarrow \prod H^{i}(Y_{\alpha\beta}) ,
\]

\[
  \cdots \longrightarrow H^{i}_{Y}(X') \longrightarrow H^{i}(X') \longrightarrow H^{i}(X) \longrightarrow ,
\]

\[
  H^{a}_{Y}(X) \Longleftarrow E_{2}^{pq} = H^{p}(Y, \underline{H}^{q}_{Y}(\mathbb{Q}_{\ell\,X'})) ,
\]

\[
  \underline{H}^{q}_{Y}(\mathbb{Q}_{\ell}) = \bigwedge^{q-1}\Bigl(\coprod_{\alpha} \mathbb{Q}_{\ell\,Y_{\alpha}}(-1)\Bigr)
  = \coprod_{\alpha_{1}<\dots<\alpha_{q-1}} \mathbb{Q}_{\ell\,Y_{\alpha_{1}\dots\alpha_{q-1}}}(-1)
\]

\[
  E_{2}^{pq} =
  \begin{cases}
    0 & \text{si } q < 2 \\
    \displaystyle\coprod_{\alpha_{1}<\dots<\alpha_{q-1}} H^{p}(Y_{\alpha_{1}\dots\alpha_{q-1}})(-(q-1)) & \text{si } q \geq 2
  \end{cases}
\]

\[
  d_{r} = 0 \quad \text{si } r \geq 3 .
\]

\[
  H^{*}(X) \Longleftarrow E_{2}^{pq} = H^{p}(X', R^{q}g_{*}(\mathbb{Q}_{\ell}))
\]

\begin{align*}
  R^{q}g_{*}(\mathbb{Q}_{\ell}) &\simeq \bigwedge^{q} \coprod_{\alpha} \mathbb{Q}_{\ell}(-1)_{Y_{\alpha}} \\
  &\simeq \coprod_{\alpha_{1}<\dots<\alpha_{q}} \mathbb{Q}_{\ell\,Y_{\alpha_{1}\dots\alpha_{q}}}(-q) .
\end{align*}

\[
  \begin{cases}
    E_{2}^{pq} = \displaystyle\coprod_{\alpha_{1}<\dots<\alpha_{q}} H^{p}(Y_{\alpha_{1}\dots\alpha_{q}})(-q) & \text{si } q \geq 1 \\
    E_{2}^{p0} = H^{p}(X')
  \end{cases}
\]

\[
  d_{r} = 0 \quad \text{si } r \geq 3 ,
\]

\[
  H^{*}(U) \Longleftarrow E_{2}^{pq} = H^{p}(X_{0}, R^{q}g_{*}(\mathbb{Q}_{\ell\,X})|X_{0})
\]

\[
  R^{q}g_{*}(\mathbb{Q}_{\ell\,X}) \simeq \bigwedge^{q} \coprod_{\alpha} (\mathbb{Q}_{\ell}(-1)_{Y_{\alpha}})
  \simeq \coprod_{\alpha_{1}<\dots<\alpha_{q}} \mathbb{Q}_{\ell\,Y_{\alpha_{1}\dots\alpha_{q}}}(-q)
\]

\[
  \begin{cases}
    E_{2}^{p,q} = \text{\struck{$H^{p}$}}\ \displaystyle\coprod_{\alpha_{1}<\dots<\alpha_{q}} H^{p}((Y_{\alpha_{1}\dots\alpha_{q}})_{0})(-q) & \text{si } q \geq 1 \\
    E_{2}^{p0} = H^{p}(X_{0})
  \end{cases}
\]

\[
  0 \longrightarrow H^{i-1}(X_{\bar\eta})_{I} \longrightarrow H^{i}(X_{\eta}) \longrightarrow H^{i}(X_{\bar\eta})^{I} \longrightarrow 0
  \tag{6}
\]