\[
  \mathfrak{f} : G \longrightarrow G
\]

\[
  \varphi : P \longrightarrow P
\]

\[
  \varphi(xg) = \varphi(x)\,\mathfrak{f}(g) .
\]

\[
  \varphi_a(g) = a\,\mathfrak{f}(g)
\]

\[
  u\,a\,\mathfrak{f}(g) = b\,\mathfrak{f}(ug) \qquad \forall g \in G(S'),\
  S' \text{ sur } S,
\]

\[
  b^{-1}a = \mathfrak{f}(u)\,u^{-1} .
\]

\[
  u \longmapsto \mathfrak{f}(u)\,u^{-1} : G \longrightarrow G
\]

\[
  G/N \xrightarrow[\ \varphi\ ]{\text{isom}} G
\]

\[
  \boxed{\varphi(gx) = \mathfrak{f}(g)\,\varphi(x)\,g^{-1}}
\]

\[
  \varphi : E^{(p)} \longrightarrow E
\]

\[
  E = \Phi \otimes_{\mathbb{F}_p} \mathcal{O}_S
\]

\[
  \mathrm{Frob}(X) : X \longrightarrow X
\]

\[
  P_S(X) = X \times_S (S, \mathrm{Frob}(S)) \qquad
  P_S : (\mathrm{Sch})_{/S} \longrightarrow (\mathrm{Sch})_{/S}
\]

\[
  \mathrm{Frob}_S(X) : X \longrightarrow P_S(X)
\]

\[
  P_S(E) = E \otimes_{\mathcal{O}_X} \mathcal{O}_{P_S(X)}
  = E \otimes_{\mathcal{O}_S} (\mathcal{O}_S, \mathrm{Frob}(S)) ,
\]

\[
  P_S(E) = E \otimes_{\mathcal{O}_S} (\mathcal{O}_S, \mathrm{Frob}(S)) .
\]

\[
  \begin{cases}
    P_S(\operatorname{Spec}(\mathcal{A})) = \operatorname{Spec}(P_S(\mathcal{A})) \\
    P_S(\operatorname{Proj}(\mathcal{A})) = \operatorname{Proj}(P_S(\mathcal{A}))
  \end{cases}
\]

\[
  \begin{cases}
    P_S(\underline{V}(E)) = \underline{V}(P_S(E)) \\
    P_S(\underline{P}(E)) = \underline{P}(P_S(E))
  \end{cases}
\]

\[
  E^{(p)} \longrightarrow \mathrm{Symm}^{p}(E) \subset \mathrm{Symm}(E)
\]

\[
  \mathrm{Symm}\,E \longrightarrow (\mathrm{Symm}\,E)_{(p)}
  \qquad (\text{degrés multiples de } p)
\]

\[
  p(L) \simeq L^{\otimes p} .
\]

\[
  P_X(X') \simeq P_S(X') \times_{P_S(X)} (X, \mathrm{Frob}_S(X)) ,
\]

\[
  P_S(X) \times_S S' \simeq P_{S'}(X \times_S S')
\]

\[
  \bigl(\mathrm{Frob}^{\nu}_S(X)\bigr)^{k} ,
\]

\[
  \mathrm{Frob}^{\nu}_S(G) : G \longrightarrow P^{\nu}_S(G)
\]

\[
  U_{\nu}(G) = \operatorname{Ker}\bigl(\mathrm{Frob}^{\nu}_S(G)\bigr) .
\]

\[
  f^{-1}(V) = f'^{-1}(V)
\]

\[
  {}_p\mathrm{Pic}(X) = {}_pH^1(X, \underline{\mathcal{O}}_X^{*})
\]

\[
  0 \to \underline{\mathcal{O}}_X^{*} \xrightarrow{\ x \mapsto x^p\ }
  \underline{\mathcal{O}}_X^{*} \to
  \underline{\mathcal{O}}_X^{*} / \underline{\mathcal{O}}_X^{*p} \to 0
\]

\[
  0 \to A^{*} \xrightarrow{\ x \mapsto x^p\ } A^{*} \to
  \Gamma(X, \underline{\mathcal{O}}_X^{*}/\underline{\mathcal{O}}_X^{*p})
  \to {}_pH^1(X, \underline{\mathcal{O}}_X^{*}) \to 0
\]

\[
  {}_p\mathrm{Pic}(X) \simeq
  \Gamma(X, \underline{\mathcal{O}}_X^{*}/\underline{\mathcal{O}}_X^{*p})
\]

\[
  \underline{\mathcal{O}}_X^{*} \longrightarrow \underline{\Omega}^1_{X/k}
  \qquad f \longmapsto \frac{df}{f}
\]

\[
  \underline{\mathcal{O}}_X^{*}/\underline{\mathcal{O}}_X^{*p}
  \longrightarrow \underbrace{Z\underline{\Omega}^1_{X/k}}_{\text{formes fermées}}
\]

\[
  0 \to \underline{\mathcal{O}}_X^{*}/\underline{\mathcal{O}}_X^{*p}
  \longrightarrow Z\underline{\Omega}^1_{X/k}
\]

\[
  C\omega = \omega
\]

\[
  \underline{R}_X^{*}/\underline{\mathcal{O}}_X^{*} \longrightarrow
  \underline{\Omega}^1(\underline{K})/\underline{\Omega}^1(\underline{\mathcal{O}}_X)
\]

\[
  \underline{\Omega}^1_{X/S} \longrightarrow\ ?\ \struck{\ill{}}(\underline{\Omega}^1_{X/S})
\]

\[
  \boxed{\mathrm{Symm}^{p}(\underline{\Omega}^1_{X/S})}\ .
\]

\[
  C : \Omega \to \mathrm{Symm}^{p}_{\mathcal{O}_S}(\Omega), \qquad
  d_1 : \Omega \to {\textstyle\bigwedge}^{2}_{\mathcal{O}_S} \Omega
\]

\[
  X \xleftarrow{\ \mathrm{Fr}_X\ } X
\]

\[
  \mathrm{Fr}_{X*}(G) \xrightarrow{\ \mathrm{Fr}_{X*}(\alpha)\ }
  \mathrm{Fr}_{X*}\mathrm{Fr}_X^{*}(F) \xrightarrow[\ \sim\ ]{\ \mathrm{can}^{-1}\ } F ,
\]

\[
  G \xrightarrow[\ \sim\ ]{\ \mathrm{can}^{-1}\ }
  \mathrm{Fr}_X^{*}\mathrm{Fr}_{X*}(G) \xrightarrow{\ \mathrm{Fr}_X^{*}(\beta)\ }
  \mathrm{Fr}_X^{*}(F) .
\]

\begin{align*}
  &(1) \qquad F \xleftarrow{\ \mathrm{Fr}_{F/X}\ } \mathrm{Fr}_X^{*}(F) \qquad \text{i.e.} \\
  &(2) \qquad \mathrm{Fr}_{X*}(F) \xrightarrow{\ \mathrm{Fr}_{F/X*}\ } F
\end{align*}

\[
  \mathrm{Fr}_{X*}(F)(U) = F(\mathrm{Fr}_X^{*}(U)) = F(U^{(p/X)})
\]

\[
  U \xrightarrow[\ \sim\ ]{\ \mathrm{Fr}_{U/X}\ } U^{(p/X)}
\]

\[
  \mathrm{Fr}_{X*}(F)(U) = F(U^{(p/X)}) \xrightarrow{\ \sim\ } F(U)
\]

\[
  \boxed{\mathrm{Fr}_{F/X*} : \mathrm{Fr}_{X*}(F) \longrightarrow F}
\]

\[
  \struck{\mathrm{Fr}_{F/X}^{*} : \ill{}\,F \to \mathrm{Fr}_X^{*}F} \qquad
  \struck{\mathrm{Fr}_X^{*}\mathrm{Fr}_{X*}(F) \nearrow}
\]

\[
  g^{*}(\mathrm{Fr}_X^{*}(F)) \xleftarrow{\ g^{*}(\mathrm{Fr}_{F/X}^{*})\ }
  g^{*}(F) \overset{\text{df}}{=} F'
\]

\[
  \mathrm{Fr}_{X'}^{*}(g^{*}(F)) = \mathrm{Fr}_{X'}^{*}(F')
  \xleftarrow{\ \mathrm{Fr}_{F'/X'}^{*}\ } F'
\]

\[
  \mathrm{Fr}_X^{*}(I_X) \simeq I_X
  \xleftrightarrow{\ \mathrm{Fr}_{F/X}^{*} = \mathrm{id}_{I_X}\ } I_X
\]

\[
  H^0(X, F) \longrightarrow H^0(X, F), \qquad
  H^0(X, F) \simeq H^0(X, \mathrm{Fr}_{X*}(F))
  \xrightarrow{\ H^0(\mathrm{Fr}_{F/X})\ } H^0(X, F)
\]

\[
  H^1(X, F) \simeq H^1(X, \mathrm{Fr}_{X*}(F))
  \xrightarrow{\ H^1(\mathrm{Fr}_{F/X})\ } H^1(X, F)
\]

\[
  R\Gamma_X(F^{\bullet}) \simeq R\Gamma_X(\mathrm{Fr}_{X*}(F^{\bullet}))
  \xrightarrow{\ R\Gamma_X(\mathrm{Fr}_{F^{\bullet}/X})\ } R\Gamma_X(F^{\bullet})
\]

\[
  H^0(X, F) \xrightarrow{\ H^0(\mathrm{Fr}_{F/X})\ } H^0(X, F)
\]

\[
  Y_{\text{ét}} \xrightarrow{\ f^{*}\ } X_{\text{ét}}
\]

\[
  \mathbf{F}_X^{*}(I_{X'}) = I_X \xrightarrow[\ \mathrm{id}_{I_X}\ ]{\ \mathbf{F}_{F/X}^{*}\ } I_X
\]

\[
  \underline{\mathrm{Syst.Lin}}_{X/S} \xrightarrow{\ \lambda\ }
  \underline{\mathrm{Pic}}_{X/S}
\]

\[
  \underline{L} \qquad\qquad
  \underline{L} \otimes_{\mathcal{O}_S} \mathcal{O}_{\check{P}}(1) \otimes \underline{N}
\]

\[
  \underline{N} \simeq \underline{L} \otimes \mathcal{O}_{\check{P}}(1)
\]

\[
  \underline{N} \otimes_{\mathcal{O}_S} \underline{L}^{-1} = \mathcal{M}
\]

\[
  k \text{ pl.\ fid} \Longrightarrow \alpha \text{ pl.\ fid}
  \Longleftrightarrow
  \bigl[\sigma\alpha : \mathrm{Progr}^{!}(C'_0) \to
  \mathrm{Gr\,qu.cpt}(\mathrm{Sch}_{/k}) \text{ pl.\ fid.}\bigr]
\]

\[
  \varphi : \varprojlim_j \varinjlim_i \mathrm{Hom}_{\mathrm{gr}}(G_i, H_j)
  \longrightarrow
  \Bigl(\varprojlim_j \varinjlim_i \mathrm{Hom}(G_i, H_j)\Bigr)_{\mathrm{mult.}}
\]

\[
  u_j : (G_i)_i \longrightarrow H_j
\]

\[
  G_{i_2} \longrightarrow X_{j_1} \longrightarrow G_{i_1} \longrightarrow G_{i_0}
  \qquad \Big/ \qquad
  X = \varprojlim G_i \longrightarrow G_{i_1} \longrightarrow G_{i_0}
\]

\[
  \Phi^{n}(X) \overset{\text{can.}}{\simeq}
  \mathbb{P}\bigl(\underline{S}^{n}_{\mathcal{O}_X}(\check{\underline{E}})^{\vee}\bigr)
\]

\[
  \Phi^{(n)}(X)(Y) = \Gamma\bigl(Y, \underline{S}^{n}(\underline{E})^{*}/\mathcal{O}_Y^{*}\bigr)
\]

\[
  \Phi^{(n)}(\mathbb{P}(\underline{E})) \xrightarrow{\ \sim\ } \mathcal{D}^{(n)}(X/Y) .
\]

\[
  0 \to \mathcal{O}_X \xrightarrow{\ \varphi\ } \mathcal{O}_X \to
  \mathcal{O}_{X'} \to 0 ,
\]

\[
  0 \to \mathcal{O}_X \otimes k(z) \xrightarrow{\ \varphi\ }
  \mathcal{O}_X \otimes k(z) \to \mathcal{O}_{X'} \otimes k(z) \to
\]

\[
  0 \to \underline{I} \otimes k(y) \to \mathcal{O}_X \otimes k(y) \to
  \mathcal{O}_{X'} \otimes k(y) \to 0 ,
\]

\[
  0 \to \mathcal{O}_X \otimes k(y) \xrightarrow{\ \varphi\ }
  \mathcal{O}_X \otimes k(y) \to \mathcal{O}_{X'} \otimes k(y) \to 0
\]

\[
  \partial(\gamma) \in H^2(X, \mathbb{Z}/n\mathbb{Z})
\]

\[
  \mathrm{GP}(m-1) \times \mathrm{GP}(n-1) \longrightarrow \mathrm{GP}(mn-1)
\]

\[
  \mathbb{B}(X) = \coprod_{n \geqslant 1} H^1(X, \mathrm{GP}(n-1))
\]

\[
  \mathcal{A} \otimes \mathcal{A}^{\circ} \simeq L(\mathcal{A}, \mathcal{A}) :
\]

\[
  \delta(ab) = i_m\,\delta(a) + i_n\,\delta(b)
\]

\[
  \mathbb{B}(X) \longrightarrow H^2(X, \mathbb{Q}/\mathbb{Z})
\]

\[
  \boxed{\mathrm{Br}(X) \xrightarrow{\ \varphi\ } H^2(X, \underline{\mathbb{C}}^{*})}
\]

\[
  0 \to H^2(\pi, \mathbb{Z}/n\mathbb{Z}) \to H^2(X, \mathbb{Z}/n\mathbb{Z})
  \to H^2(\widetilde{X}, \mathbb{Z}/n\mathbb{Z})^{\pi} \to \cdots
\]

\[
  C^{*}(T/S, M_1) \simeq C^{*}(T'/S', M')
\]

\[
  H^{*}(T/S, M_1) \simeq H^{*}(T'/S', M') ,
\]

\[
  \varinjlim_{T} H^{*}(T/S, M_1) \simeq \varinjlim_{T} H^{*}(T'/S', M') ,
\]

\[
  e \to \mathbb{G}_m \to (\mathbb{G}_m)_1 \to U \to e ,
  \qquad (\mathbb{G}_m)_1 = \underline{\mathrm{Hom}}_S(S', \mathbb{G}_m)
\]

\[
  H^1(k, U) \simeq \operatorname{Ker}\bigl(H^2(k, \mathbb{G}_m) \to
  H^2(k', \mathbb{G}_m)\bigr)
\]

\[
  H^1(k, U) \simeq \mathbb{Z}/p\mathbb{Z} \neq 0
\]

\[
  H^1(k, (\mathbb{G}_m)_1) \simeq H^1(k, \mathbb{G}_m) \times H^1(k, U)
\]

\[
  e \to V \to (\mathbb{G}_m)_1 \to \mathbb{G}_m \to e
\]

\[
  V \simeq \prod_{S'/S} (\mu_p)_{S'} .
\]

\[
  0 \to \mu_p \to V \to U \to 0
\]

\[
  Z_1 = V
\]

\[
  x = \sum_{\gamma \in \Gamma} \varphi(\gamma)\,
  \sigma(\gamma).A \qquad (A \in \underline{A}^{L})
\]

\[
  \sigma(\gamma)x = \sum_{\gamma' \in \Gamma}
    \sigma(\gamma)\varphi(\gamma')\,\sigma(\gamma\gamma').A
\]

\begin{align*}
  &= \sum_{\gamma' \in \Gamma} \sigma(\gamma)\varphi(\gamma^{-1}\gamma')\,
    \sigma(\gamma').A \\
  &= \sigma(\gamma)\varphi(\gamma^{-1}) \sum_{\gamma' \in \Gamma}
    \varphi(\gamma')\,\sigma(\gamma').A = \sigma(\gamma)\varphi(\gamma^{-1})x
\end{align*}

\[
  x = \varphi(\gamma^{-1})\,\sigma(\gamma)^{-1}x \qquad
  x = \varphi(\gamma)\,\sigma(\gamma)x
\]

\[
  \varphi(\gamma) = x\,(\sigma(\gamma)x)^{-1}
\]

\[
  P\Bigl(\sum_{\gamma} \varphi(\gamma)\bigl(\psi(\gamma).A^{\gamma}\bigr)\Bigr) = 0
\]

\[
  \bar{P}((A_\gamma)) = P\Bigl(\sum_{\gamma} \varphi(\gamma)\bigl(\psi(\gamma).
  A_{\gamma}^{\gamma}\bigr)\Bigr)
\]

\begin{gather*}
  \varphi(u^0) = 1 \\
  \varphi(u) = a \\
  \varphi(u^2) = a\,\sigma(u)a \\
  \varphi(u^3) = a\,\sigma(u)a\,\sigma(u)^2 a \\
  \ldots \\
  \varphi(u^{n-1}) = a\,\sigma(u)a \cdots \sigma(u)^{n-2}a
\end{gather*}

\[
  H^1(\Gamma, G^{L\psi}) = \lbrace e \rbrace
\]

\[
  \psi^{(u)}\psi^{(u^2)} \cdots \psi^{(u^{n-1})} = 1
\]

\[
  (*) \qquad \tilde{\psi}^{\tilde{u}}\,\tilde{\psi}^{\tilde{u}^2} \cdots
  \tilde{\psi}^{\tilde{u}^{n-1}} = 1
\]

\[
  (**) \qquad a\,a^{v}a^{v^2} \cdots a^{v^{n-1}} = 1
\]

\[
  a^{v^k} = a^{\tilde{v}^k} = b^{\tilde{v}^k}\bigl(b^{\tilde{v}^{k+1}}\bigr)^{-1}
\]

\[
  b\bigl(b^{\tilde{v}^n}\bigr)^{-1} = 1 \qquad \text{soit} \qquad
  b^{\tilde{v}^n} = b
\]

\[
  \tilde{v}^{n} = \bigl(\tilde{\psi}\tilde{\psi}^{\tilde{u}} \cdots
  \tilde{\psi}^{\tilde{u}^{n-1}}\bigr)\tilde{u}^{n} = \tilde{u}^{n}
\]

\[
  x \longmapsto x\,\bigl(\psi x^{(q)}\bigr)^{-1}
\]

\begin{align*}
  H^0(\mathbb{Z}_n, G_n) &= G_1 \quad \text{ens.\ des éléments de $G$
    invariants par $\mathbb{Z}$} \\
  H^1(\mathbb{Z}_n, G_n) &= \text{ens.\ des éléments $\alpha$ de $G_n$ tels
    que} \\
  &\qquad \alpha\,T\alpha\,T^{2}\alpha \cdots T^{n-1}\alpha = 1 ,
\end{align*}

\[
  e \to H^1(\mathbb{Z}_n, G_n) \xrightarrow{\ \mathrm{inf}\ }
  H^1(\mathbb{Z}_m, G_m) \to H^1(N_{n,m}, G_m)^{\mathbb{Z}_m} ,
  \qquad N_{n,m} \simeq \mathbb{Z}_{m/n}
\]

\[
  H^i(\mathbb{Z}_n, G_n) \to H^i(\mathbb{Z}_m, G_m)
\]

\[
  u^{-1}\alpha\,Tu\;Tu^{-1}\,T\alpha\,T^{2}u\;T^{2}u^{-1}\,T^{2}\alpha\,T^{3}u
  \cdots T^{n-1}u^{-1}\,T^{n-1}\alpha\,T^{n}u = 1
\]

\[
  H^1(T, G) = \varinjlim H^1(\mathbb{Z}_n, G_n)
\]

\[
  H^1(T, G) \to H^1(T, G'') \quad \text{est \emph{injectif}}
\]

\begin{align*}
  &= t\,\bigl(\alpha\,T(t)\,\alpha^{-1}\bigr)
  \bigl(\alpha\,T\alpha\,T^{2}(t)\,T(\alpha)^{-1}\alpha^{-1}\bigr) \cdots \\
  &\qquad \bigl(\alpha\,T\alpha \cdots T^{n-2}\alpha\,T^{n-1}(t)\,T^{n-2}\alpha^{-1}
  \cdots T\alpha^{-1}\alpha^{-1}\bigr)\,\alpha\,T\alpha \cdots T^{n-1}\alpha
\end{align*}

\[
  t\,\sigma_1 T(t)\,\sigma_2 T^{2}(t) \cdots \sigma_{n-1}T^{n-1}(t)
\]

\[
  x \longmapsto x\,\sigma_1 Tx \cdots \sigma_{n-1}T^{n-1}x
\]

\[
  H^1(X, G) \to H^1(X, G'')
\]

\[
  \gamma(gm) = (\gamma g)(\gamma m)
\]

\[
  f(\gamma\gamma') = f(\gamma).\gamma f(\gamma')
\]

\[
  \gamma_f\, m = f(\gamma).\gamma.m
\]

\begin{align*}
  [\text{car} \quad \gamma_f(\gamma'_f m) &= f(\gamma)\gamma\bigl(f(\gamma')
    \gamma' m\bigr) = f(\gamma)\,\gamma f(\gamma')\,\gamma\gamma' m \\
  &= f(\gamma\gamma')\,\gamma\gamma' m = (\gamma\gamma')_f\, m \\
  e_f\, m &= f(e).e.m = m \ ]
\end{align*}

\[
  \gamma(\sigma(g).g') = \sigma(\gamma g)\,\gamma g'
\]

\[
  \gamma_f\, g = f(\gamma)\,\gamma g\, f(\gamma)^{-1}
\]

\[
  \gamma_f(gm) = \gamma_f(g)\,\gamma_f(m)
\]

\[
  f(\gamma)\,\gamma(gm) = f(\gamma)\,\gamma g\,\gamma(m)
\]

\[
  f(\gamma)\,\gamma g\, f(\gamma)^{-1} f(\gamma)\,\gamma(m)
\]

\[
  f(\gamma)\,\gamma m = m \quad \text{pour tt } \gamma \qquad \text{i.e.}
\]

\[
  \boxed{\gamma m = f(\gamma)^{-1} m \quad \text{pour tt } \gamma \in \Gamma}
\]

\[
  u_a m = am
\]

\[
  u_a\, \gamma_{f'} m = \gamma_f\, u_a m
\]

\[
  \text{i.e.} \qquad a f'(\gamma)\,\gamma m = f(\gamma)\,\gamma(am)
\]

\[
  \text{i.e.} \qquad a a^{-1} f(\gamma)\,\gamma(a)\,\gamma(m) =
  f(\gamma)\,\gamma(am) \ ;
\]

\[
  N_f\, m = \sum_{\gamma \in \Gamma} \gamma_f\, m
  = \sum_{\gamma \in \Gamma} f(\gamma)\,\gamma.m
\]

\[
  N_f u = \sum_{\gamma \in \Gamma} f(\gamma)\,\gamma(u)
\]

\[
  x \in M_f^{\Gamma} \qquad \text{i.e.} \qquad f(\gamma)\,\gamma(x) = x
  \quad \text{si } x = N_f u
\]

\[
  \gamma(\lambda a) = (\gamma.\lambda)(\gamma.a)
\]

\[
  P\Bigl(\sum f(\gamma)\,\gamma a\Bigr) \neq 0
\]

\[
  P\Bigl(\sum f(\gamma)\,\gamma(a_\gamma)\Bigr) =
  R\Bigl(\bigl\lbrace \gamma a^1_\gamma, \ldots, \gamma a^n_\gamma
  \bigr\rbrace_{\gamma \in \Gamma}\Bigr)
\]

\[
  P\Bigl(\sum f(\gamma)\,\gamma(a_\gamma)\Bigr) = 0
\]

\[
  H^1(\Gamma, A^{*}) \longrightarrow H^1(\Gamma_d, A^{*}_{\mathfrak{m}})
  \longrightarrow H^1(\Gamma_i, A^{*}_{\mathfrak{m}})
\]

\[
  \hat{\mathcal{O}} = \bigoplus_i \hat{\mathcal{O}}_{\mathfrak{m}_i}
  \qquad\qquad
  \hat{A} \simeq \bigoplus_i \underbrace{\hat{A}_{\mathfrak{m}_i}}_{B_i}
\]

\[
  H^1(\Gamma, \hat{A}^{*}) \simeq H^1(\Gamma_d, \hat{A}_{\mathfrak{m}}^{*})
\]

\[
  g(\gamma) = u^{-1} f(\gamma)\,\gamma(u) \ :
\]

\[
  \varphi : A \longrightarrow A^{\Gamma}
\]

\[
  \varphi(u) = \bigl(u\,g(\gamma) - f(\gamma)\,\gamma(u)\bigr)_{\gamma \in \Gamma}
\]

\[
  g(\gamma) = v^{-1} f(\gamma)\,\gamma(v) .
\]

\[
  H^1(\Gamma, A^{*}) \to H^1(\Gamma_d, A_{\mathfrak{m}}^{*})
  \quad\text{et}\quad
  H^1(\Gamma, A^{*}) \to H^1(\Gamma, \hat{A}^{*})
  \quad\text{sont injectifs.}
\]

\[
  \text{si } f \sim g \ \mathrm{mod}\ \mathfrak{m}^n A
  \quad \text{alors} \quad f \sim g \ \mathrm{mod}\ \mathfrak{m}^{n+1} A
\]

\[
  e \longrightarrow N_n \longrightarrow (A/\mathfrak{m}^{n+1}A)^{*}
  \longrightarrow (A/\mathfrak{m}^{n}A)^{*} \longrightarrow 0
\]

\[
  g(\gamma) = u^{-1} f(\gamma)\,\gamma(u) \qquad \gamma \in \Gamma_i
  \qquad (*)
\]

\[
  \gamma_f(x) = f(\gamma)\,\gamma.x\, f(\gamma)^{-1}
\]

\[
  \text{on a donc} \qquad
  \gamma_f(\lambda x) = f(\gamma)\,\gamma(\lambda)\,\gamma(x)\,f(\gamma)^{-1}
  = \gamma(\lambda)\,\gamma_f(x)
\]

\[
  g(\gamma) f(\gamma)^{-1} = u^{-1} \gamma_f(u) \qquad \gamma \in \Gamma_i
\]

\[
  g(\gamma) f(\gamma)^{-1} = v^{-1} \gamma_f(v) \qquad
  \text{pour \emph{tt} } \gamma \in \Gamma .
\]

\[
  h(\gamma) = g(\gamma) f(\gamma)^{-1}
\]

\[
  e \longrightarrow H^1(\Gamma/\Gamma_i, G^{\Gamma_i}) \longrightarrow
  H^1(\Gamma, G) \longrightarrow H^1(\Gamma_i, G)
\]

\[
  H^1(\Gamma/\Gamma_i, G^{\Gamma_i}) = \lbrace e \rbrace
\]

\[
  H^1(\Gamma, A^{*}) = (e)
\]

\[
  H^1(\Gamma, A/\mathfrak{m}^n A^{*}) = \lbrace e \rbrace
\]

\[
  e \longrightarrow N_n \longrightarrow A/\mathfrak{m}^{n+1}A^{*}
  \longrightarrow A/\mathfrak{m}^{n}A^{*} \longrightarrow e
\]

\[
  H^1(\Gamma, N_n) \longrightarrow H^1(\Gamma, A/\mathfrak{m}^{n+1}A^{*})
  \longrightarrow H^1(\Gamma, A/\mathfrak{m}^{n}A)^{*}
\]

\[
  H^1(\Gamma, A/\mathfrak{m}^{n+1}A^{*})^{*} = \lbrace e \rbrace .
\]

\[
  H^1(\Gamma, A^{*})
\]

\[
  H^1(\Gamma, A^{*}) \longrightarrow H^1(\Gamma, A/\mathfrak{m}A^{*})
  \times H^1(\Lambda, A^{*})
\]

\[
  H^1(\Gamma, A^{*}) \longrightarrow H^1(\Gamma, A/\mathfrak{m}A^{*})
  \quad\text{est injectif)}
\]

\[
  H^1(\Gamma, \mathfrak{m}^{n}A/\mathfrak{m}^{n+1}A) \longrightarrow
  H^1(\Gamma, A/\mathfrak{m}^{n+1}A^{*})
\]

\[
  1 + g(\gamma)
\]

\[
  g(\gamma) \in \mathfrak{m}^{n}A \quad \text{satisfait à} \qquad
  g(\gamma\gamma') \equiv g(\gamma) + \gamma g(\gamma') \quad
  (\mathfrak{m}^{n+1}A)
\]

\[
  1 + g(\gamma) = u^{-1}\gamma(u) \qquad \text{pour } \gamma \in \Lambda
\]

\[
  \text{i.e.} \qquad \gamma(u) - u = u\,g(\gamma) \qquad
  \text{pour } \gamma \in \Lambda .
\]

\[
  g(\gamma) = \gamma(v^{-1}u) - v^{-1}u
\]

\[
  H^1(\Gamma, A) \longrightarrow H^1(\Lambda, A)
\]

\[
  g(\gamma) = \gamma(w) - w \qquad \text{pour \emph{tout} } \gamma \in \Gamma .
\]

\[
  g(\gamma) = \varepsilon(w)\,g(\gamma) = w\,g(\gamma), \quad \text{d'où}
\]

\[
  g(\gamma) = w^{-1}\gamma(w) - 1 \quad \text{d'où}
\]

\[
  1 + g(\gamma) = w^{-1}\gamma(w)
\]