\[
  T \cap U \ni u_0 , \qquad g W^{-1} \cap T \ni g u_0
\]

\[
  e \subset G^0_{\mathrm{réd}} \subset G^0 \subset G
  \qquad (G \text{ qu.-cpt} \Leftrightarrow G/G^0 \text{ cpt})
\]

\[
  f : U \longrightarrow E^n_k \qquad (U \text{ vois.\ ouvert de } x)
\]

\[
  Z \to X, \qquad \mathbb{R} \longleftarrow \mathbb{R}[x,y]/\text{---},
  \qquad 0 \longleftarrow\!\shortmid x, y
\]

\[
  X' = \varprojlim X'_i = \varprojlim_i (\pi_i)_k = (\pi)_k , \qquad
  \pi_0(X') = \pi \quad (\text{isom.\ \underline{topologique}})
\]

\[
  \left\lbrace\begin{array}{l}
    \mathrm{Proét}(k) \simeq \pi\text{-esp.\ \uncertain{prof.}\ disc.} \simeq \text{\uncertain{Alg.\ sép.}}(k) \\
    \mathrm{Locproét}(k) \simeq \pi\text{-esp.\ loc.\ cpt disc.}
  \end{array}\right.
\]

\[
      X \longrightarrow \mathrm{Spec}(A) \overset{\text{déf}}{=} \underline{\pi}_0(X/k)
    \]

\[
  \underline{\mathrm{Hom}}_{\mathrm{gr}}(G, \mathbf{G}_m) \subset
  \underline{\mathrm{Hom}}_{\underline{O}\text{-alg}}(\underline{O}_S[T, T^{-1}], \mathcal{A})
  \simeq W(\mathcal{A})^{*},
\]

\[
  \pi(u) = u \otimes u , \qquad \varepsilon(u) = 1
\]

\[
  \check{A} = \mathrm{Dist}(X)(k) ,
\]

\[
  \mathrm{Dis}(X) \quad \bigl(\,= V(A)\ \text{comme Module sur } \mathcal{O}_k\bigr)
\]

\[
  \mathrm{Dist}(X) \boxtimes \mathrm{Dist}(Y) \xrightarrow{\ \sim\ } \mathrm{Dist}(X \times Y) .
\]

\[
    X \xrightarrow{\ \alpha_X\ } \mathrm{Dist}(X) \qquad [\text{distribution ponctuelle}],
  \]

\[
  D \xrightarrow{\ \Delta\ } D \boxtimes D \qquad \bigl[V(A) \boxtimes V(B) \overset{\mathrm{def}}{\simeq} V(A \otimes B)\bigr]
\]

\[
  [\text{NB } \boxtimes = \otimes \text{ si } A \text{ ou } B \text{ proj.\ de t.f.}]
\]

\[
  \boxed{\ \Delta(x) = x \boxtimes x\ }
\]

\[
  \mathrm{Dist}(X, k) = \mathrm{Dist}(X)(k),
\]

\[
  A = k(M) \quad \text{algèbre du groupe à coeff.\ dans } k
\]

\[
  \begin{array}{lcl}
    A \otimes A \longrightarrow A & \text{provient de} & M \times M \longrightarrow M \ \text{loi de groupe} \\
    A \otimes A \longleftarrow A & \text{---} & M \times M \longleftarrow M \ \text{diagonale}
  \end{array}
\]

\[
  \mathrm{Dist}(D(M))(k) = k^M
\]

\[
  \begin{cases} 1)\ 1x = x \\ 2)\ g(g'x) = gg'x \end{cases}
  \qquad \text{i.e.}\quad G \longrightarrow \underline{\mathrm{End}}_k(M) = \underline{\mathrm{End}}_{\mathcal{O}}(W(M))
\]

\[
  u : M_A \xrightarrow[A\text{-lin}]{} M_A, \qquad M_A = A \otimes_k M,
\]

\[
  u_0 : M \longrightarrow A \otimes_k M \qquad k\text{-lin.}
\]

\[
  A \mathrel{\substack{\xrightarrow{\ g\ } \\ \xrightarrow[\ g'\ ]{}}} A \otimes A
\]

\[
  g(\lambda) = \lambda \otimes 1, \qquad g'(\lambda) = 1 \otimes \lambda ,
\]

\[
  x \longmapsto g\,x', \qquad x \longmapsto g'\,x'
\]

\[
  (gg')_M\, x' = (gg')^{\circ}_M\, x = (\mathrm{id}_M \otimes gg') \circ u_0 :
\]

\[
  g_M(g'_M x') = \ ?
\]

\[
  g_M(y \otimes \lambda \otimes \mu) = g^{\circ}_M(y)\, \lambda \otimes \mu = (\mathrm{id}_M \otimes g)(u_0(y))\, \lambda \otimes \mu ,
\]

\[
  g_M(y \otimes 1 \otimes \lambda) = \underbrace{(\mathrm{id}_M \otimes g)\, u_0(y)}_{\in\, M \otimes A \otimes 1}\, 1 \otimes \lambda ,
\]

\[
  W(M) : k' \longmapsto \mathrm{Hom}_{k\text{-}\mathrm{Mod}}(M, k')
\]

\[
    \forall M, N \qquad \mathrm{Hom}_k(M, N) \xrightarrow[\sim]{\ \alpha_{M,N}\ } \mathrm{Hom}_{\mathcal{O}}(W(N), W(M))
  \]

\[
    V(M) \otimes_{\mathcal{O}} V(N) \longrightarrow V(M \otimes_k N) ,
  \]

\[
    \bigl[\mathrm{Hom}_k(M, k') \otimes_{k'} \mathrm{Hom}_{k'}(N', k') \simeq \mathrm{Hom}_{k'}(M' \otimes_{k'} N', k')\bigr]
  \]

\[
    \mathrm{Hom}_{\mathcal{O}}(V(M \otimes_k N), V(P)) \xrightarrow{\ \sim\ } \mathrm{Hom}_{\mathcal{O}}(V(M) \otimes_{\mathcal{O}} V(N), V(P))
  \]

\[
    V(A) \times V(M) \longrightarrow V(M)
  \]

\[
  \xi : A \longrightarrow k \quad (k\text{-lin}) \in \check{A}, \qquad x \in M,
\]

\[
  \xi \cdot x = (\mathrm{id}_M \otimes \xi)(u_0(x)),
\]

\[
  \xi_M = (\mathrm{id}_M \otimes \xi) \circ u_0 .
\]

\[
  M \otimes A \simeq \mathrm{Hom}_k(\check{A}, M) ,
\]

\[
  \check{A} \longrightarrow \mathrm{Hom}_k(M, M)
\]

\[
  M \otimes_k A \simeq \mathrm{Hom\,cont}_k(\check{A}, M),
\]

\[
  \mathrm{Hom}(N, M \otimes_k A) \simeq \mathrm{Hom\,cont}_k\bigl(\check{A}, \underbrace{\mathrm{Hom}_k(N, M)}_{\text{muni de la top.\ de la conv.\ simple}}\bigr)
\]

\begin{align*}
  \mathrm{Bil\,cont}_k(\check{P}, \check{Q}; \check{R})
  &\simeq \mathrm{Bil\,cont}_k\bigl(\check{P}, \check{Q}; \varprojlim_{\gamma} \check{R}_\gamma\bigr) \\
  &\simeq \varprojlim_{\gamma} \bigl[\mathrm{Bil\,cont}_k(\check{P}, \check{Q}; \check{R}_\gamma)\bigr] \\
  &\simeq \varprojlim_{\gamma} \Bigl[\varinjlim_{\alpha,\beta} \underbrace{\mathrm{Bil}_k(\check{P}_\alpha, \check{Q}_\beta; \check{R}_\gamma)}_{\mathrm{Hom}_k(\check{P}_\alpha \otimes \check{Q}_\beta,\, \check{R}_\gamma)}\Bigr] \\
  &\simeq \varprojlim_{\gamma} \varinjlim_{\alpha,\beta} \mathrm{Hom}_k(R_\gamma, P_\alpha \otimes Q_\beta) \\
  &\simeq \varprojlim_{\gamma} \mathrm{Hom}_k(R_\gamma, P \otimes Q) \\
  &\simeq \mathrm{Hom}_k(R, P \otimes Q) \qquad !\ ]
\end{align*}

\[
  V(N') \subset V(M) \quad \text{mono},
\]

\[
  M'' \overset{i}{\hookrightarrow} M
\]

\[
    G \longrightarrow \mathrm{M}(n)_k \qquad (\text{resp.\ } G \to \mathrm{GL}(n)_k)
  \]

\[
  G \longrightarrow \struck{\ill{}}\ \underline{\mathrm{Aut}}_k(M) \quad (\simeq \mathrm{GL}(n)_k)
\]

\[
  g \longmapsto \varepsilon(g \cdot \varphi) = (g\varphi)(1) = \varphi(g)
\]

\[
  G(k') \hookrightarrow \mathrm{Dist}(G)(k')^{\times}
\]

\[
  G(k') \hookrightarrow \mathrm{Dist}(G)(k')^{*} \qquad [\text{pratiquement jamais un isom}]
\]

\[
  \begin{array}{llll}
    \mathbb{G}_m,\ \mathbb{G}_a, & \mathrm{GL}(n), & O(Q), & \text{s-groupe de } \mathrm{GL}(n) \text{ invariant des tenseurs} \\
    & \mathrm{SL}(n) & \ \cap & \quad \ldots \\
    & & \mathrm{GL}(n) & \\
    & \mathrm{Tr}(n) & & \\
    & \mathrm{Tr}_0(n) & & \\
    \mathrm{Diag}_0(n) \subset \mathrm{Diag}(n) & & & \\
    \quad \wr & & & \\
    \mathbb{G}_m & & &
  \end{array}
\]

\[
  W(M) : k' \longmapsto M \otimes_k k' \qquad (\text{un faisceau } \mathcal{O}_k\text{-Module, pas seulement un groupe})
\]

\[
  M \otimes_k \prod k'_i \longrightarrow \prod_i M \otimes_k k'_i \quad \text{bijectif.}
\]

\[
  M \otimes k^I \xrightarrow{\ \sim\ } M^I .
\]

\[
  0 \to R \to L \to M \to 0
\]

\[
  M \otimes_k k' = \mathrm{Ker}\bigl((1 - p)_{k'} : k'^n \to k'^n\bigr)
\]

\[
  W(M) = \mathrm{Ker}\bigl(E^n \xrightarrow{\ 1-p\ } E^n\bigr)
\]

\[
  V(M) = \mathrm{Hom}_{k\text{-mod}}(M, k') = \mathrm{Hom}_{k'\text{-mod}}(M \otimes_k k', k')
\]

\[
  \Bigl[\ V(M) \times W(M) \longrightarrow \mathcal{O}_k = W(k) \qquad V(M) \xrightarrow{\ \sim\ } \underline{\mathrm{Hom}}_{\mathcal{O}_k}\bigl(W(M), W(k)\bigr)
\]

\[
  \mathrm{Mod}(k) \longrightarrow \mathrm{Mod}(\mathcal{O}_k) \quad \text{est \emph{pleinement fidèle}.}
\]

\[
  \underline{\mathrm{Aut}}_{\mathcal{O}}(M) \subset \underline{\mathrm{End}}_{\mathcal{O}}(M) \qquad \text{représentable si } M \text{ est proj.\ t.f.}
\]