\[
  (\mathrm{v}) \Leftrightarrow (\mathrm{iv\ bis}) \Longrightarrow
  (\mathrm{iii\ ter}) \Longrightarrow (\mathrm{i}) \Leftrightarrow
  (\mathrm{ii}) \Leftrightarrow (\mathrm{iii}) \Leftrightarrow
  (\mathrm{iii\ bis}) \Leftrightarrow (\mathrm{iv}) \Leftrightarrow
  (\mathrm{vi}) \; ;
\]

\[
  \dim V + \mathrm{rang}\, \mathfrak{n} \leqslant \dim(G \times W)
  \leqslant \dim_{(e,a)} \phi^{-1}(a) + \dim V
  \leqslant \dim \mathfrak{m}_a + \dim V ,
\]

\[
  (\mathrm{ii}) \Leftrightarrow (\mathrm{iii}) \Leftrightarrow (\mathrm{iv})
  \Longrightarrow (\mathrm{i}) \Leftrightarrow (\mathrm{i\ bis})
  \Leftrightarrow (\mathrm{i\ ter}) ,
\]

\[
  \varphi : X \times Y \to \mathbf{G}_m \quad \text{tel que} \quad
  \varphi(x, y) = 1,
\]

\[
  \varphi(x, y) = \varphi_1(x)\, \varphi_2(y)
\]

\[
  G \longrightarrow H
\]

\[
  G/G^{\circ}_{\mathrm{réd}} \to H/H^{\circ}
\]

\[
  0 \to C^{\cdot}(e, R) \to C^{\cdot}(H, R) \longrightarrow
  \mathrm{Hom}_{\mathrm{gr}}(H^{\cdot}, R) \to 0
\]

\[
  H^{i}(\mathrm{Hom}_{\mathrm{gr}}(H^{\cdot}, R))
\]

\[
  \mathrm{Pic}(G) \simeq \mathrm{Pic}(G/B)/\mathrm{Im}\, \mathbf{D}(T)
\]

\[
  \mathbf{D}(T) \to \mathrm{Pic}(G/B), \qquad \mathbf{D}(T) \simeq
  \mathrm{Hom}(B, \mathbf{G}_m)
\]

\[
  \begin{array}{ll}
    G/R \to G/B, & T/R \subset G/R \\
    G \to G/B, & T \subset G
  \end{array}
\]

\[
  G/R = GP(n) \to \text{Dém.}, \qquad
  G = GL(n) \to \text{Dém.}, \quad T(n) \subset G
\]

\[
  \left\{
  \begin{array}{ll}
    (\mathrm{vii}) \not\Rightarrow (\mathrm{vi}) & = \text{si } \psi
      \text{ birationnel} : \mathrm{Sl}(2, k), \text{ car. } 2 \\[3pt]
    (\mathrm{x}) \not\Rightarrow (\mathrm{v}) & [\mathrm{x} \not\Rightarrow
      (\mathrm{i}),\ (\mathrm{x}) \not\Rightarrow (\mathrm{xiii}),\
      (\mathrm{vi}) \Rightarrow (\mathrm{v})] \\
      & \text{Cas d'un système de } G \text{ semi-simple} \ldots \\[3pt]
    (\mathrm{i}) \not\Rightarrow (\mathrm{x}) & [(\mathrm{v}) \not\Rightarrow
      (\mathrm{x}),\ (\mathrm{vi}) \not\Rightarrow (\mathrm{x}),\
      (\mathrm{i}) \not\Rightarrow (\mathrm{xiii})] \\
      & GP(2, k), \text{ car. } 2
  \end{array}
  \right.
\]

\[
  (\mathrm{vii}) \not\Rightarrow (\mathrm{vi}), \qquad
  (\mathrm{x}) \not\Rightarrow (\mathrm{v}), \qquad
  (\mathrm{v}) \not\Rightarrow (\mathrm{i})
\]

\[
  P_i(x, t) = \sum c_i(x)\, t^{n-i}
\]

\[
  \underbrace{t^{n} - c_1(x)\, t^{n-1} \ldots \pm c_n}_{[t - \lambda(x)]^{n}} = 0
  \qquad \lambda(x)^{n} = c_n
\]

\[
  \mathfrak{L} \subset \mathfrak{g}, \qquad \mathfrak{L} + k\,
  \mathfrak{g}_{\alpha}, \qquad \sum_{\beta \neq \alpha} \lambda_{\beta}
  X_{-\beta}
\]

\[
  \Bigl[ X_{-\alpha}, \sum_{\beta \neq \alpha} \lambda_{\beta} X_{-\beta}
  \Bigr] = \sum \lambda_{\beta} [X_{-\alpha}, X_{-\beta}], \qquad
  [X_{\alpha}, X_{\beta}]
\]

\[
  \begin{array}{l}
    (\mathrm{xiv}) \Rightarrow (\mathrm{i}) \Rightarrow (\mathrm{iii})
      \Rightarrow (\mathrm{v}) \\
    (\mathrm{xiv}) \searrow (\mathrm{x}?) \Rightarrow (\mathrm{xii})
  \end{array}
\]

\[
  \bigcup_{g \in G(k)} \mathrm{ad}(g)\, \mathfrak{L} = \mathfrak{g}
\]

\[
  \mathfrak{u} = \mathfrak{g}, \quad B \subset N \neq G, \quad \text{donc}
  \quad \dim N = \dim B = \mathrm{rang}\, \mathfrak{h},
\]

\[
  [X, Y] = 2H = 0, \qquad [H, X] = X, \qquad [H, Y] = Y.
\]

\[
  \Updownarrow
\]

\[
  [x, a] \in \mathfrak{h} \Rightarrow x \in \mathfrak{h}, \qquad
  \mathfrak{m}_a = \mathfrak{h}
\]

\[
  \Downarrow
\]

\[
  \mathfrak{h} = \mathfrak{u}
\]

\[
  \mathfrak{h} = kH + k(\lambda X + \mu Y),
\]

\[
  \bigcup \mathrm{Ker}\, \alpha_i = \mathbb{F}_p^2 .
\]

\[
  \chi_i : T = \mathbf{G}_m^2 \longrightarrow \mathbf{G}_m
\]

\[
  G = T \cdot U \struck{\mathbf{G}_a^{I}}, \qquad U = \mathbf{G}_a^{I}
\]

\[
  G_0 = F_k, \qquad G_1 = E_K .
\]

\[
  Z(\hat{A}) = \{e\}, \qquad \varprojlim Z(A_n) = F .
\]

\[
  e \longrightarrow H \longrightarrow G \longrightarrow A \longrightarrow e
\]

\[
  \rho_{\mathrm{ab}}(G) = \dim A \qquad \rho_{\mathrm{aff}}(G) = \dim H
\]

\[
  \rho_{\mathrm{r}}(G) = \rho_{\mathrm{r}}(H) = \text{dim tores maximaux}
\]

\[
  \rho_{\mathrm{s}}(G) = \rho_{\mathrm{s}}(H) =
  \rho_{\mathrm{r}}(H/\mathrm{rad}(H))
\]

\[
  d_{\mathrm{s}}(G) = d_{\mathrm{s}}(H) = \dim(H/\mathrm{rad}(H)),
\]

\[
  \rho_{\mathrm{n}}(G) = \rho_{\mathrm{u}}(V) + \rho_{\mathrm{ab}}(G) =
  \text{dimension des Cartan}
\]

\[
  \rho_{\mathrm{na}}(G) = \rho_{\mathrm{u}}(V)
\]

\[
  \rho_{\mathrm{u}}(G) = \rho_{\mathrm{u}}(V) = \rho_{\mathrm{na}}(G) -
  \rho_{\mathrm{r}}(H)
\]

\[
  \left\{
  \begin{array}{l}
    \rho_{\mathrm{rs}}(G) = \rho_{\mathrm{rs}}(H) =
      \rho_{\mathrm{r}}(H/\mathrm{rad}(H)) \\[2pt]
    \rho_{\mathrm{rad}}(G) = \rho_{\mathrm{r}}(\mathrm{rad}(H))
  \end{array}
  \right.
\]

\[
  \rho_{\mathrm{na}} = \rho_{\mathrm{r}} + \rho_{\mathrm{u}}
\]

\[
  \rho_{\mathrm{n}} = \rho_{\mathrm{na}} + \rho_{\mathrm{ab}} =
  \rho_{\mathrm{r}} + \rho_{\mathrm{u}} + \rho_{\mathrm{ab}}
\]

\[
  \left\{
  \begin{array}{l}
    \rho_{\mathrm{aff}} = \rho_{\mathrm{ss}} + \rho_{\mathrm{rad}} \\
    \dim G = \rho_{\mathrm{ab}} + \rho_{\mathrm{aff}} = \rho_{\mathrm{ab}} +
      \rho_{\mathrm{ss}} + \rho_{\mathrm{rad}}
  \end{array}
  \right.
\]

\[
  {}_n A_0 \qquad T_0 \subset C_0 \qquad
  T_0 \subset \underbrace{C_0^{H}}_{\text{\scriptsize V.A}} \subset C_0
\]

\[
  (x u) \qquad x^{n} = 1 \qquad (xu)^{n} = \,??
\]

\[
  H^2(\ , T_0) \simeq H^2(\ , {}_\infty T_0) = H^2(\ , \mathbb{Q}/\mathbb{Z})
\]

\[
  G - \qquad \rho_{\mathrm{ab}} \leqslant \rho'_{\mathrm{ab}}
\]

\[
  \begin{array}{c}
    \rho_{\mathrm{ab}} + \ldots \\
    \vee\!| \\
    \rho'_{\mathrm{ab}}
  \end{array}
  \qquad \frac{G_1}{Z_1} = Z \qquad
  \rho_{\mathrm{ab}}(Z_0) \leqslant \rho_{\mathrm{ab}}(Z_1) =
  \rho_{\mathrm{ab}}(G_1)
\]

\[
  \rho_{\mathrm{ab}} + 2\rho_{\mathrm{r}} \qquad
  \rho_{\mathrm{a}} + \qquad \rho_{\mathrm{ab}} + \qquad
  \rho_{\mathrm{rig}}.
\]

\[
  d_{\mathrm{radaff}} \geqslant \rho_{\mathrm{u}} + \tilde\rho_{\mathrm{r}}
\]

\[
  \rho_{\mathrm{ab}} + d_{\mathrm{ss}} + \underbrace{(\rho_{\mathrm{r}} -
  \rho_{\mathrm{ss}})}_{\rho_{\mathrm{r\,radaff}}} \qquad
  \rho_{\mathrm{ab}} + \rho_{\mathrm{r}} + (d_{\mathrm{ss}} -
  \rho_{\mathrm{ss}})
\]

\[
  \begin{array}{ll}
    (1)\ \rho_{\mathrm{ab}} & (5)\ \rho_{\mathrm{n}} \\
    (2)\ \rho_{\mathrm{r}} + \rho_{\mathrm{ab}} = \rho_{\mathrm{rig}} &
      (6)\ \rho_{\mathrm{u}} \\
    (3)\ \rho_{\mathrm{ss}} & (7)\ \rho_{\mathrm{naff}} \\
    (4)\ d_{\mathrm{ss}} & (8)\ d_{\mathrm{aff}} \\
     & (9)\ d_{\mathrm{rad}} \\
     & (10)\ d_{\mathrm{radaff}}
  \end{array}
\]

\[
  \begin{array}{l}
    (2) + (5) \Longrightarrow (6) \\
    (1) + (5) \Longrightarrow (7) \\
    (1) \Longleftrightarrow (8) \\
    (3) \Longleftrightarrow (9) \\
    (1) + (3) \Longrightarrow (10)
  \end{array}
\]

\[
  \left\{
  \begin{array}{l}
    \rho_{\mathrm{aff}} \geqslant \rho'_{\mathrm{aff}} \qquad
      d \geqslant d' \\[2pt]
    (5)\ \rho_{\mathrm{n}} \geqslant \rho'_{\mathrm{n}}, \quad
    (6)\ \rho_{\mathrm{u}} \geqslant \rho'_{\mathrm{u}}, \quad
    (7)\ \rho_{\mathrm{na}} \geqslant \rho'_{\mathrm{na}} \\[4pt]
    (1)\ \rho_{\mathrm{ab}} \leqslant \rho'_{\mathrm{ab}}, \quad
    (2)\ \rho_{\mathrm{ab}} + \rho_{\mathrm{r}} \leqslant \rho'_{\mathrm{ab}}
      + \rho'_{\mathrm{r}}, \quad
    (3)\ \rho_{\mathrm{s}} \leqslant \rho'_{\mathrm{s}}
  \end{array}
  \right.
\]

\[
  D_0 = \mathrm{Cent}_{G_0}(T_0) = C_0 .
\]

\[
  \begin{array}{l}
    \rho_{\mathrm{u}}(G_1) = \rho_{\mathrm{u}}(C_1) \\
    \rho_{\mathrm{ab}}(G_1) = \rho_{\mathrm{ab}}(C_1) \\
    \rho_{\mathrm{n}}(G_1) = \rho_{\mathrm{n}}(C_1)
  \end{array}
\]

\[
  G = C,
\]

\[
  \boxed{\rho_{\mathrm{n}} \geqslant \rho'_{\mathrm{n}}}
\]

\[
  \overbrace{C/\underline{\mathrm{Cent}}(C)^{\circ}_{\mathrm{réd}}}^{Z}
\]

\[
  \underline{\mathrm{Hom}}_{k\text{-gr}}(H, C) \simeq
  \underline{\mathrm{Hom}}_{k\text{-gr}}(H, Z).
\]

\[
  {}_mH(n) = H(m) \quad \text{si } m \mid n .
\]

\[
  \begin{array}{l}
    \rho_{\mathrm{ab}}(K_1) \leqslant \rho_{\mathrm{ab}}(G_1) =
      \rho'_{\mathrm{ab}} \\
    \rho_{\mathrm{r}}(K_1) \leqslant \rho_{\mathrm{r}}(G_1) =
      \rho'_{\mathrm{r}} \\
    \rho_{\mathrm{rig}}(K_1) \leqslant \rho_{\mathrm{rig}}(G_1) =
      \rho'_{\mathrm{rig}}.
  \end{array}
\]

\[
  \rho_{\mathrm{ab}}(G_0) = \rho_{\mathrm{ab}}(K_0), \qquad
  \rho_{\mathrm{r}}(G_0) = \rho_{\mathrm{r}}(K_0)
\]

\[
  \varprojlim M_n = M \simeq \mathbb{Z}_\ell^{\,2\rho_{\mathrm{ab}} +
  \rho_{\mathrm{r}}}.
\]

\[
  M_\infty \simeq \varprojlim H(n)_{\bar s_1}
\]

\[
  M_\infty \subset \varprojlim_n ({}_nG)_{\bar s_1} =
\]

\[
  \varprojlim ({}_nT)_{\bar s_1} \simeq ({}_n\mathbf{G}_m)^{r}_{\bar s_1},
\]

\[
  \pi_1(\tilde S, \bar s_0) = \pi_1(s_0, \bar s_0)
\]

\[
  (3) \Longleftrightarrow (9),
\]

\[
  H(n)_1/R_1 \cap H(n)_1 , \quad \ldots
\]

\[
  r_0 = \rho_{\mathrm{ss}}(G_0) \leqslant \dim T_0/T_0 \cap R_0
\]

\[
  \begin{array}{cc}
    G & G_1 \\
    \cup & \cup \\
    \overline{R}_1 = R & R_1
  \end{array}
  \qquad G_0/R_0
\]