\[
  F : C \longrightarrow C'
\]

\[
  F(A) = \bigl(A \times_S U,\ T_p(A),\ \mathrm{id}_{T_p(A)|U}\bigr).
\]

\[
  \underset{L_0}{\pi_0(S')} \leftleftarrows \underset{L_1}{\pi_0(S'')}
  \;\substack{\leftarrow\\[-2pt]\leftarrow\\[-2pt]\leftarrow}\;
  \underset{L_2}{\pi_0(S''')},
\]

\[
  K_0 \leftleftarrows K_1 \;\substack{\leftarrow\\[-2pt]\leftarrow\\[-2pt]\leftarrow}\; K_2
\]

\[
  T \subset S \supset U = S - T
\]

\[
  A \longmapsto \bigl(A_U,\ T_p^{\natural}(A),\ \varphi\bigr)
\]

\[
  \varphi_0 : A_{X_0} \to B_{X_0},
\]

\[
  U \sqcup S' \longrightarrow S
\]

\[
  U \longrightarrow R
\]

\[
  \begin{array}{ccc}
    & S'' & \hookleftarrow\; U'' \\
    & \downarrow & \wr \\
    R \longleftarrow & S' & \hookleftarrow\; U' \\
    & \downarrow & \wr \\
    & S & \hookleftarrow\; U
  \end{array}
\]

\[
  \begin{array}{ccccc}
    U & \longleftarrow & U' & \leftleftarrows & U'' \\
    \cap & & \cap & & \cap \\
    S & \xleftarrow{\ p\ } & S' & \leftleftarrows & S''
  \end{array}
\]

\[
  N|U \simeq {}_{\ell}(A_U).
\]

\[
  \begin{array}{ccccc}
    U & \longleftarrow & U' & \leftleftarrows & U'' \\
    \cap & & \cap & & \cap \\
    S & \longleftarrow & S' & \leftleftarrows & S''
  \end{array}
\]

\[
  A(t) \simeq A(s) \times_{S(s)} S(t),
\]

\[
  \underset{\substack{\shortparallel\\ (p^{*}A)_s}}{p^{*}(A_s)}
  \;\simeq\;
  \underset{\substack{\shortparallel\\ (M(s))_s}}{M(s)(s)} .
\]

\[
  \varphi : B' \simeq B|V'.
\]

\[
  z \in Z \Longrightarrow \exists\, s \in \overline{\{z\}} \text{ tel que }
  k(s) \text{ de car.\ } p \text{ et } s \in T
\]

\[
  A - A' \qquad \struck{\ill{}}\ A' \quad \struck{A''_1}\,,\ A''_2 \qquad
  T_p(A''_1) \simeq T_p(A''_2)
\]

\[
  S - S' \rightrightarrows S'' \qquad
  S - T = T \sqcup_{\ill{}} T = T \sqcup T \qquad
  T \sqcup T \sqcup T
\]

\[
  \uncertain{\mathfrak{Y}} - Y' \qquad Y'' = Y' \sqcup Y'
\]

\[
  \mathrm{Hom}_S(S, X) \longrightarrow \mathrm{Hom}_S(S', X)
  \rightrightarrows \mathrm{Hom}_S(S'', X)
\]

\[
  \mathcal{O}_S \longrightarrow f_*(\mathcal{O}_{S'})
\]

\[
  g' : S' \longrightarrow X
\]

\[
  g_0^{-1}(\mathcal{O}_X) \longrightarrow f_*(\mathcal{O}_{S'}) ;
\]

\[
  \mathcal{A} = \mathrm{Ker}\bigl(f_*(\mathcal{O}_{S'}) \rightrightarrows
  \struck{\ill{}}\, h_*(\mathcal{O}_{S''})\bigr)
\]

\[
  S' \longrightarrow \mathrm{Spec}(\mathcal{A}) = S'_1 \longrightarrow X .
\]

\[
  \mathrm{Hom}_S(S, X) \longrightarrow \mathrm{Hom}_S(S', X)
\]

\[
  \mathrm{Ker}\bigl(f_{2*}(\mathcal{O}_{S'_2}) \rightrightarrows
  h_{2*}(\mathcal{O}_{S''_2})\bigr)
\]

\[
  \begin{array}{ccccccccl}
    S & - & S' & = & S'' & \equiv & S''' & & \text{1-descente} \\
    \mid & & \mid & & \mid & & \mid & & \\
    S_1 & - & S'_1 & = & S''_1 & \equiv & S'''_1 & & \text{1-descente} \\
    \parallel & & \parallel & & \parallel & & \parallel & & \\
    S_2 & \underset{f_2}{-} & S'_2 & = & S''_2 & \equiv & S'''_2 & &
      \text{1-descente} \\
    \vert\vert\vert & & \vert\vert\vert & & \vert\vert\vert & &
      \vert\vert\vert & & \\
    S_3 & - & S'_3 & = & S''_3 & \equiv & S'''_3 & & \text{0-descente}
  \end{array}
\]

\[
  \begin{array}{ccccccccc}
    & & & & \hat{A}' & & \hat{A}'' & & \hat{A}''' \\
    M & \longleftarrow & S_0 & \longleftarrow & S'_0 & \leftleftarrows &
      S''_0 & & \\
    \downarrow & & & & & & & & \\
    T & \longleftarrow & \hat{S} & \longleftarrow & \hat{S}' &
      \leftleftarrows & \hat{S}'' & \Lleftarrow & \hat{S}''' \\[1ex]
    & & S_0 & - & S'_0 & = & S''_0 & \equiv & S'''_0
  \end{array}
\]

\[
  \hat{S} \longleftarrow \hat{S}' \leftleftarrows \hat{S}''
\]

\[
  P^{\alpha} \text{ sur } S ; \qquad
  \hat{P} = \coprod_{\alpha} \widehat{P^{\alpha}}
\]

\[
  H^2(X_0, \mathcal{T}_{X_0/S_0} \otimes_{\mathcal{O}_{S_0}} J) \simeq
  H^2(\hat{X}_0, \mathcal{T}_{\hat{X}_0/\hat{S}_0}
  \otimes_{\mathcal{O}_{\hat{S}_0}} \hat{J})
\]

\[
  H^1(X_0, \mathcal{T} \otimes J) \simeq H^1(\hat{X}_0, \hat{\mathcal{T}}
  \otimes \hat{J}) .
\]

\[
  V[[T_1, \ldots, T_n]] .
\]

\[
  x \notin \bigcup Z_i .
\]

\[
  V[[T_1, \ldots, T_n]] \longrightarrow V . \qquad \struck{\ill{}}
\]

\[
  V(\mathrm{Ker}\,\varphi) \subset V(f) \ \ldots \qquad
  f \in \mathrm{Ker}\,\varphi \ \ldots \qquad \varphi(f) = 0 ;
\]

\[
  f(\varphi_1, \ldots, \varphi_n) = 0 .
\]

\[
  \underline{\mathrm{NS}}_{X_n/S_n} = \underline{\mathrm{Pic}}_{X_n/S_n} /
  \underline{\mathrm{Pic}}^0_{X_n/S_n} ,
\]

\[
  P_n = \underline{\mathrm{NS}}^{+}_{X_n/S_n} \quad
  (\text{polarisations \uncertain{relatives}}) .
\]

\[
  \hat{P} = \coprod_{\alpha} \hat{P}_{\alpha} ,
\]

\[
  v_\eta : M_\eta \times G_\eta \longrightarrow G_\eta, \qquad
  (m, g) \longmapsto u_\eta(m) + g .
\]

\[
  \widetilde{v} : \widetilde{M} \times_S G \longrightarrow G, \qquad
  (\widetilde{m}, g) \longmapsto \widetilde{u}(\widetilde{m}) + g ,
\]

\[
  (1) \qquad 0 \to \operatorname{Ker} \widetilde{v} \to \widetilde{M} \times_S G
  \to G \to 0 .
\]

\[
  G_1 \xrightarrow{\;g\;} G \xrightarrow{\;f\;} G_1 .
\]

\[
  \varphi(n) : M(n) \longrightarrow G .
\]