\[
  \check{H}^i(X^{\cdot}/S, F) \simeq
  \begin{cases}
    0 & \text{si } i \neq 0 \\
    H^0(S, F) & \text{si } i = 0 .
  \end{cases}
\]

\[
  \mathcal{H}^i(X^{T}, F^{T}) = 0, \qquad
  \mathcal{H}^0(X^{T}, F^{T}) \simeq H^0(X, F).
\]

\[
  H^{*}(S, F) \Longleftarrow \check{H}^p(S'/S, \mathcal{H}^q(F))
\]

\[
  0 \to \check{H}^1(S'/S, F) \to H^1(S, F) \to
  \check{H}^0(S'/S, \mathcal{H}^1(F)) \to \check{H}^2(S'/S, F) \to
  H^2(S, F) .
\]

\[
  \check{H}^p(S'/S, \mathcal{H}^q(F)) = 0 \quad \text{pour tt } p,
  \text{ si } q \neq 0 .
\]

\[
  \check{H}^0(S'/S, \mathcal{H}^q(F)) =
  \mathrm{Ker}\bigl(H^q(S', F) \rightrightarrows H^q(S' \times_S S', F)\bigr)
  = 0
\]

\[
  A = \mathrm{Ker}\bigl(H^0(X, \mathcal{O}_X) \rightrightarrows
  H^0(R, \mathcal{O}_R)\bigr)
\]

\[
  \mathrm{Ker}\bigl(H^0(X_0, \mathcal{J}_{X_0}) \rightrightarrows
  H^0(R_0, \mathcal{J}_{R_0})\bigr) = H^0(Y_0, \mathcal{J}_{Y_0}) =
  \mathcal{J}_{Y_0} .
\]

\[
  0 \to \mathcal{J}_{Y_0} \to A \to A_0 \to 0
\]

\[
  f_0 \in \mathrm{Ker}\bigl(H^0(X_0, \mathcal{O}_{X_0}) \rightrightarrows
  H^0(R_0, \mathcal{O}_{R_0})\bigr) ;
\]

\[
  \xi \in \mathrm{Ker}\bigl(H^1(X_0, \mathcal{J}_{X_0}) \rightrightarrows
  H^1(R_0, \mathcal{J}_{R_0})\bigr)
  = \check{H}^0\bigl(X_0/Y_0, \mathcal{H}^1(\mathcal{J}_{Y_0})\bigr)
\]

\[
  y \in Z^1(X_0/Y_0, \mathcal{J}_{Y_0}) \quad \text{i.e. } dy = 0
\]

\[
  \mathcal{J}_{Y_0} X = \mathcal{J}_{X_0} \simeq
  \mathcal{J}_{Y_0} \otimes_{\mathcal{O}_{Y_0}} \mathcal{O}_{X_0}
\]

\[
\begin{array}{ccc}
 & & U_{ij\mu} \\
 & & \downarrow \\
U_i & \longleftarrow & U_{ij} \\
\downarrow & & \\
U & &
\end{array}
\]

\[
  F(S) \to F(S') \rightrightarrows F(S' \times_S S')
\]

\[
  X \xrightarrow{f_1} X^{\ast}_0 \xrightarrow{f_2} Y
\]

\[
  Y' = \mathrm{Hom}(I, X) = X^{I}
\]

\[
  Y' \times I \to X \qquad \text{i.e. un } Y'\text{-morphisme
  \uncertain{canonique}}
\]

\[
  f : \struck{\ill{}}\,Y' \times I \to X'
\]

\[
  I \times \mathrm{Spec}\,k(y') \to X' \times_{Y'} \mathrm{Spec}\,k(y')
\]

\[
  d(X_{(1)}/Y) = 1, \qquad d(X^{(1)}/Y) = n - 1 .
\]

\[
  I \times \mathrm{Spec}\,k(y') \to X' \times_{Y'} \mathrm{Spec}\,k(y')
\]

\[
X^{*} \longrightarrow Y^{*} \times I = X^{*}_0
\]

\[
X \longrightarrow X_0
\]

\[
X^{*} \longrightarrow X^{*}_0 \longrightarrow Y'^{*}
\]

\[
\varphi \colon (X \times_S T)|\mathcal{U} \xrightarrow{\ \sim\ }
(T \times_S X)|\mathcal{U}
\]

\[
\varphi' \colon X \times_S T' \xrightarrow{\ \sim\ } T' \times_S X
\]

\[
\tau \in \mathcal{U}_1 \Rightarrow V \cap p^{(3)-1}_{ij}(\tau)
\ \text{dense dans}\ p^{(3)-1}_{ij}(\tau)
\]

\[
t \in T' \Rightarrow p_i^{-1}(t) \cap \mathcal{U}
\ \text{est dense dans}\ p_i^{-1}(t) \quad \text{pour } i = 1, 2 .
\]

\[
t \in T \Rightarrow \mathcal{U} \cap p_i^{-1}(t) \ \text{dense dans}\ \ill{}
\quad \text{pour } i = 1, 2 .
\]

\[
W = p^{(3)-1}_{12}(\mathcal{U}) \cap p^{(3)-1}_{23}(\mathcal{U})
\]

\[
\pi \colon W \longrightarrow T \times T
\]

\[
\psi \colon \pi^{*}(X_1) \xrightarrow{\ \sim\ } \pi^{*}(X_2)
\]

\[
\pi^{*}(X_1) \simeq p^{(3)*}_{12}(X \times_S T | \mathcal{U}), \qquad
\pi^{*}(X_2) \simeq p^{(3)*}_{23}(T \times_S X | \mathcal{U})
\]

\[
p^{(3)*}_1(X) \xrightarrow{\ \sim\ } p^{(3)*}_2(X) \xrightarrow{\ \sim\ }
p^{(3)*}_3(X) .
\]

\[
W \times_{T \times_S T} W \subset T \times_S T \times_S T \times_S T ,
\]

\[
(t''', t', t_1),\ (t''', t', t_2),\ (t''', t'', t_1),\ (t''', t'', t_2),\
(t''', t_1, t_2) \in V .
\]

\[
\varphi_{t_2 t'} \varphi_{t' t_1}
= \varphi_{t_2 t'} (\varphi_{t' t'''} \varphi_{t''' t_1})
= (\varphi_{t_2 t'} \varphi_{t' t'''}) \varphi_{t''' t_1}
= \varphi_{t_2 t'''} \varphi_{t''' t_1}
\]

\[
\varphi_{t_2 t''} \varphi_{t'' t_1} = \varphi_{t_2 t'''} \varphi_{t''' t_1}
\]

\[
\varphi_{t_2 t'} \varphi_{t' t_1} = \varphi_{t_2 t''} \varphi_{t'' t_1} .
\]

\[
\varphi' \colon X_1 \xrightarrow{\ \sim\ } X_2 .
\]

\[
\varphi'_{t_3 t_2} \varphi'_{t_2 t_1} = \varphi'_{t_3 t_1}
\quad (\text{sur } T \times_S T \times_S T) .
\]

\[
(t_1, t'),\ (t', t_2),\ (t_2, t'''),\ (t''', t_3) \in \mathcal{U},
\quad \text{et} \quad
(t_1, t', t'''),\ (t'', t''', t_3) \in V .
\]

\[
(\varphi_{t_3 t''} \varphi_{t'' t_2})(\varphi_{t_2 t'} \varphi_{t' t_1})
\]

\[
\varphi_{t'' t_2} \varphi_{t_2 t'} = \varphi_{t'' t'''} \varphi_{t''' t'}
\qquad [= \varphi'_{t'' t'}]
\]

\[
(\varphi_{t_3 t''} \varphi_{t'' t'''})(\varphi_{t''' t'} \varphi_{t' t_1})
\]

\[
V \xrightarrow{\ p_{13}\ } \mathcal{U}
\]

\[
X \simeq Y \times_{S'} T' ,
\]

\[
\Phi = f^{-1}(s) - (t) \qquad (f(t) = s)
\]

\[
T \times_S T = \Delta_T \times (G \times G)
\]

\[
\mathcal{U}_g = (g \times e)\, \Delta_T ,
\]

\[
\varphi_g^{-1}\bigl(\mathcal{U}_g \cap (T' \times_S T')\bigr)
= g^{-1}(T') \cap T' .
\]

\[
X \times_S T' \simeq T' \times_S X
\]

\[
X | T'' \simeq g^{*}(X | T'') .
\]

\[
T \times_S T \times_S T \simeq \Delta_3 T \times (G \times G \times G) .
\]

\[
V_{g, g'} = (g \times g' \times e) \Delta_3 T ,
\]

\[
V'_{g, g'} = V_{g, g'} \cap (T' \times_S T' \times_S T')
\]

\[
g^{-1}(T') \cap g'^{-1}(T') \cap T' ,
\]

\[
g^{-1}(X) \simeq X
\]

\[
X | T'' \simeq X_0 \times_{S''} T''
\]

\[
X'_0 \times_{S - s} \bigl(T - f^{-1}(s)\bigr)
\]

\[
T' = \bigl(T - f^{-1}(s)\bigr) \cup \{t\} ,
\]