\[
\begin{array}{ccc}
F(T) & \longrightarrow & F(T_0)\\
\downarrow & & \downarrow\\
F(U) & \longrightarrow & F(U_0)
\end{array}
\]

\[
F(S')=\begin{cases}\emptyset & \text{si } E_{S'} \text{ non plat sur } S'\\ \{\emptyset\} & \text{sinon}\end{cases}
\]

\[
F(T)=\begin{cases}\emptyset & \text{si } X_T \text{ non fid.\ plat sur } T\\ \{\emptyset\} & \text{sinon}\end{cases}
\]

\[
\mathrm{Hom}\bigl(u_s'^{*}(\Omega^1_{Y'_s/k(s)}),k(s)\bigr)\longrightarrow\mathrm{Hom}\bigl(v_s^{*}(\Omega^1_{Y'_s/k(s)}),k(s)\bigr)
\]

\[
\mathrm{Hom}_{\mathcal{O}_{X'}}\bigl(u^*(\Omega^1_{Y'/S'}),\mathcal{O}_{X'}\otimes_{\mathcal{O}_{S'}}J\bigr)\simeq\mathrm{Hom}_{\mathcal{O}_{S'}}(M_u,J),
\]

\[
\mathrm{Hom}_{\mathcal{O}_{Z'}}\bigl(v^*(\Omega^1_{Y'/S'}),\mathcal{O}_{Z'}\otimes_{\mathcal{O}_{S'}}J\bigr)\simeq\mathrm{Hom}_{\mathcal{O}_{S'}}(M_v,J),
\]

\[
(*)\qquad M_v\longrightarrow M_u
\]

\[
\mathrm{Hom}_{\mathcal{O}_{X'}}\bigl(u^*(\Omega^1_{Y'/S'}),\mathcal{O}_{X'}\otimes_{\mathcal{O}_{S'}}J\bigr)\longrightarrow\mathrm{Hom}_{\mathcal{O}_{Z'}}\bigl(v^*(\Omega^1_{Y'/S'}),\mathcal{O}_{Z'}\otimes_{\mathcal{O}_{S'}}J\bigr).
\]

\[
M_v\otimes k(s)\longrightarrow M_u\otimes k(s)
\]

\[
F=F_\varphi=\underline{\mathrm{Hom}}_S(X,Y;\varphi)
\]

\[
\partial(u_0)=\xi\in\mathrm{Ext}^1_{\mathcal{O}_{P_0}}(P_0;I_{\Gamma_0},\mathcal{O}_{\Gamma_0}\otimes_{S_0}J)
\]

\[
(\ast)\qquad 0\to J\xrightarrow{\;\pi^{n}\;}J\to J_n\to 0
\]

\[
0\to\mathcal{O}_{\Gamma_0}\otimes_{S_0}J\xrightarrow{\;\pi^{n}\;}\mathcal{O}_{\Gamma_0}\otimes_{S_0}J\to\mathcal{O}_{\Gamma_0}\otimes_{S_0}J_n\to 0
\]

\[
\xi'\in\mathrm{Hom}_{\mathcal{O}_{P_0}}(P_0;I_{\Gamma_0},\mathcal{O}_{\Gamma_0}\otimes_{S_0}J_n)
\]

\[
\mathrm{Hom}_{\mathcal{O}_{P_0}}(I_{\Gamma_0}/I_{\Gamma_0}^2,\mathcal{O}_{\Gamma_0}\otimes_{S_0}J_n)\simeq\mathrm{Hom}_{\mathcal{O}_{X_0}}\bigl(u_0^*(\Omega^1_{Y_0/S_0}),\mathcal{O}_{X_0}\otimes_{S_0}J_n\bigr)
\]

\[
\simeq\mathrm{Hom}_{S_0}(M_{u_0},J_n)
\]

\[
\xi''\in\mathrm{Hom}_{A_0}(M_{u_0},J_n)/\mathrm{Im}\,\mathrm{Hom}_{A_0}(M_{u_0},J)\hookrightarrow\mathrm{Ext}^1_{A_0}(M_{u_0},J)
\]

\[
\xi''\in\mathrm{Ext}^1_{A_0}(M_{u_0},J)
\]

\[
\eta''\in\mathrm{Ext}^1_{A_0}(M_{v_0},J)
\]

\[
0\to N\to M_{v_0}\to M_{u_0}\to 0,
\]

\[
\xi'''\in\mathrm{Hom}(N,J)/\mathrm{Im}\,\mathrm{Hom}(M_{v_0},J).
\]

\[
\mathrm{Hom}(M_{v_0},J_\eta)/\mathrm{Im}\bigl(\mathrm{Hom}(M_{v_0},J)+\mathrm{Hom}(M_{u_0},J_\eta)\bigr)
\]

\[
=\mathrm{Hom}(N,J_\eta)/\mathrm{Im}\,\mathrm{Hom}(M_{v_0},J).
\]

\[
\mathrm{Hom}(N,J_n)/\mathrm{Im}\,\mathrm{Hom}(M_{v_0},J)\subset\mathrm{Hom}(N,J_\eta)/\mathrm{Im}\ldots
\]

\[
\operatorname{Hom}\bigl(\omega^{\circ}(\Omega^{1}_{Y_0/S_0}), \mathcal{O}_{X_0}\otimes G\bigr)
\longrightarrow
\operatorname{Hom}\bigl(\omega^{\circ}(\Omega^{1}_{X_0/S_0}), \mathcal{O}_{Z_0}\otimes G\bigr)
\]

\[
0 \to \mathcal{O}_{\Gamma_0}\otimes_{S_0} J \to E \to \underline{I}_{\Gamma_0} \to 0 .
\]

\[
0 \to \mathcal{O}_{\Gamma'_0}\otimes_{S_0} J \to E' \to \underline{I}_{\Gamma'_0} \to 0 .
\]

\[
L' \xleftarrow{\ \pi'\ } M' .
\]

\[
\mu\beta = (\lambda\sigma\alpha)\pi' .
\]

\[
U \longrightarrow \prod_{1\le i\le n} \text{\struck{\ill{}}} \prod_{Z_i/S} P_{Z_i}/Z_i
\]

\[
\phi : Z \longrightarrow X
\]

\[
\phi' : \underline{\operatorname{Hom}}_S(X,Y) \longrightarrow \underline{\operatorname{Hom}}_S(Z,Y) .
\]

\[
\underline{\operatorname{Hom}}_S(X,Y;\phi) \longrightarrow \underline{\operatorname{Hom}}_S(Z,Y) .
\]

\begin{align*}
f_*(X/T) &= \underline{\operatorname{Hom}}_{T/S}(T/S, X/S) \\
f_*(X/T) &= \underline{\operatorname{Hom}}_S(T,X) \times_{\underline{\operatorname{Hom}}_S(T,T)} S \\
\underline{\operatorname{Hom}}_{T/S}(X/T, Y/T) &= f_*\bigl(\underline{\operatorname{Hom}}_T(X,Y)\bigr) \\
\underline{\operatorname{Hom}}_S(X,Y) &= \varphi_*(X \times Y / X) \\
\underline{\operatorname{Hom}}_S(X,Y) &= \underline{\operatorname{Hom}}_{S/S}(X/S, Y/S)
\end{align*}

\[
X_0 = f_*(X) \times_{g_*(X \times_S T)} f_*(X) .
\]

\[
\text{\struck{$(X\otimes_k L)\otimes_K L = X\otimes_k(L\otimes_K L)$}} = X\otimes_k L
\]

\[
\boxed{\, f_*\bigl(\check{\underline{\mathbb{V}}}(\mathcal{F})\bigr) = \check{\underline{\mathbb{V}}}\bigl(f_*(\mathcal{F})\bigr) \,}
\]

\[
H \times_S T \longrightarrow X ,
\]

\[
\underline{\operatorname{Hom}}_S(S', H) \xrightarrow{\ \sim\ } \underline{\operatorname{Hom}}_T(S' \times_S T, X) .
\]

\[
\underline{\operatorname{Hom}}_S(S, H) \simeq \underline{\operatorname{Hom}}_T(T, X) ,
\]

\[
  f_{*}(T/T)=S,\qquad f_{*}(X\times_T Y/T)=f_{*}(X/T)\times_S f_{*}(Y/T),
\]

\[
  f_{*}(X\times_Z Y/T)=f_{*}(X/T)\times_{f_{*}(Z/T)}f_{*}(Y/T).
\]

\[
  f_{*}(X/T)=\bigcup_i f_{*}(U_i/T),
\]

\[
  f_{*}(X/T)\to f_{*}(Y/T).
\]

\[
  T=\operatorname{Spec}(B),
\]

\[
  \varphi_{i\lambda\mu}=0\qquad
  \begin{bmatrix}1\le i\le\nu\\ 1\le\lambda\le(n-k)\nu\\ 1\le\mu\le k\nu\end{bmatrix}
\]

\[
  B^{k}\xrightarrow{\;u_\xi\;}M
\]

\[
  X^{\nu}=\underbrace{X\times_S\cdots\times_S X}_{\nu},
\]

\[
  \bigcup U_i^{\nu}=\bigcup U^{\nu}.
\]

\[
  f_{*}(X/T)=\operatorname{Spec}\bigl(f_{*}(\underline{F})\bigr).
\]

\[
  \Bigl(\coprod f_i\Bigr)_{*}(X/T)=\prod_i f_{i*}(X_i/T_i),
\]

\[
  X\xrightarrow{g_1}Y[t_1,\dots,t_d]\xrightarrow{g_2}Y,
\]

\[
  X=Y[t_1,\dots,t_d]=Y\times_T\overline{T}[t_1,\dots,t_d]
\]

\[
  f_{*}\bigl(T[t_1,\dots,t_d]/T\bigr)
\]

\[
  \begin{cases}
    \nu=\sup_s\nu_s\\
    n=\deg T/S\\
    d=\dim.\ \text{fibres de } g\\
    \lambda=\deg X/Y
  \end{cases}
\]

\[
  f_{*}(Y/T)\times_S f_{*}(T\times G/T).
\]

\[
  f_{*}(T\times I/T)=\varphi(I)
\]

\[
  T\to\overline{T}
\]

\[
  (*)\qquad \underline{\operatorname{Hom}}_S\bigl(f^{\text{ét}}_{*}(T),S'\bigr)
  =\underline{\operatorname{Hom}}_S(T,S')
\]

\[
  \underline{\operatorname{Hom}}_S\bigl(f^{\text{ét}}_{*}(T),S'\bigr)\simeq\underline{\operatorname{Hom}}_S(T,S')
\]

\[
  T\longrightarrow f^{\text{ét}}_{*}(T)
\]

\[
  \underline{\operatorname{Hom}}_S\bigl(f^{\text{ét}}_{*}(T),S'\bigr)\longrightarrow\underline{\operatorname{Hom}}_S(T,S')
\]

\[
  T\to f^{\text{ét}}_{*}(T)
\]

\[
  \underline{\operatorname{Hom}}_S(\overline{T},S')=\underline{\operatorname{Hom}}_S(T,S')
\]

\[
  \underline{\operatorname{Hom}}_S(\overline{T},S')\simeq\underline{\operatorname{Hom}}_S(T,S'),
\]

\[
  (*) \qquad \operatorname{Hom}_{\mathbb{W}(S)}(\mathbb{W}(T), \mathfrak{X}) .
\]

\[
  \operatorname{Hom}_{S}(T, X) \xrightarrow{\ \sim\ }
  \operatorname{Hom}_{\mathbb{W}(S)}(\mathbb{W}(T), \mathfrak{X})
\]

\[
  \operatorname{Hom}_{S^{a}}(T^{a}, \mathfrak{X}) \simeq
  \operatorname{Hom}_{S}(T, \mathfrak{X})
\]

\[
  \mathbb{W}^{a}(\mathfrak{X} \times_{\mathbb{W}(S)} \mathfrak{Y}) \simeq
  \mathbb{W}^{a}(\mathfrak{X}) \times_{S} \mathbb{W}^{a}(\mathfrak{Y})
\]

\[
  \mathbb{W}^{a}(\mathfrak{X}) \times_{S} T \simeq
  \mathbb{W}_{T}^{a}(\mathfrak{X} \times_{\mathbb{W}(S)} \mathbb{W}(T))
\]

\[
  \mathbb{W}'(S) \to \mathbb{W}(S) .
\]

\[
  \mathfrak{X}' = \mathfrak{X} \times_{\mathbb{W}(S)} \mathbb{W}'(S) .
\]

\[
  (*) \qquad \operatorname{Hom}_{\mathbb{W}(S)}(\mathbb{W}(T), \mathfrak{X})
  \to \operatorname{Hom}_{\mathbb{W}'(S)}(\mathbb{W}(T)
  \times_{\mathbb{W}(S)} \mathbb{W}'(S), \mathfrak{X}')
\]

\[
  \mathbb{W}'(T) \to \mathbb{W}(T) \times_{\mathbb{W}(S)} \mathbb{W}'(S)
\]

\[
  (**) \qquad \operatorname{Hom}_{\mathbb{W}'(S)}(\mathbb{W}(T)
  \times_{\mathbb{W}(S)} \mathbb{W}'(S), \mathfrak{X}') \to
  \operatorname{Hom}_{\mathbb{W}'(S)}(\mathbb{W}'(T), \mathfrak{X}')
\]

\[
  \operatorname{Hom}_{\mathbb{W}(S)}(\mathbb{W}(T), \mathfrak{X})
  \longrightarrow
  \operatorname{Hom}_{\mathbb{W}'(S)}(\mathbb{W}'(T), \mathfrak{X}') .
\]

\[
  \mathbb{W}^{a}(\mathfrak{X}) \xrightarrow{\alpha^{*}(\mathfrak{X})}
  \mathbb{W}'^{a}(\mathfrak{X} \times_{\mathbb{W}(S)} \mathbb{W}'(S)) .
\]

\[
  \mathbb{W}'^{a}(\mathfrak{X} \times_{\mathbb{W}(S)} \mathbb{W}'(S))
\]

\[
  \mathbb{W}(B) = \operatorname{Hom}_{A\text{-}\mathrm{alg}}(\Lambda, B)
\]

\[
\begin{aligned}
  \operatorname{Hom}_{\mathbb{W}(S)}(\mathbb{W}(T), \mathfrak{X})
  &= \operatorname{Hom}_{\mathbb{W}(A)\text{-}\mathrm{alg}}(\mathcal{C},
     \mathbb{W}(B)) \\
  &\simeq \varprojlim \operatorname{Hom}_{\mathbb{W}(A)\text{-}\mathrm{alg}}
     (\mathcal{C}_i, \mathbb{W}(B)) \\
  &\simeq \varprojlim \operatorname{Hom}_{A\text{-}\mathrm{alg}}
     (\mathbb{W}^{a}(\mathcal{C}_i), B) \\
  &\simeq \operatorname{Hom}_{A\text{-}\mathrm{alg}}
     (\varinjlim \mathbb{W}^{a}(\mathcal{C}_i), B)
\end{aligned}
\]

\[
  \varinjlim \mathbb{W}^{a}(\mathcal{C}_i) = \mathcal{D},
\]

\[
  \operatorname{Hom}_{\mathbb{W}(S)}(\mathbb{W}(T), \mathfrak{X}) \simeq
  \operatorname{Hom}_{S}(T, \operatorname{Spec}(\mathcal{D}))
\]

\[
  \mathbb{W}(X) \xrightarrow{\ \varphi\ } \mathfrak{X}
  \qquad \mathfrak{U}' \subset \mathfrak{U} .
\]

\[
  \mathbb{W}(T) \xrightarrow{\ f\ } \mathfrak{X}, \qquad
  T = \bigcup_{i} T_i \ \text{(les \ill{} de $T$) tel que}
\]

\[
  f(\mathbb{W}(T_i)) \subset \mathfrak{X}_i .
\]

\[
  \begin{array}{ccccc}
  X_{ijk} & \xrightarrow{\varphi_{ji} \mid} & X_{jik} & \xrightarrow{\varphi_{kj} \mid} & X_{kij} \\
  \| & & \| & & \| \\
  X_{ij} \cap X_{ik} & & X_{ji} \cap X_{jk} & & X_{ki} \cap X_{kj}
  \end{array}
\]

\[
  X_{ijk} \simeq \mathbb{W}^{a}(\mathfrak{X}_i \cap \mathfrak{X}_j \cap
  \mathfrak{X}_k)
\]

\[
  \mathbb{W}(X) \to \mathfrak{X}
\]

\[
  \mathbb{W}(X_i) \to \mathfrak{X}_i \subset \mathfrak{X}
\]

\[
  \operatorname{Hom}_{S}(T, X) \to
  \operatorname{Hom}_{\mathbb{W}(S)}(\mathbb{W}(T), \mathfrak{X})
\]

\[
  \mathbb{W}(T) \xrightarrow{\ \psi\ } \mathfrak{X}
\]

\[
  \mathbb{W}(T) \xrightarrow{\mathbb{W}(u)} \mathbb{W}(X)
  \xrightarrow{\ \varphi\ } \mathfrak{X}
\]

\[
  \mathbb{W}(T_i) \xrightarrow{\mathbb{W}(u_i)} \mathbb{W}(X_i)
  \xrightarrow{\ \varphi_i\ } X_i
\]

\[
  |T| \xrightarrow{\ \sim\ } |\mathbb{W}(T)|
\]

\[
  T_i = \complement\, \varphi_{T}(T' - T'_i)
\]

\[
  \bigcup_{i \in I} \underbrace{\mathfrak{X}_i \times_S \cdots \times_S
  \mathfrak{X}_i}_{n\ \text{facteurs}}
  = \underbrace{\mathfrak{X} \times_S \cdots \times_S \mathfrak{X}}_{n\
  \text{facteurs}} .
\]

\[
  \mathfrak{X} \xrightarrow{\ f'\ } \mathfrak{X}' \xrightarrow{\ f''\ }
  \mathfrak{Y}
\]

\[
\begin{cases}
Z \text{ quasi-compact et} \\
\forall x \in Z,\ \exists \text{ un voisinage } U_x \text{ de } x \text{ dans } X,
\end{cases}
\]

\[
\begin{array}{ll}
X \to Y & \\
X_i & Y_i \\
X'_j & \\
X_i \cap X'_j \to Y_{ij} &
\end{array}
\]

\[
X' \times_Z Y' = \mathrm{pr}_1^{-1}(X') \cap \mathrm{pr}_2^{-1}(Y') \ \dots
\]

\[
\operatorname{Cons} X \xrightarrow{\ \sim\ } \varprojlim \operatorname{Cons}(X_i) \quad ???
\]

\[
\operatorname{Hom}(\operatorname{Cons} T, \operatorname{Cons} X) = \operatorname{App\,cons}(T, X)
= \varinjlim_{\mathfrak{P}} \operatorname{Hom}\Bigl(\text{\struck{$\Pi$}}\ \coprod_{\alpha \in \operatorname{Ind}(\mathfrak{P})} T_{\alpha}^{\mathfrak{P}}, X\Bigr)
\]

\[
= \varinjlim_{\mathfrak{P}} \varprojlim_{i} \prod_{\alpha \in \operatorname{Ind}(\mathfrak{P})} \operatorname{Hom}(T_{\alpha}^{\mathfrak{P}}, X_i)
\]

\[
\operatorname{Hom}\bigl(\operatorname{Cons}(T), \text{\struck{$\operatorname{Cons} X$}}\ \varprojlim \operatorname{Cons}(X_i)\bigr)
= \varprojlim_{i} \varinjlim_{\mathfrak{P}} \prod_{\alpha \in \operatorname{Ind}(\mathfrak{P})} \operatorname{Hom}(T_{\alpha}^{\mathfrak{P}}, X_i)
\]

\[
\operatorname{Hom}(\operatorname{Cons} X, \operatorname{Cons} Y) = \varinjlim_{\mathfrak{P}} \prod_{\alpha} \operatorname{Hom}(\text{\struck{\ill{}}}\, X_{\alpha}^{\mathfrak{P}}, Y)
\]

\[
= \varinjlim_{\mathfrak{P}} \varprojlim_{j} \text{\struck{$\operatorname{Hom}$}}\ \prod_{\alpha} \operatorname{Hom}(X_{\alpha}^{\mathfrak{P}}, Y_j)
\]

\[
X \times \coprod_i Y_i \simeq \coprod_i X \times Y_i
\]

\[
R \to S
\]

\[
X-(E-F)=V\cup F \quad \text{rétrocompact dans } X,
\]

\[
E^{\mathrm{cons}} \text{ qu.-cpte} \Longrightarrow E \text{ qu.\ cpct} \quad \text{(i)}
\]

\[
\Longrightarrow E \text{ est contenu dans une réunion finie des } U_i
\]

\[
\Updownarrow
\]

\[
\mathrm{Cons}(X,E) : (\mathrm{Cons\,Aff})^{\circ}\longrightarrow(\mathrm{Ens})
\]

\[
\mathrm{Cons}(X,E)(Y)=\mathrm{App\,cons}(Y,(X,E)).
\]

\[
\mathrm{App\,cons}((X,E),(X',E'))\xrightarrow{\ \sim\ }\mathrm{Hom}(\mathrm{Cons}(X,E),\mathrm{Cons}(X',E'))
\]

\[
A^{\mathrm{cons}}=\varinjlim_{\mathfrak{P}} A_{\mathfrak{P}}
\]

\[
A_{\mathfrak{P}} \text{ désigne } A\Bigl(\coprod X_{i\,\mathrm{red}}\Bigr)=\prod A(X_{i\,\mathrm{red}}).
\]

\[
\widetilde{X}=\mathrm{Spec}(A^{\mathrm{cons}}).
\]

\[
\mathrm{Top}(\widetilde{X})=\varprojlim_{\mathfrak{P}}\mathrm{Top}\,\mathrm{Spec}(A_{\mathfrak{P}}).
\]

\[
\mathrm{Ens}(\widetilde{X})=\varprojlim_{\mathfrak{P}}\mathrm{Ens}\underbrace{\mathrm{Spec}\,A_{\mathfrak{P}}}_{\coprod X_i}
\]

\[
\mathrm{Ens}\,\widetilde{X}\simeq\mathrm{Ens}\,X.
\]

\[
\mathrm{Hom}(K,\gamma X)\xrightarrow{\ \sim\ }\mathrm{Hom}(\gamma' K,X)
\]

\[
\mathrm{Hom}(K,\beta Y)\xrightarrow{\ \sim\ }\mathrm{Hom}(\beta' K,Y)
\]

\[
\mathrm{Hom}(K,\beta(\alpha X))=\mathrm{Hom}(\beta' K,\alpha X)
\]

\begin{align*}
\mathrm{Hom}(\overline{K},\overline{\beta}\,\overline{Y}) &\simeq \mathrm{Hom}(\overline{\beta}'(K),\overline{Y})
 \simeq \varprojlim_{\lambda}\varinjlim_{\mu}\mathrm{Hom}(\overline{K}_{\lambda},\beta(\overline{Y}_{\mu}))\\
&\simeq \varprojlim_{\lambda}\varinjlim_{\mu}\mathrm{Hom}(\beta'(\overline{K}_{\lambda}),\overline{Y}_{\mu})
\end{align*}

\[
\text{\struck{$\mathrm{Hom}(\overline{K},\overline{\gamma}(X))\simeq$}}
\]

\begin{align*}
\mathrm{Hom}(K,\overline{\gamma}(X)) &\simeq \mathrm{Hom}(\overline{\gamma}'K,X)\\
&\simeq \varprojlim_{\lambda}\varinjlim_{\mu}\mathrm{Hom}(K_{\lambda},\gamma(X_{\mu}))\\
&\simeq \varprojlim_{\lambda}\varinjlim_{\mu}\mathrm{Hom}(\gamma' K_{\lambda},X_{\mu})
\end{align*}

\[
  X'_i = X_i - X_i \cap \bigcup_{j<i} X_j ,
\]

\[
  U_j \cap X'_j \subset \bar{x}_j \qquad\qquad X'_i \longleftarrow X'_j
\]

\[
  U_i \subset \bar{x}_i
\]

\[
  x \in Z \subset \bar{x} \qquad x \in U_0 \cap Z_0 \qquad \bar{x}
\]

\[
  \bar{X}(k') \simeq X(V_{k'}) \qquad \text{\struck{$\to X$}}
\]

\[
  \bar{X}(k') \simeq X(V_{k'}) \longrightarrow X(K')
\]

\[
  \varphi : X_K \longrightarrow Y_K
\]

\[
  \bar{\varphi} : \bar{X} \longrightarrow \bar{Y}
\]

\[
  \varphi(\alpha_X(\bar{x}')) \in Y(K_{k'}) \xleftarrow{\ \sim\ } Y(V_{k'}) \overset{\alpha_Y}{\simeq} \bar{Y}(k')
\]

\[
  k(\varphi(\bar{x})) \longrightarrow \widetilde{k(\bar{x})}
\]

\[
  \Gamma : A \rightsquigarrow A((t))^{*} = \bigl(A[[t]]_t\bigr)^{*} \qquad (t \text{ une indéterminée})
\]

\[
  \mathcal{G} = \varinjlim_n (\mathcal{G}_n)
\]

\[
  \text{\struck{$A \rightsquigarrow A((t))^{*}$}}
\]

\[
  \Bigl(\sum_i a_i t^i\Bigr)\Bigl(\sum_j b_j t^j\Bigr) = 1 ,
\]

\[
  \Gamma_0 : A \rightsquigarrow A[[t]]^{*}
\]

\[
  \mathcal{G}_0 = \mathbf{G}_m \times \operatorname{Spec} \mathbf{Z}[(a_i)_{i \geq 1}]
\]

\[
  \mathcal{G}' = \mathbf{Z}_{\mathbf{Z}} \times \mathcal{G}_0
\]

\[
  \mathcal{G}' = \varinjlim_n \mathcal{G}'_n , \qquad \mathcal{G}'_n = [-n, n]_{\mathbf{Z}} \times \mathcal{G}_0 .
\]

\[
  \mathcal{G}' \longrightarrow \mathcal{G}
\]

\[
  \mathcal{G}'_n \longrightarrow \mathcal{G}_n
\]

\[
  \varphi_n^{\nu} : \mathcal{G}_0 \longrightarrow \mathcal{G}_n \qquad \nu \in [-n, +n]
\]

\[
  \varphi_n^{\nu}\Bigl(\text{\struck{$\Sigma$}}\, a_i t^i_{\ i \geq 0}\Bigr) = t^{\nu} \sum a_i t^i
\]

\[
  \mathcal{G}_n^i \longrightarrow \mathcal{G}_{n'}^i
\]

\[
  \begin{cases}
    \forall\, i \text{ grand}, X_i \text{ est qu.-compact et de présentation finie sur } S \\
    \exists\, i \text{ t.q. } j \geq i \text{ implique } u_{ji} \text{ affine}
  \end{cases}
\]

\[
  \operatorname{Hom}_{\operatorname{Pro}(\mathrm{Sch}_{/S})}(\mathcal{Y}, \mathfrak{X}) \xrightarrow{\ \sim\ } \operatorname{Hom}\bigl(\varprojlim \mathcal{Y}, \varprojlim \mathfrak{X}\bigr) ,
\]

\[
  \operatorname{Hom}_{\operatorname{Pro}(\mathrm{Sch}_{/S})}(\mathcal{Y}, \mathfrak{X}) \simeq \operatorname{Hom}_S\bigl(\varprojlim \mathcal{Y}, \varprojlim \mathfrak{X}\bigr)
\]

\[
  \begin{cases}
    f_i = a_i + \displaystyle\sum_{1 \leq j \leq n} b_{ij}\, t_j + \sum_{|p| \geq 2} c_{ip}\, t^p \\
    1 \leq i \leq n
  \end{cases}
  \qquad
  \begin{cases}
    a_i \in \mathfrak{m} \\
    \det(b_{ij}) \notin \mathfrak{m}
  \end{cases}
\]

\[
  (\lambda_1, \ldots, \lambda_n) \in \mathfrak{m}^{(n)} \xmapsto{\ \lambda_{(f = (f_1, \ldots, f_n))}\ } \bigl(f_1(\lambda_1, \ldots, \lambda_n), \ldots, f_n(\lambda_1, \ldots, \lambda_n)\bigr)
\]

\[
  \boxed{\, f_i(\lambda_1, \ldots, \lambda_n) = a_i + \sum_j b_{ij}\, \lambda_j + \sum_{2 \leq |p| \leq N} c_{ip}\, \lambda_1^{p_1} \cdots \lambda_n^{p_n} \,}
\]

\[
  \begin{array}{c}
    B \simeq A[[t_1, \ldots, t_n]] \\
    \uparrow \\
    A
  \end{array}
  \qquad
  \begin{array}{l}
    V_1 \; [\; \mathfrak{m}^{?},\ \mathfrak{m}^2 \\
    V_2 \; [\; \mathfrak{m}^3 \\
    \quad\ \vdots \\
    \phantom{V_N} \; [\; \mathfrak{m}^{N_1},\ \mathfrak{m}^{N_?} \\
    V_N \; [\; \mathfrak{m}^{N_?} = 0
  \end{array}
\]

\[
  G_{n,N} = \underline{\operatorname{Aut}}_{\text{algèbres augmentées}} \mathbf{Z}[t_1, \ldots, t_n]/(t_1, \ldots, t_n)^{N+1}
\]

\[
  \begin{cases}
    f_i(t_1, \ldots, t_n) = \displaystyle\sum_j b_{ij}\, t_j + \sum_{2 \leq |p| \leq N} c_{ip}\, t_1^{p_1} \cdots t_n^{p_n} \\
    b_{ij}, c_{ip} \in A \qquad \det(b_{ij}) \notin \mathfrak{m}
  \end{cases}
\]

\[
  \text{\struck{$T_g$}}\ b \dotplus T_g\bigl(\text{\struck{$T$}}\, a \dotplus T_f(x)\bigr)
\]

\[
  b \dotplus T_g(a) \dotplus
\]

\[
  T_{g'}(a + b) \qquad\qquad g(x + y) = g(x)
\]

\[
  \text{\struck{$T_i$}} \qquad \text{\struck{$g$}}\ g(a + b) = g(a) + g'
\]

\[
  g(x + y) =
\]

\[
\begin{cases}
  f_{i} = a_{i} + \sum_{j} b_{ij}\, t_{j} + \sum_{2 \leq |p| \leq N} c_{i,p}\, t^{p},
  \qquad a_{i},\, b_{ij},\, c_{i,p} \in A, \\
  a_{i} \in \mathfrak{m}, \quad \det(b_{ij}) \notin \mathfrak{m}.
\end{cases}
\]

\[
\begin{cases}
  f_{i} = a_{i} + \sum b_{ij}\, t_{j} + \sum_{2 \leq |p| \leq N} c_{i,p}\, t_{1}^{p_{1}} \cdots t_{n}^{p_{n}}, \\
  \det(b_{ij}) \neq 0
\end{cases}
\]

\[
a'_{0} + a'_{1}(a_{0} + a_{1} t + a_{2} t^{2}) + a'_{2}(a_{0} + a_{1} t + a_{2} t^{2})^{2}
\]

\[
\begin{aligned}
  &= (a'_{0} + a'_{1} a_{0} + a'_{2} a_{0}^{2}) \\
  &\quad + (a'_{1} a_{1} + 2 a'_{2} a_{0} a_{1})\, t \\
  &\quad + (a'_{1} a_{2} + a'_{2} a_{1}^{2} + 2 a'_{2} a_{0} a_{2})\, t^{2} \\
  &\quad + 2 a'_{2} a_{1} a_{2}\, t^{3} \\
  &\quad + a'_{2} a_{2}^{2}\, t^{4}
\end{aligned}
\]