\[
  B \;=\; A\,[T_1,\dots,T_r]\,/\,(T_1^{\,n_1}-x_1,\;\dots,\;T_r^{\,n_r}-x_r)
\]

\[
  \underline{C} \longrightarrow (\mathrm{Sch}) .
\]

\[
  \underline{r} : \mathrm{Rev}_{\underline{R}}(S) \longrightarrow (\mathrm{Sch})_{/S} .
\]

\[
  (*) \qquad \underline{r} : \underline{\mathrm{Rev}}^{\underline{R}}
  \longrightarrow \underline{\mathrm{Rev}} ,
\]

\[
  F : \mathrm{Rev}^{\underline{R}}(S) \longrightarrow (\text{Ens finis}) ,
\]

\[
  f^{\circ} : \underline{Et}_S \longrightarrow \underline{Et}_T .
\]

\[
  u(M) : \underline{s}_T(M \times_S T) \longrightarrow
  \underline{s}_S(M) \times_S T
\]

\[ \hat{A}^{(x,y)} = k[[x,y]] \]

\[ \pi_1(X_{\bar y}, \xi) \to \pi_1(X, \xi) \to \pi_1(Y, \zeta) \to 0 \]

\[ H^1(X, G) \longrightarrow \prod_i H^1(X_i, G) \]

\[ C = \bar X^n \times_{\mathbf{P}^r_k} L^{r-n+1} \]

\[ \pi_1(C) \longrightarrow \pi_1(X) \]

\[ \struck{\pi_1^t(X)} \quad \pi \longrightarrow \pi' \times \pi'' \]

\[ \pi_1'(X \times Y) \;\xrightarrow{\ \sim\ }\; \pi_1'(X) \times \pi_1'(Y). \]

\[ \pi_1(X \times Y) \;\xrightarrow{\ \mathrm{surj}\ }\;
   \pi_1(X) \times \pi_1(Y) \]

\[ \pi_1(Y) \longrightarrow \pi_1(X) \qquad (X \text{ normal}) \]

\[ \pi_1(X) \qquad\qquad
   \struck{\pi_1(Y) \to \pi_1(X) \leftarrow \pi_1(X - Y)} \]

\[ X = \bar X - \bar X \cap Y, \]

\[ X_u = X \times_{\mathbf{P}^r_k} L_u
       = X_K \times_{\mathbf{P}^r_K} L_u. \]

\[ \pi_1(U) \longrightarrow \pi_1(X) \]

\[ \pi_1(U, \Omega) \longrightarrow
   \varprojlim_{X \text{ modèle propre}} \pi_1(X, \Omega) \]

\[ \underline{\mathrm{Pic}}^{\tau}_Y \longrightarrow
   \underline{\mathrm{Pic}}^{\tau}_X \]

\[ \mathrm{Pic}(X) \xrightarrow{\ \sim\ } \mathrm{Pic}(U), \]

\[ f^{*} : H^1(Y, \mathcal{O}_Y) \longrightarrow H^1(X, \mathcal{O}_X) \]

\[ H^1(Y, \mathcal{O}_Y)^F \longrightarrow H^1(X, \mathcal{O}_X)^F \]

\[ {}_F H^1(Y, \mathcal{O}_Y) \longrightarrow {}_F H^1(X, \mathcal{O}_X) \]

\[ H^1(Y, \mathbf{Z}/p\mathbf{Z}) \longrightarrow
   H^1(X, \mathbf{Z}/p\mathbf{Z}) \]

\[ H^0(Y, \mathcal{O}_Y / \mathcal{O}_Y^{p^{\nu}}) \longrightarrow
   H^0(X, \mathcal{O}_X / \mathcal{O}_X^{p^{\nu}}) \]

\[ f^{*} : H^0(Y, \mathcal{O}_Y^{p^{-\nu}} / \mathcal{O}_Y) \longrightarrow
   H^0(X, \mathcal{O}_X^{p^{-\nu}} / \mathcal{O}_X) \]

\[ \dim \underline{\mathrm{Pic}}_{X/k} \leq n(K/k) \]

\[ \bar K = K \otimes_k \bar k \]

\[ \dim \underline{\mathrm{Pic}}_{X/k} = \tfrac{1}{2}\,
   \mathrm{rang}_{\mathbf{Z}_{\ell}}
   \left(\text{partie } \ell\text{-primaire de } \pi_1(X/k)
   \text{ rendu abélien}\right) \]

\[ 0 \to T \to A^{\sharp} \to B \to 0. \tag{\struck{1}7.2.2} \]

\[ A^{\sharp}_{n} \simeq A_{n} \tag{7.2.3} \]

\[ (A^{\sharp})^{\wedge} \simeq \hat{A}, \tag{7.\struck{3}2.4} \]

\[ T_{o \cdot X} X' \to T_{o \cdot S} S' \]

\[ T_{o \cdot X} X' = T'_{o} \]

\[ e \to I_{o} \to \pi_{1}^{t}(\mathcal{X}/D, \bar{\xi}_{1}) \to
\pi_{1}^{t}(X_{o}/D'_{o}, \bar{\xi}_{o}) \to e \]

\[ \struck{\ill{}} \to \mu^{\infty}_{X_{o}, \mathrm{tame}}(\bar{\xi}_{o}) \to
I_{o} \to o \]

\[ \mu(\bar{\xi}_{o}) \to \pi_{1}^{t}(\mathcal{X}/D) \to
\pi_{1}^{t}(\mathcal{U}) \to e \]

\[ \alpha \struck{\ill{}} \in H^{2}(\pi_{1}^{t}(\mathcal{U}), I_{o}) \]

\[ o \to H^{2}(\pi_{1}^{t}(\mathcal{U}), I) \xrightarrow{\;\rho\;} H^{2}(\mathcal{U}, I)
\to H^{2}(\tilde{\mathcal{U}}^{t}, I_{o})^{\pi_{1}^{t}} \to H^{3}(\pi_{1}^{t}, I) \]

\[ H^{2}(\tilde{\mathcal{U}}^{t}, I_{o})^{\pi_{1}^{t}} =
\mathrm{Hom}(\pi_{2}(\tilde{\mathcal{U}}^{t}), I)^{\pi_{1}^{t}} =
\mathrm{Hom}(\pi_{2}^{t}(\mathcal{U}), I)^{\pi_{1}^{t}} \]

\[ \mathcal{L}_{X_{o}/\mathcal{X}} \simeq i^{*}(\underline{L}(X_{o})). \]

\[ \rho(\alpha) = \sigma(\beta). \]

\[ H^{2}(\mathcal{U}, \mu) \to H^{2}(\mathcal{U}, I) \to
H^{2}(\tilde{\mathcal{U}}^{t}, I_{o}) = \mathrm{Hom}(\pi_{2}^{t}(\mathcal{U}),
I_{o}) \]

\[ \pi_{2}^{t}(\mathcal{U}) = H_{2}(\tilde{\mathcal{U}}) \qquad
\struck{\ill{}} \qquad \pi_{2}^{t}(\mathcal{U}) \to H_{2}(\tilde{\mathcal{U}}^{t}) \]

\[ \mathrm{Im}\, \beta \in \ker\!\left(H^{2}(\mathcal{U}, I'_{o}) \to
H^{2}(\tilde{\mathcal{U}}^{t}, I'_{o})\right) \simeq H^{2}(\pi_{1}^{t},
I'_{o}), \]

\[ o \to \pi'_{1} \to \pi_{1}^{t}(\mathcal{X}/D) \]

\[ \mu_{o} \to I_{o} \to \ill \]

\[ \beta \;(\in H^{2}(\mathcal{U}, \mu)) \quad\text{dans}\quad
H^{2}(\tilde{\mathcal{U}}^{t}, \mathcal{J}_{o}) \]

\[ H^{2}(\mathcal{X}, \mu_{o}) \to H^{2}(\mathcal{U}, \mathcal{J}_{o}). \]

\[ \beta_{o} = n\gamma_{o} + \sum \lambda_{i} \beta_{o}^{i}, \]

\[ \beta \in \ill \; \mathrm{Pic}(X_{m}) \]

\[ \mathrm{Pic}(X_{m}) \to \mathrm{Pic}(X_{o}) \]

\[ \mathrm{Pic}(\mathcal{X}) = \varprojlim \mathrm{Pic}(X_{m}) \to
\mathrm{Pic}(X_{o}), \]

\[ {}_{n}\mathrm{Pic}(\mathcal{X}) \to {}_{n}\!\left(\mathrm{Pic}(\mathcal{X})/\ill\right) \]

\[ \pi_{2}^{t}(X_{o}/D'_{o}, \bar{\xi}_{o}) \to
\mu^{\infty}_{X_{o}, t}(\bar{\xi}_{o}) \to \pi_{1}^{t}(\mathcal{X}/D,
\bar{\xi}_{1}) \to \pi_{1}^{t}(X_{o}/D'_{o}, \bar{\xi}_{o}) \to o \]

\[ \alpha \in H^{2}\!\left(X_{o} \struck{- D_{o}},\;
\mu^{\infty}_{X_{o}, \mathrm{tame}}\right) \]

\[ \pi_{1}^{t}(\mathcal{X}/D, \bar{\xi}_{1}) \simeq \pi_{1}^{t}(X_{o}/D'_{o},
\bar{\xi}_{o}). \]

\[ \left(\simeq \prod_{\ell \in D'_{o}} (\mathbb{Z}/\ell\mathbb{Z})^{*}\right), \]

\[ \mu^{\infty}_{t, X_{o}} = \prod_{\ell \in S} \mu^{\infty}(\ell)_{X_{o}}. \]

\[ \struck{\ill{}} \prod_{\ell} T_{\ell}(\Omega^{*}) \to I_{i} \]

\[ \struck{\ill{}} \; \mu_{\underline{n}} = \prod \mu_{n_{i}} \qquad
(\underline{n} = (n_{1}, \ldots, n_{r})) \quad \text{sur } X. \]

\[ 1 \to \mu_{\underline{n}} \to G \to \Gamma \to 1 \]

\[ X' \to \tilde{X}_{1} \to X \]

\[ \tilde{X}[T_{1}, \ldots, T_{r}] \big/ \left(T_{1}^{n_{1}} - a_{1}, \ldots,
T_{r}^{n_{r}} - a_{r}\right) \]

\[ \tilde{X} \times^{\Gamma} \mu \simeq \mu_{\underline{n}} \]

\[ H^{2}(B_{\Gamma}, \mu) \xrightarrow{\;\varphi\;} H^{2}(X, \mu_{\underline{n}})
\simeq \prod H^{2}(X, \mu_{n_{i}}) \]

\[ H^{2}(X, \mu_{\underline{n}}) = H^{2}\!\left(B_{\Gamma}/(\tilde{X}, \Gamma),
\mu\right) \]

\[ \xi_{i} = c\!\left(\mathcal{O}_{X}(D_{i})\right) \pmod{n_{i}}. \]

\[ X_{\underline{a}}(\underline{b}) = X^{-}_{\underline{b}}(\underline{a}),
\qquad \text{d'où} \qquad v\big(X_{\underline{a}}(\underline{b})\big) =
v\big(X^{-}_{\underline{b}}(\underline{a})\big). \]

\[ i(X_{a}, b) = i(X^{-}_{b}, a). \]

\[ i(X_{\underline{a}}, \underline{b}) = i(X^{-}_{\underline{b}},
\underline{a}). \]

\[ 1 \longrightarrow I \longrightarrow G \longrightarrow \Gamma \longrightarrow
1 \]

\[ \alpha \in H^{2}(\Gamma, I) \]

\[ \alpha \in H^{2}(X, \tilde{X} \times^{\Gamma} I), \]

\[ \tilde{D} \times^{\Gamma} I \;\simeq\; \prod_{i} \mu_{n_{i}, D} \quad
\text{sur } D \]

\[ I_{\tilde{X}} \simeq \mu_{\underline{n}} \]

\[ I_{i} \simeq \mu, \qquad c\big(\mathcal{O}_{X}(D)\big) \in H^{2}(X,
\mu_{\underline{n}}), \]

\[ c\big(T_{D/X}\big) \in H^{2}(D, \mu_{\underline{n}}), \quad \text{fibré
normal.} \]

\begin{equation*}
(6.4) \qquad f g f^{-1} = g^{q}
\end{equation*}

\[ F G F^{-1} = G^{q}. \]

\[ \lambda \longmapsto \lambda^{q}. \]

\[ H^{2}(X, M) = \operatorname{Hom}\nolimits_{\mathrm{cont}}\big(\pi_{2}(X),
M\big). \]

\[ Z = \operatorname{Spec} \hat{\mathcal{O}}_{X,a} \]

\[ X_{T'} \cap X_{T''} = p \]

\[ P = QQ', \qquad QR + Q'R' = 1, \qquad (P) = (Q)(Q') = (Q) \cap (Q') \]

\[ A[t]/(P) \longrightarrow A[t]/(Q) \times A[t]/(Q') \]

\[ A/(QQ') \longrightarrow A/Q \times A/Q' \]

\[ Q \cap Q' = Q\,Q'\,(Q + Q') \subset Q'Q + QQ' = QQ' \]

\[ A/Q\,Q' \longrightarrow A/Q \times A/Q' \quad \text{surjectif} \]

\[ Q + Q' = A \iff QQ' = Q \cap Q' \]

\[ J J' = R, \qquad J + J' = A \]

\[ (P) = (Q)\,(Q'), \]

\[ A \text{ anneau local}, \qquad \hat{A} \text{ son complété} \]

\[ X = \operatorname{Spec}(A), \qquad \hat{X} \]

\[ D \text{ partie fermée}, \qquad \hat{D} \]

\[ \hat{Y} \hookrightarrow \hat{Z}, \qquad \hat{Y} \times_{\hat{S}} \hat{Z} \]

\[ A \longrightarrow A' \longrightarrow \hat{A} = \hat{A}' \]

\[ \pi_{1}(Y_{\bar{t}}) \longrightarrow \pi_{1}(X) \]

\begin{equation*}
(*) \qquad \pi_{1}(Y_{\bar{t}}) \longrightarrow \pi_{1}(X_{\bar{t}})
\end{equation*}

\begin{equation*}
(**) \qquad \pi_{1}^{!}(X_{\bar{t}}) \longrightarrow \pi_{1}^{!}(X_{\bar{t}})
\end{equation*}

\[ \pi_{1}(Y_{\bar{t}}) \longrightarrow \pi_{1}(X) \; \big(\simeq
\pi_{1}(X_{\bar{t}}) \text{ par le théorème de changement de base}\big) \]

\[ \pi_{1}^{t}(U_{t}) \longrightarrow \pi_{1}^{t}(U) \]

\[ \pi_{1}^{t}(D - \check{D}) \longrightarrow \pi_{1}^{t}(P - \check{X}), \]

\begin{equation*}
(2.1) \qquad \phi(G_{\xi}) \subset H_{\bar{f}'(\xi)}.
\end{equation*}

\begin{equation*}
(3.1) \qquad I = \prod_{\ell \neq p} \underline{Z}_{\ell}(1)\big(k(\bar{x})
\big)^{C} = \prod_{\ell \neq p} T_{\ell}\big(k(\bar{x})\big)^{C},
\end{equation*}

\[ \bar{f}^{*}(\bar{E}_{c'}) = \sum_{c} n_{c,c'} \bar{D}_{c}, \]

\[ \underline{Z}_{\ell}(1)(\bar{y}) \xrightarrow{\;\sim\;}
\underline{Z}_{\ell}(1)(\bar{x}). \]

\[ \underline{Z}_{\ell}(1)(\bar{x}) \to G \]

\[ \underline{Z}_{\ell}(1)(\bar{x}) \to G. \]

\[ \underline{Z}_{\ell}(1)(D_{i}) \to G. \]

\[ e \to H_{\mathrm{inf}} \to H \to H_{\mathrm{sép}} \to e \]

\[ Z(G) = \varinjlim \operatorname{Hom}_{\underline{G}}(G_{i}, G) \]

\[ \pi_{1}(X/k) = \varprojlim {}_{n}X \]

\[ \varphi_{s} : X \times_{S} \operatorname{Spec}(\mathcal{O}_{s})
\xrightarrow{\;\sim\;} X(s), \]

\[ e \to U \to G \to \Gamma \to e \]

\[ (P, \xi) \simeq (P', \xi') \times_{G} G'. \]

\[ (Q, \eta) = (P, \xi) \times_{G} (G' \times G') = [(P, \xi) \times_{u} G']
\times [(P, \xi) \times_{v} G'] \]

\[
\mathfrak{z}(G \times G') = \mathfrak{z}(G) \times \mathfrak{z}(G')
\]

\[
\begin{array}{ccc}
\pi' & \pi''_{\alpha} & \pi'''_{\beta} \\
S & S' & S'''
\end{array}
\]

\[
\begin{array}{ll}
p_{ij}^{\alpha\beta} : \pi'''_{\beta} \to \pi''_{\alpha} & \alpha = p_{ij}(\beta), \quad (ij) = \{(1,2), (2,3), (1,3)\} \\
p_i^{\alpha} : \pi''_{\alpha} \to \pi' & i = \{1, 2\}
\end{array}
\]

\[
\begin{array}{l}
p_1^{\alpha} p_{12}^{\alpha\beta} = \operatorname{ind}(a'_{\beta})\, p_1^{\alpha} p_{13}^{\alpha\beta} \\
p_2^{\alpha} p_{12}^{\alpha\beta} = \operatorname{ind}(a''_{\beta})\, p_1^{\alpha} p_{23}^{\alpha\beta} \\
p_2^{\alpha} p_{13}^{\alpha\beta} = \operatorname{ind}(a'''_{\beta})\, p_2 p_{23}
\end{array}
\]

\[
\begin{array}{ll}
p_{12}^{-1}(p_1^{-1}(E)) \to p_{12}^{-1}(p_2^{-1}(E)) & (p_1 p_{12})^{-1}(E) \to (p_2 p_{12})^{-1}(E) \\
p_{23}^{-1}(p_1^{-1}(E)) \to p_{23}^{-1}(p_2^{-1}(E)) & (p_1 p_{23})^{-1}(E) \to (p_2 p_{23})^{-1}(E)
\end{array}
\]

\[
\boxed{\; A_{\alpha''}\, a''_{\beta}\, A_{\alpha'} = a'''_{\beta}\, A_{\alpha'''}\, a'_{\beta} \;}
\]

\[
g_{s''} = g_{s'''}\, g_{s'} \quad \text{i.e.} \quad g_{s''} = 1
\]

\[
\pi_1(X \times_S Y, c) \to \pi_1(X, a) \times_{\pi_1(S, d)} \pi_1(Y, b).
\]

\[
\pi_1(X \times_k Y) \simeq \pi_1(X) \times_{\pi_1(k)} \pi_1(Y)
\]

\[
\begin{array}{ccc}
X & & Y \\
& S &
\end{array}
\]

\[
\pi_s = \pi, \qquad \operatorname{Aut}(T) = \operatorname{Aut}(\pi/s) \simeq \pi
\quad \text{par } x \mapsto x g^{-1}
\]

\[
E = F(\pi_s), \qquad F(U) = \operatorname{Hom}_{\pi}(U, E)
\]

\[
\mathcal{F}^{\pi}_e \simeq \mathcal{F}^{\pi}_s \simeq \mathcal{E}^{\pi}_{\mathrm{Ens}}
\]

\[
\mathcal{F}^{\pi}_T \simeq (\mathrm{Ens}) \qquad \text{car} \qquad
\mathcal{E}_{/T} \simeq (\mathrm{Ens}) = \mathcal{E}^{\circ}
\]

\[
U = \coprod_i \pi/\pi_i \qquad \text{on a} \qquad \mathcal{F}_U \simeq \prod_i E^{\pi_i}
\]

\[
\begin{array}{lll}
H^0(F) = \Gamma(F) = F(e) & \text{correspond à} & E^{\pi} = \operatorname{Hom}(\pi_s, E) \\
H^1(F) & \simeq & H^1(\pi, E) \\
H^i(F) & \simeq & H^i(\pi, E)
\end{array}
\]

\[
X \to X/G
\]

\[
\hat{\mathcal{O}}_y \to \hat{\mathcal{O}}_{x'} \xrightarrow{\ \sim\ } \hat{\mathcal{O}}_x
\]

\[
(X \times G)/H \simeq X \times G/H,
\]

\[
X_{(\alpha)} = \bigcup_{i \in \alpha} X_i, \quad \text{stables sous } G.
\]

\[
X/G = \bigcup X_{(\alpha)}/G
\]

\[
Y \xrightarrow{\ f_i^{-1}\ } X_i \xrightarrow{\ \text{inj. can}\ } X
\]

\[
F(X/H) \simeq F(X)/H.
\]

\[
  \pi^1(S,\xi;\pi)
\]

\[
  \begin{cases} P_i \subset P_i' \\ \xi_i = \xi_i' \end{cases}
\]

\[
  P' \times_{\pi_1}(\pi'', \varphi_1) \xrightarrow{\ u\ }
  P' \times_{\pi_1}(\pi'', \varphi_2)
\]

\[
  P/\pi' \cap \pi'' \longrightarrow P/\pi' \times P/\pi''
\]

\[
  \underline{\mathrm{Isom}}_S(E_S, T).
\]

\[
  \pi \longmapsto \pi^1(S,\xi;\pi)
\]

\[
  H^1_{\mathcal{C}}(S,G) = \varinjlim_{T/S} H^1(T/S, G)
\]

\[
  H^1(T/S, G) = H^1(K_{T/S}, G) = H^1(\pi_0(K_{T/S}), G)
\]

\[
  H^1_{C_3} \simeq H^1_{C_2}
\]

\[
  H^1_{C_2} \xrightarrow{\ \sim\ } H^1_{C_1}
\]

\[
  H^1_{C_i} \simeq H^1(\widehat{V}/V, -) \qquad \text{pour } i = 1,2,3
\]

\[
  H^1_{C_5} \simeq H^1_{C_4} \simeq H^1_{C_3} \quad (\simeq H^1_{C_2})
\]

\[
  H^1_{C_i}(V,-) \simeq H^1_{C_i}(k, -) \qquad (i = 2,3,4,5)
\]

\[
  \pi^{\mathrm{I}}(X,G) \simeq H^2(\check{X},G) \simeq \varinjlim H^2(X_i,G)
  \simeq \varinjlim H^2(\widetilde{X}_i,G)
\]

\[
  H^*(\widetilde{X}_i,G) \Longleftarrow E_2^{p,q} = H^p(H_i, H^q(\widetilde{X},G))
\]

\[
  \struck{\ill{}} H^*(\widetilde{X},G) \Longleftarrow E^{p,q} =
  \varinjlim_i H^p(H_i, H^q(\widetilde{X},G))
\]

\[
  \cdots \to \varinjlim H^2(H_i,G) \to \pi^2(X,G) \to
  \varinjlim_i \pi^2(\widetilde{X}_i,G)^{H_i} \to \varinjlim H^3(H_i,G)
\]

\[
  H^*(X^k,G) \Longleftarrow H
\]

\[
  \mathrm{Ker}\,\bigl(\pi^2(X,G) \to \pi^2(X^k,G)\bigr) \simeq
  \varinjlim_i H^2(H_i,G)
\]

\[
  \mathrm{Coker}\,(\text{id}) \simeq \mathrm{Ker}\Bigl(\varinjlim H^3(H_i,G)
  \longrightarrow H^3(\widetilde{X},G)\Bigr)
\]

\[
  \pi_1(\overline{F}) \to \pi_1(X) \to \pi_1(Y) \to e
\]

\[
  \pi_1(Y) \longrightarrow G^{\pi_1(X)}
\]

\[
  H^1(\pi_1(X), G) \longrightarrow H^1(\pi_1(\overline{F}), G)^{\pi_1(Y)}
\]

\[
  H^1\bigl(\pi_1(Y), G^{\pi_1(\overline{F})}\bigr) =
  H^1(\pi_1(Y), \mathfrak{z})
\]

\[
  \pi_2(Y) \longrightarrow \mathfrak{z} \qquad H^1(\pi_1(\overline{F}), G)
\]

\[
  \pi_1(Y') \longrightarrow \mathfrak{z}
\]

\[
  H^1\bigl(Y'/Y, \mathrm{Hom}(\pi_1(\overline{F}), \mathfrak{z})\bigr) = 0
\]

\[
  G \rightsquigarrow \pi^1(S'/S, \xi; G)
\]

\[
  \pi^1(S'/S, \xi; G) = \varinjlim \mathrm{Hom}(\pi_i, G)
\]

\[
  \pi^1(S'/S, \xi; G) \simeq \mathrm{Hom}(\pi_1(S'/S, \xi), G)
\]

\[
  G \rightsquigarrow \pi^1_{\mathrm{eff}}(S'/S, \xi; G)
\]

\[
  K_i^{\xi} = \pi_0(X_i^{\xi})
\]

\[
  f(p_1(\xi)) = f(p_0(q_2))\, f(p_2(q_1)) \qquad \text{pour tt } \xi \in K
\]

\[
  h(p_1(\xi))\, f(\xi)\, h(p_0 q_1)^{-1} \simeq f(\xi)
\]

\[
  \begin{cases}
    f(i_1(q)) = \struck{1}\, e_G & \text{pour tt } q \in k_1 \\
    \struck{\ill{}}\quad f \circ p_1 = (f \circ p_0)(f \circ p_2) &
  \end{cases}
\]

\[
  h(p_1 q)\, f(q)\, h(p_0 q)^{-1} \simeq f(q)
\]

\[
  \pi_0(k_{\ast}) = e, \qquad \text{« } \pi_1(k_{\ast}) = e \text{ »}
  \qquad \text{i.e.} \quad \pi^1(k_{\ast}, G) = e
\]

\[
  \pi_0(K_{\ast}) = e
\]

\[
  F(I) \underset{?}{\simeq} C(K, I)
\]

\[
  g(p(t_1))\, f(t_1)\, g(q(t_1))^{-1}
\]

\[
  F^{*} : \quad F^{*0} \rightrightarrows F^{*1}
  \qquad (F^0, F^1 \in \underline{\mathrm{Hom}}(\mathrm{Ens}, \mathrm{Ens}))
\]

\[
  \pi^0(F^{*}) = F = \mathrm{Ker}\,(F^0 \rightrightarrows F^1)
\]

\[
  f^0 \rightrightarrows f^1 \Rrightarrow f^2
\]

\[
  \overline{W}_n(G) = G_{n-1} \times G_{n-2} \times \cdots \times G_0
\]

\[
  \varphi_n(x_1, \ldots, x_n) = \frac{1}{n}\left(1 + \frac{p_1}{n-1}
  + \frac{p_2}{\frac{(n-1)(n-2)}{2}} + \cdots + \frac{p_k}{\binom{n-1}{k}}
  + \cdots + p_{n-1}\right)
\]

\[
  U_j f = \frac{1}{2\pi i} \int_{B_j}
    f(z_1, \ldots, z_{j-1}, t, z_{j+1}, \ldots, z_n)\, \frac{dt}{t - z_j}
\]

\[
  T_j f = \frac{1}{2\pi i} \iint_{K_j}
    f(z_1, \ldots, z_{j-1}, t, z_{j+1}, \ldots, z_n)\,
    \frac{dt\, \overline{dt}}{t - z_j}
\]

\[
  P\omega = \varphi_n(U ; j_1, \ldots, j_q) \ast \sum_{\gamma=1}^{q}
  (-1)^{\gamma-1} (T_{j_\gamma} a)\;
  d\overline{z}_{j_1} \wedge \cdots \wedge
  \widehat{d\overline{z}_{j_\gamma}} \wedge \cdots \wedge
  d\overline{z}_{j_q}
\]

\[
  P d'' f \;\struck{d'' P f} \;=\; f - U_1 U_2 \ldots U_n f
\]

\[
  r : R \longrightarrow \mathrm{Fl}_S
\]

\[
  r(X' \cap X'') = r(X') \cap r(X''),
\]

\[
  r(\varphi(P)) \overset{\alpha}{\simeq} \struck{\ill{}}\; Z
\]

\[
  r(P') \simeq Z .
\]

\[
  P'/G' \simeq Z .
\]

\[
  s(C_{S'}) = Z^{\underline{a}}_{\underline{n}} , \qquad
  \mu_{\underline{n}}(S') = \prod_{i \in I} \mu_{n_i}(S) ,
\]

\[
  \mathrm{Ob}(C_{S'}) = \bigl\{\, \underline{a} = (a_i)_{i \in I}
  \;\bigm|\; \forall i,\; a_i \text{ une équation de } D_i \,\bigr\}
\]

\[
  \mathrm{Hom}(\underline{a}, \underline{b}) = \bigl\{\, \xi = (q_i)_{i \in I}
  \;\bigm|\; \forall i,\; b_i = a_i \xi_i^{n_i} \,\bigr\}
\]

\[
  s : C \longrightarrow \mathrm{Fl}_S
\]

\[
  M \longmapsto r(X \times_{S'} \varphi(M)) \big/ \underbrace{r(\varphi(M))}_{s(M)}
\]

\[
  (\ast) \qquad \alpha : R_S^{C_0} \longrightarrow
  \varprojlim_{C_0} \, (\mathcal{E}|C_0)
\]

\[
  R_S \;\simeq\; \varinjlim_{C_0} \; \varprojlim_{\text{sur } C_0} \,
  (\mathcal{E}|C_0)
\]

\[
  \alpha(X) = \emptyset \;\Longrightarrow\; X = \emptyset ,
\]

\[
  T \times \varphi(M) \xrightarrow{\ \mathrm{pr}_1\ } T
  \rightrightarrows \varphi(M)
\]

\[
  X = X_1 \sqcup X_2 ,
\]

\[
  A(M) = \emptyset , \quad B(M) = \emptyset ,
  \qquad A(M') = \emptyset , \quad B(M') = \emptyset
\]

\[
  A(M \times M') = A(M') \times_{s(M')} s(M \times M') ,
\]

\[
  F : C \longrightarrow (\mathrm{Ens}\ f) \quad \text{tel que}
  \quad F(e_i) \neq \emptyset
\]

\[
  X^E = \bigcup_P \alpha_P\bigl((X^P)^{\ast}\bigr)
  \qquad \text{(disjoint)}
\]

\[
  (f^E)^{-1}\bigl(\Delta_y^{(E)}\bigr) = \bigcup_P \alpha_P(F_P)
  \qquad \text{(réunion disjointe)}
\]

\[
  (f^3)^{-1}\bigl(\Delta_y^{(3)}\bigr) =
  \underset{n - 2d}{F_3} \;\cup\;
  \underset{n - d}{\bigcup_{1 \leq i < j \leq 3} \alpha_{ij}(F_2)} \;\cup\;
  \underset{n}{\Delta_X^{(3)}}
\]

\[
  \overline{F}_3 \cap \overline{\alpha_{ij}(F_2)} \;\cdot\;
  \overline{F}_3 \cap \alpha_{ij}(\overline{F}_2) \;\;?
\]

\[
  x_1^0 \neq x_2^0 \neq x_3^0 \qquad\qquad x_1^0 = x_3^0 \neq x_2^0
\]

\[
  \dim X = 3 , \quad \dim Y = 4 , \qquad \dim F_3 = 1 , \qquad
  \dim \alpha_{ij}(F_2) = 2
\]

\[
  X'' = X \times_Y Y'' \simeq Y'' \sqcup
  \underbrace{(X'_1 \times_{Y'} Y'')}_{Y'' \times [1,\, n-1]}
  = Y'' \times [1, n] .
\]

\[
  F : C \longrightarrow (\mathrm{Ens\ fini})
\]