\[
\mathcal{P}^{\underline{m}}_{K} = \mathcal{D}^{\underline{m}}_{K} / P^{\underline{m}}_{K}.
\]

\[
(1) \qquad 0 \to N^{\underline{m}}_{K} \to \mathcal{P}^{\underline{m}}_{K} \to \mathcal{P}_K \to 0
\]

\[
(2) \qquad 0 \to \mathbb{Z}_2^{S_\infty}/\mathrm{Im}\,E^{\underline{m}_0}_{K} \to N^{\underline{m}}_{K} \to (A_K/\underline{m}_0)^{*}/\mathrm{Im}\,E_K \to 0
\]

\[
0 \to (A_K/\underline{m}_0)^{*}/\mathrm{Im}\,E_K \to \mathcal{P}^{\underline{m}}_{K} \to \mathcal{P}_K \to 0
\]

\[
U^{\underline{m}}_{K} = \prod_{\mathfrak{p} \notin S} K^{*}_{\mathfrak{p}} \times \prod_{\mathfrak{p} \in S_0} \bigl(1 + \mathfrak{p}^{v_{\mathfrak{p}}(\underline{m})}\bigr) \times \prod_{\mathfrak{p} \in S_\infty} K_{\mathfrak{p}}^{*+}
\]

\[
C_K/\mathrm{Im}\,U^{\underline{m}}_{K} \approx \mathcal{P}^{\underline{m}}_{K}
\]

\[
\varprojlim_{\underline{m}} \mathcal{P}^{(\underline{m})}_{K} = C_K/(\text{\uncertain{composante connexe} de l'unité})\,\ill{}_K = C'_K
\]

\[
\mathcal{P}^{\underline{m}}_{K}/N_{L/K}\mathcal{P}^{\underline{m}'}_{L} \approx C_K/N_{L/K}C_L
\]

\[
\mathcal{D}^{S_0} \to G(L/K)
\]

\[
\mathcal{P}^{\underline{m}}_{K}/N_{L/K}\mathcal{P}^{\underline{m}'}_{L} \to G(L/K),
\]

\[
\mathcal{P}^{\underline{m}}_{K}/N_{L/K}\mathcal{P}^{\underline{m}'}_{L} = C_K/N_{L/K}C_L \to G(L/K)
\]

\[
\mathcal{D}_{\mathrm{abs}}(K) = \struck{\prod_{\mathfrak{p}}}\ \coprod_{\mathfrak{p}} W_{\mathfrak{p}}
\]

\[
0 \to V(K) \to J(K) \to \mathcal{D}_{\mathrm{abs}}(K) \to 0
\]

\[
0 \to K^{*} \to J(K) \to C(K) \to 0
\]

\[
0 \to U(K) \to K^{*} \to \mathcal{D}_{\mathrm{abs}}(K) \to \mathcal{P}_{\mathrm{abs}}(K) \to 0
\]

\[
\mathcal{D}_{\mathrm{abs}}(K) \simeq \mathcal{D}(K) \times \mathcal{D}_\infty(K)
\]

\[
0 \to \mathcal{D}_{\infty,0}(K) \to \mathcal{D}_{\mathrm{abs},0}(K) \to \mathcal{D}(K) \to 0
\]

\[
P \simeq K^{*}/U(K)
\]

\[
0 \to \mathcal{D}_{\infty,0}(K)/E_0(K) \to \mathcal{P}_{\mathrm{abs},0}(K) \to \mathcal{P}(K) \to 0
\]

\[
J_\infty^{\mathrm{conn}} \ \struck{=}\ \prod J^{\mathrm{conn}}, \quad \struck{\ill{}}
\]

\[
V_{\underline{m},\mathfrak{p}} \longrightarrow \ \struck{\underline{m}}\ \text{unités, à distance \uncertain{infinie}}
\]

\[
\prod_{\mathfrak{p}\ \text{fini},\ \mathfrak{p} \mid \underline{m}} A_{\mathfrak{p}} \ \struck{\ill{}}
\]

\[
\struck{\ill{}}\ \prod_{\mathfrak{p} \mid \underline{m}} K_{\mathfrak{p}}, \ \text{il}
\]

\[
\vec{x}_{\mathfrak{p}} \in \prod_{\mathfrak{p}\mid\underline{m}} K_{\mathfrak{p}} \quad \ill{}
\]

\[
a \vec{x} \in V_{\underline{m}} \ \struck{\ill{}}\ ; \quad \text{Ainsi,}
\]

\[
\struck{J/K^{*}V_{\underline{m}}}\ \simeq\ J_{\underline{m}}/(K^{*}V_{\underline{m}}) \cap J_{\underline{m}} \quad \ill{} \ \text{et} \ \ill{}
\]

\[
J/H_{\underline{m}} \supset V_{\underline{m}} \cap J_{\underline{m}}, \ \ill{} \ \ill{}
\]

\[
P_{\underline{m}}(K) = \ \struck{\ill{}}\ \text{\ill{} des diviseurs \struck{\ill{}} \add{premiers} à } \underline{m}
\qquad \ \simeq \mathcal{D}_{\underline{m}}(K)/P_{\underline{m}}(K)
\]

\[
a \equiv 1 \quad (\underline{m}).
\]

\[
C_{\underline{m}}(K) = C(K)/H_{\underline{m}}(K) \simeq J(K)/V_{\underline{m}}(K)\,K^{*} \simeq \mathcal{D}_{\underline{m}}(K)/P_{\underline{m}}(K)
\]

\[
P_{\underline{m}}(K) = \text{classes de diviseurs de la forme } (a), \ \text{avec } a \equiv 1 \ (\underline{m})
\]

\[
H_{\underline{m}}(K) = V_{\underline{m}}(K) K^{*}/K^{*}.
\]

\[
0 \to
\begin{array}{c}
\text{s.-groupe de } \struck{\ill{}}\ E(K) \\
\text{des éléments qui sont} \\
\equiv 1\ (\underline{m})
\end{array}
\longrightarrow K^{*} \longrightarrow \underbrace{\struck{\ill{}}\ \mathcal{D}_{\underline{m}}(K) \times \prod_{\mathfrak{p}\mid\underline{m}} \struck{\ill{}}}\ \underbrace{K_{\mathfrak{p}}/V_{\mathfrak{p},\underline{m}}}_{(*)} \longrightarrow C_{\underline{m}}(K) \to 0
\]

\[
\mathcal{D}_{\underline{m}}(K) \longrightarrow G/G' \qquad (\text{\uncertain{hom.\ de réciprocité}})
\]

\[
C_{\underline{m}}(K) \to G/G', \quad \text{d'où } \underline{C(K) \to G/G'}
\]

\[
N_{L/K}C(L) = N_{L_0/K}C(L_0).
\]

\[
\boxed{K_{\mathfrak{p}}^{*} \to G/G'}
\]

\[
\boxed{\prod_{\mathfrak{p}} \Bigl(\frac{x,\, L/K}{\mathfrak{p}}\Bigr) = 1 \quad \text{si } x \in K^{*}}
\]

\[
A(K_{\mathfrak{p}}) \xrightarrow{\ \text{surj.}\ } \text{groupe de décomposition } G^{d}_{\mathfrak{p}} \text{ de } G \text{ relatif à } \mathfrak{p}
\]

\[
A(K_{\mathfrak{p}}) \simeq K_{\mathfrak{p}}^{*\,\wedge}
\]

\[
\varphi_{\mathfrak{p}} \colon K_{\mathfrak{p}}^{*} \longrightarrow G^{d}_{\mathfrak{p}}
\]

\[
\boxed{\prod_{\mathfrak{p}} \varphi_{\mathfrak{p}}(x_{\mathfrak{p}}) = 1 \quad \text{si } x \in K^{*}}
\]

\[
\mathrm{Hom}(\mathbb{G}_m, G) \longleftarrow H^1(C, G) = J_{G/C}(k)
\]

\[
H^1(C, \mathbb{G}_m) = J_C(k)
\]

\[
\mathbb{G}_m(C) = k^{*}
\]

\[
\mathbf{V}_G(k) \times \mathbf{V}(k) \to G(k) \qquad \text{grâce aux symboles locaux}
\]

\[
G(C) \times \mathbf{V}(k)^{0}
\]

\[
G(K) \times J(k) \to G(k) \qquad (\text{définition de } J \text{ comme \uncertain{Albanese} \uncertain{généralisée}})
\]

\[
J_G(k) \times K^{*} \to G(k) \qquad \text{\uncertain{application} \uncertain{spéciale}}
\]

\[
U_G(k) \times \mathcal{D}(k) \to G(k) \qquad \text{\uncertain{triviale}, à l'aide des \uncertain{conjugaisons}}
\]

\[
\mathcal{D}_G(k) \times U(k) \to G(k) \qquad \text{théorème des symboles locaux}
\]

\[
J_{G,C} \times k^{*} \to G(k)
\]

\[
G(C) \times J_C(k) \to G(k)
\]

\[
\boxed{\xi \in \mathrm{Ext}^1\bigl(\underbrace{J^{0}_{G,C}(k)}_{\text{(a)}},\ \underbrace{J'_C(k)}_{\text{(b)}};\ G(k)\bigr)}
\qquad \text{\uncertain{accouplement} de Lang.}
\]

\[
H_{\mathrm{I}}(DD(G)) = E_1^{**}(G) \Longrightarrow H^{*}(C, G)
\]

\[
\underbrace{H^{0}_{\xi}(G)}_{G\text{-diviseurs principaux}} \longrightarrow \underbrace{\coprod_x H^{1}_x(G)}_{G\text{-diviseurs}}
\]

\[
\begin{array}{ccc}
\text{idèles entiers} & & \\
\prod G(\hat{\mathcal{O}}_x) \longleftarrow 0 & \qquad & G(C) = H^{0}(C, G) \qquad 0 \\
\downarrow \qquad\quad \downarrow & & \\
\bigl(\prod_{\text{local}} G(\hat{K}_x)\bigr)/G(K_\xi) \longleftarrow 0 & & H^{1}(C, G) \qquad 0 \\
\text{classes d'idèles} & & \\
H_{\mathrm{II}}\,DD(G) & & H_{\mathrm{I}} H_{\mathrm{II}}\,DD(G)
\end{array}
\]

\[
\prod J^{(0)}_x \longrightarrow J^{*}_{\xi} \qquad \ill{}
\]

\[
\struck{\ill{}}\ J^{*}_{C/k} \quad \text{jacobienne de } C
\]

\[
K^{*} \longrightarrow \text{diviseurs ordinaires} \qquad J \ \text{\ill{}} \ J^{\ill{}}_{x}(k) = \mathbb{G}_m(\hat{\mathcal{O}}_x)
\]

\[
H_{\mathrm{I}}\,DD(G) = \mathrm{Hom}\bigl(\underline{H}_{\mathrm{II}}\,\underline{DD}(\mathbb{G}_m), G\bigr)
\]

\[
\left\lbrace
\begin{array}{ll}
(f_x)_{x \in X} & G\text{-répartition} \\
(\xi_D)_{D \in \mathcal{D}iv\ \text{\uncertain{premiers}}} & \xi_D \in J^{*}_{\mathcal{O}_D/k}
\end{array}\right.
\]

\[
x \in D \qquad f_x \in G(K_x) \to \text{\uncertain{classe} de } G(K_D)\ \struck{\ill{}} = G(K_D)/G(\mathcal{O}_D)^{(1)} \simeq \mathrm{Hom}(J^{*}_{\mathcal{O}_D/k}, G(k))
\]

\[
0 \to N \to Z^{0} \to \mathrm{Alb}^{0} \to 0
\]

\[
0 \to \prod G(\mathcal{O}_x)/G(C) \to R_G/G(K) \to H^1(X, G)^{0} \to 0
\]

\[
X \to \prod_D J^{\ill{}}_{\mathcal{O}_D/k} \to J^{0}_{K/k}\ \Big|\ \struck{\ill{}}\ J^{0}_{X/k} \to 0
\]

\[
? \to \Bigl(\prod_D J^{*}_{\mathcal{O}_D/k}\Bigr)^{0} \to J^{0}_{X/k} \to 0
\]

\[
(?) \to \prod J^{*}_{k(D)/k} \to J^{*}_{X/k}
\]

\[
J_1 = \prod J^{0}_{\mathcal{O}_D/k}
\]

\[
H^1\bigl(\mathrm{Hom}(\mathcal{L}_{*} \otimes_{\mathbb{Z}} \mathcal{L}'_{*}, G)^{\Gamma}\bigr)
\]

\[
H^1\bigl(\mathrm{Hom}(\mathcal{M}_{*}, \mathcal{L}_{*})^{\Gamma}\bigr), \qquad
H^1\bigl(\mathrm{Hom}(\mathcal{M}_{*}, \mathcal{L}'_{*})^{\Gamma}\bigr)
\]

\[
\mathcal{M}_{*} \xrightarrow{\ \Gamma\text{-hom}\ } \mathcal{L}_{*}, \qquad
\mathcal{M}_{*} \xrightarrow{\ \Gamma\text{-hom}\ } \mathcal{L}'_{*}
\]

\[
\mathcal{M}_{*} \otimes_{\mathbb{Z}} \mathcal{M}_{*} \xrightarrow{\ \Gamma\text{-hom}\ } \mathcal{L}_{*} \otimes_{\mathbb{Z}} \mathcal{L}'_{*} \xrightarrow{\ \Gamma\text{-hom}\ } G
\]

\[
H^0(\Gamma, A_L) \times H^1(\Gamma, A'_L) \longrightarrow H^2(\Gamma, \struck{\ill{}}\,L^{*})
\]

\[
\mathrm{Ext}^1(A_L, A'_L; L^{*}), \qquad \mathbb{Z}(\Gamma)/\mathbb{Z}
\]

\[
\mathrm{Ext}^1_{\Gamma}(\mathbb{Z}; A_L), \qquad \mathrm{Ext}^1_{\Gamma}(\mathbb{Z}, A'_L)
\]

\[
\downarrow
\]

\[
\mathrm{Ext}^{2}_{\Gamma}(\mathbb{Z}, \mathbb{Z}; A_L, A'_L) \times \mathrm{Ext}^1_{\Gamma}(A_L, A'_L; L^{*})
\longrightarrow \mathrm{Ext}^2_{\Gamma}(\mathbb{Z}, \mathbb{Z}; L^{*}) \xrightarrow{\ \ill{}\ } \mathrm{Ext}^2_{\Gamma}(\mathbb{Z}, L^{*})
\]

\[
\mathrm{Hom}(\mathcal{L}_{*}, \mathcal{L}'_{*}; L^{*})
\]

\[
i + j + 1 = 2, \qquad i + j = 1 \qquad (i) \quad (1-j) \qquad 2 \quad -1
\]

\[
\begin{array}{ll}
\text{\uncertain{produit}} & \mathcal{L}^{p}_{*} \otimes \mathcal{L}^{q}_{*} \to P \otimes Q \\
 & \qquad \downarrow \\
\text{\uncertain{substitution}} & \mathcal{L}^{p \otimes q}_{*} \to P \otimes Q
\end{array}
\]

\[
A_K/N_{L/k}A_L \ \big/\ \text{s.-groupe de } H^1(\Gamma, A'_L)
\]

\[
K,\ \mathcal{L} \qquad \mathbb{Z} \qquad \underline{A} \qquad P \ \text{fibré principal sur } A
\]

\[
\struck{H^1}\ \ \underline{\mathrm{Pic}}_{P/S}
\]

\[
0 \to G(K)/G(C) \longrightarrow \mathcal{D}iv^{0}_{G} \ \Big|\ H^1(X, G)^{0} \to 0
\]

\[
\prod_{Z\ \text{de codim}\ 2} J_{\mathcal{O}_Z/k} \to \underbrace{\prod J^{0}_{\mathcal{O}_D/k}}_{?} \xrightarrow{\ u\ } J^{0}_{K/k} \ \Big|\ \longrightarrow \mathcal{J}^{0}_{X/k} \to 0
\qquad \text{\emph{$X$ complète}}
\]

\[
(G\text{-}\mathcal{D}iv) = \mathrm{Hom}\Bigl(\bigl(\prod \mathcal{J}^{0}_{\mathcal{O}_D/k}\bigr) / \mathrm{Im}\bigl(\prod_{Z\ \text{codim}\ 2} J_{\mathcal{O}_Z/k}\bigr)\Bigr)
\]

\[
(G\text{-}\mathcal{D}iv)^{0} \underset{\text{\uncertain{plausible}}}{=} \mathrm{Hom}\bigl(\prod \mathcal{J}^{0}_{\mathcal{O}_D/k} / \mathrm{Ker}\, u\bigr)
\]

\[
\boxed{(G\text{-}\mathcal{D}iv)/(G\text{-}\mathcal{D}iv)^{0} \subset \mathrm{Hom}\bigl(H_1(J_{*}), G(\ill{})\bigr)}
\]

\[
0 \to A' \to A \to A'' \to 0 \qquad
\boxed{\text{Ex. Néron-Severi} \subset \mathrm{Hom}\bigl(H_1(J_{*}), \mathbb{G}_m\bigr)}
\]

\[
\boxed{\text{A-t-on égalité ??}}
\]

\[
0 \to \prod G(\mathcal{O}_x)/G(C) \longrightarrow J^{0}_{G} \longrightarrow H^1(X, G)^{0} \to 0
\]

\[
0 \to ? \to \Bigl(\sum_D J^{*}_{k(D)/k}\Bigr)^{0} \to \mathcal{J}^{0}_{X/k} \to 0
\qquad G(L)
\]

\[
J^{*}_{X/k} \longrightarrow \mathrm{Pic}^{*}_{G}(Y)
\]

\[
0 \to \text{Noyau} \longrightarrow J^{0}_{K_Y/k} \longrightarrow \mathcal{J}^{0}_{Y/k} \to 0
\]

\[
\mathrm{Pic}_{X/Y} \longrightarrow \mathbb{Z}_{Y'/Y}
\]

\[
0 \to \mathrm{Pic}^{0}_{X/Y} \to \mathrm{Pic}_{X/Y} \to \mathbb{Z}_{Y'/Y} \to 0.
\]

\[
\mathbb{Z}_{Y'/Y} \longrightarrow \mathbb{Z}_Y
\]

\[
\mathrm{Pic}_{X/Y} \longrightarrow \mathbb{Z}_Y
\]

\[
\boxed{X - S \longrightarrow \mathrm{Pic}^{1}_{X/S}}
\]

\[
\Gamma(X - S/Y) \longrightarrow \mathrm{Pic}^{1}(X/S)
\]

\[
\boxed{\varphi_S \colon X - S \longrightarrow \mathrm{Pic}^{1}_{S, X/Y} \subset \mathrm{Pic}_{S, X/Y}}
\]

\[
\mathrm{Hom}_{\substack{Y\text{-schémas en groupes}\\ \mathbb{Z}\text{-augmentés}}}\bigl(\mathrm{Pic}_{S, X/Y}, P\bigr) \longrightarrow \mathrm{Hom}_Y(X - S, P^{1})
\]

\[
\mathrm{Hom}_{Y\text{-schémas en groupes}}\bigl(\mathrm{Pic}_{S, X/Y}, G\bigr) \xrightarrow{\ \sim\ } \mathrm{Hom}_Y(X - S, G)
\]

\[
\mathrm{Ext}^1\bigl(\mathrm{Pic}_{S}, G\bigr) \longrightarrow H^1(X - S, G)
\]

\[
\mathrm{Hom}(Q, Q) \longrightarrow \mathrm{Hom}\bigl(\varphi^{*}(Q), \varphi^{*}(Q)\bigr)
\]

\[
\mathrm{Hom}_{\text{gr.\ sur}\ Y}(\mathrm{Pic}_{S}, G) \quad\text{et}\quad \mathrm{Hom}_Y(X - S, G)
\]

\[
\mathrm{Ext}^1_{\substack{\text{schémas en groupes}\\ \text{sur}\ Y}}\bigl(\mathrm{Pic}_{S, X/S}, G\bigr) \longrightarrow H^1(X - S, G)
\]

\[
\longleftrightarrow\ H^1(X, G) \xrightarrow{\ d\ } \mathrm{Hom}_{Y\text{-gr}}(\mathbb{G}_m, G)
\qquad \text{[\ill{} \emph{degré}]}
\]

\[
\boxed{0 \to \mathrm{Ext}^1(\mathrm{Pic}_{X/Y}, G) \to H^1(X, G) \to \mathrm{Hom}_{Y\text{-gr}}(\mathbb{G}_m, G)}
\]

\[
\boxed{\to \mathrm{Ext}^2(\mathrm{Pic}_{X/Y}, G) \to H^2(X, G) \to \mathrm{Ext}^1_{Y\text{-gr}}(\mathbb{G}_m, G) \to \cdots}
\]

\[
\mathrm{Ext}^1(\mathbb{G}_m, G) \longrightarrow \mathrm{Ext}^{\ill{}}(\mathrm{Pic}_{X/Y}, G)
\]

\[
\xi \in \mathrm{Ext}^2_{Y\text{-groupes}}(\mathrm{Pic}_{X/Y}, \mathbb{G}_m)
\]

\[
0 \to N_S \longrightarrow \mathrm{Pic}_{S, X/Y} \longrightarrow \mathrm{Pic}_{X/Y} \to 0
\]

\[
\xi \in \mathrm{Ext}^1(\mathrm{Pic}_{X/Y}, N_S)
\]

\[
0 \to \mathbb{G}_{m,Y} \longrightarrow \mathcal{E}_{X/Y/S} \longrightarrow N_S \to 0
\]

\[
\eta \in \mathrm{Ext}^1(N_S, \mathbb{G}_{m,Y})
\]

\[
\zeta = \eta \circ \xi
\]

\[
\boxed{0 \to \mathrm{Ext}^{i}(\mathrm{Pic}_{X/Y}, G) \to H^{i}(X, G) \to \mathrm{Ext}^{i-1}(\mathbb{G}_m, G) \to 0}
\]

\[
\mathrm{Pic}_{\infty\, X/Y},
\]

\[
\varphi_\infty \colon X \longrightarrow \mathrm{Pic}^{1}_{\infty}
\]

\[
\left\lbrace
\begin{array}{l}
\mathrm{Hom}(R_{X/Y}, P^{1}) \simeq \mathrm{Hom}_{Y\text{-gr}}(\mathrm{Pic}^{1}_{\infty}, P) \\
\underline{H}^{0}(R_{X/Y}, G) \xleftarrow{\ \sim\ } \mathrm{Hom}_{Y\text{-gr}}(\mathrm{Pic}_{\infty}, G) \\
\underline{H}^{1}(R_{X/Y}, G) \xleftarrow{\ \sim\ } \mathrm{Ext}^{1}_{Y\text{-gr}}(\mathrm{Pic}_{\infty}, G) \quad (\text{et } \ill{}\ \ill{}) \\
\underline{H}^{i}(R_{X/Y}, G) \xleftarrow{\ \sim\ } \mathrm{Ext}^{i}_{Y\text{-gr}}(\mathrm{Pic}_{\infty}, G)
\end{array}\right.
\]

\[
(*) \qquad 0 \to \mathrm{Pic}^{0}_{S} \to \mathrm{Pic}_{S} \to \mathbb{Z} \to 0
\]

\[
\begin{aligned}
&\to H^0(S, G) \to \mathrm{Hom}(\mathrm{Pic}_S, G) \to \mathrm{Hom}(\mathrm{Pic}^{0}_{S}, G) \to H^1(S, G) \to \mathrm{Ext}^1(\mathrm{Pic}_S, G) \\
&\to \mathrm{Ext}^1(\mathrm{Pic}^{0}_{S}, G) \to H^2(S, G) \to \mathrm{Ext}^2(\mathrm{Pic}_S, G) \to \cdots
\end{aligned}
\]

\[
0 \to \underset{\substack{\text{faisceau de}\\ \text{Picard relatif}\\ \text{\uncertain{ordinaire}\ (?)}}}{\mathrm{Pic}(X/Y)_0} \longrightarrow \underline{\mathrm{Ext}}^1\bigl(\mathrm{Pic}^{0}_{X/Y}, \mathbb{G}_m\bigr)
\]

\[
\underline{\underline{H}}^{*}(X_S/Y, G) \xleftarrow{\ \sim\ } \underline{\underline{\mathrm{Ext}}}^{*}_{Y}(\mathrm{Pic}_S, G)
\]

\[
\begin{aligned}
\mathrm{Ext}^{*}(\mathrm{Pic}_S, G) &\Longleftarrow H^{p}\bigl(Y, \underline{\underline{\mathrm{Ext}}}^{q}(\mathrm{Pic}_S, G)\bigr) \\
\underline{H}^{*}(X_S/Y, G) &\Longleftarrow H^{p}\bigl(Y, \underline{\underline{H}}^{q}(X_S/Y, G)\bigr)
\end{aligned}
\]

\[
H^{*}(X_S/Y, G) \simeq \mathrm{Ext}^{*}_{S}(\mathrm{Pic}_S, G)
\]

\[
\mathbb{S}_\infty(A) = A[[t]]^{**}
\]

\[
(1 + at) * f(t) = f(at) \qquad \text{i.e.} \qquad \{1 + at\}\{f(t)\} = \{f(at)\}
\]

\[
F\Bigl(\sum a_i t^{i}\Bigr) = \sum a_i^{p} t^{i}
\]

\[
f(t)^{p} = F(f)(t^{p})
\]

\[
V(f)(t) = f(t^{p})
\]

\[
FV(\{f\}) = VF(\{f\}) = p\{f\}
\]

\[
(1 + x t^{p^{i}}) * f(t) = F^{i}(f)(x t^{p^{i}}) \qquad\quad (1 + at)^{p^{i}} * f = f(at)^{p^{i}}
\]

\[
= F^{i}(f)(a^{p^{i}} t^{p^{i}}) = F^{i}(f)(x t^{p^{i}}).
\]

\[
\psi \colon \mathbb{S}_\infty \longrightarrow \mathbb{W}_\infty
\qquad \text{[anneau de Witt pour le nb premier } p\text{]}
\]

\[
\psi(\{1 + at\}) = \varepsilon(a) = (a, 0, 0, \ldots)
\]

\[
0 \to K^{*} \longrightarrow \prod_{\text{local}} K_x^{*} \longrightarrow J_C^{(\bar k)} \to 0
\]

\[
\begin{array}{ll}
\hat H^{*}(\Gamma, K^{*}) = 0 & \text{par le théorème de \ill{} global} \\
\hat H^{*}(\Gamma, K_x^{*}) = 0 & \text{\quad " \qquad local}
\end{array}
\]

\[
\hat H^{*}(\Gamma, J_C^{(\bar k)}) = 0 \qquad \text{\uncertain{géométriques}}
\]

\[
C^{*}(\Gamma, J_C^{(\bar k)}) \quad \text{est une \emph{résolution} de} \quad H^0(\Gamma, J_C^{(\bar k)})
\]

\[
\begin{aligned}
H^0(\Gamma, J_C(\bar k)) &= H^0\Bigl(\prod_{\substack{\text{local}\\ x \in X}} K_x^{*}\Bigr) \Big/ \mathrm{Im}\, H^0(K^{*}) \\
&= \Bigl(\prod_{\substack{\text{local}\\ x' \in X'}} K'^{*}_{x'}\Bigr) \Big/ K'^{*} = J_{C'}(\bar k)
\end{aligned}
\]

\[
H^{*}(\pi, J_{C'}(\bar k)) \Longleftarrow H^{p}\bigl(H^{q}(\pi, \underbrace{C^{*}(\Gamma, J_C(\bar k))}_{C^{*}(\Gamma, J_C)(\bar k)})\bigr)
\]

\[
q > 0 \qquad H^{q}(\pi, C^{*}(\Gamma, J_C)(\bar k)) = H^{q}(\pi, C^{*}(\Gamma, \mathbb{Z})) = 0 \quad (\text{Lang})
\]

\[
q = 0 \qquad C^{*}(\Gamma, J_C(\bar k))^{\pi} = C^{*}(\Gamma, J_C(k))
\]

\[
H^{p}\bigl(k, \underline{\underline{H}}^{q}(\Gamma, J_{C'})\bigr) \Longleftarrow H^{p}(k, J_C) \qquad J_C(k)
\]

\[
H^{p}\bigl(H^{q}(k, \underline{\underline{C}}^{*}(\Gamma, J_{C'}))\bigr)
\]

\[
\underset{q = 0}{H^{p}}\bigl(C^{*}(\Gamma, J_{C'})\bigr) = H^{p}(\Gamma, J_{C'})
\]

\[
J_{C'}(\bar k) \qquad \underline{C}(\Gamma^{n}, J_{C'})(\bar k) \qquad \text{\emph{résolution} de } J_C(\bar k) \text{ \uncertain{par} des } \pi\text{-\uncertain{modules}}
\]

\[
H^{*}(\pi, J_C(\bar k)) \Longleftarrow H^{p}\bigl(H^{q}(\pi, \underline{C}(\Gamma^{\cdot}, J_{C'})(\bar k))\bigr)
\]

\[
\begin{array}{ll}
\| & \\
J_C(k) & H^{q}(\pi, \underline{C}(\Gamma^{\cdot}, \mathbb{Z})) \underset{q > 0}{=} 0
\end{array}
\]

\[
H^0(\Gamma, J_{C'}(\bar k)) \qquad C(\Gamma^{\cdot}, J_{C'}) \qquad H^{p}(\Gamma, J_{C'}(k))
\]

\[
\underline{C} \qquad \Gamma \qquad C(\Gamma^{n}, J^{\cdot}_{C'})
\]

\[
J_C(\bar k) \to \underline{J_{C'}(\bar k)} \longrightarrow J_{C'}(\bar k)^{\Gamma} \longrightarrow J_{C'}(\bar k)^{\Gamma^{2}}
\]

\[
(\delta\varphi)(\underset{\uparrow\,\gamma}{x}) = \varphi(\gamma x) - \varphi(x) \qquad \tau_\gamma \varphi - \varphi
\]

\[
\text{\struck{$C^{*}(\Gamma$}}\ \ \varprojlim H^{*}(\pi, J_{C',\alpha}(\bar{k})) \;=\; J_{C'}(k)
\]

\[
\begin{array}{ll}
= H^{i}(\pi, \mathbb{Z}) = 0 & \text{si } i > 0 \\
= H^{0}(\pi, J_{C',\alpha}(\bar{k})) = J_{C',\alpha}(k) & \text{\uncertain{si} \ill{} M.L.}
\end{array}
\]

\[
H^{p}\bigl(\varprojlim_{\alpha} H^{q}(\pi, C^{*}(\Gamma, J_{C,\alpha}(\bar{k})))\bigr)
\]

\[
\begin{array}{ll}
\text{\struck{$q > 0$}}\ = H^{q}(\pi, C^{*}(\Gamma, \mathbb{Z})) = 0 & \text{si } q > 0 \\
= H^{0}(\pi, C^{*}(\Gamma, J_{C,\alpha}(\bar{k}))) = C^{*}(\Gamma, J_{C,\alpha}(k)) &
\end{array}
\]

\[
C^{*}(\Gamma, J_{C}(k)).
\]

\[
\text{\struck{$\ill{}$}}\ \ltimes E^{1} \to G \qquad \mathrm{Ext}^{1}(J^{0}_{\text{\uncertain{$K$}}}, G)
\]

\[
\begin{array}{c}
H^{1}(k, G) \qquad H^{p}(\Gamma, J_{C}(k)) \\
\downarrow \\
H^{1}(K, G) \xleftarrow{\ \sim\ } \mathrm{Ext}^{1}(J_{K}, G) \\
\downarrow \\
\mathrm{Ext}^{1}(J^{0}_{K}, G) \\
\downarrow \\
H^{2}(k, G) = 0
\end{array}
\qquad\qquad \hat{\mathbb{Z}} \to 0
\]

\[
\cdots \to H^{0}(k, \Gamma) \to \mathrm{Hom}(J_{K}, \Gamma) \to \mathrm{Hom}(J^{0}_{K}, \Gamma)
\]

\[
\boxed{\mathrm{Ext}^{1}(J^{0}, \Gamma) = \mathrm{Hom}}
\]

\[
\mathrm{Ext}^{1}(J, G) \Rightarrow \mathrm{Hom}(J(k), G(k))
\]

\[
J(k) \times \mathrm{Ext}^{1}(J, G) \longrightarrow
\boxed{H^{1}(k, G) \simeq G(k)}
\]

\[
0 \to G \to E \to J \to 0, \qquad \mathbb{Z} \to J, \qquad J'
\]

\[
\pi_{1}(J) - \pi_{1}(\mathbb{Z})
\]

\[
\prod_{x \in X} \hat{\mathcal{O}}_{x}
\]

\[
\prod^{\mathrm{local}}_{x_1 \in X_1} \hat{\mathcal{O}}_{x_1} \otimes_{\underline{\mathcal{O}}_{x_1}} K
\;=\; \prod_{\mathrm{local}} K_{D}
\]

\[
\prod_{x_1} \text{\struck{$\ill{}$}} \prod^{\mathrm{local}}_{x_0 \in \overline{x_1}} \mathcal{O}_{[x_0, x_1]}
\]

\[
\prod_{x_0 x_1 (x_2)} \mathcal{O}_{[x_0 x_1 x_2]}
\;\longrightarrow\;
\mathcal{O}_{[x_0 x_1]} \otimes_{\mathcal{O}_{x_1}} K,
\qquad
\mathcal{O}_{[x_0 x_\nu]} = \hat{\underline{\mathcal{O}}}_{x_0} \otimes_{\underline{\mathcal{O}}_{x_0}} K
\]

\[
K \longrightarrow I_{x_1} \longrightarrow I_{x_0}
\]

\[
\hat{\underline{\mathcal{O}}}_{x_0} \quad \hat{\mathcal{O}}_{x_1} \quad \hat{\mathcal{O}}_{x_2}
\qquad\qquad
\mathfrak{m}_{0} \quad \mathfrak{m}_{1} \quad \mathfrak{m}_{2}
\]

\[
\prod_{x \in X} \mathcal{O}_{[x]}, \qquad
\prod^{\mathrm{local}}_{x_1 \in X_1} \mathcal{O}_{[x_1, x_2]}, \qquad \mathcal{O}.
\]

\[
\underline{\mathcal{O}}_{[x_1, x_2]}, \qquad \hat{\mathcal{O}}_{x_0}
\]

\[
\mathcal{O}_{[x_0 x_1]} \longrightarrow \prod_{x_0 x_1} \mathcal{O}_{[x_0 x_1 x_2]}
\]

\[
\text{\struck{$\mathcal{A}_{i_0 \cdots i_{p-1}}$}}\quad
\mathcal{A}_{i_0 \cdots i_p} \qquad \mathcal{A}_{i_1 \cdots i_{p-1}}
\]

\[
x_{i_0}\ x_{i_1} \cdots x_{i_p}, \qquad i_0 < i_1 < \cdots < i_p
\]

\[
\mathcal{A}^{*} = \prod_{x \in X} \mathcal{A}^{*}\pi_{x}
\]

\[
\mathcal{A}^{*}\pi_{x} = \varprojlim_{n} (\mathcal{A}^{*}\pi_{x}) /
\]

\[
\varprojlim_{i}\ \mathcal{A}^{0} \otimes_{A} \mathcal{A}^{0}/\mathfrak{m}_{i}
\qquad\qquad
\prod \hat{\underline{\mathcal{O}}}_{x} \otimes_{A} K
\]

\[
\mathcal{A}^{0} \otimes_{A} \mathcal{A}^{0}/\mathfrak{m}_{i}
\;\subset\; \prod_{x \in X} \hat{\underline{\mathcal{O}}}_{x} \otimes_{A} \mathcal{O}_{a}/\mathfrak{m}_{a}^{n}
\;=\; \prod_{x \leq a} \hat{\mathcal{O}}_{x} \otimes \mathcal{O}_{a}/\mathfrak{m}_{a}^{n}
\]

\[
\prod_{a} \varprojlim_{n} \Bigl(\prod_{x \leq a} \hat{\mathcal{O}}_{x} \otimes \mathcal{O}_{a}/\mathfrak{m}_{a}^{n}\Bigr)
\]

\[
= \text{\struck{$a \in X_{\ill{}}$}}\ \prod_{a} \varprojlim_{n} \Bigl[\bigl(\hat{\mathcal{O}}_{a} \otimes \underline{\mathcal{O}}_{a}/\mathfrak{m}_{a}^{n}\bigr) \times \Bigl(\bigl(\prod_{x < a} \hat{\mathcal{O}}_{x}\bigr) \otimes \mathcal{O}_{a}/\mathfrak{m}_{a}^{n}\Bigr)\Bigr]
\]

\[
= \prod_{a} \hat{\mathcal{O}}_{a} \times \prod_{i} \prod_{a \in X_i} \varprojlim_{n} \Bigl(\bigl(\prod_{x < a} \hat{\mathcal{O}}_{x}\bigr) \otimes \underline{\mathcal{O}}_{a}/\mathfrak{m}_{a}^{n}\Bigr)
\]

\[
\begin{array}{ll}
i = 0 & \text{\ill{}} \\
i = 1 & \bigl(\prod_{x < a} \hat{\mathcal{O}}_{x}\bigr) \otimes \mathcal{O}_{a}/\mathfrak{m}_{a}^{n} \\
i = 2 & \bigl(\prod_{x < a} \hat{\mathcal{O}}_{x}\bigr) \otimes K
\end{array}
\]

\[
\mathcal{A}^{n} = \text{\struck{$\mathcal{A}$}}\ \varprojlim_{\alpha} \mathcal{A}^{n-1} \otimes_{A} \mathcal{A}^{0}/\mathfrak{m}_{\alpha}
= \prod_{a \in X} \varprojlim_{k} \mathcal{A}^{n-1} \otimes_{A} \mathcal{O}_{a}/\mathfrak{m}_{a}^{k}
\]

\[
\cap \qquad \prod_{x_0 \leq x_1 \leq \cdots \leq x_n} \mathcal{A}_{[x_0, \ldots, x_n]}
\]

\[
\mathcal{A}^{n+1} = \prod_{a \in X} \varprojlim_{k} \mathcal{A}^{n} \otimes_{A} \mathcal{O}_{a}/\mathfrak{m}_{a}^{k}
\;\subset\; \prod_{a \in X} \varprojlim_{k} \Bigl(\prod_{x_0 \leq \cdots \leq x_n} \mathcal{A}_{[x_0 \ldots x_n]}\Bigr) \otimes \mathcal{O}_{a}/\mathfrak{m}_{a}^{k}
\]

\[
\subset \prod_{x_0 \leq \cdots \leq x_n} \mathcal{A}_{[x_0 \ldots x_n]} \otimes \underline{\mathcal{O}}_{\text{\uncertain{$a$}}}\ \ill{}
\qquad \mathcal{A}_{[x_0, \ldots, x_n, a]}
\]

\[
\begin{array}{c}
\mathcal{A}_{i_0 \cdots i_{p-1}} \\
\downarrow \\
\mathcal{A}_{i_0 \cdots i_p} \longleftarrow \mathcal{A}_{i_p}
\end{array}
\]

\[
\mathcal{A}_{\ell_1, \ell_2, \ldots, \ell_p} \qquad \mathcal{A}_{\ell_{p+1}} \qquad \mathcal{A}_{\ell}
\qquad \mathcal{A}_{1} \qquad \text{\struck{$\mathcal{O}_{[}$}}\ \mathcal{A}_{0} \otimes
\]

\[
\prod \hat{\underline{\mathcal{O}}}_{x} \otimes_{\mathcal{O}_x} K,
\qquad
\hat{\mathcal{O}}_{x} \otimes_{\underline{\mathcal{O}}_x} \text{\struck{$\ill{}$}}\ L_{n}
\]

\[
\hat{\mathcal{O}}_{x} \otimes_{\underline{\mathcal{O}}_x} (\mathfrak{p}^{n}/\mathfrak{p}^{n+1})
\to \hat{\mathcal{O}}_{x} \otimes L_{n+1} \to \hat{\underline{\mathcal{O}}}_{x} \otimes L_{n}
\]

\[
\hat{\mathcal{O}}_{x}/\mathfrak{p}^{n+1} \longrightarrow \hat{\mathcal{O}}_{x}/\mathfrak{p}^{n}
\]

\[
\hat{\mathcal{O}}_{x} \otimes_{\mathcal{O}_{\mathfrak{p}}} \mathfrak{p}^{n}\mathcal{O}_{\mathfrak{p}}/\mathfrak{p}^{n+1}\mathcal{O}_{\mathfrak{p}}
\to \hat{\mathcal{O}}_{x} \otimes_{\mathcal{O}_{\mathfrak{p}}} \mathcal{O}_{\mathfrak{p}}/\mathfrak{p}^{n}\mathcal{O}_{\mathfrak{p}}
\to \hat{\mathcal{O}}_{x} \otimes_{\mathcal{O}_{\mathfrak{p}}} L
\]

\[
\hat{\mathcal{O}}_{x} \to \hat{\mathcal{O}}_{x}\, S^{-1} \qquad \mathfrak{p} \quad \mathfrak{p}^{2}
\]

\[
\mathcal{A}_{[\mathfrak{p}_0, \ldots, \mathfrak{p}_n]} \otimes_{A} A_{\mathfrak{p}_{n+1}}/\mathfrak{p}_{n+1}^{\text{\uncertain{$n$}}} A_{\mathfrak{p}_{n+1}}
\qquad
A_{[\mathfrak{p}_0]} \mathbin{\hat{\otimes}_{A}} A_{[\mathfrak{p}_1]}
\qquad
\hat{\underline{\mathcal{O}}}_{x} \otimes K
\]

\[
A_{[x]} \otimes \qquad \varprojlim \hat{A} \otimes_{A} A/\mathfrak{m}_{i} \qquad \varprojlim A
\]

\[
\begin{array}{c}
k[[s,t]] \\
\uparrow \\
k[s,t]_{\mathfrak{m}} \longrightarrow k[s,t]_{(s)} \\
\uparrow \\
k[t]
\end{array}
\]

\[
k[[t]][s]\,(s^{-1}) \qquad k[[s]][t^{-1}]
\]

\[
a_{0}(t) + a_{1}(t)\,s + \cdots + a_{n-1}(t)\,s^{n-1}
\]

\[
a_{0}(t)\,[1 + a_{0}(t)^{-1} a_{1}(t)\,s + \cdots
\]

\[
\text{\struck{$a_0$}}\ k((t))[s]/(s^{n}) \longleftarrow k[[s,t]]
\]

\[
\underline{\underline{a_{0}(t)}} + \underline{\underline{a_{1}(t)}}\,s + \cdots + \underline{\underline{a_{n-1}(t)}}\,s^{n-1}
\]

\[
k((t))[[s]] \qquad k[[t]][[s]] \qquad k((t))/k[[t]]\ \ [[s]]
\]

\[
\prod_{x \in X_0} G(\hat{\mathcal{O}}_{x}), \qquad
\prod_{D} \prod^{\mathrm{local}}_{x \in |D|} G(\hat{\mathcal{O}}_{D,x}), \qquad
\prod_{\text{\struck{$\ill{}$}}\ D \in X_1} G(\hat{\mathcal{O}}_{D}), \qquad
\prod_{\mathrm{local}} G(\hat{K}_{D})
\]

\[
\mathrm{Hom}\bigl(K, \textstyle\coprod_{x} \text{\struck{$\ill{}$}}\, I_{x}\bigr)
\]

\[
K \to I_{D} \to I_{x}
\]

\[
\prod_{D} \hat{\mathcal{O}}_{D}^{*} \longleftarrow \Sigma\, \underline{\mathcal{O}}_{D}, \qquad K^{*}, \qquad \mathcal{O}_{D}
\]

\[
\begin{array}{c}
\hat{\underline{\mathcal{O}}}_{x} \leftarrow \mathcal{O}_{x} \\
\downarrow \\
\hat{\underline{\mathcal{O}}}_{x} \otimes_{\underline{\mathcal{O}}_x} \underline{\mathcal{O}}_{D} \leftarrow \mathcal{O}_{D} \\
\downarrow \\
\underline{\mathcal{O}}_{\text{\uncertain{$\mathfrak{q}$}}}
\end{array}
\qquad\qquad
\begin{array}{ccc}
\hat{\mathcal{O}}_{D} & \leftarrow & \mathcal{O}_{D} \\
\downarrow & & \downarrow \\
\hat{\underline{\mathcal{O}}}_{D,0} & \leftarrow & \mathcal{O}_{0}
\end{array}
\]

\[
\begin{array}{l}
A_{[\mathfrak{p}_1, \ldots, \mathfrak{p}_n]} \\
\uparrow \ \text{hom.\ local} \\
A_{[\mathfrak{p}_n]}\ \ \text{\struck{$\leftarrow A_{[\mathfrak{p}_{n-1}]}/\mathfrak{p}_{n-1}^{k} A_{[\mathfrak{p}_{n-1}]}$}} \\
\uparrow \ \text{hom.\ local} \\
A_{\mathfrak{p}_n} \longrightarrow A_{\mathfrak{p}_{n+1}}/\mathfrak{p}_{n+1}^{k} A_{\mathfrak{p}_{n+1}}
\end{array}
\]

\[
\text{\struck{$A_{[\mathfrak{p}_1]} \leftarrow A_{[\mathfrak{p}_2]} \to A_{[\mathfrak{p}_1]}$}}
\qquad
A_{[\mathfrak{p}_1]}, \qquad
A_{\mathfrak{p}_1} \longrightarrow \text{\struck{$A$}}\,A_{\mathfrak{p}_2}/\mathfrak{p}_2^{k} A_{\mathfrak{p}_2}, \qquad
\underline{A}
\]

\[
\text{\struck{$D(A) = \coprod_{\mathfrak{p}} D_{\mathfrak{p}}(A)$}}
\qquad
\text{\struck{$DD(A) = \prod_{\mathfrak{p}} \mathrm{Hom}(I_{\mathfrak{p}},$}}
\]

\[
D(M) = \mathrm{Hom}(M, K_{*}) = \coprod_{\mathfrak{p}} \mathrm{Hom}(M, K_{\mathfrak{p}})
\]

\[
DD(M) = \prod_{\mathfrak{p}} \mathrm{Hom}(D_{\mathfrak{p}}(M), K)
\]

\[
D_{\mathfrak{p}}(M) = \varinjlim_{i} U_{i}
\]

\[
\mathrm{Hom}(D_{\mathfrak{p}}(M), K) = \varprojlim_{i} \mathrm{Hom}(U_{i}, K)
\]

\[
\mathrm{Hom}(U_{i}, K_{*}) = \coprod_{\mathfrak{q}} \mathrm{Hom}(U_{i}, K_{\mathfrak{q}})
\]

\[
\mathrm{Hom}(U_{i}, K_{\mathfrak{q}}) = \varinjlim_{j} \mathrm{Hom}(U_{i}, K_{\mathfrak{q},j})
\]

\[
\mathrm{Hom}(U_i, K_{*}) = \coprod_{\mathfrak{q} \supset \mathfrak{p}} \mathrm{Hom}(U_i, K_{\mathfrak{q}})
\]

\[
DD(M) = \prod_{\mathfrak{p}} \mathrm{Hom}\Bigl(D_{\mathfrak{p}}(M), \coprod_{\mathfrak{q} \supset \mathfrak{p}} K_{\mathfrak{q}}\Bigr)
= \prod_{\mathfrak{p}} \prod^{\mathrm{local}}_{\mathfrak{q} \supset \mathfrak{p}} \mathrm{Hom}(D_{\mathfrak{p}}(M), K_{\mathfrak{q}})
\]

\[
A_{[\mathfrak{p} \supset \mathfrak{q}]}
\]

\[
\begin{array}{c}
A_{[\mathfrak{p}]} = \widehat{A_{\mathfrak{p}}} \\
\uparrow \\
A_{\mathfrak{p}} \longrightarrow A_{\mathfrak{q}}/\mathfrak{q}^{n} A_{\mathfrak{q}}
\end{array}
\]

\[
A_{[\mathfrak{p}]} = \widehat{A_{\mathfrak{p}}} \longleftarrow A_{\mathfrak{p}}
\]

\[
\begin{aligned}
A_{[\mathfrak{p}, \mathfrak{q}]} &= \varprojlim_{n} A_{[\mathfrak{p}]} \otimes_{A} A_{\mathfrak{q}}/\mathfrak{q}^{n} A_{\mathfrak{q}} \\
&= A_{[\mathfrak{p}]} \otimes_{A_{\mathfrak{p}}} A_{\mathfrak{q}}/\mathfrak{q}^{n} A_{\mathfrak{q}} \\
&= A_{[\mathfrak{p}]}/\mathfrak{q}^{n} A_{[\mathfrak{p}]} \otimes_{\text{\struck{$A_{\mathfrak{q}}$}}\, A_{\mathfrak{p}}} A_{\mathfrak{q}}/\mathfrak{q}^{n} A_{\mathfrak{q}} \\
&= \text{\struck{$(A/\mathfrak{q}^{n} A$}} \\
&= A_{[\mathfrak{p}]}/\mathfrak{q}^{n} A_{[\mathfrak{p}]} \otimes_{A_{\mathfrak{p}}} A_{\mathfrak{q}} \\
&= A_{[\mathfrak{p}]}/\mathfrak{q}^{n} A_{[\mathfrak{p}]} \otimes_{A} A_{\mathfrak{q}}
\end{aligned}
\]

\[
\widehat{A_{\mathfrak{p}}/\mathfrak{q}^{n} A_{\mathfrak{p}}} = (A/\mathfrak{q}^{n} A)_{[\mathfrak{p}]}
\qquad\qquad
\text{\struck{$\ill{}\ (A/\mathfrak{q}^{n} A)_{[\mathfrak{p}]}$}}
\]

\[
\mathfrak{p} \longrightarrow \mathfrak{m}, \qquad \mathfrak{p}' \quad \mathfrak{m}'
\]

\[
\mathrm{Hom}(U_{\mathfrak{p}}, K_{\mathfrak{m}}) = A_{\mathfrak{m},\mathfrak{p}}
\]

\[
\mathfrak{p} \subset \mathfrak{p}' \qquad \mathfrak{q}' \subset \mathfrak{q} = \mathfrak{m}
\qquad\qquad
A_{\mathfrak{m}}/\mathfrak{p}^{n} A_{\mathfrak{m}} = \mathbf{A}
\]

\[
\mathfrak{p} \subsetneq \mathfrak{p}' \subsetneq \mathfrak{p}'' \subset \mathfrak{q}' \subsetneq \mathfrak{q},
\qquad
\mathfrak{p} \subsetneq \mathfrak{p}' \subseteq \mathfrak{q}' \subsetneq \mathfrak{q},
\qquad
\mathfrak{p} \subset \mathfrak{q}'' \subsetneq \mathfrak{q}' \subsetneq \mathfrak{q}
\]

\[
\mathrm{Hom}(A_{\mathfrak{p}}, K_{\mathfrak{q}}), \qquad
\mathrm{Hom}(K_{*}, K_{*})
\]

\[
A_{\mathfrak{q},\mathfrak{p}} = \Bigl(\coprod_{\mathfrak{p}'} A_{\mathfrak{q},\mathfrak{p}'}\Bigr) \amalg \Bigl(\coprod_{\mathfrak{q}'} A_{\mathfrak{q}',\mathfrak{p}}\Bigr) \Big/
\]

\[
\begin{aligned}
A_{[\mathfrak{p},\mathfrak{q}]} &= \varprojlim A_{[\mathfrak{p}]} \otimes_{A_{\mathfrak{p}}} A_{\mathfrak{q}}/\mathfrak{q}^{n} A_{\mathfrak{q}} \\
&= \varprojlim A_{[\mathfrak{p}]}/\mathfrak{q}^{n} A_{[\mathfrak{p}]} \otimes_{A_{\mathfrak{p}}} A_{\mathfrak{q}}/\mathfrak{q}^{n} A_{\mathfrak{q}}
\end{aligned}
\]

\[
x_0 \leq x_1 \leq \cdots \leq x_n
\]

\[
\text{\struck{Ah}}\quad S_X^{n} \xrightarrow{\ u^{*}\ } S_X^{m}
\]

\[
\varphi(\sigma, u) : A_\sigma \longleftarrow A_{u^{*}(\sigma)},
\]

\[
(x_0 - \cdots - x_n), \qquad
(x_0 - \overset{\downarrow}{x_p} - x_n), \qquad
x_0\ x_1\ x_2, \qquad x_0 - x_2
\]

\[
\partial_i^n = 0 \quad \text{pour} \quad 1 \leq i \leq n
\]

\[
\begin{array}{ll}
\partial_i^n K^{n-1} & 1 \leq i \leq n \\
u_2\, \partial_i^{n-1} K^{n-2} & 1 \leq i \leq n \\
\cdots & \\
u\ K^{0} &
\end{array}
\]

\[
K'^{n} \subset K^{n},
\]

\[
\text{\struck{$x_0 \leq x_1 \cdots \leq x_n$}} \qquad
(x_0, x_1, x_2) \longrightarrow (x_0, x_2)
\]

\[
A_{[x_0 \ x_1}\ \cdots \qquad x_0, x_2 \qquad
A_{[x_0, x_2]} \qquad \partial^{1}_{1} A_{[x_0, x_2]}
\]

\[
\cdots \to K'' \to K' \to K \to 0
\]

\[
- \quad A_{[x_1 x_2, \ldots, x_n]} \longrightarrow
\prod_{\substack{x_0 \\ \text{précède} \\ x_1}} A_{[x_0, x_1, \ldots, x_n]}
\xrightarrow{\ d_0^{n}\ }
\prod_{\substack{x'_0, x_0 \\ (x'_0 \text{ précède } x_0)}} A_{[x'_0, x_0, x_1, \ldots, x_n]}
\]

\[
\left\lbrace
\begin{array}{l}
d'^{2} = 0 \\
d''^{2} = 0 \\
d'd'' = d''d'
\end{array}
\right.
\]

\[
A_{[x_0, x_n]} \longrightarrow \text{\struck{$\prod$}}\ \prod A_{[x_0 x_i x_n]}
\]

\[
\begin{array}{ccc}
A_{[x_0 x_{n-1}]} & & A_{[x_1, x_n]} \\
\swarrow \ \searrow & & \swarrow \ \searrow \\
A_{[x_0, x_{n-1}, x_n]} & & A_{[x_0, x_1, x_n]}
\end{array}
\]

\[
\begin{array}{llll}
X & \prod \hat{\underline{\mathcal{O}}}_{x} & & \prod G(\hat{\underline{\mathcal{O}}}_{x}) \longleftarrow \cdot \\
C & \prod_{\mathrm{loc}} \hat{K}_{x} & K & \prod_{\mathrm{loc}} G(\hat{K}_{x}) \longleftarrow G(K)
\end{array}
\]

\[
\begin{array}{c}
G(\hat{A}) \\
\downarrow \\
G(\hat{K}) \longleftarrow G(K)
\end{array}
\qquad\qquad H
\]

\[
K \to K/A, \qquad G(K) \qquad G(K)/G(A)
\]

\[
\begin{array}{l}
A_{[x_0 x_1 x_2 x_3]} \\
A_{[x_0 x_1 x_2]} \quad A_{[x_1 x_2 x_3]} \quad \text{\struck{$A_{[x_1 x_2 x_3]}$}} \quad A_{[x_0 x_2 x_3]} \quad A_{[x_0 x_1 x_3]} \\
A_{[x_0 x_1]} \quad A_{[x_1 x_2]} \quad A_{[x_2 x_3]} \quad \text{\struck{$A_{[x_2 x_3]}$}} \quad A_{[x_0 x_2]} \quad A_{[x_0 x_1]} \quad \text{\struck{$A_{[x_2 x_3]}$}} \quad A_{[x_1 x_3]} \quad \text{\struck{$A_{[x_1 x_3]}$}} \quad \text{\struck{$A_{[x_2 x_3]}$}} \\
A_{[x_0]} \quad A_{[x_1]} \quad A_{[x_2]} \quad A_{[x_3]}
\end{array}
\]

\[
\overline{A_{[x_0 x_2 x_3]}} \cap \overline{A_{[x_0 x_1 x_3]}}
\]

\[
\overline{\overline{A_{[x_0, x_3]}}}
\]

\[
0 \qquad (A_{[x_0 x_2 x_3]} + A_{[x_0 x_1 x_3]})
\]

\[
A_{[x_0 x_2]} \times A_{[x_1, x_3]} \times \bigl(A_{[x_0 x_2 x_3]} \times A_{[x_0 x_1 x_3]}\bigr)
\]

\[
0 \qquad A_{[x_0 x_2]} \qquad A_{[x_{\text{\uncertain{$2$}}} x_3]} \qquad A_{[x_{\text{\uncertain{$0$}}} x_3]}
\]

\[
\mathrm{Im}\, A_{[x_0 x_3]} + \mathrm{Im}\, A_{[x_1, x_3]}
\qquad
\alpha \quad \beta \quad -\bar{\alpha} \quad -\bar{\beta},
\qquad \alpha \circ \beta
\]

\[
\begin{array}{llllll}
S_0 & S_1 & S_2 & S_{n+1} \xleftarrow{\ d_i^{n}\ } S_n & & \\
 & & & d_i^{n}\sigma \quad \ \sigma & & \\
 & & & A_{d_i^{n}\sigma} \quad A_\sigma & &
\end{array}
\]

\[
K \qquad d_i^{n} \quad 0 \leq i \leq n+1 \qquad\qquad d' \pm d''
\]

\[
\bigcap_{1 \leq i \leq n+1} \mathrm{Ker}\, d_i^{n}
\qquad
\text{\struck{$\ill{}$}}\ \mathrm{Coker}\, d_i^{n} \quad \underline{1 \leq i \leq n}
\]

\[
d_i^{n+1} d_0^{n} x = 0 \quad \text{si} \quad 1 \leq i \leq n+2
\]

\[
A_{[x_0, x_2]} \xleftarrow{\ d\ } A_{[x_0, x_1, x_2]}
\]

\[
\mathrm{Hom}(M, A), \quad \mathrm{Ext}^{1}(M, A) \cdots
\qquad\qquad
\mathrm{Hom}(K_{\mathfrak{p}}, K_{\mathfrak{q}}) = A_{\mathfrak{q},\mathfrak{p}} \quad (\mathfrak{q} \supset \mathfrak{p})
\]

\[
\begin{array}{l}
S_{A}(M) \\
\mathrm{Hom}_{A}(M, \mathcal{A}) \\
\mathrm{Ext}^{*}(M, A) \\
\Uparrow \\
H^{*}\, \mathrm{Ext}^{*}(M, \mathcal{A}) \\
\mathrm{Ext}(\hat{M}, A) \qquad \hat{A}\ \hat{K}\ K
\end{array}
\]

\[
A \ \ill{} \ill{}
\qquad
0 \to A \to \mathcal{A} \to \mathcal{A}_1 \to 0
\]

\[
\mathrm{Ext}^{1}(M, A) \to \mathrm{Ext}^{\text{\uncertain{$1$}}}(M, \mathcal{A}) \to \ill{}
\]

\[
\Theta^{2} = \sum_{\substack{(x_0, x_1) \\ (y_0, y_1) \\ \text{chaînes saturées}}} 1_{[x_0, x_1]} \cdot 1_{[y_0, y_1]}
= \sum_{\substack{[x_0, x_1, x_2] \\ \text{chaînes saturées}}} 1_{[x_0, x_1, x_2]}
\]

\[
\sum_{x_1} \varphi \qquad \Bigl(\ill{}\ \delta(a, b) = 2\Bigr)
\]

\[
\begin{array}{ccc}
A_{[a - \alpha, \delta, \beta, \ldots, b]} & A_{[a - \alpha]} & A_{[\beta, \ldots, b]} \\
\uparrow \varphi_{\sigma} & \uparrow & \uparrow \\
A_{[\alpha - a]} \otimes A_{[\beta \ldots \beta]} & A_{[\alpha]} & A_{[\beta]}
\end{array}
\qquad
[A_{\alpha \ \ill{} \ b}]
\qquad
x_0 \longrightarrow x_n
\]

\[
c = (x_0 \geq x_1 \geq \cdots \geq x_n),
\]

\[
A_c = A_{[x_0, \ldots, x_n]}
\]

\[
\begin{array}{ll}
u_1 : A_{[x_0, \ldots, x_{n-1}]} \longrightarrow A_{[x_0, \ldots, x_n]} & [\text{i.e.\ } (1_{x_n}\Theta)_0] \\
u_2 : A_{[x_1, \ldots, x_n]} \longrightarrow A_{[x_0, \ldots, x_n]} & [\text{i.e.\ } d(\Theta 1_{x_0})]
\end{array}
\]

\[
u_1, u_2 : A_{c'} \longrightarrow \prod_{\substack{c \supset c' \\ \lg c = \lg c' + 1}} A_c
\]

\[
u_{c,c'} : A_{c'} \longrightarrow A_{c} \quad \text{si} \quad c \supset c'
\]

\[
A_{c'} \times A_{c} \longrightarrow A_{c'c}
\]

\[
\pi_{x_n} \Theta \cdots \Theta \pi_{x_2} \Theta \pi_{x_1}
\qquad
\mathfrak{p} \supset \mathfrak{q} \supset \mathfrak{r}
\]

\[
\begin{array}{ll}
A_{[\mathfrak{p}, \mathfrak{q}]} & \\
\uparrow \text{\ill{} \uncertain{plat} \ill{} \ill{}} & \\
A_{\mathfrak{q}} & A_{\mathfrak{q}} / \mathfrak{q} A_{\mathfrak{q}}
\end{array}
\]

\[
A_{[\mathfrak{p}, \mathfrak{q}]} \otimes_{A_{\mathfrak{q}}} A_{\mathfrak{r}} / \mathfrak{r}^{n} A_{\mathfrak{r}}
\qquad
A_{[\mathfrak{p}, \mathfrak{q}]} / \mathfrak{r}^{n} A_{[\mathfrak{p}, \mathfrak{q}]} \otimes A_{\mathfrak{r}}
\]

\[
A_{[x_n]} \Theta A_{[x_{n-1}]} \cdots \Theta A_{[x_1]} \Theta A_{[x_0]}
\]

\[
K_c \longrightarrow \prod_{c'} K_{c'}
\qquad
\mathrm{Hom}(K_{*}, K_{*})
\qquad
H_{I}\bigl(DD(A)\bigr) = \underline{H}^{**}(A)
\qquad
A_x
\]

\[
(u_1 - u_2)^{2} = u_1 u_1 - u_1 u_2 - u_2 u_1
\qquad
\sum 1_{x_0, x_1}
\qquad
u_1
\]

\[
A_{c'} \times A_{c} \longrightarrow A_{c'c}
\quad \text{si \emph{$c'$ et $c$ composables} (\uncertain{bout} à \uncertain{bout})}
\]

\[
c = (x_0, \ldots, x_n), \qquad c' = (x_n, \ill{}, x_m)
\]

\[
\text{\ill{}} \qquad
\begin{array}{l}
A_{c'} \longrightarrow A_{c'c} \\
A_{c} \longrightarrow A_{c'c}
\end{array}
\]

\[
f, g, h \longmapsto \bigl(i_{c'', c''c'}(f)\, i_{c', c''c'}(g),\ h\bigr)
\longmapsto
i_{c''c', c''c'c}(\,\cdots\,)\, i_{c, c''c'c}(h)
= i_{c'', c''c'c}(f)\, i_{c', c''c'c}(g)\, i_{c, c''c'c}(h)
\]

\[
\underbrace{\prod_{\mathfrak{p}_1 \supset \mathfrak{p}_2 \supset \cdots \supset \mathfrak{p}_n} A_{[\mathfrak{p}_1, \mathfrak{p}_2, \ldots, \mathfrak{p}_n]}}_{\mathcal{A}^{n}}
\longleftarrow
\prod_{\mathfrak{q}_1 \supset \cdots \supset \mathfrak{q}_{n-1}} A_{[\mathfrak{q}_1, \ldots, \mathfrak{q}_{n-1}]}
\]

\[
\mathfrak{p}_1 \supset \mathfrak{p}_2 \supset \cdots \supset \mathfrak{p}_n
\qquad
x_1 \leq x_2 \leq \cdots \leq x_n
\qquad
A_{\mathfrak{p}, \mathfrak{q}}
\qquad
\prod_{\mathfrak{q} \subset \mathfrak{p}} A_{[\mathfrak{q}, \mathfrak{p}]}
\qquad
\Theta^{2} = 0
\]

\[
\Sigma\, 1_{\mathfrak{q}, \mathfrak{p}}
\qquad
[\mathfrak{p}_1 \supset \cdots \supset \mathfrak{p}_n]
\qquad
\coprod_{\mathfrak{q}} 1_{\mathfrak{q}, \mathfrak{p}_1}
\qquad
\downarrow 1_{\mathfrak{q}\, \mathfrak{p}}
\]

\[
\prod_{\mathfrak{q}} [\mathfrak{q} \supset \mathfrak{p}_1 \supset \cdots \supset \mathfrak{p}_n]
\qquad\qquad
\prod_{\mathfrak{q}} [\mathfrak{p}_1 \supset \cdots \supset \mathfrak{p}_n \supset \mathfrak{q}]
\]

\[
\hat{A}_{\mathfrak{r}} \otimes_{A} K
\qquad
\widehat{\hat{A} \otimes A_{\mathfrak{r}}}
\qquad
\widehat{A/\mathfrak{r}^{n}A} \otimes A
\qquad
\underline{A}_{\mathfrak{r}}
\qquad
A/\mathfrak{r}^{n}A
\]

\[
\begin{array}{l}
A/\mathfrak{m} = B_0 = L_0 \\
\widehat{A/\mathfrak{r}} = B_1 \subset L_1 = \text{\uncertain{anneau} \uncertain{local} \uncertain{des} \uncertain{fractions} \uncertain{de} } \widehat{A/\mathfrak{r}} = B_1 \\
\varprojlim L_1 \qquad = \hat{A}_{\mathfrak{r}} \otimes_{A} A_{\mathfrak{r}} / \mathfrak{r} A_{\mathfrak{r}}
\end{array}
\]

\[
A \qquad \mathfrak{p}_1 \supset \mathfrak{p}_2 \supset \cdots \supset \mathfrak{p}_n
\qquad
A_{[\mathfrak{p}_1, \mathfrak{p}_2, \ldots, \mathfrak{p}_n]}
\]

\[
A \to \hat{A}_{\mathfrak{p}} \quad \text{\ill{} \uncertain{local} \uncertain{complété}}
\qquad
A_{[\mathfrak{p}]} = \hat{A}_{\mathfrak{p}}
\]

\[
A_{[\mathfrak{p}_1 \cdots \mathfrak{p}_n]}
\]

\[
A_{[\mathfrak{p}, \mathfrak{q}]} = \varprojlim_{n} \hat{A}_{\mathfrak{p}} \otimes_{A} A_{\mathfrak{q}} / \mathfrak{q}^{n} A_{\mathfrak{q}}
\qquad (\mathfrak{p} \supset \mathfrak{q})
\qquad
\widehat{A_{\mathfrak{p}}/\mathfrak{q}A_{\mathfrak{p}}} \otimes
\]

\[
\widehat{A_{\mathfrak{p}}/\mathfrak{q}A_{\mathfrak{p}}} = \hat{A}_{\mathfrak{p}} / \mathfrak{q} \hat{A}_{\mathfrak{p}}
\]

\[
A \longrightarrow \mathcal{B}
\qquad
\Theta_{x x'} =
\qquad
\Theta
\qquad
\Theta^{2} = 0
\]

\[
\pi_{x'} \Theta \pi_{x} = \Theta_{x x'}
\qquad
x_1 \geq x_2 \geq x_3 \geq \cdots \geq x_n \geq x_{n+1} \geq x_{n+2}
\qquad
\Theta_{x_1, x_2}\ \Theta
\]

\[
\pi_{x_m} \Theta \cdots \pi_{x_{n+2}} \Theta \pi_{x_{n+1}}\
\underbrace{\varphi_{x_n x_{n+1}}}\
\underbrace{\pi_{x_n} \Theta \cdots \pi_{x_3} \Theta \pi_{x_2} \Theta \pi_{x_1}}
\]

\[
\varphi_{x} \quad \varphi_{x'}
\qquad
\left\lbrace
\begin{array}{l}
\Theta\ \text{\uncertain{$A$-linéaire}} \\
\Theta^{2} = 0 \\
\Theta\ \text{\uncertain{de} \uncertain{degré} 1}
\end{array}
\right.
\qquad
\Theta
\]

\[
\left\lbrace
\begin{array}{l}
\mathcal{A}^{0} = \prod_{x \in X} \hat{\mathcal{O}}_{x}
\quad \text{\uncertain{d'où} \uncertain{des}}\ \pi_{x} \in \mathcal{A}_0 \quad (x \in X) \\
\qquad \text{(\uncertain{comme} \uncertain{A-algèbre} \uncertain{topologique})} \\[4pt]
\Theta \in \mathcal{A}^{1}, \quad \text{avec}\ \underline{\Theta^{2} = 0}, \quad
\underline{\pi_{x'} \Theta \pi_{x} = 0\ \text{si}\ x'\ \text{\uncertain{ne} \uncertain{précède} \uncertain{pas}}\ x}
\end{array}
\right.
\]

\[
\underbrace{\mathrm{Hom}(K_{*}, K_{*})}_{\substack{\text{\uncertain{$A$-algèbre} \uncertain{graduée}} \\ \text{\uncertain{par} \uncertain{la} \uncertain{composition} \uncertain{des} \uncertain{Hom}}}}
= \prod_{x} \mathrm{Hom}\bigl(K(x), K\bigr)
\supset \prod_{x} \coprod_{y} \mathrm{Hom}\bigl(K(x), K(y)\bigr)
\]

\[
\prod_{x} \mathrm{Hom}\bigl(K(x), K(x)\bigr)
\qquad
\prod_{x}
\]

\[
\mathrm{Hom}^{0}(K, K)^{0} = \prod_{x} \hat{\mathcal{O}}_{x}
\qquad
\text{\ill{} \uncertain{composantes}} \quad (\pi_{x})
\]

\[
\underset{\text{(\uncertain{l'application} \uncertain{nulle})}}{\Theta}
\in \mathrm{Hom}(K, K)^{1} = \prod_{x} \mathrm{Hom}\Bigl(K(x), \coprod_{\underline{\underline{x'\ \text{précède}\ x}}} K_{x'}\Bigr)
\simeq \prod_{\substack{\text{\uncertain{restreint}} \\ x'\ \text{\uncertain{précède}}\ x}} \hat{\mathcal{O}}_{x, x'}
\]

\[
\pi_{x'} \Theta \pi_{x} \qquad \hat{\mathcal{O}}_{x, x'}
\]

\[
\begin{array}{cc}
\mathcal{O}_{x'} & \longrightarrow \ \mathcal{O}_{x} \\
\| & \| \\
A & A_{\mathfrak{p}}
\end{array}
\qquad\qquad
\Sigma\, \varphi_{x'} \Theta \varphi_{x}
\qquad \pi_{x'}
\]

\[
A_{\mathfrak{p}} \rightsquigarrow I_{A}
\qquad
\hat{A} \otimes_{A} A_{\mathfrak{p}} / \mathfrak{p}^{n} A_{\mathfrak{p}}
\qquad
D(A_{\mathfrak{p}}/\mathfrak{p}^{n}A_{\mathfrak{p}}) \to I_{A_{\mathfrak{p}}/\mathfrak{p}^{n}A_{\mathfrak{p}}}
\qquad
\hat{A} \otimes_{A} \hat{A}_{\mathfrak{p}}
\qquad
\widehat{A_{\mathfrak{p}}}
\qquad
A_{\mathfrak{p}, \mathfrak{p}'}
\]

\[
\Gamma(u\, v) =
\qquad
[\Theta, x]
\qquad
\mathrm{Hom}(K, K)
\]

\[
\lbrace \Theta, u \rbrace = \Theta u + (-1)^{\deg \Theta \deg u} u \Theta = \Theta u + (-1)^{\deg u} u \Theta
\]

\[
K(x) \longrightarrow \coprod_{y} K(y)
\qquad
K(x) \longrightarrow \prod_{x'\ \text{\uncertain{précède}}\ x}
\]

\[
\left\lbrace
\begin{array}{c}
x \\ \downarrow \\ x' \\ \mid \\ x''
\end{array}
\right.
\qquad
\begin{array}{c}
K(x) \longrightarrow \coprod_{x'\ \text{précède}\ x} K(x') \\
\downarrow \\
\coprod_{x''\ \text{biprécède}\ x} K(x'')
\end{array}
\]

\[
\underbrace{K(x) \longrightarrow \coprod_{\substack{x' \\ \text{\uncertain{pts} \uncertain{intermédiaires}} \\ \text{\uncertain{entre} } x \text{ et } x''}} K(x') \longrightarrow K(x'')}_{\text{\uncertain{nul}}}
\]

\[
\mathrm{Hom}(K, K) = \prod_{x} \mathrm{Hom}\bigl(K(x), K\bigr)
= \prod_{x} \prod_{y}^{\text{\uncertain{restreint}}} \underbrace{\mathrm{Hom}\bigl(K(x), K(y)\bigr)}_{H(x, y)}
\]

\[
H(x, x)\ \text{\uncertain{un} \uncertain{anneau}}
\qquad
H(x
\qquad
K \rightsquigarrow I
\qquad
\mathrm{Hom}(K,
\]

\[
K(\mathfrak{p}) \longrightarrow I
\qquad
\mathrm{Hom}\bigl(K, D(A)\bigr)\ \text{(\uncertain{anneau} \uncertain{des} \uncertain{fractions})}
\qquad
\mathfrak{q} \supset \mathfrak{p}
\]

\[
H(x, y) \longrightarrow H(x', y')
\]

\[
y' \rightsquigarrow y \longrightarrow x \rightsquigarrow x'
\]

\[
H_s \quad s\ \text{de longueur}\ 0
\qquad
H_s \quad s\ \text{de longueur}\ 1
\]

\[
\Bigl(\coprod_{s \in C_2(x, y)} H_s\Bigr) \Bigl(\coprod_{t \in C_{1}(x, y)} H_t\Bigr)
\qquad
\uparrow
\qquad
\underset{s \in C_0(x, y)}{H}
\]

\[
K_n = \coprod H_s
\qquad
\prod_{x} H\bigl(x, \textstyle\coprod y\bigr)
\qquad
\prod_{x} \prod_{y\ \text{\uncertain{local}}} H(x, y)
\]

\[
\begin{array}{ll}
H^{i}(X, G) = 0 & \text{si } i > 0 \\
H^{i}(X - a, G) = 0 & \text{si } i \geq \dim A
\end{array}
\]

\[
0 \to H^{0}(X/X - a, G) \to H^{0}(X, G) \to H^{0}(X - a, G) \to H^{1}(X/X - a, G) \to 0
\]

\[
H^{i}(X/X - a, G) \overset{\partial}{\longleftarrow} H^{i-1}(X - a, G) \qquad (i \geq 2)
\]

\[
H^{0}_{A}(G) =
\left\lbrace
\begin{array}{ll}
G(A) = G(K) & \text{si } \dim A = 0 \\
0 & \text{si } \dim A > 0
\end{array}
\right.
\]

\[
H^{1}_{A}(G) =
\left\lbrace
\begin{array}{ll}
0 & \text{si } \dim A \neq 1 \\
G(K)/G(A) & \text{si } \dim A = 1
\end{array}
\right.
\]

\[
H^{2}_{A}(G) = H^{1}(X - a, G) =
\left\lbrace
\begin{array}{l}
0 \quad \text{si } \dim A \neq 2 \\
\text{\ill{}}
\end{array}
\right.
\]

\[
H^{0}_{A}(G) = \mathrm{Ker}\bigl[G(\operatorname{Spec}(A)) \to G(\operatorname{Spec}(A) - a)\bigr]
=
\left\lbrace
\begin{array}{ll}
G(A) & \text{si } \dim A = 0 \\
0 & \text{si } \dim A \neq 0
\end{array}
\right.
\]

\[
H^{1}_{A}(G) = \mathrm{Coker}\bigl(G(\operatorname{Spec}(A)) \to G(\operatorname{Spec}(A) - a)\bigr)
=
\left\lbrace
\begin{array}{ll}
G(K)/G(A) & \text{si } \dim A = 1 \\
0 & \text{si } \dim A \neq 1
\end{array}
\right.
\]

\[
H^{2}_{A}(G) = H^{1}_{?}(\operatorname{Spec}(A) - a, G)
\]

\[
H^{1}(X - a, G) = 0 \qquad (\text{\uncertain{loc.}\ \uncertain{trivial}} = \text{\uncertain{trivial}}),
\]

\[
H^{1}(X - a, G) = \mathrm{Im}\bigl(H^{1}(X - a, L) \to H^{1}(X - a, G)\bigr)
\]

\[
\mathrm{Im}\bigl(\mathcal{H}^{1}(X - a, L) \to \mathcal{H}^{1}(X - a, G)\bigr) = H^{1}(X - a, L) / \mathrm{Im}\, H^{0}(X - a, G/L)
\]

\[
0 \to \underline{O}_{x}(G) \to \underline{R}_{x}(G) \to \underline{D}_{x}(G) \to 0
\]

\[
H^{i}\bigl(\underline{O}_{x}(G)\bigr) = H^{i-1}\bigl(\underline{D}_{x}(G)\bigr) \quad \text{si } i \geq 2
\]

\[
H^{i}(X - a, G) \simeq H^{i}(X - a, L)
\qquad
\Bigl(i \geq 2,\ \text{\uncertain{et}}\ = \text{\ill{} \ill{} \uncertain{pour}}\ i \geq 1\Bigr)
\]

\[
H^{i}(X - a, G) = H^{i}(X - a, U) \qquad (i \geq 1)
\]

\[
H^{i}(X - a, \mathbb{G}_a) = 0 \quad \text{si } i \neq \dim A - 1
\]

\[
H^{i}(X - a, U) = 0 \quad \text{si } i \neq \dim A - 1
\]

\[
\mathrm{Hom}_{\text{k-gr}}(J_{A/k}, G) \simeq H^{i}_{A}(G) \qquad i = \dim A
\]

\[
\begin{array}{lll}
H^{0}_{A}(G) = G(K) & & \text{si } A = K \text{ de dim.\ } 0 \\
H^{1}_{A}(G) = G(K)/G(A) & & \text{si } A \text{ de dim.\ } 1,\ \text{de corps des fractions } K \\
H^{i}_{A}(G) = H^{i-1}(\operatorname{Spec}(A) - \mathfrak{m}_{A}, G) & & \text{si } A \text{ de dim.\ } \geq 2
\end{array}
\]

\[
H^{i}(A, G) = 0 \quad \text{si } i > 0.
\]

\[
H^{i}(L, G) = H^{i}(L, G/L_{0})
\]

\[
H^{1}_{A}(G) = \mathrm{Coker}\bigl(H^{0}(X) \to H^{0}(X - a)\bigr)
\]

\[
\mathrm{Ker}\bigl(H^{1}(X, G) \to H^{1}(X - a, G)\bigr)
\quad \text{\uncertain{par}} \quad G(K)/G(A).
\]

\[
\mathrm{Ker}\bigl(H^{1}(X, G) \to H^{1}(X - a, G)\bigr) = 0
\]

\[
\begin{array}{ccc}
0 & & \\
\downarrow & & \\
H^{1}(X, G) & \longrightarrow & H^{1}(X - a, G) \\
\downarrow & & \downarrow \\
H^{1}(X, G/L) & \longrightarrow & H^{1}(X - a, G/L)
\end{array}
\qquad \text{\uncertain{si}}
\]

\[
\to H^{1}(A, G) \to H^{1}(K, G) \xrightarrow{\ \partial\ } H^{2}(A \bmod K, G) \to H^{2}(A, G)
\]

\[
\to H^{2}(K, G) \xrightarrow{\ \partial\ } H^{3}(A \bmod K, G)
\]

\[
\begin{array}{ccc}
H^{i}(A, G) & \longrightarrow & H^{i}(K, G) \\
\| & & \\
H^{i}(L, G) & &
\end{array}
\]

\[
\left.
\begin{array}{l}
H^{i}(A, G) = H^{i}(A, G/L_{0}) \\
H^{i}(K, G) = H^{i}(K, G/L_{0})
\end{array}
\right| \ i \geq 1
\]

\[
\begin{array}{ccc}
\pi_{1}(L) = \pi_{1}(A) & \pi' & B_{\pi'} \\
\uparrow & & \uparrow \\
\pi_{1}(K) & \pi & B_{\pi} \\
\uparrow & & \uparrow \\
\pi_{1}(A \bmod K) & \pi'' & B_{\pi''} \\
\uparrow & & \uparrow \\
e & & \ill{}
\end{array}
\]

\[
\underline{\underline{\mathrm{Ext}}}^{*}(J_{*}, G)
\Longleftarrow H^{p}\bigl(\mathrm{Ext}^{q}(J_{*}, G)\bigr)
\]

\[
\qquad\qquad\qquad\ \Longleftarrow \mathrm{Ext}^{p}\bigl(H_{q}(J_{*}), G\bigr)
\]

\[
H_{0}(J_{*}) = J^{\bullet}_{X/k}
\]

\[
H_{1}(J_{*}) =
\left\lbrace
\begin{array}{ll}
\mathbb{G}_m & \text{si } X \text{ complète} \\
0 & \text{si } X \text{ non complète}
\end{array}
\right.
\]

\[
H^{*}(X, G) = \underline{\underline{\mathrm{Ext}}}^{*}(J_{*}, G) = \mathrm{Ext}^{*}(J^{\bullet}_{X/k}, G)
\]

\[
\cdots \to \mathrm{Ext}^{p}(J^{\bullet}_{X/k}, G) \to H^{p}(X, G) \to \mathrm{Ext}^{p-1}(\mathbb{G}_m, G) \to \mathrm{Ext}^{p+1}(J^{\bullet}_{X/k}, G)
\]

\[
\boxed{\mathrm{Ext}^{*}(J^{\bullet}_{K/k}, G) \simeq H^{*}(K, G)}
\]

\[
\coprod_{L/k} \mathbb{S}^{*}_{L}
= \underbrace{\coprod_{L/k} \mathbb{Z}_{L}}_{J^{*}_{L/k}}
\times \underbrace{\coprod_{L/k} \mathbb{G}_{m\,L}}_{\mathbb{G}_{m\,k}}
\times \coprod_{L/k} \mathbb{S}'_{L}
\]

\[
\mathcal{U} = \coprod_{L/k} \mathbb{S}'_{L}
\quad \text{\uncertain{est} \uncertain{un} \uncertain{groupe}
\emph{\uncertain{unipotent}} \emph{\uncertain{connexe}}.}
\]

\[
\mathrm{Hom}_{\text{k-gr}}(\mathcal{U}, G)
= \varinjlim_{n} \mathrm{Hom}_{\text{L-gr}}(\mathbb{S}'^{(n)}_{L}, G_{L})
= \varinjlim_{n} \varinjlim_{C'} \mathrm{Hom}_{\text{C'-gr}}(\mathbb{S}'^{(n)}_{C'}, G_{C'})
\]

\[
= \varinjlim_{C'} \varinjlim_{n} \mathrm{Hom}_{\text{C'-gr}}(\mathbb{S}'^{(n)}_{C'}, G_{C'})
\]

\[
\mathrm{Hom}_{\substack{\text{C'-schémas} \\ \text{\uncertain{à} \uncertain{sections}} \\ \text{\uncertain{marquées}}}}\bigl(E^{1}_{C'}, G_{C'}\bigr)
\quad \text{\uncertain{puis} \uncertain{à} Rosenlicht}
\]

\[
\|
\]

\[
\mathrm{Hom}^{\text{\uncertain{pointés}}}_{E^{1}\text{-schémas}}\bigl(C'_{E^{1}}, G_{E^{1}}\bigr)
\]

\[
\|
\]

\[
\mathrm{Hom}^{\text{\uncertain{pointés}}}_{E^{1}\text{-gr}}\bigl(\underbrace{J^{\bullet}_{C'_{E^1}/E^1}}_{J^{\bullet}_{C'/k} \times E^{1}}, G_{E^{1}}\bigr)
\]

\[
\|
\]

\[
\mathrm{Hom}^{\text{\uncertain{restreint}}}\bigl(J^{\bullet}_{C'/k} \times E^{1}, G\bigr)
\]

\[
\mathrm{Hom}_{\text{k-gr}}(\mathcal{U}, G)
= \mathrm{Hom}^{\text{\uncertain{restreint}}}\bigl(J^{\bullet}_{L/k} \times E^{1}, G\bigr)
\]

\[
0 \to \mathcal{U}_{0} \to \mathcal{U} \to \mathbb{S}'_{k} \to 0,
\quad \text{\uncertain{donc}}
\]

\[
\mathrm{Hom}_{k\text{-}\mathrm{gr}}(U_0, G) = \mathrm{Hom}_{\ill{}}\bigl(J^{0}_{L/k} \otimes E^{1}, G\bigr)
\]

\begin{align*}
&(\mathrm{i}) \quad F(S) = \varinjlim_i G_i(S) \\
&(\mathrm{ii}) \quad F'(S) = \Bigl(\varprojlim G_i\Bigr)(S)
\end{align*}

\[
\boxed{F' = u^{*} u_{*}(F)}
\]

\[
\alpha_p \quad \mu_p \quad \mathbb{Z}/p\mathbb{Z} \quad \mathbb{Z}/\ell\mathbb{Z} \quad \mathbb{G}_a,\ \mathbb{G}_m,\ \mathbb{Z},\ \mathbf{A}
\]

\[
H^{i}(U, F) = H^{i}\bigl(X, \mathbb{R}i_{*}(F|U)\bigr) \quad \text{dual de}
\]

\[
U \xrightarrow{\ i\ } X, \qquad U \xrightarrow{\ i'\ } (X - a) \qquad
H^{-i}_{a}\bigl(X, \mathbb{R}i_{!}(\hat{F}|U)\bigr) \simeq H^{-i-1}\bigl(X - a, \mathbb{R}i'_{!}(\hat{F}|U)\bigr)
\]

\[
Y - Z \xrightarrow{\ i'\ } X - Z, \qquad Y - Z \xrightarrow{\ k\ } Y - a, \qquad
Y \xrightarrow{\ i\ } X, \qquad Z \xrightarrow{\ j\ } X
\]

\[
\begin{cases}
H^{i}_{Y-Z}(F) = H^{i}\bigl(Y - Z, \mathbb{R}i'^{!}(F|X - Z)\bigr) \\
\quad \text{dual de } H^{-i-1}\bigl(Y - a, \mathbb{R}k_{!}(\hat{F}|Y - Z)\bigr)
\end{cases}
\]

\[
\begin{cases}
H^{i}_{Y}(F) = H^{i}\bigl(Y, \mathbb{R}i^{!}(F)\bigr) \quad \text{dual de} \\
H^{-i}_{a}\bigl(Y, \mathbb{L}i^{*}(\hat{F})\bigr) = H^{-i}_{a}(Y, \hat{F}|Y)
\end{cases}
\]

\[
H^{i}_{Z}(F) \quad \text{dual de} \quad H^{-i}_{a}(Z, \hat{F}|Z)
\]

\[
\begin{array}{ccc}
H^{i}_{Z}(F) & \times & H^{-i}_{a}(\hat{F}|Z) \\
\downarrow & & \uparrow \\
H^{i}_{Y}(F) & \times & H^{-i}_{a}(\hat{F}|Y) \\
\downarrow & & \uparrow \\
H^{i}_{Y-Z}(F) & \times & H^{-i-1}\bigl(Y - a, \mathbb{R}k_{!}(\hat{F}|Y - Z)\bigr) \\
\downarrow & & \uparrow \\
H^{i+1}_{Z}(F) & \times & H^{-i-1}_{a}(\hat{F}|Z) \\
\downarrow & & \uparrow \\
H^{i+1}_{Y}(F) & \times & H^{-i-1}_{a}(\hat{F}|Y)
\end{array}
\]

\[
A \rightsquigarrow \mathbb{W}(A)_{\pi} = \mathbb{K}(A)
\]

\[
\left\lbrace
\begin{array}{l}
\text{catégorie des $S'$ plats de type fini sur $S$.} \\
\text{descente fid.\ plate [\uncertain{ou} fid.\ \uncertain{plate} \ill{} ???]}
\end{array}\right.
\]

\[
\overline{G}(S') = \mathrm{Hom}_{S}(S', G).
\]

\[
\begin{cases}
H^{i}(S, F) = \mathrm{Ext}^{i}(S; \mathbb{Z}_{S}, F) \\[4pt]
\mathbb{E}\mathrm{xt}^{*}(S; F, G) \Leftarrow H^{p}\bigl(S, \underline{\mathrm{Ext}}^{q}(F, G)\bigr)
\end{cases}
\]

\[
i_{*} \colon \mathcal{C}(U) \to \mathcal{C}(S)
\]

\[
\begin{cases}
H^{i}(k, F) \simeq \mathrm{Ext}^{i}(k; \mathbb{Z}_{k}, F) \\[4pt]
\mathbb{E}\mathrm{xt}^{*}(k; F, G) \Leftarrow H^{p}\bigl(k, \underline{\mathrm{Ext}}^{q}_{k}(F, G)\bigr)
\end{cases}
\]

\[
\mathcal{N} \colon \mathcal{C}(S) \to \mathcal{C}'(k)
\]

\[
\mathcal{N}(F)(A) = \varinjlim_{\mathcal{P}} F\Bigl(\operatorname{Spec}\bigl(\mathbb{V}\bigl(\textstyle\prod_{i \in I_{\mathcal{P}}} A^{\mathcal{P}}_{i}\bigr)\bigr)\Bigr)
= \varinjlim_{\mathcal{P}} \prod_{i \in I_{\mathcal{P}}} F\bigl(\mathbb{V}(A^{\mathcal{P}}_{i})\bigr)
\]

\[
\mathcal{N}(F) = \underline{H}^{0}_{S/k}(F)
\]

\[
\mathrm{R}^{i}\mathcal{N}(F) = \underline{H}^{i}_{S/k}(F)
\]

\[
H^{0}_{S} \simeq H^{0}_{k} \circ \underline{H}^{0}_{S/k}
\]

\[
H^{*}(S, F) \Leftarrow H^{p}\bigl(k, \underline{H}^{q}_{S/k}(F)\bigr)
\]

\[
\underline{\mathrm{Hom}}_{S/k}(F, G) = \underline{H}^{0}_{S/k}\bigl(\underline{\mathrm{Hom}}_{S}(F, G)\bigr)
\]

\[
\underline{\mathrm{Ext}}^{i}_{S/k}(F, G)
\]

\[
\underline{\mathrm{Ext}}^{*}_{S/k}(F, G) \Leftarrow \underline{H}^{p}_{S/k}\bigl(\underline{\mathrm{Ext}}^{q}_{S}(F, G)\bigr)
\]

\[
\mathrm{Hom}_{S}(F, G) \simeq H^{0}\bigl(k, \underline{\mathrm{Hom}}_{S/k}(F, G)\bigr)
\]

\[
\mathrm{Ext}^{*}(S; F, G) \Leftarrow H^{p}\bigl(k, \underline{\mathrm{Ext}}^{q}_{S/k}(F, G)\bigr)
\]

\[
\underline{H}^{i}_{S/k}(F) \simeq \underline{\mathrm{Ext}}^{i}_{S/k}(\mathbb{Z}_{S}, F)
\]

\[
\underline{\mathrm{Ext}}^{i}_{K\text{-}\mathrm{gr}}(G, S\gamma) \xrightarrow{\ \sim\ } \underline{\mathrm{Ext}}^{i+n}_{k\text{-}\mathrm{gr}}(T^{*}G, \gamma)
\]

\[
\underline{\mathrm{Ext}}^{i}_{K/k}(G, S\gamma) \to \underline{\mathrm{Ext}}^{2i}(T^{*}G, \gamma)
\]

\[
\underline{\mathrm{Ext}}^{i}_{K/k}(G, S\gamma) \to \underline{\mathrm{Ext}}^{i}
\]

\[
E^{*}_{K/k}(F, G) \qquad \mathbf{E}_{K/k}(G)
\]

\[
H^{i}(K, \gamma_{K}) \simeq \mathrm{Ext}^{i}_{k\text{-}\mathrm{gr}}(J_{K/k}, \gamma)
\]

\[
k \longleftarrow V \longrightarrow K
\]

\[
T(G) = T^{\circ}\mathcal{C}(G)
\]

\[
\Gamma^{*}_{K} G \simeq \Gamma^{\circ}_{K}\mathcal{C}(G) \simeq \Gamma^{\circ}_{k} T^{\circ}\mathcal{C}(G) \simeq \Gamma^{*}_{k} T(G)
\]

\[
\underline{\underline{H}}^{*}(K, G^{\cdot}) \Longleftarrow H^{p}\bigl(k, \underline{\underline{T}}^{q}(G^{\cdot})\bigr)
\qquad
\underline{\underline{H}}^{*}(K, G^{\cdot}) \simeq \underline{\underline{H}}^{*}\bigl(k, T(G)\bigr)
\]

\[
\Bigl[\ \underline{\underline{\mathrm{Ext}}}^{\cdot}_{K\text{-}\mathrm{gr}}(F, G) \Longleftarrow H^{p}\bigl(k, \underline{\mathrm{Ext}}^{q}_{K/k}(F, G)\bigr) \Longleftarrow H^{p}\bigl(K, \underline{\mathrm{Ext}}^{q}_{K}(F, G)\bigr)
\]

\[
\mathrm{Ext}^{i}_{k\text{-}\mathrm{gr}}(G, \mathbb{Z}/n\mathbb{Z}) \simeq \mathrm{Ext}^{i-1}_{\mathrm{gr.top.}}(SG, \mathbb{Z}/n\mathbb{Z}) = \mathrm{Ext}^{i}\bigl(SG, \mathbb{Z}/n\mathbb{Z}(-1)\bigr)
\]

\[
G \xrightarrow{\ \deg i\ } \mathbb{Z}/n\mathbb{Z}
\qquad
SG \xrightarrow{\ \deg i\ } S(\mathbb{Z}/n\mathbb{Z}) \xrightarrow{\ \deg 0\ } \mathbb{Z}/n\mathbb{Z}(-1)
\]

\[
\underline{\underline{\mathrm{Ext}}}^{1}_{\mathrm{gr.top}}\bigl(S(\mathbb{Z}/n\mathbb{Z})(-1), \mathbb{Z}/n\mathbb{Z}(-1)\bigr) = \underline{\underline{\mathrm{Ext}}}^{1}_{k\text{-}\mathrm{gr}}\bigl(S(\mathbb{Z}/n\mathbb{Z}), \mathbb{Z}/n\mathbb{Z}\bigr)
\]

\[
= \mathrm{Hom}\bigl(H^{0}S(\mathbb{Z}/n\mathbb{Z}), \mathbb{Z}/n\mathbb{Z}\bigr) \simeq \mathbb{Z}/n\mathbb{Z} \quad !
\]

\[
\boxed{H^{i}(A) \times H^{2-i}(B) \longrightarrow H^{2}(K, \mathbb{G}_{m})}
\]

\[
H^{1}({}_{n}A) \longrightarrow
\qquad\qquad
\underline{\underline{\varinjlim}}\ H^{1}(K, A)
\]

\[
\mathrm{Ext}^{i}_{k\text{-}\mathrm{gr}}(G, H) \simeq \mathrm{Ext}^{i-2}(\mathrm{R}\Gamma G, H) = \mathrm{Hom}(H^{2-i}\ldots
\]

\[
\mathrm{Ext}^{i}_{k\text{-}\mathrm{gr}}(G, \mathbb{Z}/n\mathbb{Z}) \simeq \mathrm{Hom}\bigl(H^{1-i}(K, G), \mathbb{Z}/n\mathbb{Z}\bigr)
\]

\[
\mathrm{Ext}^{i}_{k\text{-}\mathrm{gr}}(G, \mathbb{Q}/\mathbb{Z}) \simeq \mathrm{Hom}\bigl(H^{1-i}(K, G), \mathbb{Q}/\mathbb{Z}\bigr)
\]

\[
H^{i}(K, \mathbb{Z}/n\mathbb{Z}) \times H^{1-i}(K, \mathbb{Z}/n\mathbb{Z}) \longrightarrow \mathbb{Q}/\mathbb{Z}
\]

\[
\mathbb{Z}/n\mathbb{Z} \qquad \mathbb{Z}/n\mathbb{Z}
\]

\[
H^{1}(k, \mu_{p}) \qquad K^{*}/K^{*p}
\]

\[
\boxed{H^{i}(k, \mathbb{G}_{m}) = \qquad k^{*} \xrightarrow{\ p\ } k^{*}}
\]

\[
\underline{\mathrm{Ext}}^{i}(\mathbb{G}_{m}, \mathbb{Q}/\mathbb{Z}) = \lbrace 0,\ (\text{racines de l'unité})^{\vee},\ 0,\ 0
\]

\[
\underline{\underline{\mathrm{Ext}}}^{i}_{k\text{-}\mathrm{gr}}(\mathbb{G}_{m}, \mathbb{Q}/\mathbb{Z}) = H^{i-1}(k, \text{racines de l'unité}) \quad \lbrace \text{racines de l'unité},\ 0,\ 0,\ 0
\]

\[
H^{i}(k, \mathbb{G}_{m}) = \lbrace k^{*},\ 0,\ 0,\ 0
\]

\[
\mathrm{Ext}^{1}(\mathbb{G}_{m}, \mathbb{Z}/n\mathbb{Z}) = \mathbb{Z}/n\mathbb{Z}
\]

\[
\mathrm{Ext}^{1}(\mathbb{G}_{m}, \mathbb{Z}/n\mathbb{Z}) \simeq {}_{n}\check{\mu}
\qquad
\mathrm{Ext}^{1}(\mathbb{G}_{m}, \mathbb{Q}/\mathbb{Z}) \simeq \varinjlim_{n}\ {}_{n}\check{\mu}
\]

\[
\mathrm{Hom}\bigl(T_{\cdot}(\mathbb{G}_{m}), \mathbb{Q}/\mathbb{Z}\bigr)
\]

\[
\boxed{\mathrm{Ext}^{1}_{k\text{-}\mathrm{gr}}(\mathbb{G}_{m}, \mathbb{Q}/\mathbb{Z}) \quad \ldots\ k^{*}}
\qquad \simeq \mathrm{Hom}\bigl(\mathbb{G}_{m}(k), \mathbb{Q}/\mathbb{Z}\bigr)
\]

\[
\textstyle\sum_{i} \operatorname{long} F_{0\,x_{i}} \geqslant \sum_{j} \operatorname{long} F_{1\,x'_{j}}
\]

\[
0 \to N \to F \to G \to 0
\]

\[
0 \to N/tN \to F/tF \to G/tG \to 0
\]

\[
\varphi_{x} \colon \mathrm{Gal}(\overline{K}_{x}/K_{x}) \xrightarrow{\ \text{épim.}\ } \mathrm{Gal}(K'/K^{D}_{x}) \subset \mathrm{Gal}(K'/K)
\]

\[
\mathrm{Gal}(\overline{K}_{x}/\overline{K}^{\,i}_{x}) \xrightarrow{\ \text{épim.}\ } \mathrm{Gal}(K'/K^{i}_{x}) \subset \mathrm{Gal}(K'/K)
\]

\[
\boxed{H^{i}\bigl(K, D(G)\bigr) \simeq \mathrm{Ext}^{1-i}\bigl(f_{*}(G), \mathbb{Q}/\mathbb{Z}\bigr)}
\]

\[
\underline{\mathrm{Ext}}^{i}( \qquad
\mathrm{Ext}^{0}\bigl(A(k), \mathbb{Q}/\mathbb{Z}\bigr)^{1-i}
\]

\[
\boxed{f_{*}D(G) \simeq \bigl(T f_{*}G\bigr)(1)}
\qquad
= \mathrm{Ext}^{i}(A, \mathbb{Q}/\mathbb{Z})(k)^{1-i}
\]

\[
H^{i}\bigl(K, D(G)\bigr) \simeq \mathrm{Ext}^{i}\bigl(f_{*}(G), \mathbb{Q}/\mathbb{Z}\bigr)
\qquad
\mathrm{Ext}^{1+h-i}(A, \mathbb{Q}/\mathbb{Z})^{1+h-i}
\]

\[
\boxed{H^{i}\bigl((T f_{*}G)(1)\bigr) = H^{i-1}(T f_{*}G) = \underline{\underline{\mathrm{Ext}}}^{i-1}(f_{*}G, \mathbb{Q}/\mathbb{Z})}
\]

\[
\mathrm{Ext}^{i-1}\bigl(A(-1), \mathbb{Q}/\mathbb{Z}\bigr) = \mathrm{Ext}^{i}\bigl(M(\mathbb{Q}/\mathbb{Z})(1)\bigr)^{i-1} = \mathrm{Ext}^{i-1}\bigl(H^{1-i}(f_{*}G)\bigr)\ \ \mathrm{Ext}\ (\ldots
\]

\[
\boxed{H^{*}(K, DG) \Longleftarrow \mathrm{Ext}^{p}\bigl(H^{-q}(K, G), \mathbb{Q}/\mathbb{Z}\bigr)}
\]

\[
0 \to \mathrm{Ext}^{1}_{k\text{-}\mathrm{gr}}\bigl(H^{1-i}(K, G), \mathbb{Q}/\mathbb{Z}\bigr) \to H^{i-1}(K, DG) \to \mathrm{Hom}_{k\text{-}\mathrm{gr}}\bigl(H^{-i}(K, G), \mathbb{Q}/\mathbb{Z}\bigr)
\]

\[
H^{*}\bigl(\Delta^{2}T(A)\bigr) \Longleftarrow (H^{p}\Delta^{2})\bigl(H^{q}(K, A)\bigr)
\]

\[
E^{pq}_{2} = \ \Bigl\lbrace\ 0 \text{ si } q \neq 0,\ p = 1
\]

\[
\begin{cases}
(H^{-1}\Delta^{2})\bigl(H^{0}(K, A)\bigr) \subset \pi_{1}(A) \\
(H^{0}\Delta^{2})\bigl(H^{0}(K, A)\bigr) \\
(H^{0}\Delta^{2})\bigl(H^{1}(K, A)\bigr) = H^{1}(K, A)
\end{cases}
\]

\[
\boxed{\underline{\underline{\mathrm{Ext}}}^{\circ}(P, \mathbb{Q}/\mathbb{Z}) = 0}
\]

\[
\Bigl|\ T(P') \Longleftrightarrow \Delta T(P_{\cdot})
\]

\[
(3)\ \begin{cases}
H^{1}(K, A) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(B, \mathbb{Q}/\mathbb{Z}) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(B'^{\circ}, \mathbb{Q}/\mathbb{Z}) \times \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(V, \mathbb{Q}/\mathbb{Z}) \\
H^{1}(K, B) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(A, \mathbb{Q}/\mathbb{Z}) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(A'^{\circ}, \mathbb{Q}/\mathbb{Z}) \times \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(U, \mathbb{Q}/\mathbb{Z})
\end{cases}
\]

\[
0 \to U \to A \to A' \to 0
\]

\[
\mu(A) = 2\alpha(A^{\circ}) + \mu(A^{\circ})
\qquad
0 \to \mathrm{Ext}^{1}(A'^{\circ}, \mathbb{Q}/\mathbb{Z}) \to \mathrm{Ext}^{1}(A, \mathbb{Q}/\mathbb{Z}) \to \underline{\mathrm{Ext}}^{1}(U, \mathbb{Q}/\mathbb{Z}) \to 0
\]

\[
(\mathbb{Q}/\mathbb{Z})^{\mu(A)} \qquad \mathrm{Ext}^{1}(A, \mathbb{Q}/\mathbb{Z}) \simeq H^{1}(K, B)
\]

\[
\underline{\mathrm{Ext}}^{i}(A, \mu_{\infty}) = \lbrace 0,\ {}_{\infty}B,\ 0,\ 0
\]

\[
\underline{\underline{\mathrm{Ext}}}^{i}(A, \mu_{\infty}) \simeq H^{i-1}(K, {}_{\infty}B) = \lbrace H^{0}(K, {}_{\infty}B),\ H^{1}(K, {}_{\infty}B)^{\times},\ 0
\]

\[
H^{i}(K, A) = \lbrace A(K)^{\times},\ H^{1}(K, A),\ 0,\ 0,\ 0
\]

\[
\to H^{0}(K, B) \otimes \mathbb{Q}/\mathbb{Z} \to H^{1}(K, {}_{\infty}B) \to H^{1}(K, B) \to 0
\]

\[
0 \to H^{0}(K, {}_{\infty}B) \to DH^{1}(K, A) \to Q \to 0
\]

\[
0 \to N \to H^{1}(K, {}_{\infty}B) \to DH^{0}(K, A) \to 0
\]

\[
Q \subset N \qquad H^{0}(K, B) \to H^{0}(K, B) \otimes \mathbb{Q}/\mathbb{Z}
\]

\[
\underline{\mathrm{Ext}}^{i}(A, {}_{n}\mu) = \lbrace 0,\ {}_{n}B,\ 0,\ 0
\]

\[
\underline{\underline{\mathrm{Ext}}}^{i}(A, {}_{n}\mu) \simeq H^{i-1}(K, {}_{n}B)
\simeq \mathrm{Ext}^{i-1}\bigl(T(A), \mathbb{Z}/n\mathbb{Z}\bigr)
\]

\[
H^{i}(K, {}_{n}B) \xrightarrow{\ \sim\ } \mathrm{Ext}^{i}_{k\text{-}\mathrm{gr}}\bigl(T(A), \mathbb{Z}/n\mathbb{Z}\bigr)
\]

\[
H^{i}(K, {}_{\infty}B) \simeq \mathrm{Ext}^{i}_{k\text{-}\mathrm{gr}}\bigl(T(A), \mathbb{Q}/\mathbb{Z}\bigr)
\]

\[
\Delta\bigl(H^{1}(K, A)'\bigr)_{n} = \angle\bigl({}_{n}H^{1}(K, A)\bigr)
\]

\[
0 \to \mathbb{Z}/n\mathbb{Z} \to \mathbb{Q}/\mathbb{Z} \xrightarrow{\ n\ } \mathbb{Q}/\mathbb{Z} \to 0
\]

\[
\begin{array}{ll}
H^{i}_{Y}(X, M) & \mathrm{Ext}^{i}(X; M, A) \\
H^{i}(X, M) & \mathrm{Ext}^{i}_{Y}(X; M, A) \\
H^{i}(U, M) & \mathrm{Ext}^{i}(U; M, A)
\end{array}
\]

\[
\mathrm{Ext}^{i}_{\mathfrak{m}}(M, A) \overset{?}{=} D(M)
\]

\[
0 \to M' \to M \to M'' \to 0
\]

\[
\ill{}\ (M, A)
\]

\[
\begin{array}{ccc}
\mathrm{Ext}^{i}_{Y}(X; M, N) & \cdot & \mathrm{Ext}^{n-i}(X; N, M) \\
\downarrow & & \uparrow \\
\mathrm{Ext}^{i}(X; M, N) & & \mathrm{Ext}^{n-i}_{Y}(X; N, M) \\
\downarrow & & \uparrow \\
\mathrm{Ext}^{i}(U; M, N) & & \mathrm{Ext}^{n-i}(U; N, M)
\end{array}
\]

\[
\mathrm{Ext}^{i}_{Y}(X; F, G) \to \mathrm{Ext}^{i}_{Z}(X; F, G) \to \mathrm{Ext}^{i}_{Z-Y}(X; F, G)
\]

\[
\begin{split}
\mathrm{Ext}^{r-i}(X_{/Y} - Y, G_{/Y}, F_{/Y}) &\overset{?}{\longleftarrow} \mathrm{Ext}^{r-i}_{a}(X_{/Y}; \hat{G}_{/Y}, \hat{F}_{/Y}) \longleftarrow \mathrm{Ext}^{r-i}_{a}(X_{/Z}; G_{/Z}, F_{/Z}) \\
&\overset{?}{\longleftarrow} \mathrm{Ext}^{r-i-1}(X_{/Z} - Y, G_{/Z}, F_{/Z})
\end{split}
\]

\[
X_{/Y} \longrightarrow X_{/Z} \qquad Y \subset Z, \quad Z \supset Y \supset \lbrace a \rbrace
\]

\[
\ldots \to H^{\cdot}_{a}(X_{/Z}; G_{/Z}, F_{/Z})
\]

\[
\mathrm{Ext}^{i}_{Y}(X, F, G) = \varinjlim_{n} \mathrm{Ext}^{i}(F/J^{n}F, G)
\]

\[
\simeq \varprojlim_{n} \mathrm{Ext}^{r-i}_{a}(X; G, F/J^{n}F) \Longleftarrow H^{p}_{a}\bigl(X, \underline{\mathrm{Ext}}^{q}_{\mathcal{O}_{X}}(G, F_{n})\bigr)
\]

\[
\begin{array}{c}
\uparrow{\scriptstyle \wr} \\
\varprojlim_{n} \mathrm{Ext}^{r-i}_{a}(\hat{X}, \hat{G}, \hat{F}_{n}) \Longleftarrow H^{p}_{a}\bigl(\hat{X}, \underline{\mathrm{Ext}}^{q}_{\mathcal{O}_{\hat{X}}}(\hat{G}, \hat{F}_{n})\bigr) \\
\uparrow \\
\varprojlim \mathrm{Ext}^{r-i}_{a}(\hat{X}, \hat{G}, \hat{F}_{n})
\end{array}
\]

\[
\mathrm{Ext}^{i}_{Y-Z}(X, F, G) = \varinjlim_{n} \mathrm{Ext}^{i}(X - Z, F/J^{n}F, G)
\]

\[
\simeq \varprojlim \ \ldots \qquad \varprojlim \mathrm{Ext}^{r-i}(\hat{X}, \hat{G}, \hat{F})
\]

\[
H^{i}(X - Z, F
\]

\[
T \subset Z \subset \mathfrak{X}
\]

\[
H^{i}_{T}(\mathfrak{X}, F) \to H^{i}_{T}(\mathfrak{X}, F_{/Z}) \overset{?}{\longrightarrow} H^{i}(\mathfrak{X}
\]

\[
f_{*} \colon \mathrm{Gr}_{C} \longrightarrow \mathrm{Gr}_{k} \longrightarrow \mathrm{QuGr}_{k}
\]

\[
\underline{H}^{i}_{f}(G) = \struck{\ill{}}\ H^{i}(Rf_{*})(G)
\]

\[
\underline{\mathrm{Ext}}^{i}_{f}(F, G) = H^{i}\bigl(\underline{E}^{\cdot}_{f}(F, G)\bigr) = H^{i}\bigl(\bigl(R\,\underline{\mathrm{Hom}}_{f}(F, -)\bigr)(G)\bigr)
\]

\[
\mathrm{Ext}^{*}(C; F, G) \Longleftarrow H^{p}\bigl(k; \underline{\mathrm{Ext}}^{q}_{f}(F, G)\bigr)
\]

\[
H^{*}(C, G) \Longleftarrow H^{p}\bigl(k; \underline{H}^{q}_{f}(G)\bigr)
\]

\[
\dim B_{\mathfrak{m}} = \dim A + n
\]

\[
\dim B_{\mathfrak{m}} = \dim A + n - \deg \mathrm{tr}\, k(\mathfrak{m})/k(\mathfrak{m}')
\]

\[
\nu = \dim \bigl[B_{\mathfrak{m}} \otimes k = k[t_{1}, \ldots, t_{n}]_{\mathfrak{m}}\bigr]
\]

\[
\mathrm{prof}\, B_{\mathfrak{m}} = \mathrm{prof}\, A + \nu \qquad
\mathrm{coprof}\, B_{\mathfrak{m}} = \mathrm{coprof}\, A
\]

\[
G \to S\gamma,
\]

\[
Rf_{*}G \to Rf_{*}S\gamma,
\]

\[
Rf_{*}S\gamma = Rf_{*}\bigl(\gamma \otimes \mu_{n}(1)\bigr) \longrightarrow \gamma
\]

\[
\underline{E}_{f}(G, S\gamma) \longrightarrow \underline{E}_{f}(Rf_{*}G, Rf_{*}S\gamma)
\]

\[
\gamma \in H^{0}(Rf_{*}S\gamma, \gamma)
\]

\[
\underline{E}_{f}(Rf_{*}G, Rf_{*}S\gamma) \longrightarrow \underline{E}_{f}(Rf_{*}G, \gamma)
\]

\[
\underline{E}_{f}(G, S\gamma) \longrightarrow \underline{E}_{f}(Rf_{*}G, \gamma)
\]

\[
\underline{E}_{f}(G, S\gamma) \xrightarrow{\ \sim\ } \underline{E}(Rf_{*}G, \gamma)
\]

\[
\underline{E}_{f}(G, S\gamma) \longrightarrow \underline{E}(Rf_{*}G, \gamma) \ \ldots
\]

\[
0 \to \mu_{n} \to \mathbb{G}_{m} \xrightarrow{\ n\ } \mathbb{G}_{m} \to 0
\]

\[
H^{1}(C, \mathbb{G}_{m}) \xrightarrow{\ n\ } H^{1}(C, \mathbb{G}_{m}) \to H^{2}(C, \mu_{n}) \to H^{2}(C, \mathbb{G}_{m})
\]

\[
0 \to H^{1}(k, \mathbb{G}_{m}) \to H^{1}(C, \mathbb{G}_{m}) \to H^{0}\bigl(k, \underline{H}^{1}_{f}(\mathbb{G}_{m})\bigr) \to H^{2}(k, \mathbb{G}_{m})
\]

\[
\underline{\mathrm{Ext}}^{i}_{f}(G, S\gamma) \longrightarrow \underline{\mathrm{Ext}}^{i}_{k}(Rf_{*}G, \gamma)
\]

\[
  \Delta({}_{p}J) \simeq H^{1}(C, \underline{\mathcal{O}}_{C})^{F} \quad \bigl(H^{1}(C, \mathbb{Z}/p\mathbb{Z})\bigr)
  \]

\[
  \mathbb{Z},\ 0,\ H^{2}(C, \mathbb{Q}/\mathbb{Z}) = \mathrm{Hom}\bigl(T.(J), \mathbb{Q}/\mathbb{Z}\bigr),\ H^{3}(C, \mathbb{Q}/\mathbb{Z}) \simeq \varinjlim_{(n,p)=1} \check{\mu}_{n},\ 0
  \]

\[
H^{*}(C, \gamma) \simeq \underline{\underline{\mathrm{Ext}}}^{*}\bigl(Rf_{*}(\mathbb{G}_{m})(-1), \gamma\bigr) \Longleftarrow \underline{\underline{\mathrm{Ext}}}^{p+q}\bigl(R^{-q}f_{*}(\mathbb{G}_{m}), \gamma\bigr)
\]

\[
H^{*}(C, \mathbb{Q}/\mathbb{Z}) \simeq \underline{\underline{\mathrm{Ext}}}^{*}\bigl(Rf_{*}(\mathbb{G}_{m})(1), \gamma\bigr) \Longleftarrow \underline{\underline{\mathrm{Ext}}}^{p-q}\bigl(R^{-q}f_{*}(\mathbb{G}_{m}), \mathbb{Q}/\mathbb{Z}\bigr).
\]

\[
H^{0}(C, \mathbb{Q}/\mathbb{Z}) \simeq \mathbb{Q}/\mathbb{Z}, \quad H^{1}(C, \mathbb{Q}/\mathbb{Z}) \simeq \mathrm{Ext}^{1}(J, \mathbb{Q}/\mathbb{Z}), \quad H^{2}(C, \mathbb{Q}/\mathbb{Z}) \simeq \ill{} = \check{\mu}
\]

\[
T(G) \simeq G_{x}
\]

\[
\mathrm{Ext}^{i}(C; G, \gamma \otimes \mu_{n}) \simeq \mathrm{Ext}^{i-2}(k; G_{x}, \gamma)
\]

\[
\mathrm{Ext}^{i}(U; F, S\gamma) \simeq \mathrm{Ext}^{i}_{\ill{}}\bigl(k; Rf_{U*}(i_{!}F), \gamma\bigr)
\]

\[
\boxed{H^{i}(U; \hat{F}) \simeq \mathrm{Ext}^{i-2}\bigl(Rf_{*}(i_{!}F_{U}), \mathbb{Q}/\mathbb{Z}\bigr)}
\]

\[
\ldots \longleftarrow H^{i}(U; \hat{F}) \longleftarrow \mathrm{Ext}^{i-2}\bigl(Rf_{*}(F_{!}), \mathbb{Q}/\mathbb{Z}\bigr) \longleftarrow \mathrm{Ext}^{i-2}(Rf_{*}F_{Y}, \mathbb{Q}/\mathbb{Z}) \longleftarrow
\]

\[
F = i_{*}(G)
\]

\[
\left\lbrace
\begin{aligned}
&H^{0}_{x}(F) = H^{1}_{x}(F) = 0 \\
&H^{2}_{x}(F) = H^{1}(K, F)
\end{aligned}
\right.
\]

\[
\underline{\underline{\mathrm{Ext}}}^{i}\bigl(H^{1}(K, F), \mathbb{Q}/\mathbb{Z}\bigr) \simeq \mathrm{Ext}^{i}\bigl(S, i_{*}(G), \ill{} \otimes \mathbb{G}_{m}\bigr)
\]

\[
H^{i}_{x}(F) \to H^{i}(X, F) \to H^{i}(U, F)
\]

\[
\longleftarrow H^{3}(U, \hat{F})
\]

\[
\underline{H}_{x}(F) \to \underline{H}_{X}(F) \to \underline{H}_{U}(F)
\]

\[
E(X; F, \mu_{\infty}) \longleftarrow E_{x}(X; F, \mu_{\infty}) \longleftarrow E(U; F, \mu_{\infty})
\]

\[
\left\lbrace
\begin{aligned}
E(U; F, \mu_{\infty})^{n} &\simeq E\bigl(k, \underline{H}_{U}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-1} \\
E(X; F, \mu_{\infty})^{n} &\simeq E\bigl(k, \underline{H}_{X}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-2} \\
E_{x}(X; F, \mu_{\infty})^{n} &\simeq E\bigl(k, \underline{H}_{x}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-2}
\end{aligned}
\right.
\]

\[
\begin{array}{ll}
E_{U}(X; F, \mu_{\infty})^{i} & E\bigl(k, \underline{H}_{U}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{i-1} \\
E_{X}(X; F, \mu_{\infty}) & E\bigl(k, \underline{H}_{X}(F), \mathbb{Q}/\mathbb{Z}\bigr) \\
E_{x}(X; F, \mu_{\infty})^{i} & E\bigl(k, \underline{H}_{x}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{i-2}
\end{array}
\]

\[
E(U; F, \mu_{\infty})^{n} \simeq E(X, F_{U}, \mu_{\infty})^{n}
\overset{\wr}{\simeq} E\bigl(k, \underline{H}_{X}(F_{U}), \mathbb{Q}/\mathbb{Z}\bigr)^{n-2}
\overset{?}{\simeq} E\bigl(k, \underline{H}_{U}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-1}
\]

\[
\left\lbrace
\begin{aligned}
&\underline{H}^{m}_{X}(F_{U})^{n} \simeq \underline{H}^{m}_{x}(F)^{n} \\
&\boxed{\underline{H}^{m}_{U}(F) \simeq \underline{H}^{m+1}_{x}(F_{U})}
\end{aligned}
\right.
\]

\[
E_{x}(X; F, \mu_{\infty})^{n} \simeq E(X, F_{x}, \mu_{\infty})^{n}
\overset{\wr}{\simeq} E\bigl(k, \underline{H}_{X}(F_{x}), \mathbb{Q}/\mathbb{Z}\bigr)^{n-2}
\overset{?}{\simeq} E\bigl(k, \underline{H}_{x}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-2}
\]

\[
\overset{?}{\simeq}\ \underline{H}^{i+1}_{S/k}(\tilde{G})
\]

\[
\underline{H}^{i}_{U/k}(F) \qquad \mathrm{Ext}^{i}
\]

\[
\underline{\mathrm{Ext}}^{i}_{U/k}(G, \tilde{\gamma}) \simeq \underline{\mathrm{Ext}}^{i}_{k}\bigl(\mathcal{N}^{\cdot}(G), \gamma\bigr)
\]

\[
\begin{array}{cc}
\underline{\mathrm{Ext}}^{i}_{S/k}(G, \tilde{\gamma}) & \underline{\mathrm{Ext}}^{i}_{k}\bigl(\mathcal{N}^{\cdot}(G), \gamma\bigr) \\
\downarrow & \uparrow \\
\underline{\mathrm{Ext}}^{i}_{U/k}(G, \tilde{\gamma}) & \underline{\mathrm{Ext}}^{i}_{k}\bigl(\mathcal{N}^{\cdot}(G_{U}), \gamma\bigr)
\end{array}
\]

\[
0 \to G_{U} \to G \to G^{Y} \to 0
\]

\[
\underline{\mathrm{Ext}}^{*}_{U/k}(G, \tilde{\gamma}) = \underline{\mathrm{Ext}}^{*}_{S/k}(G_{U}, \tilde{\gamma}) = \underline{\mathrm{Ext}}^{*}_{k}\bigl(\mathcal{N}^{\cdot}(i_{!}(G)), \gamma\bigr)
\]

\[
\underline{H}^{*}_{U/k}(\hat{G})
\]

\[
\boxed{\underline{H}^{*}_{Y/k}\bigl(i_{!}(G)\bigr) \overset{?}{\simeq} \underline{H}^{*}_{U/k}(G)} \quad ???
\]

\[
\underline{H}^{*}_{Y/k}\bigl(i_{!}(G)\bigr) \simeq \ill{}
\]

\[
0 \to \widehat{\coprod_{p \in S} H^{2}(k_{p}, \tilde{M})} \to \widehat{H^{2}(O, \tilde{M})} \to \prod_{p \in S} \hat{H}^{0}(k_{p}, M) \to \widehat{H^{2}(O, \tilde{M})}
\]

\[
\to H^{1}(O, M) \to \prod_{p \in S}{}' H^{1}(k_{p}, M) \to \widehat{H^{2}(O, \tilde{M})}
\]

\[
\to H^{2}(O, M) \to \coprod_{p \in S} H^{2}(k_{p}, M) \to 0
\]

\[
\boxed{H^{r}(O, M) \simeq \prod_{p \in S}{}' H^{r}(k_{p}, M)} \qquad r > 2
\]

\[
\underline{\mathrm{III}}(O, M) = \mathrm{Ker}\Bigl(H^{1}(O, M) \to \prod_{p \in S} H^{1}(k_{p}, M)\Bigr)
\]

\[
\pi_{1}(S, \xi) \longrightarrow \mathrm{Aut}(V).
\]

\[
M \rightsquigarrow M \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell} = \bigl(M_{n} = M,\ M_{m} \xrightarrow{\ \ell^{m-n}\ } M_{n}\bigr)
\]

\[
\mathcal{V}_{\ell}(S) \longrightarrow \mathcal{M}_{\ell}(S)
\]

\[
\pi' = \pi_{1}(S', \xi') \longrightarrow \pi = \pi_{1}(S, \xi)
\]

\[
\mathcal{V}_{\ell}(\mathcal{S}) = \text{Pseudo}\varinjlim \mathcal{V}_{\ell}(S_{i})
\]

\[
\pi = \pi_{1}(S, \xi) \xrightarrow{\ \sim\ } \varprojlim_{i} \pi_{i} \qquad \pi_{i} = \pi_{1}(S_{i}, \xi_{i}).
\]

\[
\varinjlim_{\substack{k \\ (k \geqslant i, j)}} \mathrm{Hom}_{S_{k}}\bigl(V_{i} \times_{S_{i}} S_{k},\ W_{j} \times_{S_{j}} S_{k}\bigr)
\]

\[
\pi_{j} \longrightarrow \mathrm{Aut}\,V_{\xi}
\]

\[
\mathfrak{g} \subset \mathfrak{gl}(V_{\xi}) = \mathrm{End}_{\mathbb{Q}_{\ell}}(V_{\xi})
\]

\[
A = \varinjlim_{i} A_{i}
\]

\[
A = \varinjlim_{\alpha} B_{\alpha}
\]

\[
\mathcal{V}((A_{i})) = \text{Pseudo}\varinjlim \mathcal{V}(A_{i}).
\]

\[
\pi_{1}(S, \xi) = \varprojlim \pi_{1}(S_{i}, \xi_{i}) \longrightarrow \mathbf{G} \qquad \mathfrak{g}
\]

\[
A \to A_{i}, \quad A' \qquad S \qquad A \to B, \quad S \leftarrow T, \quad S_{i} \leftarrow T_{i}, \quad \pi_{1}(S_{i}) \leftarrow \pi_{1}(T)
\]

\[
S \leftarrow S' \qquad \mathbf{G} \longleftrightarrow \mathbf{G}' \qquad \Gamma \subset G \times G' \qquad G \to G'
\]

\[
\Updownarrow
\]

\begin{align*}
&1) \quad \mathrm{Ext}^{i}_{Y}(X; F, G) \longleftarrow \varinjlim \mathrm{Ext}^{i}(X; F/\mathcal{J}^{n+1}F, G) \\
&2) \quad \underline{\mathrm{Ext}}^{i}_{Y}(F, G) \longleftarrow \varinjlim \underline{\mathrm{Ext}}^{i}(F/\mathcal{J}^{n+1}F, G)
\end{align*}

\[
H^{p}(X, \underline{\mathrm{Ext}}^{q}_{Y}(F, G)) \Longrightarrow \mathrm{Ext}^{*}_{Y}(X; F, G).
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c|c}
\underline{\mathrm{Hom}}(F,G) & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p & \mu_p & \alpha_p & \mathbb{G}_m & \mathbb{G}_a & \mathbb{V} & \mathbb{Z} & A \\ \hline
\mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/\ell\mathbb{Z} & & 0 & 0 & \mu_\ell & 0 & 0 & 0 & {}_\ell A \\
\mu_p & 0 & & \mathbb{Z}/p\mathbb{Z} & & & & & &
\end{array}
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c|c}
 & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mathbb{G}_m & \mathbb{G}_a & \mathbb{Z} & \mathbb{V} & A \\ \hline
\mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/\ell\mathbb{Z} & 0 & 0 & 0 & \mu_\ell & 0 & 0 & 0 & {}_\ell A \\
\mathbb{Z}/p\mathbb{Z} & 0 & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mu_p & \mathbb{G}_a & 0 & \mathbb{V} & {}_p A \\
\alpha_p & 0 & 0 & \mathbb{G}_a & \alpha_p & \alpha_p & \mathbb{G}_a & 0 & (\mathrm{gros}) & ? \\
\mu_p & 0 & 0 & 0 & \mathbb{Z}/p\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & 0 & 0 & 0 & ? \\
\mathbb{G}_m & 0 & 0 & 0 & 0 & \mathbb{Z} & 0 & 0 & 0 & 0 \\
\mathbb{G}_a & 0 & 0 & (\mathrm{gros}) & \mathbb{V} & \mathbb{V} & ? & 0 & (\mathbb{G}_a ?) & \mathrm{gros} \\
\mathbb{Z} & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mathbb{G}_m & \mathbb{G}_a & \mathbb{Z} & \mathbb{V} & A \\
\mathbb{V} & 0 & 0 & \mathbb{G}_a & \mathbb{G}_a & \mathbb{G}_a & \mathbb{G}_a ? & 0 & (\mathrm{Imm}) & \underline{V}(\omega_A) \\
A & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ? & \mathbb{Z}^n
\end{array}
\]

\[
\begin{pmatrix}
\mathbb{Z}/p\mathbb{Z} & 0 & 0 \\
\alpha_p & \mathbb{G}_a & 0 \\
\mu_p & \alpha_p & \mathbb{Z}/p\mathbb{Z}
\end{pmatrix}
\]

\[ \underline{\mathrm{Hom}}(G,H) = 0 \]

\[ \underline{\mathrm{Hom}}(\mathbb{Z}/n\mathbb{Z}, G) \simeq {}_nG \]

\[ \underline{\mathrm{Hom}}(\mathbb{Z}, G) \simeq G \]

\[
\underline{\mathrm{Hom}}(\alpha_p, G) \simeq \mathrm{Ker}\bigl(\underline{V}(\omega_G) \xrightarrow{\ \text{puiss. $p$-ième}\ } \underline{V}(\omega_G)\bigr)
\;\simeq\; \underline{\mathrm{Hom}}(\alpha_p, \underline{\mathfrak{L}}^{(1)}_G)
\]

\[
\underline{\mathrm{Hom}}(\mu_p, G) \simeq \mathrm{Ker}\bigl(\underline{V}(\omega_G) \xrightarrow{\ \text{id $-$ puiss. $p$-ième}\ } \underline{V}(\omega_G)\bigr)
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c}
 & \mu_p & W_{11} & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \mathbb{G}_m & \mathbb{G}_a & Ab & \mathbb{Z} \\ \hline
{}_p\mathbb{G}_m = \mu_p & \mathbb{Z}/p & 0 & 0\ \ast & \ast\ 0 & ? & 0 & [a] & 0\ \ast \\
{}_F\mathbb{G}_a = W_{11} & 0 & \mathbb{G}_a^2\ (?) & 0\ \ast & \ast\ 0 & 0 & (?) & [b] & 0\ \ast \\
\mathbb{Z}/\ell\mathbb{Z} & 0\ \times & 0\ \times & \mathbb{Z}/\ell\mathbb{Z} & 0 & \emptyset\ 0 & \emptyset\ 0 & \emptyset\ 0 & \emptyset\ \mathbb{Z}/\ell \\
\mathbb{Z}/p\mathbb{Z} & 0\ \ast & 0\ \times & 0 & \mathbb{Z}/p\mathbb{Z} & \emptyset\ 0 & \mathbb{G}_a & \emptyset\ 0 & \emptyset\ \mathbb{Z}/p\mathbb{Z} \\
\mathbb{G}_m & \mathbb{Z}/p\mathbb{Z} & 0 & [c] & 0 & 0 & 0 & [d] & \\
\mathbb{G}_a & 0 & (?) & 0\ 0 & (?) & 0 & (?) & [e] & \\
Ab & [f] & \longleftarrow & \longrightarrow & & (Ab)' & H^1(Ab, \underline{O}) & & \\
\mathbb{Z} & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast
\end{array}
\]

\[
k[[t]] \longleftarrow k[t]/(1-t)^p
\]

\[
1 + a_1 t + a_2 t^2 + \cdots + a_n t^n \qquad a_i^p = 0
\]

\[
P(t+t') = P(t)\,P(t')
\]

\[
(*)\qquad \mathrm{Ext}^1_{S\text{-gr}}(G, \Gamma) \longrightarrow H^1(G, \Gamma_G)
\]

\[
0 \to \Gamma \xrightarrow{\ \alpha\ } E \xrightarrow{\ \beta\ } G \to 0
\]

\[
\mathrm{Ext}^1_{S\text{-gr}}(G, \Gamma) \simeq \mathrm{Ext}^1_{S_0\text{-gr}}(G_0, \Gamma_0)
\]

\[
\mathrm{Ext}^1_{S\text{-gr}}(\mathbb{G}_a, \mathbb{Z}/p\mathbb{Z}) \simeq \Gamma(S, \mathcal{O}^{p^{-\infty}})
\]

\[
\mathcal{O}^{p^{-\infty}} = \varinjlim_n (\mathcal{O}, n), \ldots
\]

\[
\begin{array}{c|c|c|c}
\text{multiplicatif} & \mathbb{G}_m & \mathbb{Z}/\ell\mathbb{Z} & \mu_p \\ \hline
\text{unipotent} & \mathbb{G}_a & \mathbb{Z}/p\mathbb{Z} & \alpha_p
\end{array}
\]

\[
\mu_p,\ (\mathbb{G}_a, \alpha_p),\ \mathbb{Z}/p\mathbb{Z},\ \mu_p,\ (\alpha_p, \mathbb{V}),\ \mathbb{Z}/p\mathbb{Z}
\]

\[
\mathbb{G}_m \quad \bigl[\,(\mu_p),\ (\mathbb{G}_a, \alpha_p, \mathbb{V}),\ (\mathbb{Z}/p\mathbb{Z})\,\bigr] \quad (\mathbb{Z})
\]

\[
\mu_p + (\mathbb{G}_a, \alpha_p) \qquad (\alpha_p, \mathbb{V}) + \mathbb{Z}/p\mathbb{Z}
\]

\[
\mu_p \quad \alpha_p \quad \mathbb{Z}/p
\]

\[
\mathbb{G}_a \quad \alpha_p \quad \mathbb{V}
\]

\[
\mu_p \times \mathrm{Unip.}, \quad \mathbb{V}
\]

\[
A = \varprojlim_n \mathbb{W}_n[F, \mathbb{V}]/(V^n, FV - p)
\]

\[
C_1' \simeq C_2' .
\]

\[ H^0_{\mathfrak{m}}(X) \xrightarrow{\ \sim\ } D\,H^0_{\mathfrak{m}}(Y) \]

\[ X \to X', \qquad Y \leftarrow Y' \]

\[ \mathbb{Z} \times A \]

\[ \mathbb{W}_{\infty\,\sigma}[F,V]/(FV-p) \ ?? \]

\[ 0 \to \mu_n \to \mathbb{G}_m \xrightarrow{\ n\ } \mathbb{G}_m \to 0 \]

\[ 0 \to \mathbb{Z} \xrightarrow{\ n\ } \mathbb{Z} \to
\mathrm{Ext}^1(\mathbb{G}_m, \mu_n) \to 0 \]

\[ \mathbb{W}_{\sigma}[F,V]/(V^n, FV-p) \]

\[ \mathbb{W}_{\sigma}\{F,V\}/(FV-p) \longrightarrow
\mathbb{W}_{\sigma}[[F,V]]/(FV-p) \]

\[ \mathbb{W}_\infty \to \mathbb{W}_\infty
 = \varinjlim \mathrm{Hom}(\mathbb{W}_\infty, \mathbb{W}_n) \]

\[ D(\mathbb{W}_\infty) \subset \mathbb{W}_{\infty 1} \]

\[ D(\mathbb{W}_\infty), \qquad D(\mathbb{W}_\infty) \longrightarrow
\mathbb{W}_\infty, \qquad \underline{D(\mathbb{W}_\infty)} \times
\underline{D(\mathbb{W}_\infty)} \]

\[ \mathrm{Hom}(\mathbb{W}_\infty, \mathbb{G}_m), \qquad \mathbb{W}_\infty \times \]

\[
\begin{array}{ccc}
\mathbb{G}_m & \mathbb{Z}/\ell & \mu_p \\[4pt]
F(x) = x^p & F(x) = x^{?} & F = 0 \\
V(x) = x & V(x) = x^p & V = \mathrm{id}
\end{array}
\]

\[
\left.\begin{array}{l}
D(\mathbb{G}_m) = \mathbb{Z} \\
D(\mathbb{Z}/\ell\mathbb{Z}) \simeq \mathbb{Z}/\ell\mathbb{Z} \\
D(\mathbb{Z}/p^n\mathbb{Z}) \simeq \mu_{p^n} \\
D(\mu_{p^n}) \simeq \mathbb{Z}/p^n\mathbb{Z} \\
D(\mathbb{G}_a) = \ \cdots
\end{array}\right.
\]

\[ \mathrm{Pro}(\underline{C}) \subset
\underline{\mathrm{Hom}}(\underline{C}, (\mathrm{Ens})) \]

\[ Y \mapsto \mathrm{Hom}_{\mathrm{Pro}(\underline{C})}(Y, \mathfrak{X}) \]

\[ \mathfrak{Y} = (Y_j)_{j\in J}, \qquad
\mathrm{Hom}_{\mathrm{Pro}(\underline{C})}(\mathfrak{Y}, \mathfrak{X})
= \varprojlim_{i} \mathrm{Hom}(\mathfrak{Y}, \mathfrak{X}_i) = \]

\[ \mathrm{Pro}(\underline{C}) \longrightarrow
\underline{\mathrm{Hom}}(\underline{C}^{\circ}, (\mathrm{Ens})) \]

\[ X = \varprojlim_{\underline{C}} \mathfrak{X} \longrightarrow \mathfrak{X} \]

\[ \underline{C} \longrightarrow \mathrm{Pro}(\underline{C}) \]

\[ \mathrm{Pro}(\underline{C}) \longrightarrow \underline{C}' \]

\[
\begin{array}{ll}
\mathbb{G}_m,\ \mathbb{Z}/\ell\mathbb{Z},\ \mu_p\ \ \mathbb{G}_a,\ \alpha_p,\ \mathbb{Z}/p\mathbb{Z}
 & \qquad \mu_p,\ \alpha_p,\ \mathbb{Z}/p\mathbb{Z} \\[4pt]
\mu_p,\ \alpha_p,\ \mathbb{V},\ \mathbb{Z}/p\mathbb{Z},\ \mathbb{Z}/\ell\mathbb{Z},\ \mathbb{Z} &
\end{array}
\]

\[ \mathbb{V} = D(\mathbb{G}_a) \]

\[ \sum_{-\infty}^{+\infty} w_i F^i + \sum_{j}^{\infty} w_j V^j \]

\[
\begin{aligned}
F^{-1} w_i &= \sigma^{-1}(w_i)\, F^{-1} \\
F^{-1} w_j &= \sigma^{-1}(w_j)\, F^{-1} \\
w_i F &= F \sigma^{-1}(w_i)
\end{aligned}
\qquad\qquad
\begin{aligned}
FV &= p \\
V &= pF^{-1}
\end{aligned}
\]

\[ W_{n,k} = \mathrm{Ker}\bigl(W_n \xrightarrow{\ F^k\ } W_n\bigr) \]

\[ \overline{W}_n = \bigcup_k W_{n,k} \qquad
W_{1,k} \overset{V}{\hookrightarrow} W_{2,k} \hookrightarrow W_{3,k}
\hookrightarrow \cdots \]

\[ \overline{W}_1 \to \overline{W}_2 \to \cdots \overline{W}_n \to \cdots
\qquad \mathcal{U} = \varinjlim_i \overline{W}_i \]

\[ F : \mathcal{U} \to \mathcal{U}, \qquad V \]

\[ \overline{W}_1 \qquad \underline{\mathcal{U}} \subset V \]

\[ w_1 \quad \sigma^{-1}(w_2) \quad \sigma^{-2}(w_2) \quad \sigma^{-3}(w_3) \]

\[ \overline{W}_1 \subset \overline{W}_2 \subset \overline{W}_3 \subset
\overline{W}_4 \subset \cdots \qquad \overleftarrow{R} \]

\[ A = W_\infty[[F, V]] \qquad
\left\lbrace
\begin{aligned}
FV &= VF = p \\
Fw &= \sigma(w)\, F \\
wV &= V \sigma(w)
\end{aligned}
\right. \]

\[ w + \sum_{1}^{\infty} w'_i F^i + \sum_{1}^{\infty} w''_j V^j \]

\[ \sum_{0}^{\infty} \sigma(w'_i) F^{i+1} +
\sum_{0}^{\infty} \sigma(w'_i) F^{i+1} V \]

\[ \sum_{0}^{\infty} \sigma^{-1}(w''_j)\, V F^i +
\sum_{0}^{\infty} \sigma^{-1}(w''_j)\, V F^i V \]

\[ w + \sum_{1}^{\infty} w'_i F^i + \sum_{1}^{\infty} w''_j V^j \]

\[
\begin{aligned}
&p\sigma(w''_1) + \Bigl(\sigma(w) F + \sum_{2}^{\infty} \sigma(w'_{i-1}) F^{i}\Bigr)
+ \Bigl(\sum_{1}^{\infty} p\,\sigma(w''_{j+1}) V^j\Bigr) \\
&p\sigma^{-1}(w'_1) + \Bigl(\sum_{1}^{\infty} p\,\sigma^{-1}(w'_{i+1}) F^i\Bigr)
+ \Bigl(\sigma^{-1}(w) V + \sum_{1}^{\infty} \sigma^{-1}(w''_j) V^{j+1}\Bigr)
\end{aligned}
\]

\[ \underline{\mathbf{W}} \to \underline{A}_{\mathbf{F}} = \mathcal{A}
\qquad W_{\sigma}[F, \mathbf{V}] \qquad \mathbf{W}_n \]

\[ \left\lbrace
\begin{aligned}
F(wx) &= F(w)\, F(x) \\
V(wx) &= F^{-1}(w)\, V(x)
\end{aligned}
\right. \]

\[
C_1/C_0 \simeq C_F/C_0 \times C'_F/C_0
\]

\[
\Bigl[\ \text{donc}\quad A \times
\underbrace{\underline{\mathrm{Pic}}^{*}_{A/S}/
\underline{\mathrm{Pic}}^{0}_{A/S}}_{\text{Néron--Severi}}
\longrightarrow \underline{\mathrm{Pic}}^{0}_{A/S} = A'\ \Bigr]
\]

\[
H^{*}(S, \mathbb{G}_m) \underset{p}{\Longleftarrow}
H^p(S, \underline{H}^q_{\mathrm{loc}}(\mathbb{G}_m))
\]

\[
H^n(S, \mathbb{G}_m) \longrightarrow
H^0(S, \underline{H}^n_{\mathrm{loc}}(\mathbb{G}_m))
\quad \ldots\ ].
\]

\[
\boxed{\;H^1(S, A) \times H^0(S, A') \longrightarrow H^2(S, \mathbb{G}_m)\;}
\]

\[
H^1(S'/S, A) \times H^0(S, A') \longrightarrow H^2(S'/S, \mathbb{G}_m)
\]

\[
\hat{H}^i(S'/S, A) \times \hat{H}^{1-i}(S'/S, A') \longrightarrow
H^2(S'/S, \mathbb{G}_m)
\]

\[
\boxed{\;(f,g)_x = \frac{f^{v_x(g)}}{g^{v_x(f)}}(x)\,
(-1)^{v_x(f)\,v_x(g)}\;}
\]

\[
\mathrm{I}(X) \times \mathrm{I}(X) \longrightarrow K^{*}
\qquad (\mathrm{I}(X),\ \text{idèles})\ \text{par}
\]

\[
(\vec{f}, \vec{g}) = \prod_x (f_x, g_x)_x
\]

\[
\boxed{\;K^{*} \times \mathcal{C}(X) \longrightarrow k^{*}\;}
\]

\[
\{f, \vec{g}\} = f\langle D \rangle
\prod_{x \in \ldots} g_x(x)^{-v_x(f)}
\]

\[
\boxed{\;K^{*}/k^{*} \times C_0(X) \longrightarrow k^{*}\;}
\]

\[
\vec{f} = (f_x)_{x \in X}
\]

\[
(f_x)_x = [D]_x \quad \text{pour tout } x.
\]

\[
C = \mathrm{I}/K^{*} \longrightarrow \mathrm{Pic}(X)
\quad (\text{classes de diviseurs}),
\]

\[
C_0 \longrightarrow \hat{A}
\]

\[
N \times C_0 \longrightarrow k^{*}
\]

\[
N \times \mathrm{I}_0 \longrightarrow k^{*}
\]

\[
\mathfrak{z} = \sum a_i x_i \in N, \qquad
\vec{f} = (f_x)_{x \in X} \in \mathrm{I}_0
\]

\[
\lambda = \Theta\langle \mathfrak{z}, \mathfrak{z}' \rangle
\]

\[
(g) = \Theta\langle \mathfrak{z} \rangle,
\]

\[
\lambda = g\langle \mathfrak{z}' \rangle \qquad
(\text{indépendant de } g \text{ car } \deg \mathfrak{z}' = 0)
\]

\[
\mu = \prod_i f_{x_i}(x_i)^{a_i}
\]

\[
\langle \mathfrak{z}, \bar{\mathfrak{z}} \rangle = \lambda/\mu =
\Theta\langle \mathfrak{z}, \mathfrak{z}' \rangle \Big/
\prod_i f_{x_i}(x_i)^{a_i}
\]