\[
H^{*}(G, \mathbb{Z})^{+} \simeq
\mathrm{Sym}_{\mathbb{Z}/n\mathbb{Z}}(\check{G})^{+} \simeq
\mathbb{Z}/n\mathbb{Z}[\xi]^{+}
\]

\[
c(r_{G}) = \prod_{\xi \in \check{G}} (1 + \xi)
= \prod_{\lambda \in \mathbb{Z}/n\mathbb{Z}} (1 + \lambda \xi_{0}) .
\]

\[
\begin{cases}
c_{p}(r_{G}) = 0 \\
c_{p-1}(r_{G}) = \prod_{\xi \in \check{G} - \{0\}} \xi
= \Bigl(\prod_{\lambda \in \mathbb{F}_{p}^{*}} \lambda\Bigr) \xi_{0}^{p-1}
= \xi_{0}^{p-1}
\end{cases}
\]

\[
\begin{cases}
c_{i}(N r_{G}) = 0 & \text{si } i > N(p-1) \\
c_{N(p-1)}(N r_{G}) = \xi^{N(p-1)} & (\xi \in \check{G})
\end{cases}
\]

\[
(c_{i}(\uncertain{n}\, r_{G}))|H = c_{i}(N n' r_{H}) ,
\]

\[
\boxed{H^{Nk}(G)(p) \neq 0 \text{ pour tout entier } N, \text{ où } k = 2n'(p-1)}
\]

\[
\boxed{X = X_{0}^{(2)}}
\qquad \text{caractère de } X_{0} : e_{1} + e_{-1} ,
\qquad \text{caractère de } X : \struck{\ill{}}\ e_{2} + e_{-2} ,
\qquad X = X_{0}^{2} - 2 .
\]

\[
\widetilde{c}_{1} = 2 c_{1} .
\]

\[
\gamma_{1} = EP(X) = 2(1-g) , \quad \ill{}
\]

\[
c_{1} = c_{1}^{\mathbb{R}}(\Sigma_{0}) = (1-g)
\]

\[
\Gamma \subset G \quad \text{d'où :}
\]

\[
\boxed{\gamma_{1}^{X} \in H^{2}(X,\mathbb{Z}) \simeq \mathbb{Z}}
\]

\[
\boxed{\tfrac{1}{2}\gamma_{1}^{X} = c_{1}^{X} \in H^{2}(X,\mathbb{Z}) = \mathbb{Z}}
\]

\[
\boxed{\gamma_{1}^{X} = 2(1-g) = 2 - 2g}
\qquad \{= EP(X) \neq 0
\]

\[
H^{2}(\Gamma,\mathbb{Z}) \simeq H^{2}(X,\mathbb{Z}) .
\]

\[
H^{2}(B_{G}, H) \quad \ill{}
\]

\[
B_{H} \to B_{E} \to B_{G} \qquad \ill{}\ H^{2}(B_{S},\mathbb{Z})\ \ill{}\
\text{Chern}.
\]

\[
\gamma_{1}^{X} = \struck{\ill{}}\ \ill{}\ H^{2}(X,\mathbb{Z}) \simeq
H^{2}(\Gamma,\mathbb{Z}) .
\]

\[
H^{2}(\Gamma, \mathbb{Z}/2\mathbb{Z}) = \mathbb{Z}/2\mathbb{Z}
\quad \struck{\ill{}}
\]

\[
0 \to \underline{\mathbb{Z}}/\delta''\underline{\mathbb{Z}} \to
\mathrm{III}'(Y,\underline{A}) \to \mathrm{III}'(Y,\underline{B}) \to 0 .
\tag{4.38}
\]

\[
\mathrm{Br}(X) \xrightarrow{\ \sim\ } H^{1}(Y,\underline{B}) ;
\tag{4.39}
\]

\[
0 \to \underline{\mathbb{Z}}/\delta\underline{\mathbb{Z}} \to
H^{1}(Y,\underline{A}) \to H^{1}(Y,\underline{B}) \to 0 ,
\tag{4.40}
\]

\[
0 \to \mathrm{Br}(X) \to \mathrm{III}(Y,\underline{B}) \to
\underline{\mathbb{Z}}/\Delta\underline{\mathbb{Z}} \to 0 ,
\qquad \text{où } \Delta \mid \delta .
\tag{4.41}
\]

\[
0 \to \underline{\mathbb{Z}}/\delta''\underline{\mathbb{Z}} \to
\mathrm{III}(Y,\underline{A}) \to \mathrm{III}(Y,\underline{B}) \to 0 ,
\tag{4.42}
\]

\[
\bar{\delta}_{y} = \delta_{y}
\tag{4.43}
\]

\[
i : \eta \longrightarrow Y
\]

\[
\underline{B} = i_{*}(i^{*}(P)) ,
\tag{4.8}
\]

\[
P \longrightarrow \underline{B} ,
\tag{4.9}
\]

\[
\underline{E} = \mathrm{Ker}(P \to \underline{B}) , \qquad
\underline{F} = \mathrm{Coker}(P \to \underline{B}) .
\tag{4.10}
\]

\[
\add{\mathrm{Pic}(\widetilde{X}) =}\ \struck{\mathrm{Pic}}\
\mathrm{Pic}(\widetilde{X}/\widetilde{Y}) \longrightarrow
\mathrm{Pic}(\widetilde{X}_{\tilde{\eta}}/\tilde{\eta}) ,
\]

\begin{align}
\struck{\ill{}}\quad \underline{F}_{\bar{y}} &=
\mathrm{Pic}(\widetilde{X}_{\eta}/\tilde{\eta}) /
\mathrm{Pic}(\widetilde{X}_{\tilde{\eta}}) \tag{4.11} \\
\underline{E}_{\bar{y}} &= \mathrm{Ker}\bigl(\mathrm{Pic}(\widetilde{X})
\longrightarrow \mathrm{Pic}(\widetilde{X}_{\tilde{\eta}})\bigr) . \tag{4.12}
\end{align}

\[
\cdots\ \mathrm{Pic}(X_{\eta}/\eta) \to \coprod_{y}
H^{1}(y,\underline{E}_{y}) \longrightarrow H^{1}(Y,P) \longrightarrow
\mathrm{III}(Y,\underline{B}) \longrightarrow 0 ,
\tag{4.17}
\]

\[
X_{y} = \sum_{i} a_{y}^{i} C_{y}^{i} ,
\tag{4.18}
\]

\[
\mu_{y}^{i} , \ \nu_{y}^{i}
\tag{4.19}
\]

\[
Z_{y}^{i} = \mathrm{Spec}(k(y)^{i}) , \qquad Z_{y} = \coprod Z_{y}^{i} .
\tag{4.20}
\]

\[
0 \to \underline{\mathbb{Z}}_{y} \longrightarrow
p_{y*}(\underline{\mathbb{Z}}_{Z_{y}}) \longrightarrow \underline{E}_{y}
\longrightarrow 0 ,
\tag{4.21}
\]

\[
p_{y}^{*}(\underline{\mathbb{Z}}_{y}) = \underline{\mathbb{Z}}_{Z}
\longrightarrow \underline{\mathbb{Z}}_{Z}
\tag{4.22}
\]

\begin{align}
H^{1}(y,\underline{E}_{y}) &= \mathrm{Ker}\bigl(H^{2}(y,\underline{\mathbb{Z}}_{y})
\longrightarrow \textstyle\prod H^{2}(Z_{y}^{i},\underline{\mathbb{Z}})\bigr)
\tag{4.23} \\
&= \mathrm{Ker}\bigl(H^{1}(y,\underline{\mathbb{Q}}/\underline{\mathbb{Z}})
\longrightarrow \textstyle\prod H^{1}(Z_{y}^{i},\underline{\mathbb{Q}}/
\underline{\mathbb{Z}})\bigr) , \notag
\end{align}

\[
d_{y} = \mathrm{pgcd}(a_{y}^{i}\nu_{y}^{i}) .
\tag{4.24}
\]

\[
\begin{aligned}
C_{(0)}^{p,q} &= \Omega^{q}(G^{p+1}/G) \\
&\ \ \cup \\
C_{(1)}^{pq} &= \omega^{(q)}(G^{p+1}/G) \\
&\ \ \downarrow \\
C_{(2)}^{pq} &= \omega_{\mathrm{inv}}^{(q)}(G^{p+1}/G)
\end{aligned}
\]

\[
H^{*}(B_{G},\mathbb{R}) = H^{*}(C_{\bullet}^{pq}) \simeq H^{*}(C_{(1)}^{pq})
\simeq \sum_{p+q} \bigl\{ H^{p}(\nu \mapsto \omega_{\mathrm{inv}}^{(q)}
(G^{\nu+1}/G)) \bigr\}
\]

\[
\qquad = E_{2}^{pq}
\qquad \text{(isomorphismes canoniques)}
\]

\[
T \xrightarrow{\ \varphi\ } E_{2}^{1*} = \omega_{\mathrm{inv}}^{*}(G \times
G/G) \subset \omega_{\mathrm{inv}}^{*}(G \times G) =
\omega_{\mathrm{inv}}^{*}(G) \otimes \omega_{\mathrm{inv}}^{*}(G)
\]

\[
\varphi(T) = T \otimes 1 - 1 \otimes T
\]

\[
\mathrm{cyc}(D) = \sum_{x \in X^{(1)}} \mathrm{long}(\mathcal{O}_{Y(D),x})
\cdot \overline{\{x\}}
\tag{21.6.5.1}
\]