\[
  q' + q'' \;\uncertain{+}\; q''' = -1 \qquad\qquad q''' = q'q''\,\ill{}
\]

\[
  \mathrm{Ext}^{1}(\pi_0, \pi_0, \pi_0 ; \pi_1)
  \xleftarrow{\ \delta^{(1)}\ }
  \mathrm{Ext}(\pi_0, \pi_0 ; \pi_1)
\]

\[
  P' = W^{*}(L^{\otimes 2}) \qquad
  L_{P'}^{\otimes 2} \simeq \mathcal{O}_{P'}
\]

\[
  \eta \in H^{1}(P', \mathbb{F}_2)
\]

\[
  \eta^{2} + w_1 \eta + w_2 = 0
\]

\[
  w_1 = \alpha \overset{\mathrm{df}}{=} \partial(-1), \qquad
  w_2 \equiv c_1 \bmod 2
\]

\[
  \partial : H^{0}(S, \mathbb{G}_m) \to H^{1}(S, \mu_2)
\]

\[
  E = L + L^{-1} \qquad
  Q(x, x') = \struck{\ill{}}\ x x'
\]

\[
  \prod_i (1 + \alpha + \xi_i)
\]

\[
  (1 + y)(1 + y + \alpha) = 1 + \alpha + \underbrace{y(y + \alpha)}_{\overset{\shortparallel}{c_1\,?}}
\]

\[
  A \to B, \quad M, \ N \qquad
  E_2^{p,q} = \mathrm{Ext}_B^{p}(\mathrm{Tor}_q^{A}(M, B), N)
\]

\[
  \lambda^{i}(xy) = P_i(x, \struck{\ill{}}, \ldots, \lambda^{i-1}x ; y, \struck{\ill{}}, \ldots, \lambda^{i-1}y) \qquad (i \geqslant 2)
\]

\[
  \lambda^{i}(\lambda^{j} x) = Q_{i,j}(x, \lambda^{2}x, \ldots, \lambda^{ij}(x)) \qquad (i, j \geqslant 2)
\]

\[
  K_s(S) \xrightarrow{\ \alpha\ } WG(S)
\]

\[
  K_s(S) \xrightarrow{\ \alpha\ } WG(S) \longrightarrow W(S) \longrightarrow 0
\]

\[
  (M, q) \oplus (M', -q') + \alpha(F) \simeq \alpha(E)
\]

\[
  (M, q) \oplus (M', -q') \simeq \alpha(G) \ ]
\]

\[
  WG(S) \xrightarrow{\ w\ } H^{*}(S, \mathbb{F}_2)
\]

\[
  WG(S) \longrightarrow H^{0}(S, \mathbb{Z}) \times (1 + \hat{H}^{*+}(S, \mathbb{F}_2))
\]

\[
  K_n(G) = K(G \times \mathfrak{S}_n)
\]

\[
  \lambda_t(E) = \sum \lambda^{i}(E)\, t^{i} \qquad
  \lambda^{i}(E) = \mathrm{cl}(\mathfrak{P}_i(E))
\]

\[
  \sigma_t(E) = \sum \sigma^{i}(E)\, t^{i} \qquad
  \sigma^{i}(E) = \mathrm{cl}(\mathrm{Sym}^{i}(E))
\]

\[
  \struck{\sigma_t(E) = \lambda_{\frac{t}{1-t}}(E)} \qquad
  \boxed{\sigma_{-t}(E)\, \lambda_t(E) = 1}
\]

\[
  \sigma^{i}(E) = \struck{\ill{}}\ Q_i(\lambda^{j}(E), 1 \leqslant j \leqslant i)
\]

\[
  \sigma_t(L) = \sum L^{i} t^{i} = (1 - tL)^{-1} = \lambda_{-t}(L)^{-1}
\]

\[
  \lambda_t(L) = 1 + tL
\]

\[
  \boxed{\mathcal{E}(G) \hookrightarrow R(G)} \longrightarrow R_{\uncertain{\mathbb{Q}}}(G)
\]

\[
  \sigma_t = [1 - (tE - t^{2}\lambda^{2}(E) + \ldots)]^{-1}
\]

\[
  \struck{1 + tE}\quad 1 + tE - t^{2}\lambda^{2}E, \ \ldots \ + t^{2}E^{2}
\]

\[
  1 + tE + t^{2}(E^{2} - \lambda^{2}(E)) \ \ldots
\]

\[
  \sigma^{2}(E) = E^{2} - \lambda^{2}(E)
\]

\[
  E \times E \simeq \sigma^{2}(E) + \underline{\lambda^{2}(E)}
\]

\[
  \sigma^{2}(E) = \lambda^{2}(E) + E
\]

\[
  \boxed{2\lambda^{2}(E) + E = E^{2}}
  \qquad
  2\,\tfrac{d(d-1)}{2} + d \quad d^{2}
\]

\[
  \lambda^{i}(xy) = \ ?
\]

\[
  \mathfrak{P}_{\alpha, \beta}(E) \qquad (\alpha_i)_{i \in I}
\]

\[
  k_n(G) \struck{\ill{}} = k(\underbrace{G \times \mathfrak{S}_n}_{G(n)})
\]

\[
  k_*(G) \struck{\ill{}} = \bigoplus_{n \geqslant 0} k_n(G)
\]

\[
  \mathrm{cl}_m(E) * \mathrm{cl}_n(F) = \mathrm{cl}_{m+n}\big((E \times F) \wedge^{\mathfrak{S}_m \times \mathfrak{S}_n} \mathfrak{S}_{m+n}\big)
\]

\[
  \tau^{n}(E) = \mathrm{cl}_n(\mathrm{Mon}(I_n, E)) \in k_n(G) \qquad (n \geqslant 0)
\]

\[
  \tau^{*}(E) = \sum_{n \geqslant 0} \tau^{n}(E) \in 1 + \widehat{k_*(G)^{+}}
\]

\[
  \tau^{*}(E \sqcup F) = \tau^{*}(E) . \tau^{*}(F)
\]

\[
  \tau^{n}(E \sqcup F) = \sum_{i+j=n} \tau^{i}(E)\, \tau^{j}(F)
\]

\[
  \mathrm{Mon}(I_n, E \sqcup F) = \coprod_{H \subset I_n} \mathrm{Mon}(H, E) \times \mathrm{Mon}(\complement_{I_n} H, F)
\]

\[
  = \struck{\ill{}} \coprod_{0 \leqslant i \leqslant n} \ \underbrace{\coprod_{J \in \mathfrak{P}_i(I_n)} \mathrm{Mon}(J, E) \times \mathrm{Mon}(\complement_{I_n} J, F)}_{\text{st.\ par } \mathfrak{S}_n}
\]

\[
  \mathrm{cl}_n\Big(\coprod_{J \in \mathfrak{P}_i(I_n)} \mathrm{Mon}(J, E) \times \mathrm{Mon}(\complement_{I_n} J, F)\Big) = \tau^{i}(E) * \tau^{j}(F)
\]

\[
  k_0(G) \xrightarrow{\ \tau^{*}\ } 1 + \widehat{k_*(G)}^{+}
\]

\[
  \tau^{*}(\mathrm{cl}_0(E)) = \tau^{*}(E)
\]

\[
  q_H : k_n(G) \longrightarrow k_0(G)
\]

\[
  q_H(\mathrm{cl}_n(E)) = \mathrm{cl}_0({}^{H}\!E)
\]

\[
  q^{n}_H : k_0(G) \longrightarrow k_0(G)
\]

\[
  q^{n}_H(x) = q_H\, \tau^{n}(x)
\]

\[
  T^n(E) = \mathrm{cl}_n\bigl(\mathrm{Hom}(I_n, E)\bigr) \in k_n(G)
  \qquad (E \in \uncertain{\mathrm{Ob}\,\mathrm{Ens}(G)})
\]

\[
  T^*(E) = \sum_n T^n(E) \in 1 + \widehat{k_*(G)}^{+} ;
\]

\[
  T^* : k_0(G) \to 1 + \widehat{k_*(G)}^{+}
\]

\[
  \mathrm{Hom}(I_n, E) \simeq \coprod_{Q \text{ quotient de } I_n}
  \mathrm{Mon}(Q, E) \simeq
  \coprod_{\substack{\alpha = (\alpha_1, \ldots, \alpha_n) \in \mathbf{N}^n \\
  \sum i\alpha_i = n}}
  \struck{\ill{}}
  \underbrace{\coprod_{Q \text{ quotient de } I_n \text{ de type } \alpha}
  \mathrm{Mon}(Q, E)}_{\varphi_{\alpha}(\tau^{i}(E))}
\]

\[
  T^n(E) = \sum_{1 \leqslant i \leqslant n} \varphi_{n,i}\bigl(\tau^i(E)\bigr),
\]

\[
  T^*(E) = \sum_{n \geqslant 0} T^n(E)
  = 1 + \sum_{1 \leqslant i \leqslant n} \varphi_{n,i}\,\tau^i(E)
  = 1 + \sum_{i \geqslant 1} \varphi_{*i}\,\tau^i(E)
\]

\[
  \begin{array}{ll}
    \alpha_1 \text{ classes de 1 él.} & \alpha_1! \\
    \alpha_2 \text{ — 2 él.} & \alpha_2!\,2^{\alpha_2} \\
    \alpha_i \text{ — } i \text{ él.} & \alpha_i!\,(i!)^{\alpha_i} \\
    \alpha_n \text{ — } n \text{ él.} & \alpha_n!\,(n!)^{\alpha_n}
  \end{array}
\]

\[
  \sum_{\alpha = (\alpha_1, \alpha_2, \ldots) \in \mathbf{N}^{(\mathbf{N}^*)}}
  \varphi_{\nu(\alpha), \mu(\alpha)}\bigl(\tau^{\mu(\alpha)}(x)\bigr)
\]

\[
  h_I : E \mapsto \mathrm{Hom}_{\mathcal{E}}(I, E) = \mathrm{Mon}(I, E)
\]

\[
  I \mapsto h_I \qquad \mathcal{E}^{\circ} \to
  \underline{\mathrm{Hom}}(\mathcal{E}, (\mathrm{Ensf}))
\]

\[
  H_I : E \mapsto \mathrm{Hom}_{(\mathrm{Ens})}(I, E) = E^I = H(I, E)
\]

\[
  E \mapsto \mathrm{Mon}(I, E) / H
\]

\[
  H(I, E)/H = \mathrm{Hom}_{\mathrm{Ens}}(I, E)/H .
\]

\[
  E \mapsto \coprod_{n \geqslant 0} \mathrm{Mon}(I_n, E)
  \wedge^{\mathfrak{S}_n} X_n
\]

\[
  \underline{\Omega} = \underline{\mathrm{End}}(\mathcal{E}, \mathcal{E})
  \subset \underline{\mathrm{Hom}}(\mathcal{E}, \mathrm{Ens}) =
  \widehat{\mathcal{E}^{\circ}}
\]

\[
  \begin{cases}
    (F + G)(E) = F(E) \sqcup G(E) \\
    (F \times G)(E) = F(E) \times G(E) \\
    (F \circ G)(E) = F(G(E))
  \end{cases}
\]

\[
  K(X, *) = \sum_{n \geqslant 0} K(B_{\mathfrak{S}_n X})
\]

\[
  \tau^*(\xi) = \sum \tau^n(\xi) \in 1 + K(X, *)^{\wedge +}
  = 1 + \xi + \tau^2\xi + \cdots
\]

\[
  \tau^n(\xi) =
  \sum_{\substack{(\alpha_1, \ldots, \alpha_n) \in \mathbf{N}^n \\
  1\alpha_1 + 2\alpha_2 + \cdots + n\alpha_n = n}}
  \Sigma_{\alpha_1, \ldots, \alpha_n}(\lambda^1\xi, \ldots, \lambda^n\xi)\,
  \rho_1^{\alpha_1} \cdots \rho_n^{\alpha_n}
\]

\[
  R(1, *) = K(e, *) = \sum_{i \geqslant 0} R(\mathfrak{S}_i)
\]

\[
  \Sigma_{\alpha_1, \ldots, \alpha_n} \in \mathbf{Z}[S_1, \ldots, S_n]
\]

\[
  \sum (x_1 \cdots x_{\alpha_1})(x_{\alpha_1 + 1} \cdots
  x_{\alpha_1 + \alpha_2})^2 \cdots
  (x_{\alpha_1 + \cdots + \alpha_{n-1} + 1} \cdots
  x_{\alpha_1 + \cdots + \alpha_n})^n
  = \sum x_1^{\nu_1} \cdots x_m^{\nu_m}
\]

\[
  R_k(1, *) = \sum_{i \in \mathbf{Z}} R_k(\mathfrak{S}_i)
\]

\[
  \boxed{N = P + Q} \quad \boxed{N' = P' + Q'} \quad \text{i.e.} \quad
  \boxed{\varepsilon \overset{\text{déf}}{=} N - N' = 0 \text{ ou } 1}
\]

\[
  \boxed{P + P' + T + T' = \nu^2}
\]

\begin{align*}
  D \ (= S - S') &= (Q' + T) - (Q + T') = (Q' - Q) + (T - T') \\
  &= (P - P') + (T - T') - \varepsilon
\end{align*}

\[
  D + \nu^2 = 2(P + T) - \varepsilon \qquad \text{ou} \qquad
  \nu^2 - D' = 2(P' + T') + \varepsilon \quad
  (D' \overset{\text{déf}}{=} -D = S' - S)
\]

\[
  \boxed{D = 2(P + T) - \nu^2 - \varepsilon}
\]

\[
  (D' = 2(P' + T') - \nu^2 + \varepsilon)
\]

\[
  n = p + q \qquad n' = p' + q' \qquad n = n'
\]

\[
  d = s - s' = (t - t') + (q' - q) = (t - t') + (p - p')
\]

\[
  \nu^2 + d = 2(p + t) \quad \text{i.e.} \quad
  \boxed{d = 2(p + t) - \nu^2}
\]

\[
  D = d - \varepsilon \qquad \varepsilon = 0, 1 \qquad
  (\text{NB } p + t = P + T)
\]

\[
  \left\{\begin{array}{l}
    \text{existe} \\
    \text{stabilité par ext.\ corps de base} \\
    \text{indépendance vis-à-vis sous-groupes} \\
    \text{sur groupes quotients}
  \end{array}\right.
\]

\[
  \begin{bmatrix}
    L_1 = \mathfrak{Z}(G) \\
    D_1 = D(G) \\
    L_0 = G/\mathfrak{Z}(G) \\
    D_0 = G/D(G)
  \end{bmatrix}
  \qquad
  \begin{cases}
    D_1^{D_0} = \Pi_1 \\
    D_{1\,D_0} = \{e\}
  \end{cases}
  \qquad
  \begin{array}{l}
    \Pi_1 = L_1 \cap D_1 \\
    \Pi_0 = G/L_1 D_1
  \end{array}
\]

\[
  \underbrace{\Pi_1, \ \mathrm{Ind} \times \mathrm{Ind}\,\delta, \ \Pi_0}
  \qquad
  D' = D_1 \amalg_{\Pi_1} L_1 \qquad L' = L_0 \times_{\Pi_0} D_0
\]

\[
  1 \to \Pi_1 \to G_1 \xrightarrow{d} G_0 \to \Pi_0 \to 1, \qquad
  G_0 \overset{\theta}{\dashrightarrow} \mathrm{Aut}(G_1, \Pi_1)
\]

\[
  (\mathrm{Cent}\,G_0) \cap \mathrm{Ker}\,\theta \xrightarrow{\varphi_G}
  Z^1(\Pi_0, \Pi_1)
\]

\[
  [\,G_1 = D' = L_1 . D_1 = \mathrm{Cent}\,G . \mathrm{Dér}\,G, \quad
  G_0 = L' = G / L_1 \cap D_1 = G/\mathrm{Cent}\,G \cap \mathrm{Dér}\,G
  \ \text{--}\,]
\]

\[
  \mathrm{Cent}(D_0) \cap \mathrm{Ker}\,\theta \xrightarrow{\varphi_G}
  \mathrm{Hom}(\Pi_0, \Pi_1)
\]

\[
  \boxed{\mathrm{Cent}\,D_0 \cap \mathrm{Ker}\,\theta \text{ est
  quasi-unipotent !}}
\]

\[
  \left\{
  \begin{array}{l}
    \boxed{\mathrm{Cent}(\mathrm{Dér}\,G) \text{ unipotent}} \\
    \boxed{\struck{\ill{}}\ (G/\mathbf{Z})_{\mathrm{ab}}
      \ \struck{\text{a-tori}\ill{}} \ \uncertain{\text{linéaire}}} \\
    \boxed{\mathrm{Cent}(G/\mathbf{Z}) \text{ unipotent}}
  \end{array}
  \right.
\]

\[
  (\mathrm{Dér}\,G)_{\mathrm{ab}} \text{ unipotent ?} \longrightarrow
  \text{voir fin}
\]

\[
  \underbrace{Z \cap DG \to DG \to Z.DG}
  \to G
\]

\[
  1 \to \mathfrak{z} \to D \to \hat{H} \to 1
\]

\[
  \begin{array}{c} [Z \to A] \\ \wr\!\wr \\ {[D \to L]} \end{array}
  \qquad
  \begin{array}{c} [\mathfrak{Z} \to A] \\ \wr \\ {[\hat{D} \to \hat{L}]}
  \end{array}
  \qquad
  \begin{array}{c} \mathfrak{Z} \to A \\ \wr\ \ \ \wr \\ \hat{D} \to \hat{L}
  \\ \mathfrak{Z} \to A \\ D \to L \end{array}
\]

\[
  \Phi(x, y) = \underbrace{\varphi(x, y)}_{\ill{}} +
  \underbrace{\psi(x + y) - \psi(x) - \psi(y)}
\]

\[
  1 \to L \to G \to A \to 1 \qquad (*)
\]

\[
  1 \to Z \to G \to M \to 1
\]

\[
  \mathrm{Ext}^{1}(A, \mathfrak{z}) = \mathrm{Hom}(D(\mathfrak{z}), \underline{\mathrm{Pic}}_{A/k})
\]

\[
  \varphi \colon D(\mathfrak{z}) \longrightarrow \underline{\mathrm{Pic}}_{A/k}
\]

\[
  \varphi_{0} \colon \struck{\mathfrak{z}_{u}} \ t \longrightarrow t_{\underline{\mathrm{Pic}}_{A/k}} \simeq H^{1}(A, \underline{O}_{A})
\]

\[
  \varphi_{1} \colon \Gamma \longrightarrow \underline{\mathrm{Pic}}_{A/k}
\]

\[
  \varphi_{0} \colon D(\mathfrak{z}_{u}) \longrightarrow \underline{\mathrm{Pic}}_{A/k}
  \quad (\text{se factorise par } \underline{\mathrm{Pic}}^{0} = A^{*} \text{ et même par } \widehat{A^{*}})
\]

\[
  \varphi_{1} \colon \Gamma \longrightarrow \underline{\mathrm{Pic}}_{A/k}
\]

\[
  D(\mathfrak{z}) = \tau \times \Gamma \hookrightarrow \underline{\mathrm{Pic}}_{A/k}
\]

\[
    D(\underline{\mathrm{Cent}}(L)) \longrightarrow \underline{\mathrm{Pic}}_{A/k}
  \]

\[
  N' \xrightarrow{\;d'\;} M'
\]

\[
  \mathfrak{Z} = \struck{\underline{\mathrm{Cent}}(M)}\ [\underline{\mathrm{Cent}}(K)/(N'/\pi_1)] \cap \operatorname{Ker}\Theta
  \qquad (N' \subset \underline{\mathrm{Cent}}(G) \subset \overline{N'})
\]

\[
  M \longrightarrow \underline{\mathrm{Hom}}_{\mathrm{gr}}(\mathfrak{Z}, N') \qquad \struck{Z^{1}(\ill{}, \pi_1)}
\]

\[
  \varphi_G \colon \pi_0 \longrightarrow \underline{\mathrm{Hom}}_{\mathrm{gr}}(\mathfrak{Z}, \pi_1) \qquad \struck{Z^{1}}
\]

\[
  \Psi_G \colon \mathfrak{Z} \longrightarrow \underline{\mathrm{Hom}}_{\mathrm{gr}}(\pi_0, \pi_1) \qquad \struck{\ill{}}
\]

\[
  \overline{N'}_G = \underline{\mathrm{Cent}}(G)
\]

\[
  \boxed{\mathfrak{Z}_G = \operatorname{Ker}\Psi_G = 1 \quad \text{i.e.\ } \Psi_G \text{ injectif}}
\]

\[
  1 \to \pi_1 \to E \to \pi_0 \to 1
\]

\[
  c_E = c_{G, G_1} \colon \pi_0 \times \pi_0 \longrightarrow \pi_1
\]

\[
  \Psi_{G_1} = \Psi_G + \struck{\beta}\, \tilde{c}_E\, \alpha
\]

\[
  \underbrace{\mathfrak{Z} \to M \to \pi_0}_{\alpha}
\]

\[
  \Psi_G + \struck{\beta}\, \tilde{c}_E\, \alpha \colon \mathfrak{Z} \longrightarrow \underline{\mathrm{Hom}}(\pi_0, \pi_1)
\]

\[
  \mathfrak{Z} \xrightarrow{\;\Psi_G\;} \underline{\mathrm{Hom}}(\pi_0, N') \longrightarrow \underline{\mathrm{Hom}}(\pi_0, N'/\pi_1)
\]

\[
  \varphi_0 \colon \mathfrak{Z} \longrightarrow \underline{\mathrm{Hom}}(\pi_0, N'/\pi_1)
\]

\[
  M \longrightarrow \underline{\mathrm{Hom}}(\mathfrak{Z}, N'/\pi_1)
\]

\[
  \mathfrak{Z} \xrightarrow{\;\varphi_0\;} \underline{\mathrm{Hom}}(\pi_0, N'/\pi_1).
\]

\[
  \mathfrak{Z}_0 = \operatorname{Ker}(\mathfrak{Z} \to \pi_0)
\]

\[
  \Psi_{0\struck{G}} \colon \mathfrak{Z}_0 \longrightarrow \underline{\mathrm{Hom}}(\pi_0, \pi_1).
\]

\[
  \overline{\Psi}_{G_1} \colon \mathfrak{Z}/\mathfrak{Z}_0 = \pi_0^{*} \longrightarrow \underline{\mathrm{Hom}}(\pi_0, \pi_1)/\mathfrak{Z}_0
\]

\[
  G \longrightarrow \underline{\mathrm{Hom}}(M', \mathcal{D})
\]

\[
  M' \longrightarrow \underline{\mathrm{Hom}}(M', \mathcal{D})
\]

\[
  M' \times M' \xrightarrow{\;\lambda_G\;} \mathcal{D}
\]

\[
  \pi_0 \otimes \pi_0 \xrightarrow{\;\lambda_G\;} \mathcal{D} = (N_{\mathrm{comm}})_{M}
\]

\[
  N/DG \simeq \mathcal{D}/\lambda_G(\pi_0 \otimes \pi_0)
\]

\[
  \lambda_{G_1} = \lambda_G + \beta c_E
\]

\[
  \beta \colon \pi_1 \longrightarrow \mathcal{D} = (N_{\mathrm{comm}})_{M}
\]

\[
  \lambda_0 \colon \pi_0 \otimes \pi_0 \longrightarrow \mathcal{D}_0
\]

\[
  \overline{\lambda}_G \colon P \twoheadrightarrow \pi_{1*} = \operatorname{Im}(\pi_1 \to \mathcal{D}) \subset \mathcal{D}
\]

\[
  \overline{\lambda}_{G_1} = \overline{\lambda}_G + p\, c_E\, i
\]