\[
  R_0 . R_0 \subset R_0, \qquad R_1 R_1 \subset R_0, \qquad R_1 . R_0 = R_0 . R_1 \subset R_1 .
\]

\begin{align*}
  x, y \in R_0 &\Longrightarrow x \wedge y = x . y \\
  x, y \in R_1 &\Longrightarrow x \wedge y = \beta(x.y) \qquad (\ldots : \beta(x)\beta(y)) \\
  x \in R_1,\ y \in R_0 &\qquad x \wedge y = y \wedge x = x\, \alpha(y)
\end{align*}

\[
  \text{(A)}\quad \beta(\alpha(x)u) = x\,\beta(u), \qquad
  \text{(B)}\quad \alpha\beta(x) = x.\eta, \quad \eta = \alpha\beta(1) = \alpha(\zeta), \quad \zeta = \beta(1)
\]

\[
  \beta\alpha(x) = x\zeta \quad \text{OK}, \qquad (\beta x)(\beta y) = \zeta\, \beta(x.y).
\]

\[
  w_1(E) = \bigwedge_{1 \le i \le n} (L_i, 4\delta_i, 0) = \Bigl(\bigotimes L_i,\ 4^{n} \textstyle\prod \delta_i,\ 0\Bigr),
\]

\[
  \mathrm{Arf}(E) = \Bigl(\bigotimes L_i,\ 2^{\nu} \textstyle\prod \delta_i,\ 0\Bigr),
  \qquad \mathrm{Arf}(E) = w_1(E).\,\xi^{\ldots}
\]

\[
  c_1(E) = \mathrm{Arf}(E).\,\xi^{(n - \nu') = \nu''}, \qquad \nu'' = \begin{cases} \tfrac{n}{2} & \text{si } n \text{ pair} \\ \tfrac{n-1}{2} & \text{si } n \text{ impair} \end{cases}
\]

\[
  R_1 \times R_1 \to R_0, \quad R_0 \times R_0 \to R_0, \quad R_0 \times R_1 \to R_1, \quad R_1 \times R_0 \to R_1 .
\]

\begin{align*}
  (L, \delta)(L', \delta') &= (LL', 4\delta\delta', 0) \\
  (L, \delta)(L', \delta', T'_0) &= (LL', 2\delta\delta') \\
  (L, \delta, T_0)(L', \delta') &= (LL', \delta\delta') \\
  (L, \delta, T_0)(L', \delta', T'_0) &= (LL', \delta\delta', T_0 T'_0)
\end{align*}

\[
  A_1 \otimes A_2 \otimes A_3 \overset{\mathrm{df}}{=} A .
\]

\[
\begin{array}{lll}
  k & 1 & \sigma_1, \sigma_2, \sigma_3, \sigma_2\sigma_3, \sigma_3\sigma_1, \sigma_1\sigma_2, \sigma_1\sigma_2\sigma_3 = \tau \\
  A_1 & 1, u_1 & \sigma_2, \sigma_3, \sigma_2\sigma_3 \\
  A_2 & 1, u_2 & \sigma_3, \sigma_1, \sigma_3\sigma_1 \\
  A_3 & 1, u_3 & \sigma_1, \sigma_2, \sigma_1\sigma_2 \\
  B_1 & 1, u_2u_3 & \sigma_1, \sigma_2\sigma_3, \sigma_1\sigma_2\sigma_3 = \tau \\
  B_2 & 1, u_3u_1 & \sigma_2, \sigma_3\sigma_1, \sigma_2\sigma_3\sigma_1 = \tau \\
  B_3 & 1, u_1u_2 & \sigma_3, \sigma_1\sigma_2, \sigma_3\sigma_1\sigma_2 = \tau \\
  B_0 & 1, u_1u_2u_3 & \sigma_2\sigma_3, \sigma_3\sigma_1, \sigma_1\sigma_2
\end{array}
\]

\[
  B_1 \simeq A_2 \wedge A_3, \quad B_2 \simeq A_3 \wedge A_1, \quad B_3 \simeq A_1 \wedge A_2, \quad B_0 \simeq A_1 \wedge A_2 \wedge A_3 .
\]

\[
\begin{array}{lll}
  A'_1 = (A_2, A_3, B_1) & 1, u_2, u_3, u_2u_3 & \sigma_1 \\
  A'_2 = (A_3, A_1, B_2) & 1, u_3, u_1, u_3u_1 & \sigma_2 \\
  A'_3 = (A_1, A_2, B_3) & 1, u_1, u_2, u_1u_2 & \sigma_3 \\
  B'_1 = (A_1, B_1, B_0) & 1, u_1, u_2u_3, u_1u_2u_3 = u & \sigma_2\sigma_3 \\
  B'_2 = (A_2, B_2, B_0) & 1, u_2, u_3u_1, u_2u_3u_1 = u & \sigma_3\sigma_1 \\
  B'_3 = (A_3, B_3, B_0) & 1, u_3, u_1u_2, u_3u_1u_2 = u & \sigma_1\sigma_2 \\
  B'_0 = (B_1, B_2, B_3) & 1, u_2u_3, u_3u_1, u_1u_2 & \sigma_1\sigma_2\sigma_3
\end{array}
\]

\begin{align*}
  A_1 &\subset A'_2, A'_3, B'_1 &  B_1 &\subset A'_1, B'_1, B'_0 \\
  A_2 &\subset A'_3, A'_1, B'_2 &  B_2 &\subset A'_2, B'_2, B'_0 \\
  A_3 &\subset A'_1, A'_2, B'_3 &  B_3 &\subset A'_3, B'_3, B'_0 \\
      &                          &  B_0 &\subset B'_1, B'_2, B'_3
\end{align*}

\[
  N(\bar u + \alpha) = N(\bar u) \quad (= N(u)), \qquad
  \mathrm{Tr}(\bar u + \alpha) = \mathrm{Tr}\,\bar u \quad (= \mathrm{Tr}\,u)
\]

\[
  N(\bar u) + \alpha\,\mathrm{Tr}(\bar u) + \alpha^2 = N(\bar u)
\]

\[
  \begin{cases} \alpha(\mathrm{Tr}\,\bar u + \alpha) = 0 & (\mathrm{Tr}\,\bar u = b) \\ 2\alpha = 0 \end{cases}
  \Longrightarrow \alpha = 0 \text{ si } 2 \text{ inv.}
\]

\[
  N(v) = 1 \Longrightarrow v = u\,\bar u^{-1} \;(= u^2 N(u)^{-1}), \qquad u^2 + bu + c = 0, \quad \mathrm{Tr}\,u = -b .
\]

\begin{align*}
  v &= \alpha + \beta u, & N(v) &= \alpha^2 - \alpha\beta b + \beta^2 c = 1 \\
  \bar v &= (\alpha - \beta b) - \beta u, & &= \alpha(\alpha - \beta b) \quad \text{si } c = 0 \\
  v \bar v &= \alpha(\alpha - \beta b) - \beta^2 c \ldots
\end{align*}

\[
  v = \alpha + \beta u, \quad \bar v = \alpha + \beta \bar u, \qquad
  v\bar v = \alpha^2 + \beta^2 + \alpha\beta\,(u + \bar u)
\]

\[
  N(w) = \lambda^2 - \lambda\mu b = \lambda(\lambda - \mu b) ; \qquad \alpha(\alpha - \beta b) = 1 .
\]

\[
  \underline{\mathrm{Quad}}(E) \simeq \underline{\mathrm{Hom}}(\Gamma^2(E), E)
\]

\[
  \underline{\mathrm{Quad}}(E) \simeq \underline{\mathrm{Sym}}^2(E^\vee),
\]

\[
  \underline{\mathrm{Sym}}^2(E^\vee) \longrightarrow \underline{\mathrm{Quad}}(E),
\]

\[
  \underline{\mathrm{Sim}}(E) \simeq \underline{\mathrm{Quad}}(E)/\underline{O}^* .
\]

\[
  a \longmapsto a.Q : \underline{O}_X \ (\text{sans } * \,!) \longrightarrow \underline{\mathrm{Quad}}(E)
\]

\[
  \Delta = \Delta_P \subset \underline{\mathrm{Quad}}(E),
\]

\[
  P \longmapsto \Delta_P
\]

\[
  \Delta \hookrightarrow \underline{\mathrm{Quad}}(E)
\]

\[
  \Delta \xrightarrow{\ i\ } \underline{\mathrm{Quad}}(E)
\]

\[
  Q = Q_E : E \longrightarrow \underline{\mathrm{Quad}}(E)^\vee :
\]

\[
  Q(x) = \{ f \longmapsto f(x) \} \qquad \text{i.e.} \quad f(x) = \langle f, Q(x) \rangle
\]

\[
  (x, y) \longmapsto Q(x + y) - Q(x) - Q(y) = \{ f \longmapsto \underbrace{f(x+y) - f(x) - f(y)}_{\varphi_f(x,y)} \}
\]

\[
  \underline{\mathrm{Quad}}(E)^\vee \xrightarrow{\ i^\vee\ } \Delta^\vee ,
\]

\[
  Q_i = Q_\Sigma : E \longrightarrow \Delta^\vee .
\]

\[
  L = \underline{\det} E = \textstyle\bigwedge^2 E
\]

\[
  E \xrightarrow{\ \sim\ } \check E \otimes L
\]

\[
  E \times E \longrightarrow L = \textstyle\bigwedge^2 E : \quad (x, y) \longmapsto x \wedge y .
\]

\[
  \begin{cases}
    \check E \simeq E \otimes L^\vee \\
    \underbrace{\underline{\mathrm{Sym}}^2(E^\vee)}_{\underline{\mathrm{Quad}}(E)} \simeq \underbrace{\underline{\mathrm{Sym}}^2(E)}_{\underline{\mathrm{Quad}}(E^\vee)} \otimes \underbrace{L^{\vee \otimes 2}}_{L^{\otimes(-2)}}
  \end{cases}
\]

\[
  \underline{\mathrm{Sim}}(E) \xrightarrow[\ \sim\ ]{\ \mathrm{can}_E\ } \underline{\mathrm{Sim}}(\check E)
\]

\[
  \Delta \otimes L^{\otimes 2} \subset \underline{\mathrm{Quad}}(\check E) \otimes L^{\otimes 2}
  \simeq \underline{\mathrm{Quad}}(\check E) \otimes L^{\otimes(-2)} \otimes L^{\otimes 2}
  \simeq \underline{\mathrm{Quad}}(\check E).
\]

\[
  \check E \simeq E \otimes L^\vee ,
\]

\[
  \check E \otimes E \simeq E \otimes E \otimes L^\vee .
\]

\[
  0 \to \textstyle\bigwedge^2 E \longrightarrow E \otimes E \longrightarrow \underline{\mathrm{Sym}}^2(E) \to 0 ,
  \qquad x \wedge y \longmapsto x \otimes y - y \otimes x
\]

\[
  0 \to L \longrightarrow E \otimes E \longrightarrow \underline{\mathrm{Sym}}^2(E) \to 0 ,
\]

\[
  0 \to \underline{O} \longrightarrow (E \otimes E) \otimes L^\vee \longrightarrow \underline{\mathrm{Sym}}^2(E) \otimes L^\vee \to 0
\]

\[
  \underline{\mathrm{Sym}}^2(E) \otimes L^\vee \simeq \underline{\mathrm{Quad}}(E) \otimes L \qquad (\simeq \underline{\mathrm{Quad}}(E, L))
\]

\[
  \wr\!\downarrow
\]

\[
  \underline{\mathrm{End}}(E)/\underline{O}
\]

\[
  \Delta \simeq (A/\underline{O}) \otimes L^\vee \qquad (\text{où } L = \textstyle\bigwedge^2 E = \det E)
\]

\[
  \check\Delta \simeq (A/\underline{O})^\vee \otimes L ,
\]

\[
  Q_\Sigma : E \longrightarrow \Delta^\vee \overset{\mathrm{df}}{=} \Delta' \quad (\simeq A/\underline{O} \otimes L^\vee)
\]

\[
  Q : E \longrightarrow \Delta' \qquad \bigl(\text{i.e. } \Delta'^\vee \to \underline{\mathrm{Quad}}(E) \text{ un homom. de Modules}\bigr)
\]

\[
  \underline{\mathrm{End}}(E)/\underline{O}.1_E \simeq \underline{\mathrm{Quad}}(E) \otimes \underline{L} \qquad (\simeq \underline{\mathrm{Quad}}(E, L))
\]

\[
  \boxed{\,Q_u(x) = u(x) \wedge x\,}
\]

\[
  Q_u = 0 \iff u \in \Gamma\, \underline{O}.\mathrm{id}_E
\]

\[
  u = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

\[
  Q_u(\lambda, \mu) = c\lambda^2 + (d - a)\lambda\mu \ \ldots\ b\mu^2
\]

\[
  Q = \alpha\lambda^2 + \beta\lambda\mu + \gamma\mu^2 ,
\]

\[
  u_a = \begin{pmatrix} a & -\gamma \\ \alpha & a + \beta \end{pmatrix}, \qquad a \in \Gamma\,\underline{O} \text{ arbitraire},
\]

\[
  u_0 = \begin{pmatrix} 0 & -\gamma \\ \alpha & \beta \end{pmatrix} \quad \ldots \quad ]
\]

\[
  \boxed{\,\delta(u) \;\Bigl(\overset{\mathrm{df}}{=} (\mathrm{Tr}\,u)^2 - 4N(u)\Bigr) = -\underbrace{\delta(Q_u)}_{\text{discr. de la forme } Q_u}\,}
\]

\[
  \delta(Q) \in \Gamma(\Delta'^{\otimes 2} \otimes L^{\otimes 2})
\]

\[
  \varphi_Q : E \otimes E \longrightarrow \Delta' ,
\]

\[
  E \longrightarrow \underline{\mathrm{Hom}}(E, M) \simeq E^\vee \otimes \Delta' ,
\]

\[
  \underbrace{\det E}_{L} \longrightarrow \underbrace{\det(E^\vee)}_{L^\vee} \otimes \Delta'^{\otimes 2}
\]

\[
  \delta(Q) \in \Gamma\, \underbrace{L^{\vee \otimes 2} \Delta'^{\otimes 2}}_{(\Delta' \otimes L^\vee)^{\otimes 2}}
\]

\[
  Q : E \longrightarrow \Delta' \qquad Q \in \Gamma\, \underline{\mathrm{Quad}}(E, \Delta')
\]

\[
  \Delta \longrightarrow \underline{\mathrm{Quad}}(E) \qquad (\simeq \underline{\mathrm{End}}(E)/\underline{O} \otimes L^\vee)
\]

\[
  u_Q : \Delta \otimes L \longrightarrow \underline{\mathrm{End}}(E)/\underline{O}.1
\]

\[
  A_Q \longrightarrow \underline{\mathrm{End}}(E)
\]

\[
  0 \to \underline{O}_X \longrightarrow \underline{\mathrm{End}}(E) \longrightarrow \underline{\mathrm{End}}(E)/\underline{O}.1_E \to 0
\]

\[
  N_Q(x) = N(u^!_Q(x))
\]

\[
  N_Q(1_Q) = 1 ,
\]

\[
  (A_Q, 1_Q, N_Q)
\]

\[
  \mathrm{Tr}(u^!_Q(x)) = \mathrm{Tr}_{A_Q/\underline{O}}(x)
\]

\[
  u^!_Q(\bar x) = \overline{u^!_Q(x)}
\]

\[
  v \longmapsto \bar v = (\mathrm{Tr}\,v).1 - v
\]

\[
  A_Q \to \underline{\mathrm{End}}(E)
\]

\[
  A_Q \simeq A_{\lambda Q}
\]

\[
  A_Q = \underline{\mathcal{O}} \times \Lambda \qquad (\Lambda = \Delta \otimes L)
\]

\[
  A_Q \simeq \mathrm{D}\ill{}^{*}_{\underline{\mathcal{O}}}(\Lambda) .
\]

\[
  u_Q^{!} \colon A_Q \to \underline{\mathrm{End}}(E)
\]

\[
  Q(\lambda, \mu) = \alpha \lambda^{2} + \beta \lambda \mu + \gamma \mu^{2}, \qquad \alpha, \beta, \gamma \in \Gamma(\underline{\mathcal{O}}),
\]

\[
  \left|\begin{array}{l}
    u^{2} - \beta u + \alpha\gamma = 0 \\
    \mathrm{Tr}\, u = \beta \\
    \mathrm{N}\, u = \alpha\gamma
  \end{array}\right.
\]

\[
  u_Q = \begin{pmatrix} 0 & -\gamma \\ \alpha & \beta \end{pmatrix} .
\]

\[
  A_{\alpha, \beta, \gamma} = A[u]/(u^{2} - \beta u + \alpha\gamma)
\]

\[
  A_{\alpha', \beta', \gamma'} = A[u]/(u^{2} - \beta' u + \alpha'\gamma'),
\]

\[
  (E, \Delta),
\]

\[
  \Delta \subset \underline{\mathrm{Quad}}(E)
\]

\[
  (E, \Delta', Q),
\]

\[
  Q \colon E \to \Delta'
\]

\[
  \left\{\begin{array}{l}
    E = p_{*}(\Lambda'), \quad \text{où } p \colon X' \to X \text{ morph.\ struct.} \\
    \Delta' = \mathrm{N}_{X'/X}\, \Lambda' \\
    Q \colon p_{*}(\Lambda') \to \Delta' = \mathrm{N}_{X'/X}(\Lambda')
  \end{array}\right.
\]

\[
  (A, P),
\]

\[
  W(A)^{*} = \{ S/X \mapsto \Gamma(S, A_S^{*}) \}
\]

\[
  P \overset{W(A)^{*}}{\wedge} \mathbb{G}_m ,
\]

\[
  W(A)^{*} \xrightarrow{\mathrm{N}_{A/\underline{\mathcal{O}}}} \mathbb{G}_m ,
\]

\[
  1 \to \mathcal{U}_A \to W(A)^{*} \xrightarrow{\mathrm{N}} \mathbb{G}_m \to 1,
\]

\[
  \mathrm{Spec}(A) \hookrightarrow \mathbb{P}^{1}(E)
\]

\[
  M = \underline{\mathrm{End}}(E),
\]

\[
  M / \underline{\mathcal{O}}\, 1_M \overset{\mathrm{df}}{=} \Sigma(M),
\]

\[
  \check{\mathbb{P}}(A) \hookrightarrow \check{\mathbb{P}}(M)
\]

\[
  Q \subset \check{\mathbb{P}}(M)
\]

\[
  \Lambda \otimes p^{*}(\Theta),
\]

\[
  \mathrm{N}(\Lambda \sigma(\Lambda)^{-1}) \simeq \mathrm{N}(\Lambda)\bigl(\mathrm{N}(\sigma(\Lambda))\bigr)^{-1} \simeq \underline{\mathcal{O}} ,
\]

\[
  \mathrm{N}(\sigma(\Lambda)) \simeq \mathrm{N}(\Lambda) \quad \text{canoniquement.}
\]

\[
  E \simeq \underline{\mathrm{End}}(E)/A
\]

\[
  \text{\struck{$\check{\mathbb{P}}(E) \simeq \mathbb{P}(\underline{\mathrm{End}}(E)^{+}/A)$}}
\]

\[
  \text{\struck{$\mathbb{P}(E) \simeq \{$droites vect.\ $D$ dans $\underline{\mathrm{End}}(E)^{+}$ telles que $\mathrm{N}(D) = 0\}$,}}
\]

\[
  \text{\struck{$D \mapsto D^{\perp} \otimes D$),}}
\]

\[
  \mathbb{P}(\underline{\mathrm{End}}(E)/A) \simeq C,
\]

\[
  u \mapsto \mathrm{Tr}(u) - u \qquad (\mathrm{Tr} = \text{trace réduite})
\]

\[
  \underline{\mathrm{End}}(E)/A \simeq \det\nolimits_A \underline{\mathrm{End}}(E) .
\]

\[
  \underline{\mathrm{End}}(E) \simeq E \otimes \check{E}
\]

\[
  \det\nolimits_A \underline{\mathrm{End}}(E) \simeq E^{\otimes_A 2} \otimes (\underbrace{\det\nolimits_h \check{E}}_{\check{L}})^{2}
\]

\[
  \det\nolimits_A \underline{\mathrm{End}}(E) \simeq (\underbrace{\det\nolimits_h(E)}_{L})^{2} \otimes \check{E}^{\otimes_A 2},
\]

\[
  \underline{\mathrm{End}}(E)/A \simeq E^{\otimes_A 2} \otimes_h \check{L} \simeq \check{E}^{\otimes_A 2} \otimes_h L
\]

\[
  \check{E}^{\otimes_A 2} \otimes_h \check{L} \simeq \check{E}^{\otimes_A 2} \otimes_h L
\]

\[
  \check{E}^{\otimes_A 2} \simeq E^{\otimes_A 2} \otimes_h L^{\otimes_h (-2)}
\]

\[
  \check{E} \simeq E \otimes_h L^{\otimes_h (-1)}
\]

\[
  E \simeq \underline{\mathrm{End}}(E)/A \qquad (\simeq E^{\otimes_A 2} \otimes_h \check{L})
\]

\[
  E \otimes_h \check{L},
\]

\[
  E \otimes_A (\check{E})^{\otimes_A -1} \simeq M/\text{\struck{\ill{}}}\,M^{\natural}
\]

\[
  \varphi_u \colon M \to M, \qquad \text{\struck{$\theta$}}\ \theta \mapsto u\theta - \theta\bar{u},
\]

\[
  M^{\natural} = \mathrm{Im}\, \varphi_u, \quad \text{donc } M/M^{\natural} = \mathrm{Coker}\, \varphi_u .
\]

\[
  M \otimes \check{\Lambda} \to M \to \Delta \to 0
\]

\[
  \mathrm{N}_{A/\underline{\mathcal{O}}}(\Delta) \simeq \underline{\mathcal{O}} .
\]

\[
  \langle \theta, \theta' \rangle \overset{\mathrm{def}}{=} \mathrm{Tr}\, \theta\bar{\theta}' = \mathrm{Tr}\, \theta'\bar{\theta}
\]

\[
  \lambda_u \theta = u.\theta, \qquad \gamma_u \theta = \theta u,
\]

\[
  {}^{t}\lambda_u = \lambda_{\bar{u}}, \qquad {}^{t}\gamma_u = \gamma_{\bar{u}}
\]

\[
  \varphi_u \colon \theta \mapsto u\theta - \theta\bar{u}
\]

\[
  {}^{t}\varphi_u = \varphi_{\bar{u}} \colon \theta \mapsto \bar{u}\theta - \theta u ,
\]

\[
  \varphi_{\bar{u}} = -\varphi_u \quad \text{car } \bar{u} \equiv -u \bmod \underline{\mathcal{O}}.1 \text{ et } \varphi_{\mathcal{O}} = 0 .
\]

\[
  B \subset M, \qquad B = \{\theta \in \Gamma M \mid \bar{u}\theta - \theta u = 0\}
\]

\[
  B \cap M^{*} = \{\theta \in \Gamma M^{*} \mid \theta \text{ normalise } A^{*} \text{ (ou $A$), et induit sur $A$ l'invol.\ can.\ } \sigma\}
\]

\[
  B \cap M^{*} \text{ est un \add{pseudo-}torseur sous } \mathrm{Cent}_{M^{*}}(A) = A^{*} ;
\]

\[
  \mathrm{Tr}\,\bar{u} = \mathrm{Tr}\, u = b, \qquad \mathrm{N}(\bar{u}) = \mathrm{N}(u) = c,
\]

\[
  u' = \bar{u} + \alpha.1_A \qquad \alpha \in \Gamma\underline{\mathcal{O}}
\]

\[
  \left\{\begin{array}{l}
    \mathrm{Tr}(u') = \mathrm{Tr}(u) \ (= -b) \\
    \mathrm{N}(u') = \mathrm{N}(u) \ (= c)
  \end{array}\right.
\]

\[
  \left\{\begin{array}{l}
    2\alpha = 0 \\
    \alpha(\alpha - b) = 0
  \end{array}\right.
  \qquad (\text{car } \mathrm{Tr}\, \alpha.1_A = 2\alpha)
\]

\[
  f \colon A \to E, \qquad u \mapsto u.e
\]

\[
  Q(f(u)) = Q(u.e) = \mathrm{N}(u)\, \underbrace{Q(e)}_{1} = \mathrm{N}(u)
\]

\[
  \sigma \colon A \to A, \qquad \sigma(u) = \mathrm{Tr}(u) - u .
\]

\[
  \sigma_e(ue) = \bar{u}.e
\]

\[
  \sigma_e(x) = \varphi_Q(e, x)\,e - x
\]

\[
  \mathrm{Tr}(u) = \varphi_{\mathrm{N}}(u, 1) .
\]

\[
  \sigma_e(x) = 2(e.x)\,e - x
\]

\[
  \Pi = \{\theta \in \Gamma\, \underbrace{\mathrm{S}M^{*}}_{\substack{\text{sections de}\\ \text{Norme } 1\\ (\mathrm{N}(\theta) = 1)}} \mid \theta \text{ \struck{\ill{}} normalise } A \text{ et y induit } u \mapsto \bar{u}\}
\]

\[
  \sigma_D \colon E \xrightarrow{\sim} E
\]

\[
  Q \colon E \to \underline{\mathcal{O}}
\]

\[
  Q(e) = 1 .
\]

\[
  u' = \underbrace{\bar{u}}_{-b-u} + b = \text{\struck{$(\mathrm{Tr}(u)1 - u)$}}\ -u = u
\]

\[
  \theta_{\lambda, \alpha} \colon u \mapsto \lambda u + \alpha \qquad \lambda \in \Gamma\, \mathcal{O}^{*},\ \alpha \in \Gamma\, \underline{\mathcal{O}}
\]

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

\[
  \alpha^{2} = 0 .
\]

\[
  u \mapsto -u + \alpha \qquad (\text{avec } 2\alpha = 0,\ \underline{\underline{\alpha^{2} = 0}}),
\]

\[
  \underbrace{\mathrm{int}(\sigma_D)}_{\substack{\text{autom.\ de } \underline{\mathrm{End}}(E)\\ \theta \mapsto \sigma_D \theta \sigma_D^{-1}}} \text{ transforme } \underset{\underline{\mathrm{End}}(E)}{\overset{}{A}} \text{ en lui-même},
\]

\[
  \boxed{\sigma_D u \sigma_D^{-1} = \bar{u}} \qquad \text{pour } u \in \Gamma A .
\]

\[
  \sigma \mu_u \sigma^{-1} = \mu_{\bar{u}}
\]

\[
  \overline{u\bar{x}} = \bar{u}\, x \qquad \text{OK.}
\]

\[
  \Pi = W(\underline{\mathrm{End}}(E))^{*}
\]

\[
  \overset{\shortparallel}{\{}\theta \text{ sections locales de } \underline{\mathrm{End}}(E) \text{ dans } \mathrm{Sch}_{/X} \mid \theta \text{ normalise } A, \text{ induit } \sigma \text{ sur } A \text{ et } \det\theta = -1\},
\]

\[
  M = W(\underline{\mathrm{End}}(E))
\]

\[
  \overset{\shortparallel}{\{}\theta \mid \theta u = \bar{u}\theta \text{ pour toute section } u \text{ de } A\}
\]

\[
  u\theta = \theta\bar{u}, \qquad \text{pour } \theta \in \Gamma M,\ u \in \Gamma A
\]

\[
  \mathrm{N}(\theta) = -1 .
\]

\[
  \underset{\overset{\cup}{X'}}{P} = \mathbb{P}(E), \qquad P^{*} = P - \underset{\mathrm{Spec}(A)}{\overset{\shortparallel}{X'}}
\]

\[
  \begin{array}{rcl}
    P^{*} & \xrightarrow{f} & \Pi \\
    D & \longmapsto & \sigma_D
  \end{array}
\]

\[
  f(u.D) = u\bar{u}^{-1} f(D) \qquad D \in \Gamma P^{*},\ u \in \Gamma A^{*}
\]

\[
  W(A)^{*} \longrightarrow SW(A), \qquad u \longmapsto u\bar{u}^{-1}
\]

\[
  u \longmapsto u^{2}, \qquad SW(A) \longrightarrow SW(A)
\]

\[
  0 \to \mathcal{O} \to A \xrightarrow{\ \Psi\ } \check{L} \to 0
  \qquad E = \check{A}
  \qquad 0 \to L \to E \xrightarrow{\ \psi\ } \mathcal{O} \to 0
\]

\[
  \begin{array}{ll}
    P = \psi^{-1}(1) \quad \text{tors.\ sous } L \qquad & \psi(x) = \langle 1, x\rangle = x(1)\\[2pt]
    P_{\lambda} = \psi^{-1}(\lambda) \quad \text{id.} &
  \end{array}
\]

\[
  \varphi_{N} : A \times A \to \mathcal{O} \quad\text{d'où}\quad A \to \check{A} \overset{\mathrm{df}}{=} E
\]

\[
  \textstyle\bigwedge^{2} A \to \bigwedge^{2} E
  \qquad\text{d'où}\qquad \delta \in \Gamma L^{\otimes 2}
\]

\[
  \textstyle\bigwedge^{2} A \simeq \check{L}, \qquad \bigwedge^{2} E \simeq L,
  \qquad L^{\otimes 2} \simeq \mathrm{Hom}(\check{L}, L)
\]

\[
  \boxed{T^{2} - 4N = +\delta} \qquad\text{i.e.}\qquad \boxed{4N = T^{2} - \delta}
\]

\[
  \text{\struck{$x'(a,b)$}} \qquad x'(a,b) = a, \quad x''(a,b) = b,
  \qquad \mathrm{Tr}(a,b) = a + b, \quad N(a,b) = ab
\]

\[
  \boxed{\begin{array}{l} T = x' + x'' \\ N = x'x'' \\ \delta = (x' - x'')^{2} \end{array}}
  \qquad \text{\struck{\ill{}}}
\]

\[
  x \longmapsto \underbrace{(x - x')}_{\in L}\,\underbrace{(x - x'')}_{\in L} \in L^{\otimes 2}
\]

\[
  x \longmapsto x^{2} - Tx\,\text{\struck{$\psi(x)$}} + N
\]

\[
  Q : x \longmapsto x^{2} - \psi(x)Tx + \psi(x)^{2}N, \qquad E \to L^{\otimes 2} \subset \mathrm{Sym}^{2}(E)
\]

\[
  x \longmapsto Q(x) = x^{2} : L \to L^{\otimes 2}
\]

\[
  \left\{\begin{array}{l}\text{rev.\ quadratiques}\\ \text{de } S\end{array}\right\}
  \;\simeq\;
  \left\{\begin{array}{l}
    \text{\struck{Triples} $(\text{\struck{$P_{1}$}}\ L, P_{1}, Q)$ sur $S$}\\
    \text{a) $L$ Module inv.}\\
    \text{b) $P_{1}$ $L$-torseur}\\
    \text{c) $Q : P_{1} \to L^{\otimes 2}$ fonct.\ pol.\ de degré 2}\\
    \qquad\text{à partie homogène $x \mapsto x^{2} : L \to L^{\otimes 2}$}
  \end{array}\right.
\]

\[
  \varphi_{N} : A \longrightarrow \check{A} = E
  \qquad\text{via forme bil.\ ass.\ à forme qu.\ $N$}
\]

\[
  A \xrightarrow{\ \varphi_{N}\ } E \xrightarrow{\ Q\ } L^{\otimes 2}
\]

\[
  \boxed{Q(\varphi_{N}(x)) = - N(x)\,\delta}
\]

\[
  Q(\varphi_{N}(x)) = 0 \,..
\]

\[
  x \longmapsto \boxed{2x - T = z} \quad (z \in \Gamma L), \qquad P_{1} \longrightarrow L
\]

\[
  2x = z + T \;\Longrightarrow\; Q(z + T) = 4Q(x)
\]

\[
  \text{\struck{$Q(T+z)$}} \qquad Q(z + T) = z^{2} - \delta
\]

\[
  z^{2} + 2Tz + T^{2} - 2(Tz + T^{2}) + 4N = z^{2} - (T^{2} - 4N) = z^{2} - \delta
\]

\[
  \left\{\begin{array}{l}
    Q(x) = 0 \ \text{ i.e.\ } x \in \Gamma(X'/X) \;\Longrightarrow\; \boxed{z^{2} = \delta}\\
    \text{(et iso si 2 inv.\ sur $X$)}
  \end{array}\right.
\]

\[
  \left\{\begin{array}{ll}
    x \longmapsto \text{\struck{\ill{}}}\ \bar{x} = T - x & \text{sur } P_{1} \quad (\text{auto})\\
    x + \bar{x} = T & \text{si } x \in \Gamma P_{1}
  \end{array}\right.
\]

\[
  \left\{\begin{array}{ll}
    x \longmapsto \bar{x} = \psi(x)T - x & \text{sur } E\\
    \text{i.e.\ } x + \bar{x} = \psi(x)T &
  \end{array}\right.
\]

\[
  \left\{\begin{array}{l}
    \psi(T) = T\\
    \psi(N) = N\\
    Q(\text{\struck{$\psi(x)$}}\ \bar{x}) = Q(x)
  \end{array}\right.
\]

\[
  x \longmapsto \bar{x} = \mathrm{Tr}(x) - x \qquad\text{i.e.}\qquad x + \bar{x} = \mathrm{Tr}(x).1
\]

\[
  T_{0} \in \Gamma L_{0} \quad (\Gamma E_{0})
\]

\[
  T_{0}^{2} = \delta_{0} \ (= -\delta_{0}) \quad\text{sur } X_{0}
\]

\[
  (\text{p.\ ex.\ } \tau_{1} = T|X_{1} \overset{\mathrm{df}}{=} T_{1})
  \qquad\text{on a}\qquad
  \tau_{1}^{2} = \delta_{1} \quad (\overset{\mathrm{df}}{=} \delta|X_{1})
\]

\[
  \boxed{T_{0}^{2} \equiv \delta \quad (4)}
\]

\[
  \left\{\begin{array}{l} z_{0} = T_{0} \\ z^{2} = \delta \end{array}\right.
\]

\[
  \mathfrak{X}' : \mathrm{Sch}_{/X} \to \mathrm{Ens}
\]

\[
  \mathfrak{X}'(X) = \Bigl\{ x \in \Gamma L \Bigm| \begin{array}{l} x^{2} = \delta \\ x|X_{0} = T_{0} \end{array} \Bigr\}
\]

\[
  \mu'_{2} \subset \mu_{2,X} : \mathrm{Sch}_{/X} \to \mathrm{Groupes}, \quad\text{comme}
\]

\[
  \mu'_{2}(X) = \bigl\{ \lambda \in \Gamma \mathcal{O}_{X}^{*} \bigm| \lambda^{2} = 1,\ \lambda_{0} = 1 \bigr\}
\]

\[
  X' \xrightarrow{\ f\ } \mathfrak{X}', \qquad (\mathbb{Z}/2\mathbb{Z})_{X} \longrightarrow \mu'_{2X}
\]

\[
  f(\bar{x}) = -f(x) \qquad \text{\struck{\ill{}}}
\]

\[
  z \longmapsto \bar{z} \ : \ \mathfrak{X}' \to \mathfrak{X}'
\]

\[
  \left\{\begin{array}{l}
    L,\ \delta \in \Gamma L^{\otimes 2},\ T_{0} \in \Gamma(L_{0}) \quad (\text{où } L_{0} = L|X_{0})\\
    \text{tels que, satisfaisant } T_{0}^{2} \equiv \delta \ (4)
  \end{array}\right.
\]

\[
  \Gamma(X'/X) \hookrightarrow \Gamma(\mathfrak{X}'/X) \qquad (\text{injectif})
\]

\[
  \text{\struck{\ill{}}} \qquad T + z \equiv 0 \quad (2)
\]

\[
  x = \frac{T + z}{2},
\]

\[
  \Gamma E = \Bigl\{ (\lambda, \xi) \Bigm| \lambda \in \Gamma \mathcal{O},\ \xi \in \Gamma\Bigl(\frac{\lambda\tau}{2} + L\Bigr) \Bigr\}
  \qquad
  \left\{\begin{array}{l} \lambda = \psi(x) \\ \xi = x - \lambda\frac{T}{2} \end{array}\right.
\]

\[
  \boxed{\Gamma E \simeq \bigl\{ (\lambda, \zeta) \bigm| \lambda \in \Gamma \mathcal{O},\ \zeta \in \Gamma(\lambda\tau + 2L) \subset \Gamma(L) \bigr\}}\ {}^{\ast}
\]

\[
  \boxed{\left\{\begin{array}{l} \lambda = \psi(x) \\ \zeta = 2x - \psi(x)T \end{array}\right.}
\]

\[
  0 \to L \longrightarrow E \xrightarrow{\ \psi\ } \mathcal{O} \to 0
  \qquad
  \left\{\begin{array}{l} \psi(\lambda, \zeta) = \lambda \\ i(\xi) = (0, 2\xi) \end{array}\right.
\]

\[
  N = (T^{2} - \delta)/4,
\]

\[
  Q(x) = x^{2} - Tx + N
\]

\[
  4Q(x) = Q(2x)
\]

\[
  2x = T + z
\]

\[
  Q(T + z) = T^{2} + 2zT + z^{2} - 2T(T + z) + 4N
  \qquad\text{i.e.}
\]

\[
  \boxed{Q(T + z) = \text{\struck{\ill{}}}\ z^{2} - \delta}
\]

\[
  A \longrightarrow (\mathcal{O} \times \check{L}) \qquad x \longmapsto (\mathrm{Tr}(x), \varphi(x))
\]

\[
  A \subset \mathcal{O} \times \check{L}
\]

\[
  \Gamma A \simeq \bigl\{ (b, \gamma) \in \Gamma \mathcal{O} \times \Gamma \check{L} \bigm| \boxed{b_{0} = \gamma_{0} T_{0}} \bigr\}
\]

\[
  b \in \Gamma \mathcal{O}_{X}, \qquad \text{relation}\quad
  \underset{\Gamma \check{L}_{0}}{\gamma_{0}} \,.\, \underset{\Gamma L_{0}}{T_{0}} \in \Gamma \mathcal{O}_{X_{0}}
\]

\[
  \text{i.e.\ } \xi \mapsto 2\xi \qquad (b, \gamma) \longmapsto \gamma
\]

\[
  0 \to \mathcal{O}_{X} \longrightarrow A \longrightarrow \check{L} \to 0 \quad (\text{surj})
\]

\[
  \left\{\begin{array}{l}
    \text{\struck{N.B.}}\quad T(b, \gamma) = b\\
    \delta(b, \gamma) = \text{\struck{\ill{}}}\ \gamma^{2}.\delta\\[2pt]
    \text{donc}\quad N(b, \gamma) = (T^{2} - \delta)(b, \gamma)/4 = (b^{2} - \gamma^{2}\delta)/4
  \end{array}\right.
\]

\[
  b \equiv \gamma T \ (2) \qquad\text{donc}\qquad b^{2} \equiv \gamma^{2}T^{2} \ (4)
\]

\[
  b^{2} - \gamma^{2}\delta \equiv \gamma^{2}(\underbrace{T^{2} - \delta}_{\equiv 0\ (4)}) \equiv 0 \quad (4)
\]

\[
  x^{2} - bx + \frac{b^{2} - \gamma^{2}\delta}{4} = 0 \qquad\text{i.e.}
\]

\[
  x^{2} = bx - \frac{b^{2} - \gamma^{2}\delta}{4}
  = \Bigl( b^{2} - \frac{b^{2} - \gamma^{2}\delta}{2},\ b\gamma \Bigr)
  = \Bigl( \frac{b^{2} + \gamma^{2}\delta}{2},\ b\gamma \Bigr)
\]

\[
  x = (b, \gamma), \qquad x' = (b', \gamma')
\]

\[
  xx' = \tfrac{1}{2}\bigl( (x + x')^{2} - x^{2} - x'^{2} \bigr)
\]

\begin{align*}
  2xx' &= \Bigl( \frac{(b + b')^{2} + (\gamma + \gamma')^{2}\delta}{2}
          - \frac{b^{2} + \gamma^{2}\delta}{2} - \frac{b'^{2} + \gamma'^{2}\delta}{2},\\
       &\qquad (b + b')(\gamma + \gamma') - b\gamma - b'\gamma' \Bigr)\\
       &= (bb' + \gamma\gamma'\delta,\ b\gamma' + b'\gamma)
\end{align*}

\[
  \boxed{xx' = (b, \gamma)(b', \gamma') = \bigl( \tfrac{1}{2}(bb' + \gamma\gamma'\delta),\ \tfrac{1}{2}(b\gamma' + \gamma b') \bigr)}
\]

\[
  A \times E \longrightarrow \mathcal{O} \qquad (x, f) \longmapsto \langle x, f\rangle = f(x)
\]

\[
  \boxed{\langle (b, \gamma), (\lambda, \zeta) \rangle = \tfrac{1}{2}(\lambda b + \gamma.\zeta)}
\]

\[
  \lambda_{0}\underset{\gamma_{0}T_{0}}{b_{0}} + \gamma_{0}.\underset{\lambda_{0}T_{0}}{\zeta_{0}}
  = \underbrace{\lambda_{0}\gamma_{0}T_{0}}_{} + \underbrace{\gamma_{0}\lambda_{0}T_{0}}_{}
  = 2\gamma_{0}\lambda_{0}T_{0} = 0
\]

\[
  \varphi_{N}\bigl((\underset{\Gamma\mathcal{O}}{b}, \underset{\Gamma\check{L}}{\gamma}), (\underset{\Gamma\mathcal{O}}{b'}, \underset{\Gamma\check{L}}{\gamma'})\bigr)
  = \tfrac{1}{2}(bb' - \gamma\gamma'\delta)
  = \bigl\langle (b, \gamma), (b', \underset{\in \Gamma L}{-\gamma'\delta}) \bigr\rangle,
  \qquad\text{donc}
\]

\[
  \boxed{\varphi_{N}(\underset{\Gamma\mathcal{O}}{b}, \underset{\Gamma\check{L}}{\gamma}) = (\underset{\Gamma\mathcal{O}}{b}, \underset{\Gamma L}{-\gamma\delta})}
\]

\[
  Q(\underset{\lambda}{b}, \underset{\zeta}{-\gamma\delta}) = - (b^{2} - \gamma^{2}\delta)/4\ \delta
  = - (\lambda^{2} + \gamma\zeta)/4\ \delta
\]

\[
  \text{\struck{\ill{}}}\ \text{donc} \qquad \gamma = -\zeta\delta^{-1}
\]

\[
  Q(\lambda, \zeta) = - (\lambda^{2} - \zeta^{2}\delta^{-1})/4\ \delta
  = \text{\struck{\ill{}}}\ (\zeta^{2} - \lambda^{2}\delta)/4
\]

\[
  \boxed{Q(\lambda, \zeta) = (\zeta^{2} - \lambda^{2}\delta)/4}
\]

\[
  \zeta^{2} - \lambda^{2}\delta \equiv 0 \quad (4)
\]

\[
  \zeta^{2} - \lambda^{2}\delta \equiv \lambda^{2}(\underbrace{T^{2} - \delta}_{4N}) = 4\lambda^{2}N \equiv 0 \quad (4) \qquad \text{OK}
\]

\[
  Q(y) = 0 \quad\text{équivaut bien à}\quad (\zeta^{2} - \delta)/4 = 0 \quad\text{i.e.}\quad \zeta^{2} = \delta
\]

\[
  Q'(\lambda, \zeta) = (\zeta^{2} - \lambda^{2}\delta)/4
\]

\[
  y = T, \qquad\text{i.e.}\qquad y = (2, 0)
\]

\[
  \text{\struck{$Q(T) = T^{2} - 2$}}
\]

\[
  Q(y) = y^{2} - Ty\,\psi(y) + N\psi(y)^{2} = T^{2} - 2T^{2} + 4N = 4N - T^{2} = -\delta
\]

\[
  Q'(y) = (0 - 4\delta)/4 = -\delta \qquad \text{OK.}
\]

\[
  (\mathbb{Z}/2\mathbb{Z})_{X} \longrightarrow \mu_{2X}
\]

\[
  \mathrm{Ass}'(L, \delta, T_{0}) = (L, \delta)
\]

\[
  \mathrm{Oub}'(L, \theta) = (L, 4\theta, 0)
\]

\[
  \mathrm{Tors}(\mu_{2X}) \longrightarrow \mathrm{Rev\,quad}(X)
\]

\[
  (L, \delta, T_{0}) \wedge (L', \delta', T'_{0}) = (L \otimes L', \delta\delta', T_{0}T'_{0})
\]

\[
  (L, \theta) \wedge (L', \theta') = (L \otimes L', \theta\theta')
\]

\[
  (n, \mathfrak{X}) \qquad\text{où}\qquad
  \begin{array}{l}
    n \in \mathbb{Z}/2\mathbb{Z} \text{ est une « parité »}\\
    \mathfrak{X}/X \text{ rev.\ quadratique}\\
    \text{avec } \mathfrak{X} \in \underline{\mathrm{Rev\,quad}}^{!} \text{ si } \underline{n = 1}
  \end{array}
\]

\[
  (n, \mathfrak{X}) \wedge (n', \mathfrak{X}') =
  \left\{\begin{array}{ll}
    (n + n', \mathfrak{X} \wedge \mathfrak{X}') & \text{si $n$ ou $n'$ pair}\\
    (n + n', \mathfrak{X} \overset{!}{\wedge} \mathfrak{X}') & \text{si $n$ \emph{et} $n'$ impairs}
  \end{array}\right.
\]

\[
  (L, \delta, 0) \overset{!}{\wedge} (L', \delta', 0) = (L \otimes L', \tfrac{1}{4}\delta\delta', 0)
\]

\[
  \begin{array}{ll}
    \delta = 4\Delta & \Delta \text{ défini mod } h \text{ avec } 4h = 0\\
    \delta' = 4\Delta' & \Delta' \text{ — } h' \text{ — } 4h' = 0
  \end{array}
\]

\[
  \delta\delta' = 16\Delta\Delta' \qquad \tfrac{1}{4}\delta\delta' \overset{\mathrm{def}}{=} 4\Delta\Delta'
\]

\[
  L \rightrightarrows L^{\otimes 2} \qquad
  \left\{\begin{array}{l} u \longmapsto u^{2} \\ u \longmapsto Tu \end{array}\right.
  \qquad T \in \Gamma L \ \text{la section trace}
\]

\[
  L \xrightarrow{\ f\ } L^{\otimes 2} : \qquad f(u) = u^{2} + Tu
\]

\[
  \check{V}(L) \longrightarrow \check{V}(L^{\otimes 2})
\]

\[
  \mathcal{O}_{X} \to \mathcal{O}_{X} : \quad u \mapsto u^{2} + u
\]

\[
  x'' - x' = x'' + x' = T \qquad\text{i.e.}\qquad x'' = x' + T
\]

\[
  x''^{(2)} = x'^{(2)} + T^{2} \qquad\text{sections de}\quad P_{1} \wedge_{L} L^{2} \simeq P_{1}^{(2)}
\]

\[
  P_{1}^{(2)} = \psi^{(2)-1}(\{1\})
\]

\[
  u \in \Gamma(P_{1}^{(2)} \wedge T.P_{1}) \iff u : P_{1}^{(2)} \simeq T P_{1} \quad \text{iso de $L^{\otimes 2}$-torseurs}
\]

\[
  \iff P_{1} \xrightarrow{\ u''\ } P_{1}^{(2)} \quad \text{compatible avec}\quad L \to L^{\otimes 2},\ u \mapsto T.u
\]

\[
  Q(x) = 0 \qquad x^{2} + Tx + N = 0
\]

\[
  Q(x + u) = 0 \quad (u \in \Gamma L) \qquad x^{2} + u^{2} + Tx + Tu + N = 0, \quad\text{i.e.}\quad u^{2} + Tu = 0
\]

\[
  (L, \overset{P_{1}}{\text{\struck{$\ast$}}}, T, \alpha), \quad\text{où}\quad
  \left\{\begin{array}{l}
    L \text{ module inv.\ sur } X, \ P_{1} \ L\text{-tors.}\\
    \quad \text{(i.e.\ $P_{1}$ un fibré en droites affine sur $X$)}\\
    T \text{ section de } L \text{ (module des translations de $P_{1}$)}\\
    \alpha \text{ section de } P_{1} \wedge_{L} (L^{\otimes 2}, f_{T} : L \to L^{\otimes 2})
  \end{array}\right.
\]

\[
  f_{T} : L \to L^{\otimes 2} \quad\text{défini par}\quad f_{T}(u) = u^{2} + T.u
\]

\[
  X'_{\Sigma} = f_{T, P_{1}}^{-1}(V(\alpha))
\]

\[
  (L, T, P_{1}, \alpha) \wedge (L', T', P'_{1}, \alpha') \overset{?}{=} (L \otimes L', TT', ?, ?)
\]

\[
  \begin{array}{lll}
    G_{T} \xrightarrow{\ d_{T'}\ } G_{TT'} & \text{induit par} & d_{T'} : L \to L \otimes L', \ u \mapsto uT'\\
    G_{T'} \longrightarrow G_{TT'} & \text{—} & d_{T} : L' \to L \otimes L', \ u' \mapsto Tu'
  \end{array}
\]

\[
  G_{T} \times G_{T'} \longrightarrow G_{TT'} \quad\text{hom.\ de groupes}
\]

\[
    N(e) = 1
  \]

\[
    P \longrightarrow L^{\otimes 2}
  \]

\[
    Q(x + u) = Q(x) + u^{2} + \mathrm{Lin}_{u}(x)
  \]

\[
    \mathrm{Lin}_{u} : P \to L_{S}^{\otimes 2}
  \]

\[
  0 \subset \underbrace{L^{\otimes 2}}_{L^{\otimes 2}} \subset \underbrace{L \otimes E}_{L} \subset \underbrace{\mathrm{Sym}^{2}(E)}_{\mathcal{O}_{X}}
\]

\[
  Q(x) - x^{2} \ : \ P \to \mathrm{Sym}^{2}(E)
\]

\[
  L \to L^{\otimes 2}
\]

\[
  \left\{\begin{array}{l} x = e + u, \\ u = x - e \end{array}\right.
\]

\[
  Q(e + u) = u^{2} + bu + c
  \qquad
  \begin{array}{l} u \in \Gamma L\\ b \in \Gamma L\\ c \in \Gamma L^{\otimes 2} \end{array}
\]

\begin{align*}
  Q(\underbrace{e + u}_{x}) - (\underbrace{e + u}_{x})^{2}
  &= bu + c - 2eu - e^{2} = \underset{x - e}{u}(b - 2e) + c - e^{2}\\
  &= x(b - 2e) \ \text{\struck{\ill{}}} + (e^{2} + c - be)
\end{align*}

\[
  \left\{\begin{array}{l}
    Q(x) = x^{2} - xT + N\\[2pt]
    \text{où}\quad \boxed{T = 2e - b \in E, \quad N = e^{2} + c - be \in \mathrm{Sym}^{2}(E)}\\[2pt]
    \text{avec}\quad \boxed{\psi(T) = 2, \ \varepsilon(N) = 1}
  \end{array}\right.
\]

\[
  0 \to L^{\otimes 2} \to L^{\otimes 2} \otimes \check{E} \to L \to 0
\]

\[
  L^{\otimes 2} \otimes \check{E} \simeq L \otimes E
\]

\[
  \text{i.e.}\qquad E \simeq \check{E} \otimes L \ (\simeq \mathrm{Hom}(E, L)), \qquad L = \det E
\]

\[
  P_{1} \longrightarrow L^{\otimes 2}
\]

\[
  \xi \in L.E \simeq L \otimes E \subset \mathrm{Sym}^{2}E
\]

\[
  x \longmapsto x \wedge \xi
  \qquad\qquad
  \begin{array}{l}
    E \otimes E \to L, \quad (x \otimes y) \mapsto x \wedge y\\
    L \otimes E \otimes E \to L \otimes L = L^{\otimes 2}\\
    (\xi, x) \mapsto \xi \wedge x = -x \wedge \xi
  \end{array}
\]

\[
  T \in E, \qquad N \in \mathrm{Sym}^{2}(E),
\]

\[
  \varphi_{1}(T) = 2, \qquad \varphi_{2}(N) = 1
\]

\[
  N(x + \lambda e) = N(x) + \lambda T(x) + \lambda^{2}
  \qquad\text{(on aura $N(x + \lambda e) = N(x) + \lambda T(x) + \lambda^{2}$)}
\]

\[
  T(x) = \varphi_{N}(x, e)
\]

\[
  Q_{T,N}(x) \overset{\mathrm{def}}{=} x^2 - xT + N \qquad (x \in \Gamma P_1)
\]

\[
  Q_{T,N}(x) = x^2 - \psi(x)\,xT + \psi(x)^2 N \qquad (x \in \Gamma E)
\]

\[
  4N - T^2 \in \Gamma L^{\otimes 2} \subset \Gamma\,\mathrm{Sym}^2 E
\]

\[
  \mathrm{Sym}^2 E \simeq L^2 \oplus L \oplus \mathcal{O},
  \qquad
  L = x_0 \cdot L, \quad \mathcal{O} = x_0^2\,\mathcal{O}_X,
\]

\[
  x_0 = (0,1), \qquad T = (t,2) = 2x_0 + t, \quad t \in \Gamma L,
\]

\[
  N = c - x_0 b + x_0^2 \qquad (\text{car } \psi_2(N) = 1),
  \qquad
  \left\{\begin{array}{l} t \in \Gamma L \\ b \in \Gamma L \\ c \in \Gamma L^{\otimes 2} \end{array}\right.
\]

\[
  Q_{T,N}(x_0+u) = Q_{T,N}(x) = (x_0+u)^2 - (x_0+u)(2x_0+t) + c - bx_0 + x_0^2
\]

\[
  \text{\struck{$= \ldots u^2 - (\ldots)$}}
\]

\[
  = u^2 - ut + (-bx_0 + c - x_0 t),
  \qquad u^2,\ ut \in L^{\otimes 2},
\]

\[
  \text{\struck{$x_0^2 - x_0(x_0$}}\quad
  \underbrace{x_0(-b - t)}_{\in\, \Gamma(E \otimes L)} + c \in \Gamma L^{\otimes 2},
  \qquad c \in \Gamma L^{\otimes 2} \subset \Gamma(E \otimes L).
\]

\[
  \boxed{N = x_0^2 - bx_0 + c
  \quad (b \in \Gamma L,\ c \in \Gamma L^{\otimes 2})
  \Longrightarrow T = 2x_0 - b}
\]

\[
  \boxed{Q(x_0+u) = u^2 + bu + c},
  \qquad
  \text{d'où } \delta = T^2 - 4N = b^2 - 4c .
\]

\[
  \mathbb{L}(X_1 \wedge X_2) \simeq \mathbb{L}(X_1) \otimes \mathbb{L}(X_2)
\]

\[
  \delta_{X_1 \wedge X_2} = \delta_{X_1} \cdot \delta_{X_2}
  \qquad \in \Gamma\,\mathbb{L}(X_1 \wedge X_2).
\]

\[
  (x_1, x_2) \longmapsto x_1 \wedge x_2 : P_{X_1} \times P_{X_2} \to P_{X_1 \wedge X_2}
\]

\[
  \begin{array}{lll}
  b_1 \in \Gamma L_1 & b_2 \in \Gamma L_1^{\otimes 2} & b_1 = 2x_1 - T_1 \\
  \uncertain{c_1} \in \Gamma L_2 & c_2 \in \Gamma L_2^{\otimes 2} & b_2 = 2x_2 - T_2
  \end{array}
\]

\[
  \boxed{b \overset{\mathrm{def}}{=} b_{X_1 \wedge X_2} = b_1 b_2}
\]

\[
  4c = b^2 - \delta = (b_1 b_2)^2 - \delta_1 \delta_2 = 4(c_1\delta_2 + c_2\delta_1 + 4c_1c_2)
\]

\[
  (b_1b_2)^2 = b_1^2 b_2^2 = (\delta_1 + 4c_1)(\delta_2 + 4c_2)
\]

\[
  \boxed{c = c_1\delta_2 + c_2\delta_1 + 4c_1c_1}
  = b_2^2 c_1 + b_1^2 c_2 - 4c_1c_2,
  \qquad \delta_2 = b_2^2 - 4c_2,\ \delta_1 = b_1^2 - 4c_1.
\]

\[
  (x_1+u_1) \wedge (x_2+u_2) \overset{?}{=} x_1 \wedge x_2 + \underbrace{f_{x_1,x_2}(u_1,u_2)}_{?},
  \qquad u_1 \in \Gamma L_1,\ u_2 \in \Gamma L_2,
\]

\[
  \begin{array}{lll}
  f_{x_2} : L_1 \to L_1 \otimes L_2 & & u_1 \mapsto u_1 b_2 \\
  f_{x_1} : L_2 \to L_1 \otimes L_2 & & u_2 \mapsto u_2 b_1 \\
  f_{x_1,x_2} : L_1 \times L_2 \to L_1 \otimes L_2 & & (u_1,u_2) \mapsto u_1 \otimes u_2
  \end{array}
\]

\[
  \boxed{(x_1+u_1) \wedge (x_2+u_2) = x_1 \wedge x_2
  + \underbrace{(u_1b_2 + u_2b_1 + 2u_1u_2)}_{\in\, \Gamma(L_1 \otimes L_2)}}
\]

\[
  P_1 \times P_2 \to P \quad \text{i.e.} \quad L_1 \times L_2 \to L_1 \otimes L_2
\]

\[
  \left.\begin{array}{l}
  \overbrace{u_1^2 + b_1u_1 + c_1}^{Q_1(u_1)} = 0 \\
  \underbrace{u_2^2 + b_2u_2 + c_2}_{Q_2(u_2)} = 0
  \end{array}\right\}
  \Longrightarrow
  \underbrace{u^2 + bu + c}_{Q(u)} = 0
\]

\[
  \left\{\begin{array}{l}
  u = u_1b_2 + u_2b_1 + 2u_1u_2 \\
  b = b_1b_2 \\
  c = b_2^2 c_1 + b_1^2 c_2 - 4c_1c_2
  \end{array}\right.
\]

\[
  [u_1b_2 + u_2b_1 + 2u_1u_2]^2 + b_1b_2(u_1b_2 + u_2b_1 + 2u_1u_2)
  + b_2^2c_1 + b_1^2c_2 - 4c_1c_2 \overset{?}{=} 0 .
\]

\[
  b_2^2(u_1^2 + b_1u_1 + c_1) + b_1^2(u_2^2 + b_2u_2 + c_2) + 4u_1^2u_2^2
  + 4b_1b_2u_1u_2
\]

\[
  + 4u_1^2b_2u_2 + 4u_2^2b_1u_1 - 4c_1c_2
  \quad \text{\struck{$+ 4u_1u_2b_1b_2$}}
\]

\[
  = b_2^2 Q_1(u_1) + b_1^2 Q_2(u_2) + 4(Q(u_1) - c_1)(Q(u_2) - c_2)
\]

\[
  \text{\struck{$- 4b_1b_2u_1u_2$}} \quad - 4c_1c_2
\]

\[
  = b_2^2 Q_1(u_1) + b_1^2 Q_2(u_2)
  + 4[Q(u_1)Q(u_2) - c_1Q(u_2) - c_2Q(u_1)]
  \quad \text{\struck{$-4b_1b_2u_1u_2$}}
\]

\[
  \boxed{\begin{array}{l}
  Q(u_1 * u_2) = b_2^2 Q_1(u_1) + b_1^2 Q_2(u_2) \\
  \qquad + 4[Q_1(u_1)Q_2(u_2) - c_1Q_2(u_2) - c_2Q_1(u_1)]
  \end{array}}
\]

\[
  \underbrace{Q(x_1 * x_2)}_{= c} = b_1^2 c_2 + b_2^2 c_1
  + 4\underbrace{(c_1c_2 - c_1c_2 - c_1c_2)}_{-4c_1c_2}
\]

\[
  \boxed{\underbrace{Q(x_1 * x_2)}_{c} = \underbrace{Q_1(x_1)}_{c_1}\delta_2
  + \underbrace{Q_2(x_2)}_{c_2}\delta_1 + 4Q_1(x_1)Q_2(x_2)}
\]

\[
  c_1\delta_2 + c_2\delta_1 + 4c_1c_2
\]

\[
  \chi \in \Gamma\mathcal{O}.
\]

\[
  (M, E, i, \psi, T)
\]

\[
  (*) \qquad 0 \to M \xrightarrow{i} E \xrightarrow{\psi} \mathcal{O} \to 0
\]

\[
  \psi(T) = \chi
\]

\[
  (L_1, P_1, T_1) \wedge (L_2, P_2, T_2) = (L_1 \otimes L_2,\ P_1 \wedge P_2,\ T),
  \qquad T \overset{?}{=} T_1 \wedge T_2,
\]

\[
  \mathcal{O} \xrightarrow{\ \psi' \ (= \psi'_T)\ } E
  \qquad (\text{nécess.\ de la forme } \lambda \mapsto \lambda T, \text{ où } T \in \Gamma E,\ T = \psi'(1))
\]

\[
  \psi\psi'(\lambda) = \chi\lambda
\]

\[
  E \xrightarrow{\ \pi\ (= \pi_T)\ } M
\]

\[
  \pi \circ i = \chi\, \mathrm{id}_M .
\]

\[
  i(\pi(x)) = \underset{\psi(T)}{\chi}\, x - T\psi(x) \quad \text{i.e.} \quad \pi = \chi\,\mathrm{id} - \psi \otimes T
\]

\[
  \bigl(= \psi(T)x - T\psi(x) \overset{\mathrm{def}}{=} x \underset{\psi}{\wedge} T\bigr)
\]

\[
  \psi(\chi x - T\psi(x)) = \chi\psi(x) - \psi(T)\psi(x) = 0 \quad \text{car } \psi(T) = \chi,
\]

\[
  i\,\pi(x) = \chi x \qquad \text{\struck{$i\,\pi(x) = \chi$}}
\]

\[
  x \longmapsto \chi x - \pi x : E \to \text{\struck{$M$}}\ E ;
\]

\[
  E/M \simeq \mathcal{O} \xrightarrow{\psi'} E, \qquad \lambda \mapsto \lambda T,
\]

\[
  \chi x - \pi x = \psi(x) T \quad \text{i.e.} \quad \boxed{\pi(x) = \chi x - \psi(x) T}
\]

\[
  \chi\psi(x) - \psi(x)\psi(T) = 0 \quad \text{i.e.} \quad \psi(x)(\chi - \psi(T)) = 0
\]

\[
  \begin{array}{lll}
  0 \to L_1 \xrightarrow{i_1} E_1 \xrightarrow{\psi_1} \mathcal{O} \to 0 & \quad T_1 \in E_1 & \psi_1(T_1) = \chi \\
  0 \to L_2 \xrightarrow{i_2} E_2 \xrightarrow{\psi_2} \mathcal{O} \to 0 & \quad T_2 \in E_2 & \psi_2(T_2) = \chi
  \end{array}
\]

\[
  0 \to L_1 \otimes L_2 \xrightarrow{i} E \xrightarrow{\psi} \mathcal{O} \to 0,
  \qquad T \in E,\ \psi(T) = \chi,
\]

\[
  E \overset{\mathrm{def}}{=} (E_1, T_1) \underset{\chi}{\wedge} (E_2, T_2).
\]

\[
  0 \subset \underbrace{L_1 \otimes L_2}_{L_1 \otimes L_2}
  \subset \underbrace{L_1 \otimes E_2 + E_1 \otimes L_2}_{L_1 \oplus L_2}
  \subset \underbrace{E_1 \otimes E_2}_{\mathcal{O}},
\]

\[
  0 \to E_1 *_0 E_2 \to E_1 \otimes E_2 \to \mathcal{O} \to 0
\]

\[
  E_1 *_0 E_2 \overset{\mathrm{def}}{=} L_1 \otimes E_2 + E_1 \otimes L_2 \supset L_1 \otimes L_2
\]

\[
  0 \to L_1 \otimes L_2 \to E_1 * E_2 \to L_1 \oplus L_2 \to 0 .
\]

\[
  E_1 * E_2 \xrightarrow{\ \mu = \mu_{E_1,E_2}\ } L_1 \otimes L_2
\]

\[
  \alpha = \mathrm{id}_{L_1} \otimes \pi_{T_2}, \qquad \beta = \pi_{T_1} \otimes \mathrm{id}_{L_2},
\]

\[
  \alpha(u_1 \otimes x_2) = u_1 \otimes \pi_{T_2}(x_2) = u_1 \otimes (\chi x_2 - T_2\psi(x_2)),
  \qquad u_1 \in \Gamma L_1,\ x_2 \in \Gamma E_2,
\]

\[
  \text{\struck{$\beta$}}\ \alpha_2(x_1 \otimes u_2) = \pi_{T_1}(x_1) \otimes u_2
  = (\chi x_1 - T_1\psi(x_1)) \otimes u_2,
\]

\[
  x_1 \in \Gamma E_1,\ u_2 \in \Gamma L_2 ;
\]

\[
  \alpha_1(u_1 \otimes u_2) = \chi\, u_1 \otimes u_2, \qquad
  \alpha_2(u_1 \otimes u_2) = \chi\, u_1 \otimes u_2 \qquad \text{OK !}
\]

\[
  E_1 * E_2 \to L_1 \otimes L_2
\]

\[
  E_1 * E_2 = E_1 \otimes E_2 \underset{E_1 *_0 E_2}{\amalg} (L_1 \otimes L_2),
\]

\[
  \rho : E_1 \otimes E_2 \to L_1 \otimes L_2, \qquad \rho = \pi_{E_1} \otimes \pi_{E_2},
\]

\[
  \rho(x_1 \otimes x_2) = (\underbrace{\chi x_1 - \psi_1(x_1)T_1}_{\pi_{T_1}(x_1)})
  \otimes (\underbrace{\chi(x_2) - \psi_2(x_2)T_2}_{\pi_{T_2}(x_2)}).
\]

\[
  \text{\struck{$\rho(u_1 \otimes x_2 + x_1 \otimes u_2) = \chi u_1 \otimes \chi(x_2)$}}
\]

\[
  \rho(u_1 \otimes x_2) = \chi u_1 \otimes (\chi(x_2) - \psi_2(x_2)T_2)
\]

\[
  \mu_{E_1,E_2}(u_1 \otimes x_2) \overset{\mathrm{def}}{=} u_1 \otimes (\chi x_2 - \psi_2(x_2)T_2)
\]

\[
  \chi T = \mu'(T_1 \otimes T_2) \overset{\mathrm{def}}{=} T_1 * T_2
\]

\[
  T = \frac{1}{\chi}\, T_1 * T_2 \qquad (!)
\]

\[
  \pi_T : E \to L_1 \otimes L_2, \qquad x \mapsto \chi x - T\psi(x),
  \qquad \text{d'où } T\psi(x) = \chi x - \pi_T(x).
\]

\[
  \chi^2 T = \chi T_1 * T_2 - \underbrace{\pi_T(T_1 * T_2)}_{\pi_{T_1}(T_1) \otimes \pi_{T_2}(T_2) = 0}
\]

\[
  \text{\struck{$T = \chi\, x_1 * x_2 - \pi_T(x_1 * x_2)$,}}
  \qquad
  \text{\struck{$\pi_T(x_1 * x_2) = (\chi x_1 - T_1)(\chi x_2 - T_2)$}}
\]

\[
  \text{\struck{$T = \chi\bigl(\tfrac{1}{\chi}T_1\bigr) * \bigl(\tfrac{1}{\chi}T_2\bigr) - \ldots$}}
  \qquad
  \text{\struck{$= \tfrac{1}{\chi}\, T_1 * T_2$}}
\]

\[
  \chi^2 T = \chi\, T_1 * T_2
\]

\[
  \boxed{\chi T = T_1 * T_2}
\]

\[
  E_1 \times E_2 \to E_1 \otimes E_2 \to E_1 * E_2,
  \qquad (x_1, x_2) \mapsto x_1 \otimes x_2,\ x_1 \otimes x_2 \mapsto x_1 * x_2
\]

\[
  P_1 \times P_2 \to P = \psi^{-1}(1).
\]

\[
  (x_1 + u_1) * (x_2 + u_2) = \underbrace{x_1 * x_2}_{\in\, \Gamma P}
  + \underbrace{u_1 * x_2 + x_1 * u_2 + u_1 * u_2}_{\in\, \Gamma(L_1 \otimes L_2)},
  \qquad u_1 * u_2 = \chi\, u_1 \otimes u_2,
\]

\[
  = x_1 * x_2 + \bigl[u_1 \otimes (\overbrace{\chi x_2 - \psi(x_2)T_2}^{-b_2})
  + (\overbrace{\chi x_1 - \psi(x_1)T_1}^{-b_1}) \otimes u_2
  + \chi\, u_1 \otimes u_2\bigr]
\]

\[
  \boxed{\begin{array}{l}
  (x_1 + u_1) * (x_2 + u_2) = x_1 * x_2 + (-u_1 \otimes b_2 - b_1 \otimes u_2 + \chi\, u_1 \otimes u_2) \\
  \qquad = x_1 * x_2 + (-u_1 \otimes b_2 - b_1 \otimes u_2 + \chi\, u_1 \otimes u_2)
  \end{array}}
\]

\[
  T_1 = \chi x_1 + b_1, \qquad T_2 = \chi x_2 + b_2 .
\]

\[
  T = \text{\struck{$\ldots$}}\ \chi(x_1 * x_2) - b_1 \otimes b_2
\]

\[
  \left\{\begin{array}{lll}
  b_{T_1,x_1} = -b_1 & \text{i.e.} & b_{T_1,x_1} = \chi x_1 - T_1 = \pi_{T_1}(x_1) \\
  b_{T_2,x_2} = -b_2 & \text{i.e.} & b_{T_2,x_2} = \chi x_2 - T_2 = \pi_{T_2}(x_2) \\
  b_{T,x_1 * x_2} = -b & \text{i.e.} & b = \chi x_1 * x_2 - T = \pi_T(x) \quad (x = x_1 \otimes x_2)
  \end{array}\right.
\]

\[
  \text{\struck{$b$}}\ b = -b_{T,x_1 * x_2} = -b_1 \otimes b_2 = -b_{T_1,x_1} \otimes b_{T_2,x_2}
\]

\[
  \boxed{b_{T,x_1 * x_2} = b_{T_1,x_1} \otimes b_{T_2,x_2}}
\]

\[
  T = \chi \cdot (x_1 * x_2) - \underbrace{b_{T,x_1 * x_2}}_{b_{T_1,x_1} \otimes b_{T_2,x_2}}
\]

\[
  (x_1, x_2) \mapsto x_1 * x_2 : E_1 \times E_2 \to E_1 * E_2,
\]

\[
  \pi_i : P_i \to L_i
\]

\[
  \pi_i(x_i + u_i) = \pi_i(x_i) + \chi u_i, \qquad x_i \in \Gamma P_i,\ u_i \in \Gamma L_i .
\]

\[
  E_i \to L_i
\]

\[
  \pi_i(u_i) = \chi \cdot u_i \qquad \text{si } u_i \in L_i \subset E_i .
\]

\[
  E_1 \xrightarrow{\psi_1} \mathcal{O}, \qquad
  E_2 \xrightarrow{\psi_2} \mathcal{O}, \qquad
  E = E_1 * E_2 \xrightarrow{\psi} \mathcal{O}
\]

\[
  P_1 \times P_2 \to P, \quad (x_1, x_2) \mapsto x_1 * x_2
  \qquad (\text{où } P_i = \psi_i^{-1}(\{1\}),\ P = \psi^{-1}(\{1\}))
\]

\[
  \boxed{\begin{array}{l}
  (x_1 + u_1) * x_2 = x_1 * x_2 + u_1 \otimes \pi_2(x_2) \\
  x_1 * (x_2 + u_2) = x_1 * x_2 + \pi_1(x_1) \otimes u_2
  \end{array}}
  \qquad x_i \in \Gamma E_i,\ u_i \in \Gamma L_i
\]

\[
  \boxed{\begin{array}{l}
  (x_1 + u_1) * (x_2 + u_2) = x_1 * x_2 + u_1 \otimes \pi_2(x_2) + \pi_1(x_1) \otimes u_2 \\
  \qquad + \chi\, u_1 \otimes u_2
  \end{array}}
\]

\[
  \boxed{\begin{array}{l}
  u_1 * x_2 = u_1 \otimes \pi_2(x_2) \\
  x_1 * u_2 = \pi_1(x_1) \otimes u_2
  \end{array}}
\]

\[
  \boxed{u_1 * u_2 = \chi\, u_1 \otimes u_2},
\]

\[
  \boxed{\pi_i(x) = \chi x - T_i}
\]

\[
  \boxed{\pi_E(x_1 * x_2) = \pi_1(x_1) \otimes \pi_2(x_2)}
\]

\[
  \chi x = \chi\, \pi_1(x_1) \otimes u_2,
\]

\[
  \pi_E(x_1 * x_2) = (\underbrace{\chi x_1 - T_1}_{b_1}) \otimes (\underbrace{\chi x_2 - T_2}_{b_2})
  = \chi\, x_1 \otimes x_2 - T
\]

\[
  \boxed{T = \chi(x_1 * x_2) - b_1 \otimes b_2}
\]

\[
  b_i = \pi_E(x_i) = \chi x_i - T_i .
\]

\[
  \boxed{\chi T = T_1 * T_2}, \qquad T_1 = \chi x_1 - b_1,\quad T_2 = \chi x_2 - b_2 .
\]

\[
  \chi^2 x_1 * x_2 - \chi\,\underbrace{\pi_1(x_1)}_{(\chi x_1 - T_1) = b_1} \otimes b_2
  - \chi\, b_1 \otimes \underbrace{\pi_2(x_2)}_{(\chi x_2 - T_2) = b_2}
  + \underbrace{b_1 * b_2}_{\chi\, b_1 \otimes b_2}
\]

\[
  = \chi^2 x_1 * x_2 - \chi\, b_1 \otimes b_2
  = \chi(\overbrace{\chi\, x_1 * x_2 - b_1 \otimes b_2}^{T}),
\]

\[
  P_1 \times P_2 \xrightarrow{f} \Pi
\]

\[
  (*) \qquad \exists\ L_1 \times L_2 \xrightarrow{\alpha} M, \text{ application bilinéaire,}
\]

\[
  \left\{\begin{array}{l}
  f(x_1 + u_1, x_2) = \alpha(x_1, x_2) + \alpha(u_1, \pi_2(x_2)) \\
  f(x_1, x_2 + u_2) = \alpha(x_1, x_2) + \alpha(\pi_1(x_1), u_2)
  \end{array}\right.
\]

\[
  f(x_1 + u_1, x_2 + u_2) = \alpha(x_1, x_2) + \alpha(u_1, \pi_2(x_2)) + \alpha(\pi_1(x_1), u_2)
  + \chi\, \alpha(u_1, u_2)\text{)}
\]

\[
  (P_1 * P_2,\ P_1 \times P_2 \xrightarrow{*} P_1 * P_2)
\]

\[
  \pi_{\Pi}(f(x_1, x_2)) = \pi_1(x_1) \otimes \pi_2(x_2)
\]

\[
  P \xrightarrow{f} P', \qquad M \xrightarrow{f_0} M' .
\]

\[
  \left\{\begin{array}{ll}
  f(x + m) = f(x) + f_0(m) & (x \in \Gamma P,\ m \in \Gamma M) \\
  \pi_{P'}(f(x)) = f_0(\pi_P(x))
  \end{array}\right.
\]

\[
  f(T) = T' .
\]

\[
  P \simeq L, \qquad u \mapsto x_0 + u
\]

\[
  E \simeq L + \mathcal{O}, \qquad (u, \lambda) \mapsto \lambda x_0 + u,
\]

\[
  b\ (= b_{x_0}) = \pi(x_0) = \chi x_0 - T
\]

\[
  T = \chi x_0 - b,
\]

\[
  T = (-b, \chi).
\]

\[
  \pi(\underbrace{\lambda x_0 + u}_{(u, \lambda)}) = \lambda b + \chi u .
\]

\[
  b = b_1 \otimes b_2 .
\]

\[
  E_1 * E_2 \simeq E_2 * E_1
\]

\[
  (E_1 * E_2) * E_3 \simeq E_1 * (E_2 * E_3)
\]

\[
  b_0 = \pi(x_0)
\]

\[
  P_0 \simeq \mathcal{O}_X, \quad x_0 = 1, \qquad
  E_0 = \mathcal{O} \times \mathcal{O} = \{(u, \lambda) \mid u, \lambda \in \Gamma\mathcal{O}\} \xrightarrow{\psi} \lambda
\]

\[
  \left|\begin{array}{l}
  \pi(u, \lambda) = \lambda + \chi u \\
  T = (-1, \chi)
  \end{array}\right.
\]

\[
  \begin{array}{rcl}
  P_0 * P_1 & \simeq & P_1 \\
  \mathcal{O}_0 \otimes L_1 & \simeq & L_1
  \end{array}
  \qquad
  \begin{array}{l}
  x_1 \longmapsto x_0 * x_1 \\
  u_1 \longmapsto b_0 \otimes u_1
  \end{array}
\]

\[
  \chi = 2 .
\]

\[
  0 \to L \to E \xrightarrow{\psi} \mathcal{O} \to 0
\]

\[
  \sigma_E : x \longmapsto \bar{x} = T\psi(x) - x
  \qquad (= T - x \text{ si } x \in \Gamma P_1, \text{ i.e. } \psi(x) = 1).
\]

\[
  \psi(\bar{x}) = \underbrace{\psi(T)}_{2}\psi(x) - \psi(x) = \psi(x),
\]

\[
  \bar{\bar{x}} = x \qquad \text{i.e.} \quad \sigma^2 = \mathrm{id}_E,
\]

\[
  \bar{u} = -u \qquad \text{si } u \in L,
\]

\[
  E \simeq L + \text{\struck{$M$}}\ E_0, \qquad E_0 \subset E,
\]

\[
  E_0 = E^{\sigma} = \mathrm{Ker}(1 - \sigma) = \{x \in E \mid x = \bar{x}\}
  = \mathrm{Im}(1 + \sigma) = \{x + \bar{x} \mid x \in E\}
\]

\[
  E_0 \simeq \mathcal{O}_X \quad \text{\struck{via $\psi$}}
\]

\[
  \overline{(u, \lambda)} = (-u, \lambda)
\]

\[
  T = \text{\struck{$\ldots$}}\ x + \bar{x} \quad \text{pour } \psi(x) = 1,
  \quad \text{d'où} \quad \boxed{T = (0, 2)},
\]

\[
  \boxed{\pi(u, \lambda) = (2u, 0)}
\]

\[
  b_{x_0,T} = 2x_0 - T = 0 ,
\]

\[
  \begin{cases}
    \overline{x_1 * x_2} = \overline{x_1} * x_2 = x_1 * \overline{x_2} \\
    \overline{x_1} * \overline{x_2} = x_1 * x_2
  \end{cases}
\]

\[
  \overline{x_1} * x_2 = \overline{x_1 * x_2} ;
\]

\[
  \overline{x_1} * x_2 = (T_1 - x_1) * x_2 = \underbrace{T_1}_{2x_1 - b_1} * x_2 - x_1 * x_2 = x_1 * x_2 - b_1 * x_2
\]

\[
  \overline{x_1 * x_2} = T - x_1 * x_2 = x_1 * x_2 - b_1 \otimes b_2
\]

\[
  T = 2x_1 * x_2 - b_1 \otimes b_2 ,
  \qquad
  \overline{x_1 * x_2} = x_1 * x_2 - b_1 \otimes b_2
\]

\[
  E_1 \otimes E_2 \to E_1 * E_2 \text{ induit } E_1 \otimes E_2 / \sigma_1 \otimes \sigma_2 \xrightarrow{\ \sim\ } E_1 * E_2 .
\]

\[
  L_1 \otimes E_2 \oplus_{L_1 \otimes L_2} E_1 \otimes L_2 \longrightarrow L_1 \otimes L_2 ,
\]

\[
  \operatorname{Coker} \pi_1 \otimes \operatorname{Coker} \pi_2 \qquad \pi_i : E_i \to L_i
\]

\[
  b_1 \in \Gamma L_1 , \quad b_2 \in \Gamma L_2 ,
\]

\[
  \operatorname{Im}(\pi_i : E_i \to L_i) = \mathcal{O}_X . b_i + 2 L_i
\]

\[
  (b_1, b_2, \underset{\substack{\| \\ 2 \text{ en l'occurrence}}}{\chi}) . \mathcal{O}_X = \mathcal{O}_X
\]

\[
  E_i \simeq L_i \oplus \mathcal{O}_i \qquad \text{où } \sigma_i | \mathcal{O}_i = \mathrm{id}_{\mathcal{O}_i}
\]

\[
  E_1 \otimes E_2 \simeq L_1 \otimes L_2 \oplus L_1 \oplus L_2 \oplus \mathcal{O}
\]

\[
  \sigma_1 \otimes \sigma_2 \text{ est }
  \begin{cases}
    \mathrm{id} \text{ sur } L_1 \otimes L_2 \text{ et sur } \mathcal{O} \\
    -\mathrm{id} \text{ sur } L_1 \oplus L_2
  \end{cases}
\]

\[
  \mathrm{id} - \sigma_1 \otimes \sigma_2 \text{ est donc }
  \begin{cases}
    0 \text{ sur } L_1 \otimes L_2 \text{ et sur } \mathcal{O} \\
    2\,\mathrm{id} \text{ sur } L_1 \oplus L_2
  \end{cases}
\]

\[
  E_1 \otimes E_2 / \sigma_1 \otimes \sigma_2 \simeq (L_1 \otimes L_2) \oplus \mathcal{O} \xrightarrow{\ \sim\ } E_1 * E_2
\]

\[
  E_1 \simeq L_1 \oplus \mathcal{O} x_1 , \quad E_2 \simeq L_2 \oplus \mathcal{O} x_2
\]

\[
  E_1 \otimes E_2 \simeq (L_1 \otimes L_2) \oplus \underbrace{L_1 \otimes x_2}_{\simeq L_1} \oplus \underbrace{x_1 \otimes L_2}_{\simeq L_2} + \underbrace{\mathcal{O} . x_1 \otimes x_2}_{\simeq \mathcal{O}}
\]

\[
  \overline{u_1 \otimes x_2} - u_1 \otimes x_2 = -u_1 \otimes \underset{\substack{\| \\ T_2 - x_2 \\ \| \\ x_2 - b_2}}{\overline{x_2}} - u_1 \otimes x_2 = -2u_1 \otimes x_2 + u_1 \otimes b_2
\]

\[
  \overline{x_1 \otimes u_2} - x_1 \otimes u_2 = -2x_1 \otimes u_2 + b_1 \otimes u_2
\]

\[
  \overline{x_1 \otimes x_2} - x_1 \otimes x_2 = \underset{(x_1 - b_1) \otimes (x_2 - b_2)}{\overline{x_1} \otimes \overline{x_2}} - x_1 \otimes x_2 = -x_1 \otimes b_2 - b_1 \otimes x_2 + b_1 \otimes b_2
\]

\[
  \begin{cases}
    -2u_1 + u_1 \otimes b_2 \\
    -2u_2 + b_1 \otimes u_2 \\
    -b_1 - b_2 + b_1 \otimes b_2
  \end{cases}
  \qquad u_1 \in \Gamma L_1 , \ u_2 \in \Gamma L_2
\]

\[
  \begin{cases}
    u_1 \otimes b_2 , \ b_1 \otimes u_2 \in \Gamma L_1 \otimes L_2 \\
    -b_1 - b_2 - b_1 \otimes b_2
  \end{cases}
\]

\[
  L_1 \otimes L_2 + \mathcal{O} . \xi
\]

\[
  \xi = \overline{x_1 \otimes x_2} - x_1 \otimes x_2 \equiv \text{\struck{$b_2$}}\ b_1 \oplus b_2 \quad (L_1 \otimes L_2)
\]

\[
  \operatorname{rang} L_1 \otimes L_2 + 1 = n + 1
\]

\[
  \operatorname{rang}(\operatorname{Ker} *) =
  \underset{\substack{\| \\ \operatorname{rang} E_1 \cdot \operatorname{rang} E_2 \\ \| \\ 2 . (n+1)}}{\operatorname{rang}(E_1 \otimes E_2)}
  - \underset{\substack{\| \\ 1 + \operatorname{rang}(L_1 \otimes L_2) \\ \| \\ 1 + n}}{\operatorname{rang} E_1 * E_2}
  = n + 1
\]

\[
  E_1 \otimes E_2 / \sigma_1 \otimes \sigma_2 \xrightarrow{\ \sim\ } E_1 * E_2 ,
\]

\[
  2 \mathcal{O}_X + b_1 L_1^{\vee} + b_2 L_2^{\vee} = \mathcal{O}_X \;) .
\]

\[
  \underline{\operatorname{Hom}}_{\mathcal{O}}(E_1 * E_2, M) \xrightarrow{\ \sim\ } \underline{\operatorname{Hom}}(E_1 \otimes E_2, M)^{\sigma_1 \otimes \sigma_2}
\]

\[
  \mathcal{A} \longrightarrow \mathcal{A}_1 \otimes \mathcal{A}_2
\]

\[
  \mathcal{A} \xrightarrow{\ \sim\ } (\mathcal{A}_1 \otimes \mathcal{A}_2)^{\sigma_1^{\vee} \otimes \sigma_2^{\vee}} ,
\]

\[
  \pi(x) = -T \overset{\text{déf}}{=} b
\]

\[
  E_1 \otimes E_2 \to E_1 * E_2
\]

\[
  (*) \qquad E_1 \otimes E_2 / L_1 \otimes L_2 \longrightarrow E_1 * E_2 ,
\]

\[
  0 \to L_1 \otimes L_2 \to E_1 \overset{\leftrightarrows}{\otimes} E_2 \to \mathcal{O} \to 0
\]

\[
  P_1 \times P_2 \quad \text{sous} \quad L_1 \times L_2
\]

\[
  E_1 \overset{\leftrightarrows}{\otimes} E_2 \simeq E_1 \underset{\mathcal{O}}{\times} E_2 = \{ (x_1, x_2) \in E_1 \times E_2 \mid \varphi_1(x_1) = \varphi_2(x_2) \}
\]

\[
  L_1 \times L_2 \longrightarrow L_1 \otimes L_2
\]

\[
  (u_1, u_2) \longmapsto u_1 \otimes b_2 + b_1 \otimes u_2
\]

\[
  Q_1 : E_1 \to L_1^{\otimes 2} , \qquad Q_2 : E_2 \to L_2^{\otimes 2}
\]

\[
  P_1 \to L_1^{\otimes 2} , \qquad P_2 \to L_2^{\otimes 2} ,
\]

\[
  \delta_1 \in \Gamma L_1^{\otimes 2} , \quad \delta_2 \in \Gamma L_2^{\otimes 2}
\]

\[
  \delta_1 = b_1^2 - 4c_1 , \qquad \delta_2 = b_2^2 - 4c_2
\]

\[
  b_1 = \pi_1(x_1) = 2x_1 - T_1 \in \Gamma L_1 , \qquad b_2 = \pi_2(x_2) = 2x_2 - T_2 \in \Gamma L_2
\]

\[
  c_1 = Q(x_1) \in \Gamma L_1^{\otimes 2} , \qquad c_2 = \Gamma L_2^{\otimes 2}
\]

\[
  \begin{cases}
    Q_1(\lambda x_1 + u_1) = u_1^2 + \lambda b_1 u_1 + \lambda^2 c_1 ; \\
    Q_2(\lambda x_2 + u_2) = u_2^2 + \lambda b_2 u_2 + \lambda^2 c_2
  \end{cases}
\]

\[
  Q_1(x_1 + u_1) = u_1^2 + b_1 u_1 + c_1 , \qquad Q_2(x_2 + u_2) = u_2^2 + b_2 u_2 + c_2
\]

\[
  \boxed{Q(x_1 * x_2) = Q_1(x_1) \delta_2 + Q_2(x_2) \delta_1 + 4 Q_1(x_1) Q_2(x_2)}
\]

\[
  \boxed{c = c_1 \delta_2 + c_2 \delta_1 + 4 c_1 c_2}
\]

\[
  c = c_1 (b_2^2 - 4c_2) + c_2 (b_1^2 - 4c_1) + 4 c_1 c_2 \quad \text{i.e.}
\]

\[
  \boxed{c = c_1 b_2^2 + c_2 b_1^2 - 4 c_1 c_2}
\]

\[
  \boxed{b = b_1 b_2} ,
\]

\[
  Q(\lambda\, x_1 * x_2 + u) = u^2 + \lambda b u + \lambda^2 c
  \qquad (\lambda \in \Gamma \mathcal{O} ,\ u \in \Gamma(L_1 \otimes L_2)) .
\]

\[
  \delta = \delta_1 \delta_2 \qquad (\text{en plus on a } b = b_1 b_2)
\]

\[
  \delta = b^2 - 4c = b_1^2 b_2^2 - 4(c_1 b_2^2 + c_2 b_1^2 - 4 c_1 c_2) = (b_1^2 - 4c_1)(b_2^2 - 4c_2)
\]

\[
  \boxed{\begin{array}{l} b = b_1 b_2 \\ \delta = \delta_1 \delta_2 \end{array}}
\]

\[
  \delta = b^2 - 4c = \delta_1 \delta_2 \qquad \text{d'où, comme } b = b_1 b_2 ,
\]

\[
  -4c = \delta_1 \delta_2 - b^2 = (b_1^2 - 4c_1)(b_2^2 - 4c_2) - b_1^2 b_2^2 = 4(-b_1^2 c_2 - c_1 b_2^2 + 4 c_1 c_2)
\]

\[
  4c = 4(b_1^2 c_2 + b_2^2 c_1 - 4 c_1 c_2) = 4(c_1 \delta_2 + c_2 \delta_1 + 4 c_1 c_2)
\]

\[
  X_1 \times X_2 \longrightarrow X_1 \star X_2
\]

\[
  E_1 \simeq L_1 \oplus \underset{\substack{\| \\ \mathcal{O}.x_1}}{\mathcal{O}} , \qquad E_2 \simeq L_2 \oplus \underset{\substack{\| \\ \mathcal{O}.x_2}}{\mathcal{O}}
\]

\[
  \mathcal{A}_1 = \mathcal{O} \oplus L_1^{\vee} , \qquad \mathcal{A}_2 \simeq \mathcal{O} \oplus L_2^{\vee}
\]

\[
  L_1 = \operatorname{Ker}(x_1 : \mathcal{A}_1 \to \mathcal{O}) , \qquad L_2 = \operatorname{Ker}(x_2 : \mathcal{A}_2 \to \mathcal{O})
\]

\[
  b_1 = 2x_1 - T_1 \quad \text{i.e.} \quad \text{\struck{$b_1(u_1')$}}\ b_1 . u_1' = \operatorname{Tr}(u_1')
\]

\[
  N(u_1') = u_1'^2 . c_1
\]

\[
  u_1' \in L_1^{\vee} ,
\]

\[
  u_1'^2 + b_1 u_1' + c_1 = 0
\]

\[
  N(\underbrace{u_1' + \lambda e_1}_{\xi_1}) = N(u_1') + \lambda \operatorname{Tr}(u_1') + \lambda^2
\]

\[
  = \text{\struck{$u_1'^2 . \ldots$}}\ u'^2 . c_1 + \lambda u' . b_1 + x_1^2(\xi_1)
\]

\[
  = u'^2 . c_1 + x_1(\xi_1) \ldots
\]

\[
  E \simeq L \oplus \underset{\substack{\wr \\ \mathcal{O}}}{\mathcal{O}.x_0} , \qquad \mathcal{A} \simeq \mathcal{O} \oplus L^{\vee}
\]

\[
  \boxed{\begin{array}{c} b = 2x_0 - T \\ \cap \\ \Gamma L \end{array}}
  \qquad
  \text{donc pour } u \in L^{\vee} ,\quad
  \boxed{u . b \overset{\text{déf}}{=} \langle u, b \rangle = -\operatorname{Tr}(u)}
\]

\[
  N = x_0^2 - b x_0 + c \in \operatorname{Sym}^2(E) , \qquad c \in \Gamma L^{\otimes 2}
\]

\[
  N(u) = u^2 . c \quad (\overset{\text{déf}}{=} \langle u^{\otimes 2}, c \rangle)
\]

\[
  u^2 + \underbrace{\langle u, b \rangle}_{b(u)} + \underbrace{\langle u^{\otimes 2}, c \rangle}_{c(u)} = 0
\]

\[
  N(\underbrace{u + \lambda}_{\xi}) = N(u) + \lambda \operatorname{Tr}(u) + \lambda^2 \qquad \text{où } \lambda = x_0(\xi)
\]

\[
  = x_0(\xi)^2 - x_0(\xi)\, \underset{\substack{\| \\ b(\xi)}}{b(u)} + \underset{\substack{\| \\ c(\xi)}}{c(u)}
\]

\[
  N = x_0^2 - x_0 b + c
\]

\[
  v^2 + b v + c = 0 \quad \text{i.e.} \quad Q(v) = 0
\]

\[
  \underset{\substack{\cap \\ \Gamma L^{\otimes 2}}}{v^2 + b v + c}
\]

\[
  E \xrightarrow{\ \sim\ } \underline{\operatorname{Hom}}(E, \underbrace{\det E}_{\overset{\text{déf}}{=} \Delta}) \simeq \underset{\substack{\| \text{déf} \\ \mathcal{A}}}{\check{E}} \otimes \Delta = \mathcal{A} \otimes \Delta
\]

\[
  f_x : E \to \det E = \textstyle\bigwedge^2 E , \qquad y \mapsto x \wedge y
\]

\[
  0 \to L \to E \xrightarrow{\ \Psi\ } \mathcal{O} \to 0 ,
\]

\[
  \underset{\substack{\| \\ \Delta}}{\det E} \overset{\alpha}{\xleftarrow{\ \sim\ }} L
  \qquad \alpha(u) = x_0 \wedge u , \quad \text{où } x_0 \in \Gamma P_1
\]

\[
  E \xrightarrow{\ \sim\ } \underset{\substack{\| \\ \check{E}}}{\mathcal{A}} \otimes L
\]

\[
  f_x : E \to L
\]

\[
  x_0 \wedge f_x(y) = x \wedge y
\]

\[
  f_x(y) = \Psi(x) y - \Psi(y) x
\]

\[
  \Psi(x) y - \Psi(y) x = \lambda\, \alpha^{-1}(x \wedge y) \quad \text{i.e.}
\]

\[
  \alpha(\Psi(x) y - \Psi(y) x) = \lambda\, x \wedge y ,
\]

\[
  \alpha(-u) = \lambda \underbrace{u \wedge x_0}_{-\alpha(u)}
\]

\[
  -u = \lambda u
\]

\[
  0 \to \mathcal{O} \to \mathcal{A} \xrightarrow{\ \varphi\ } L^{\vee} \to 0
\]

\[
  0 \to L \to \mathcal{A} \otimes L \xrightarrow{\ \Psi\ } \mathcal{O} \to 0
\]

\[
  \varphi \otimes \mathrm{id}_L (\underset{\substack{\cap \\ \mathcal{A} \otimes L = \underline{\operatorname{Hom}}(E, L)}}{g(x)}) = \Psi(x) \qquad \forall x \in \Gamma(E)
\]

\[
  g(x) | L = (u \mapsto \Psi(x) u)
\]

\[
  \underset{\substack{\| \\ \Psi(x) . u - \Psi(u) x \\ \hphantom{\Psi(x) . u - {}} \| \\ \hphantom{\Psi(x) . u - {}} 0}}{g(x)(u)} = \Psi(x) u \qquad \text{pour tout } u \in \Gamma L \subset \Gamma E \quad \text{c'est OK.}
\]

\[
  g(x) : E \to L \quad \text{est nulle sur } L, \text{ et provient de } \mathcal{O} \text{ par}
\]

\[
  \lambda \mapsto -\lambda x
\]

\[
  g(x)(y) = -\Psi(y) x ,
\]

\[
  \operatorname{Quad}(E, L^{\otimes 2}) \simeq \operatorname{Quad}(\mathcal{A}, \mathcal{O})
\]

\[
  \text{\struck{$\operatorname{Quad}(\check{E}) \otimes L^{\otimes 2} \simeq \operatorname{Sym}^2 \ldots$}}
\]

\[
  E \text{ scindé en } E \simeq L \oplus \mathcal{O}.x_0
\]

\[
  \mathcal{A} \simeq \mathcal{O} \oplus L^{\vee} \qquad \text{i.e.} \quad \mathcal{A} \otimes L \simeq L \oplus \underset{\mathcal{O}_X}{\underline{L \otimes L^{\vee}}}
\]

\[
  Q(\underset{\substack{\cap \\ \Gamma L}}{u} + \lambda x_0) = u^2 + \lambda \langle u, \underset{\substack{\cap \\ \Gamma L}}{b} \rangle + \lambda^2 \underset{\substack{\cap \\ \Gamma L^{\otimes 2}}}{c}
\]

\[
  N(\underset{\substack{\cap \\ \Gamma L^{\vee}}}{v} + \lambda 1) = \lambda^2 - \lambda \langle v, b \rangle + \langle v^2, c \rangle
\]

\[
  \boxed{Q_N(\varphi_N(x)) = -N(x)\, \delta_N}
\]

\[
  E \otimes L^{\vee} \longleftrightarrow (E_1 \otimes L_1^{\vee}) \otimes (E_2 \otimes L_2^{\vee})
\]

\[
  L^{\vee} \simeq L_1^{\vee} \otimes L_2^{\vee} ,
\]

\[
  E \xrightarrow{\ i\ } E_1 \otimes E_2
\]

\[
  \alpha \circ \mathrm{inj} = 2\beta
\]

\[
  E_1 \otimes_0 E_2 \longrightarrow L_1 \otimes L_2 \subset E_1 \otimes E_2
\]

\[
  \begin{array}{c} \cap \\ E_1 \otimes E_2 \end{array} \ \overset{?}{\dashrightarrow}
  \qquad (\text{pour } \boxed{\chi = 2 \cdot 1}) ,
\]

\[
  \xi \mapsto \overline{\xi} .
\]

\[
  \xi \longmapsto \xi + \overline{\xi}
\]

\[
  x_1 \otimes x_2 \longmapsto x_1 \otimes x_2 + \overline{x_1} \otimes \overline{x_2} = x_1 \otimes x_2 + (T_1 - x_1) \otimes (T_2 - x_2)
\]

\[
  u_1 \otimes x_2 \longmapsto u_1 \otimes x_2 + (-u_1)(T_2 \Psi_2(x_2) - x_2)
\]

\[
  \begin{array}{ccc}
    x_1 \otimes x_2 & \longmapsto & x_1 * x_2 \\
    E_1 \otimes E_2 & \longrightarrow & E_1 * E_2 = E \\
    \wr & & \wr \\
    (\mathcal{A}_1 \otimes L_1) \otimes (\mathcal{A}_2 \otimes L_2) & & \mathcal{A} \otimes L , \quad \text{où } L = L_1 \otimes L_2
  \end{array}
\]

\[
  \mathcal{A}_1 \otimes \mathcal{A}_2 \xrightarrow{\ \tau_{\mathcal{A}}\ } \mathcal{A}
\]

\[
  \xi_1 \otimes \xi_2 \longmapsto \xi_1 * \xi_2
\]

\[
  \mathcal{X} = X_1 \wedge X_2
\]

\[
  X_1 \times X_2 \longrightarrow \mathcal{X}
\]

\[
  \mathcal{A} \xrightarrow{\ i_{\mathcal{A}}\ } \mathcal{A}_1 \otimes \mathcal{A}_2
\]

\[
  E_1 \otimes E_2 \xrightarrow{\ \tau_E\ } E \qquad (x_1 \otimes x_2 \mapsto x_1 * x_2)
\]

\[
  \mathcal{A}_1 \underset{\mathcal{O}}{\oplus} \mathcal{A}_2 \longrightarrow \mathcal{O} .
\]

\[
  \mathcal{A} \longrightarrow \mathcal{A}_1 \otimes \mathcal{A}_2
\]

\[
  \mathcal{O} \longrightarrow \mathcal{A}_1 \otimes \mathcal{A}_2 \qquad (\text{on prendra l'inclusion canonique } \lambda \mapsto \lambda . e_{\mathcal{A}_1} \otimes e_{\mathcal{A}_2})
\]

\[
  \mathcal{A}_1 \otimes \mathcal{A}_2 \xrightarrow{\ \varphi\ } \mathcal{A}_1 \otimes \mathcal{A}_2 , \qquad x_1 \otimes x_2 \longmapsto i(x_1 * x_2)
\]

\[
  \xi^{\natural} = \xi + \bar{\xi} \quad \text{pour } \xi \in A_1 \otimes A_2 ,
\]

\[
  x_1 \mapsto \bar{x}_1 = \text{\struck{$\xi_1(x_1)$}}\ \mathrm{Tr}_1(x_1) - x_1 ,
  \qquad
  x_2 \mapsto \bar{x}_2 = \mathrm{Tr}_2(x_2)\, e_2 - x_2
\]

\[
  \xi \mapsto \bar{\xi} , \qquad
  x_1 \otimes x_2 \mapsto \overline{x_1 \otimes x_2} \overset{\mathrm{déf}}{=} \bar{x}_1 \otimes \bar{x}_2
\]

\[
  \xi^{\natural} = (x_1 \otimes 1)^{\natural} = (x_1 + \bar{x}_1) \otimes e_2 = \mathrm{Tr}_1(x_1)\, e_1 \otimes e_2
\]

\[
  (A_1 \otimes A_2)^{\natural} = (A_1 \otimes A_2)^{\sigma_1 \otimes \sigma_2} .
\]

\[
  (x_1, x_2) \mapsto x_1 * x_2 : A_1 \times A_2 \to A , \qquad
  x_1 \otimes x_2 \mapsto x_1 * x_2 : A_1 \otimes A_2 \to A
\]

\[
  1 \mapsto 2 , \qquad U_1 \mapsto -b_1 , \qquad U_2 \mapsto -b_2 .
\]

\[
  U_1 \otimes U_2 + \underbrace{\bar{U}_1 \otimes \bar{U}_2}_{(-b_1 - U_1) \otimes (-b_2 - U_2)}
  = \underbrace{2 U_1 U_2 + b_1 U_2 + b_2 U_1}_{\text{image de } \mathcal{U}} + b_1 b_2 ,
\]

\[
  U_1 * U_2 = \mathcal{U} + b_1 b_2 .
\]

\[
  N_{A_1 \otimes A_2 / A} : A_1 \otimes A_2 \to A ,
\]

\[
  (u, v) \mapsto u \bar{v} + v \bar{u} = \mathrm{Tr}_N(u \bar{v})
\]

\[
  N_A(x_1 \otimes x_2) = N_1(x_1)\, N(x_2) \in \Gamma\mathcal{O} \subset \Gamma A
\]

\[
  = (x_1 \otimes x_2)(\bar{x}_1 \otimes \bar{x}_2) = (x_1 \bar{x}_1) \otimes (x_2 \bar{x}_2) .
\]

\[
  N(1) = 1 , \quad N(U_1) = N_1(U_1) = c_1 , \quad N(U_2) = N_2(U_2) = c_2 ,
\]

\[
  N(U_1 \otimes U_2) = c_1 c_2 .
\]

\[
  A_1 = \mathcal{O}[U_1] / (U_1^2 + b_1 U_1 + c_1) , \qquad
  A_2 = \mathcal{O}[U_2] / (U_2^2 + b_2 U_2 + c_2) ,
\]

\[
  \begin{cases}
    A_1 \otimes A_2 = \mathcal{O}[U_1, U_2] / (U_1^2 + b_1 U_1 + c_1 ,\ U_2^2 + b_2 U_2 + c_2) \\
    A \simeq \mathcal{O}[\mathcal{U}] / (\mathcal{U}^2 + b\, \mathcal{U} + c)
  \end{cases}
\]

\[
  \begin{cases}
    b = b_1 b_2 \\
    c = b_1^2 c_2 + b_2^2 c_1 - 4 c_1 c_2
  \end{cases}
\]

\[
  \mathcal{U} \mapsto 2 U_1 U_2 + b_1 U_2 + b_2 U_1 .
\]

\[
  A_1 \otimes A_2 \xrightarrow{\ *\ } A , \qquad x_1 \otimes x_2 \mapsto x_1 * x_2 ,
\]

\[
  x_1 * e_2 = \mathrm{Tr}_1(x_1) \cdot 1_A , \qquad
  e_1 * x_2 = \mathrm{Tr}_2(x_2) \cdot 1_A
\]

\[
  A_1 \otimes A_2 \xrightarrow{\ \mathrm{Tr}_{A_1 \otimes A_2 / A}\ } A
\]

\[
  A_1 \oplus_{\mathcal{O}} A_2 \xrightarrow{\ \mathrm{Tr}\ } \mathcal{O}
\]

\[
  A \to A_1 \otimes A_2
\]

\[
  \xi \mapsto \xi + \bar{\xi} , \qquad \text{i.e.} \qquad
  i\bigl(\mathrm{Tr}_{A_1 \otimes A_2 / A}\, \xi\bigr) = \xi + \bar{\xi}
\]

\[
  \bar{\xi} = \sigma_1 \otimes \sigma_2 (\xi)
\]

\[
  x_1 \mapsto \mathrm{Tr}_{A_1}(x_1) - x_1 , \qquad x_2 \mapsto \mathrm{Tr}_{A_2}(x_2) - x_2 .
\]

\[
  N = N_{A_1 \otimes A_2 / A} ,
\]

\[
  N(\xi + \eta) = N(\xi) + N(\eta) + \mathrm{Tr}(\xi \bar{\eta})
\]

\[
  \begin{cases}
    i(N(\xi)) = \xi \bar{\xi} \\
    N(\sigma_1 \otimes \mathrm{id}_{A_2}(\xi)) = N(\mathrm{id}_{A_1} \otimes \sigma_2(\xi)) = \overline{N(\xi)} \\
    N(\bar{\xi}) = N(\xi)
  \end{cases}
\]

\[
  N_{A/\mathcal{O}}\bigl(N_{A_1 \otimes A_2 / A}(\xi)\bigr) = N_{A_1 \otimes A_2 / \mathcal{O}}(\xi)
\]

\[
  \mathrm{Tr}_{A/\mathcal{O}}\bigl(\mathrm{Tr}_{A_1 \otimes A_2 / A}(\xi)\bigr) = \mathrm{Tr}_{A_1 \otimes A_2 / \mathcal{O}}(\xi)
\]

\[
  \sigma_1 = \sigma_{A_1} \otimes \mathrm{id}_{A_2} , \qquad
  \sigma_2 = \mathrm{id}_{A_1} \otimes \sigma_{A_2} , \qquad
  \sigma_3 = \sigma_1 \sigma_2
\]

\[
  N_{A_1 \otimes A_2 / \mathcal{O}}(\xi) = \xi_0\, \xi_1\, \xi_2\, \xi_3 .
\]

\[
  \xi_0 \bar{\xi}_0\, \xi_1 \bar{\xi}_1 , \qquad \bar{\xi}_0 = \xi_3 , \quad \bar{\xi}_1 = \xi_2 ,
\]

\[
  \xi_0 \bar{\xi}_0 \overset{!}{=} N_{A_1 \otimes A_2 / A}(\xi_0)
\]

\[
  \xi_1 \bar{\xi}_1 = N_{A_1 \otimes A_2 / A}(\xi_1) = \overline{N_{A_1 \otimes A_2 / A}(\xi_0)} ,
  \qquad \xi_1 = \sigma_1(\xi_0)
\]

\[
  N \bar{N} = N_{A/\mathcal{O}}(N) \qquad \text{OK.}
\]

\[
  \alpha : k \to A , \qquad \lambda \mapsto \lambda \cdot 1_A
\]

\[
  t_{A/k} = t : A \to k \quad (\text{trace relative}) , \qquad
  n_{A/k} = n : A \to k \quad (\text{norme relative}) ,
\]

\[
  t(x) = \varphi_n(x, 1) \qquad (\text{la forme bil. ass. à } n)
\]

\[
  n(x + \lambda 1_A) = n(x) + \lambda^2 n(1) + \lambda t(x) , \qquad n(1) = 1 \ \text{cf. ci-dessous b)}
\]

\[
  \sigma : f \mapsto \bar{f} = \tau(f) - f
\]

\[
  \tau(\bar{f}) = \tau(f) \underbrace{\tau(1)}_{2} - \tau(f) = \tau(f)
\]

\[
  (\bar{f})^{-} = \tau(\bar{f}) - \bar{f} = f
\]

\[
  \sigma(xy) = \sigma(x)\, \sigma(y) \qquad x, y \in A
\]

\[
  2xy = \underbrace{\tau(xy)}_{\varphi_n(x, \bar{y})} - \tau(x)\tau(y) + x\tau(y) + y\tau(x)
\]

\[
  \alpha(n(x)) = x \bar{x}
\]

\[
  n(\bar{x}) = n(x)
\]

\[
  n(\bar{x}) = n(\lambda 1_A - x) \overset{(a)}{=} n(x) + \lambda^2 - \lambda t(x) \qquad \text{où } \lambda = t(x)
\]

\[
  \alpha(n(x)) = x \bar{x}
\]

\[
  \underbrace{\alpha(n(y+z))}_{n(y) + n(z) + \varphi_n(y,z)}
  = \text{\struck{$\ldots$}}\ (y+z)\overline{(y+z)}
\]

\[
  = \underbrace{y\bar{y}}_{\alpha(n(y))} + \underbrace{z\bar{z}}_{\alpha(n(z))}
  + \underbrace{y\bar{z} + z\bar{y}}_{\tau(y\bar{z})} ,
  \qquad z\bar{y} = \overline{y\bar{z}}
\]

\[
  \alpha(\varphi_n(y, z)) = \tau(y \bar{z}) \qquad \text{i.e. } \alpha(t(y\bar{z}))
\]

\[
  \varphi_n(y, z) = t(y \bar{z})
\]

\[
  n(y + z) = n(y) + n(z) + t(y\bar{z}) , \qquad t(y\bar{z}) = t(\bar{y} z)
  \quad \text{car } t(x) = t(\bar{x})
\]

\[
  \begin{cases}
    \mathrm{Tr}_{k/k_0}(t(x)) = \mathrm{Tr}_{A/k_0}(x) \\
    N_{k/k_0}(n(x)) = N_{A/k_0}(x)
  \end{cases}
\]

\[
  \overline{xy} = \bar{x}\,\bar{y} \qquad \text{i.e.} \qquad
  2xy = \bigl(\tau(xy) - \tau(x)\tau(y)\bigr) + \tau(y)\,x + \tau(x)\,y
\]

\[
  \begin{cases}
    n(\lambda x) = \lambda^2 n(x) \\
    n(x + y) = n(x) + n(y) + t(x\bar{y})
  \end{cases}
  \qquad \lambda \in k , \ x \in A
\]

\[
  \varphi_n(x, y) = t(x \bar{y})
\]

\[
  \begin{cases}
    \mathrm{Tr}_{k/k_0}(t(x)) = \mathrm{Tr}_{A/k_0}(x) \\
    N_{k/k_0}(n(x)) = N_{A/k_0}(x)
  \end{cases}
  \qquad x \in A
\]

\[
  \bar{x} = t_i(x) - x
\]

\[
  \text{\struck{$\sigma_i(x) = T_i(x) - x$}}
\]

\[
  \text{\struck{$\sigma_1 \sigma_2 = \sigma_3 , \quad \sigma_2 \sigma_3 = \sigma_1 , \quad \sigma_3 \sigma_1 = \sigma_2$}}
\]

\[
  T_2\bigl(\alpha_1(x_1)\, \text{\struck{$\alpha_2(x_2)$}}\bigr) = t_1(x_1)\, \text{\struck{$\alpha_2(x_2)$}}
\]

\[
  \sigma_2(\alpha_1(x)) = \alpha_1(\bar{x}) ,
\]

\[
  N_2(\alpha_1(x_1)) = n_1(x_1)
\]

\[
  t_1(x_1) \cdot x_2^2
\]

\[
  \text{\struck{$(x,y) \mapsto$}}\ \varphi_{n_1}(x, y) = t_1(x \bar{y}) \quad \text{sur } A_1 \times A_1 ,
\]

\[
  \begin{cases}
    \alpha_1(\bar{y}) = \sigma_2(\alpha_1(y)) \\
    t_1(x \bar{y}) = T_2(\alpha_1(x\bar{y})) \quad \text{dans } A_2
  \end{cases}
\]

\[
  \varphi_{n_1}(x, y) = T_2\bigl(x' \sigma_2(y')\bigr) = \varphi_{N_2}(x', y') \qquad \text{OK.}
\]

\[
  \sigma_1 \sigma_2 \sigma_3 = 1
\]

\[
  \begin{cases}
    \sigma_1 \sigma_2 = \sigma_3 \\
    \sigma_2 \sigma_3 = \sigma_1 \\
    \sigma_3 \sigma_1 = \sigma_2
  \end{cases}
\]

\[
  \sigma\tau = \tau\sigma \iff (\sigma\tau)^2 = 1 \quad \text{i.e. } \sigma\tau\sigma\tau = 1
\]

\[
  2x = T_1(x) + T_2(x) + T_3(x) - t_2 T_2(x) ,
\]

\[
  \sigma_2 \sigma_3 \sigma_1 = 1 \quad \text{ou} \quad \sigma_3 \sigma_1 \sigma_2 = 1 ,
\]

\[
  2x = T_1(x) + T_2(x) + T_3(x) - t_3 T_3(x)
\]

\[
  2x = \cdots - t_1 T_1(x) ,
\]

\[
  t_1 T_1(x) = t_2 T_2(x) = t_3 T_3(x)
\]

\[
  2x = T_1(x) + T_2(x) + T_3(x) - T_{A/k}(x) .
\]

\[
  \text{\struck{$\alpha_1 T_1(x) = 2x + T_{A/k}(x)$}}
\]

\[
  \text{\struck{${} - \alpha_2 T_2(x) - \alpha_3 T_3(x)$}}
\]

\[
  \text{\struck{$N_{A/k}(x)\, x^2 = N_1(x)\, N_2(x)\, N_3(x)$}}
\]

\[
  \text{\struck{$n_1(N_1(x)) = n_2(N_2(x)) = n_3(N_3(x))$}}
\]

\[
  x_0 = x , \quad x_1 = T_1(x) , \quad x_2 = T_2(x) , \quad x_3 = T_3(x)
\]

\[
  \begin{aligned}
    \sigma_1 &: x_0 \leftrightarrow x_1 , \quad x_2 \leftrightarrow x_3 \\
    \sigma_2 &: x_0 \leftrightarrow x_2 , \quad x_1 \leftrightarrow x_3 \\
    \sigma_3 &: x_0 \leftrightarrow x_3 , \quad x_1 \leftrightarrow x_2
  \end{aligned}
\]

\[
  n_1(N_1(x)) \overset{!}{=} \underbrace{N_1(x)}_{x\, \sigma_1(x)} \overline{N_1(x)}
  = (x_0 x_1)\, \underbrace{\sigma_2(x_0 x_1)}_{x_2 x_3} = x_0 x_1 x_2 x_3
\]

\[
  N_1(x)\, N_2(x)\, N_3(x) = (x_0 x_1)(x_0 x_2)(x_0 x_3)
\]

\[
  = x_0^2 \underbrace{(x_0 x_1 x_2 x_3)}_{N_{A/k}(x_0)} = x^2 N_{A/k}(x) ,
\]

\[
  A_1 \otimes A_2 \otimes A_3 ,
\]

\[
  \textstyle\bigotimes A_i \to A ,
\]

\[
  \Gamma \subset \mathbb{F}_2^3 , \qquad
  \Gamma = \mathrm{Ker}\bigl(\mathbb{F}_2^3 \to \mathbb{F}_2\bigr) , \quad (\varepsilon_i) \mapsto \textstyle\sum \varepsilon_i
\]

\[
  \text{\struck{$(0,0,1)$}}, \quad (0,1,1) , \quad (1,0,1) , \quad (1,1,0)
\]

\[
  \begin{aligned}
    \mathrm{id}_{A_1} \otimes \sigma_{A_2} \otimes \sigma_{A_3} &\overset{\mathrm{déf}}{=} \tau_1 \\
    \sigma_{A_1} \otimes \mathrm{id}_{A_2} \otimes \sigma_{A_3} &\overset{\mathrm{déf}}{=} \tau_2 \\
    \sigma_{A_1} \otimes \sigma_{A_2} \otimes \mathrm{id}_{A_3} &\overset{\mathrm{déf}}{=} \tau_3
  \end{aligned}
\]

\[
  \textstyle\bigotimes A_i \to A
\]

\[
  X_1 \wedge X_2 \wedge X_3 , \qquad \text{d'algèbre } B = A_1 \wedge A_2 \wedge A_3 ,
\]

\[
  X_1 \times X_2 \times X_3 \to X_1 \wedge X_2 \wedge X_3 , \qquad
  (x_1, x_2, x_3) \mapsto x_1 * x_2 * x_3 ,
\]

\[
  B \to A_1 \otimes A_2 \otimes A_3 ,
\]

\[
  B \to (A_1 \otimes A_2 \otimes A_3)^{\Gamma} \qquad \text{invariant sous } \Gamma
\]

\[
  B \to A_1 \otimes A_2 \otimes A_3 \to A , \quad \text{\uncertain{i.e.}}
\]

\[
  \mathcal{X} \to X_1 \times_X X_2 \times_X X_3 \to X_1 \wedge X_2 \wedge X_3 ,
\]

\[
  f(B) \subset A^{\Gamma} ,
\]

\[
  A^{\Gamma} = k \cdot 1_A .
\]

\[
  T_i(x) - x = x \quad \text{i.e.} \quad 2x = T_i(x) ,
\]

\[
  6x = \underbrace{T_1(x) + T_2(x) + T_3(x)}_{= 2x + T_{A/k}(x) \ \text{cf. plus haut}}
\]

\[
  4x = T_{A/k}(x)
\]

\[
  x = \tfrac{1}{4} T_{A/k}(x) \in k \qquad \text{OK.}
\]

\[
  \mathcal{X} = \mathrm{Spec}\, A \to X_1 \times_k X_2 \times_k X_3
\]

\[
  \mathcal{X} \to X_1 \times X_2 , \quad \mathcal{X} \to X_2 \times X_3 , \quad \mathcal{X} \to X_3 \times X_1
\]

\[
  J \subset I_1 \times I_2 \times I_3 , \qquad
  J = \{ (a_1, a_2, a_3) \in I_1 \times I_2 \times I_3 \mid a_1 \wedge a_2 \wedge a_3 = \varepsilon \}
\]

\[
  \mathcal{X} \subset J_X
\]

\[
  k \to \Gamma(X_J, \mathcal{O}_{X_J}) \simeq k^J .
\]

\[
  \mathcal{X}_j = X_j \qquad \text{donc} \qquad \mathcal{X} = X_J \qquad \text{OK.}
\]

\[
  \mathcal{X} = \pi^{-1}(\varepsilon) \qquad
  \bigl(\pi : X_1 \times_X X_2 \times_X X_3 \to X_1 \wedge X_2 \wedge X_3\bigr)
\]

\[
  X_{1\,\mathfrak{X}_0},\quad X_{2\,\mathfrak{X}_0},\quad X_{3\,\mathfrak{X}_0}
  \qquad (X_{i\,\mathfrak{X}_0} = X_i \times_{\mathfrak{X}} \mathfrak{X}_0)
\]

\[
  X_1 \times X_2 \times X_3 \longrightarrow \mathfrak{X}_0 = X_1 \wedge X_2 \wedge X_3
\]

\[
  a_1, a_2, a_3 \longmapsto x_1 \wedge \alpha_2 \wedge x_3
\]

\[
  B = A_1 \otimes_{O} A_2 \otimes_{O} A_3
  \;\rightleftarrows\;
  A_1 \otimes_{O} B_0 = B_1,
  \qquad B_0 = A_1 * A_2 * A_3
\]

\[
  \alpha\,\pi(x) = \pi'\beta(x)
  \;\overset{?}{\Longleftrightarrow}\;
  \beta\varpi(y) = \varpi'\gamma(y)\ ?
\]

\[
  \text{\struck{$\beta$}}\ \pi(x) = \pi'\beta(x)
  \;\Longleftrightarrow\;
  \beta\varpi(x) \equiv
\]

\[
  x_1 \otimes x_2 \longmapsto \beta_1(x_1) \otimes \beta_2(x_2)
  \longrightarrow \beta_1(x_1) * \beta_2(x_2)
\]

\[
  = \pi_1'\beta_1(x_1) \otimes \alpha_2(\ldots)
\]

\[
  \pi_1(x_1) \otimes u_2 \longrightarrow
  \underbrace{\alpha_1\pi_1(x_1)}_{\pi_1'\beta_1(x_1)} \otimes \alpha_2(x_2)
\]

\[
  \mathcal{E}_{(p_1,q_1)(p_2,q_2)} = (M_1 \otimes M_2) \oplus (L_1 \otimes L_2)
  \xrightarrow{\ T_u\ }
  \mathcal{E}_{(p_1',q_1')(p_2',q_2')}
\]

\[
  b' = b + \chi u
\]

\[
  u - \sigma b' = u - \sigma b - \sigma\chi u = -u - \sigma b
\]

\[
  E_1 \vee E_2 \simeq E_1 \wedge E_2
\]

\[
  E_1' \vee E_2' \;\simeq\; E_1' \wedge E_2'
\]

\[
  \tilde b_i = \varpi_i - \chi q_i\,; \qquad
  b_i + \tilde b_i = \chi\,\mathrm{id} - \chi\,\mathrm{id} = 0
\]

\[
  \tilde b_i = -b_i
\]

\[
  q_i' = \underbrace{q_i}_{\mathrm{id} - p_i} - \tilde u_i
  = \text{\struck{$\chi$}}\,\mathrm{id} - \underbrace{(p_i + \tilde u_i)}_{p_i'} =
\]

\[
  \tilde u_i = -u_i
\]

\[
  \tilde u = \tilde b_1 \otimes \tilde u_2 + \tilde u_1 \otimes \tilde b_2 + \chi\, \tilde u_1 \otimes \tilde u_2
  = b_1 \otimes u_2 + u_1 \otimes b_2 + \chi\, u_1 \otimes u_2 = u
\]

\[
  \tilde b = \tilde b_1 \tilde b_2 = b_1 b_2
\]

\[
  (= (\eta,\xi) \longmapsto (\eta,\ \xi + v(\eta))),
\]

\[
  M_1 \otimes M_2 \xrightarrow{\ b = b_1 b_2\ } L_1 \otimes L_2
\]

\[
  b + \chi v = \underset{\displaystyle -b}{\overset{\sim}{b}}
  \quad\text{i.e.}\quad v \overset{!}{=} -\frac{2}{\chi}\, b
\]

\[
  \boxed{v = -\sigma b}
\]

\[
  0 \to L_i \to E_i \to M_i \to 0, \qquad \pi_i : E_i \to L_i
\]

\[
  \Theta_{p_i} : E_i \simeq M_i \oplus L_i
\]

\[
  b_i := \pi_i - \chi p_i : M_i \to L_i
\]

\[
  b = b_1 \otimes b_2 : M_1 \otimes M_2 \to L_1 \otimes L_2
\]

\[
  \mathcal{E}_{(p_1,p_2)} = (M_1 \otimes M_2) \oplus (L_1 \otimes L_2)
\]

\[
  \pi_{(p_1,p_2)} - \chi\, p = b_1 \otimes b_2
\]

\[
  \pi_{(p_1,p_2)} : (\eta,\xi) \longmapsto \text{\struck{\ill{}}}\ \chi\xi + b_1 \otimes b_2(\eta)
\]

\[
  p_i' = p_i - u_i, \quad u_i : M_i \to L_i \qquad (p' = (p_1', p_2'))
\]

\[
  b_1' = b_1 + \chi u_1, \qquad b_2' = b_2 + \chi u_2
\]

\[
  \mathcal{E}_{p} \xrightarrow{\ \Theta_{p,p'}\ } \mathcal{E}_{p_1',p_2'},
  \qquad \Theta_{p,p'} = T_{-u} :
  (\eta,\xi) \longmapsto (\eta,\ \xi - u\eta)
\]

\[
  u = b_1 \otimes u_2 + u_1 \otimes b_2 + \chi\, u_1 \otimes u_2
\]

\[
  \pi_{p_1',p_2'} : (\eta,\xi) \longmapsto \text{\struck{\ill{}}}\ \chi\xi + (b_1' \otimes b_2')(\eta)
  = \pi_{(p_1,p_2)}(\eta,\xi) + u(\eta)
\]

\[
  \text{\struck{$\Theta_{p+u}(x) =$}}
  \qquad
  \Theta_p(x) = \psi(x) \oplus p(x)
\]

\[
  \Theta_{p+u}(x) = \psi(x) \oplus \bigl(p(x) + u\,\psi(x)\bigr)
\]

\[
  \boxed{b_i = (\pi_i - \chi p_i)}
\]

\[
  \mathrm{Hom}(M_i, L_i) \ni b_i,
  \qquad
  (\eta,\xi) \longmapsto (\eta,\ \xi - u\eta)
\]

\[
  \mathrm{Hom}(M_i, L_i) \ni b_i' = b_i + \chi u_i,
  \qquad
  (\eta,\xi) \longmapsto (\eta,\ \xi - u'\eta)
\]

\[
  b_i'' = b_i' - \chi u_i'
\]

\[
  b_1 \otimes b_2 \in \mathrm{Hom}(M, L),
  \qquad
  (\pi - \chi\, p_1 * p_2)^{\Theta_{p_1 * p_2}}
\]

\[
  b_1' \otimes b_2' = (b_1 - \chi u_1) \otimes (b_2 - \chi u_2) = b_1 \otimes b_2 - \chi u
\]

\[
  b_1'' \otimes b_2'' = b_1' \otimes b_2' - \chi u' = b_1 \otimes b_2 - \chi\underbrace{(u + u')}_{u''}
\]

\[
  \left\{
  \begin{aligned}
    p_1' * p_2' &= p_1 * p_2 - u \\
    p_1'' * p_2'' &= p_1' * p_2' - u' = p_1 * p_2 - u'' \\
    &\Longrightarrow u'' = u + u'
  \end{aligned}
  \right.
\]

\[
  \underbrace{b_1, b_2;\ u_1, u_2}_{\mathrm{I}}
  \qquad
  \underbrace{u_1',\ u_2'}_{\mathrm{I}}
\]

\[
  \begin{array}{c}
    (p_1,p_2) \\ \downarrow \\ (p_1',p_2') \\ \downarrow \\ (p_1'',p_2'') \\[2pt] \hline
    (p_1,p_2) \\ \downarrow \\ (p_1'',p_2'')
  \end{array}
  \quad
  \left\{
  \begin{aligned}
    u &= b_1 \otimes u_2 + u_1 \otimes b_2 + \chi\, u_1 \otimes u_2 \\
    u' &= b_1' \otimes u_2' + u_1' \otimes b_2' + \chi\, u_1' \otimes u_2' \\
    u'' &= b_1 \otimes u_2'' + u_1'' \otimes b_2 + \chi\, u_1'' \otimes u_2'' \\
    &\quad (\text{si } u_1'' = u_1 + u_1',\ u_2'' = u_2 + u_2')
  \end{aligned}
  \right.
\]

\[
  \left\{
  \begin{aligned}
    b_1' &= b_1 + \chi u_1, & b_2' &= b_2 + \chi u_2 \\
    b_1'' &= b_1' + \chi u_1', & b_2'' &= b_2 + \chi u_2'
  \end{aligned}
  \right.
  \qquad
  \begin{aligned}
    b &= b_1 \otimes b_2 \\
    b' &= b_1' \otimes b_2' \\
    b'' &= b_1'' \otimes b_2''
  \end{aligned}
\]

\[
  u'' = b_1 \otimes (u_2 + u_2') + (u_1 + u_1') \otimes b_2
  + \chi\, (u_1 + u_1') \otimes (u_2 + u_2')
\]

\[
  = u + b_1 \otimes u_2' + \underbrace{u_1' \otimes b_2 + \chi\, u_1' \otimes u_2}_{u_1' \otimes (b_2 + \chi u_2)}
  + \chi\, u_1 \otimes u_2' + \chi\, u_1' \otimes u_2'
\]

\[
  \ldots + (b_1 + \chi u_1) \otimes u_2' + \chi\, u_1' \otimes u_2'
\]

\[
  \Lambda_u : (\eta,\xi) \longmapsto (\eta,\ \xi + u(\eta)),
  \qquad b_1, b_2 \quad\leadsto\quad b_1', b_2'
\]

\[
  \Theta_{p_1' * p_2'} = \Theta_{p_1 * p_2}\ \ill{}\ u
\]

\[
  u = b_1(p_1) \otimes u_2 + u_1 \otimes b_2(p_2) + \chi\, u_1 \otimes u_2
\]

\[
  b_1' = b_1 - \chi u_1, \qquad b_2' = b_2 - \chi u_2
\]

\[
  p_1' = p_1 - u_1\psi_1, \qquad p_2' = p_2 - u_2\psi_2,
  \qquad u_1 : M_1 \to L_1,\quad u_2 : M_2 \to L_2
\]

\[
  p_1' * p_2' = p_1 * p_2 - \Bigl[\underbrace{(\pi_1 - \chi p_1)}_{b_1(p_1)} \otimes u_2
  + u_1 \otimes \underbrace{(\pi_2 - \chi p_2)}_{b_2(p_2)}\Bigr]
  - \chi\, u_1 \otimes u_2
\]

\[
  u_1 * u_2 = -\chi\, u_1 \otimes u_2
\]

\[
  \Phi(x)(y) = -x\,\psi(y) + y\,\psi(x)
\]

\[
  \mathcal{H}om(\underline{O}, \mathcal{H})
  = \text{\struck{$\mathcal{H}om \ldots f \mapsto \lambda \ldots \exists \lambda$}}
  \ \bigl\{ f : \mathcal{E} \to \mathcal{H} \bigm| f \circ \alpha = \alpha(\lambda)\,\mathrm{id}_{\mathcal{H}} \bigr\}
\]

\[
  \xi_1,\ \eta_1 \longmapsto \chi\eta_1 + b_1(\xi_1),
  \qquad \xi_1 \in M_1 \otimes M_2,\ \eta_1 \in L_1 \otimes L_2
\]

\[
  p_1' = p_1 - u_1', \qquad p_2' = p_2 - u_2', \qquad p_1' * p_2' = p_1 * p_2 - u'
\]

\[
  \left\{
  \begin{aligned}
    u_1' &= -u_1 \\ u_2' &= -u_2 \\ u_3' &= -u_3
  \end{aligned}
  \right.
  \quad\leadsto\ldots\quad
  u' = -b_1 \otimes u_2 - u_1 \otimes b_2 + \chi\, u_1 \otimes u_2
\]

\[
  \boxed{u' = b_1 \otimes u_2' + u_1' \otimes b_2 + \chi\, u_1' \otimes u_2'}
\]

\[
  x \longmapsto \bigl[\, y \longmapsto (-x\,\psi(y) + y\,\psi(x)) \,\bigr]
\]

\[
  E \xrightarrow{\ \psi(x)\ } L
\]

\[
  {}_{\lambda_1}E_1 * {}_{\lambda_2}E_2 \simeq
\]

\[
  \boxed{
  \begin{aligned}
    (f_1 * f_2)(x_1 * x_2) &= \psi_1(f_1)\,\pi_1(x_1) \otimes f_2(x_2)
    + f_1(x_2) \otimes \psi_2(f_2)\,\pi_2(x_2) \\
    &\quad - \chi\, f_1(x_1) \otimes f_2(x_2) \\
    (f_1 * f_2)(u_1 \otimes u_2) &= \lambda_1\lambda_2\, u_1 \otimes u_2
  \end{aligned}}
\]

\[
  (f_1 * u_2)(x_1 * x_2) = \psi_1(f_1)\,\pi_1(x_1) \otimes \underbrace{u_2(x_2)}_{u_2(\psi_2(x_2))}
  - \chi\, f_1(x_1) \otimes u_2(x_2)
\]

\[
  \text{\struck{$(u_1 * f_2)(x_1 * x_2)$}}
\]

\[
  (f_1 * u_2)(x_1 * x_2) = \bigl(\underbrace{\psi_1(f_1)\,\pi_1 - \overset{\psi_1(\pi_1)}{\chi}\, f_1}_{\text{nul sur } L_1}\bigr) \otimes u_2\,(x_1 \otimes x_2)
\]

\[
  \bigl( [\pi_1', f_1']' \otimes u_2 \bigr)(x_1 \otimes x_2)
\]

\[
  \left\{
  \begin{aligned}
    (u_1 * f_2)(x_1 * x_2) &= \bigl(u_1 \otimes [\pi_2', f_2']'\bigr)(x_1 \otimes x_2) \\
    (u_1 * u_2)(x_1 * x_2) &= -\chi\, u_1(x_1) \otimes u_2(x_2)
  \end{aligned}
  \right.
\]

\[
  (x_1 + u_1) * (x_2 + u_2) = x_1 * x_2
  + \underbrace{x_1 * u_2}_{\underbrace{\pi_1(x_1)}_{b_1} \otimes u_2}
  + \underbrace{u_1 * x_2}_{u_1 \otimes \underbrace{\pi_2(x_2)}_{b_2}}
  + \underbrace{u_1 * u_2}_{\chi\, u_1 \otimes u_2}
\]

\[
  \text{\struck{$(f_1 * f_2)(x_1 * x_2) =$}}
  \qquad
  (f_1 * f_2) \mid E_1 \otimes E_2 : E_1 \otimes E_2 \to L_1 \otimes L_2
\]

\[
  \boxed{
  \begin{aligned}
    f_1 * f_2\,(x_1 * x_2) &= \lambda_1\pi_1(x_1) \otimes f_2(x_2)
    + f_1(x_1) \otimes \lambda_2\pi_2(x_2) - \chi\, f_1(x_1) \otimes f_2(x_2)
  \end{aligned}}
\]

\[
  f_1 * f_2 \mid E_1 \otimes E_2 \ \bigl[E_1 \otimes E_2 \to L_1 \otimes L_2\bigr]
  \doteq \lambda_1\pi_1 \otimes f_2 + f_1 \otimes \lambda_2\pi_2 - \chi\, f_1 \otimes f_2
\]

\[
  f_1 * f_2\,(u_1 \otimes u_2) = \lambda_1\lambda_2\,(u_1 \otimes u_2)
\]

\[
  f_1 * f_2\,(x_1 * u_2) = \lambda_1\lambda_2\, \pi_1(x_1) \otimes u_2
\]

\[
  \lambda_1\pi_1(x_1) \otimes \lambda_2 u_2
  + \text{\struck{$f_1(x_1) \otimes \lambda_2\chi\, u_2$}}
  - \chi\, \underbrace{f_1(x_1) \otimes \lambda_2 u_2}
\]

\[
  (\pi_1 * \pi_2)(x_1 * x_2) = \chi\,\{\pi_1(x_1) \otimes \pi_2(x_2)\}
\]

\[
  (\pi_1 * \pi_2)(u_1 \otimes u_2) = \chi^2\, u_1 \otimes u_2
  \qquad
  \frac{1}{\chi}\,\pi_1 \otimes \pi_2 = \pi
\]

\[
  x_1 * x_2 \equiv \text{\struck{$\psi_1(x_1)$}}
\]

\[
  \begin{array}{ll}
    L_i & p_i \\ M_i & q_i
  \end{array}
  \qquad
  1 + \underbrace{p_1 q_1},\quad 1 + \underbrace{p_2 q_2}
  \qquad 1 + (p_1 p_2)(q_1 q_2)
\]

\[
  \text{\struck{$[x,y] =$}}\quad
  x \longrightarrow \text{\struck{$\psi(x) = \ldots$}}\ \psi(x)\,T - \chi
\]

\[
  \begin{array}{l}
    M_1 \\ \downarrow\ \ q_1 \\ E_1 \\ \alpha_1 \downarrow\ \ p_1 \\ L_1
  \end{array}
\]

\[
  q(m_1 \otimes m_2) = q(m_1) * q(m_2)
\]

\[
  p(x_1 * x_2) = x_1 * x_2 - q(m_1 \otimes m_2)
\]

\[
  = x_1 * x_2 - q_1(m_1) * q_2(m_2)
\]

\[
  q(m_1) = x_1 - p_1(x_1), \qquad q(m_2) = x_2 - p_2(x_2)
\]

\[
  p(x_1 * x_2) = x_1 * x_2 - \bigl(x_1 - p_1(x_1)\bigr) * \bigl(x_2 - p_2(x_2)\bigr)
\]

\[
  = x_1 * p_2(x_2) + p_1(x_1) * x_2 - \underline{p_1(x_1) * p_2(x_2)}
\]

\[
  \text{\struck{$= \chi\,(x_1 \otimes p_2(x_2) + p_1(x_1) \otimes x_2$}}
\]

\[
  \text{\struck{$-\ \chi\, p_1(x_1) \otimes p_2(x_2))$}}
\]

\[
  u_1 * u_2 = \chi\, u_1 \otimes u_2,
  \qquad
  \text{\struck{$p(x_1 *$}}\ \ p(u_1 * u_2) = \chi\, u_1 \otimes u_2
\]

\[
  \boxed{
  (f_1 * f_2)(x_1 * x_2) = \lambda_1 x_1 * f_2(x_2) + f_1(x_1) * \lambda_2 x_2
  - \chi\, f_1(x_1) \otimes f_2(x_2)}
\]

\[
  2\lambda_1 \ldots\, \chi\,(x_1 \otimes x_2) - \chi\, \lambda_1\lambda_2\, x_1 \otimes x_2
\]

\[
  = \text{\struck{$(2 - \chi)\,\lambda_1\lambda_2\, u_1 \otimes u_2$}}
  \qquad 2 - \chi = 1
\]

\[
  (f_1 * f_2)(u_1 \otimes u_2) = \lambda_1\lambda_2\, u_1 \otimes u_2
  = f_1(u_1) \otimes f_2(u_2)
\]

\[
  \underbrace{xy - (x - x)(y - y)}_{\displaystyle xy + yx - xy} = xy
\]

\[
\boxed{\begin{gathered}
\chi\,\mathrm{id}-(\pi'_1+\pi'_2)+\gamma\,\pi'_1\otimes\pi'_2\\
\gamma\chi=2 \qquad \text{e.g. } \chi=2,\ \gamma=1
\end{gathered}}
\]

\[
\left\{\begin{aligned}
F_{11}(\text{\struck{\ill{}}}\,x)&=\bar x+x\\
\bar x_i&=x_i-\pi_i(x_i)
\end{aligned}\right.
\qquad
\bar x_1\otimes\bar x_2=\bar x_1\otimes\bar x_2
\]

\begin{align*}
&f_1\otimes f_2 \text{ sur } E_1\otimes E_2\\
&f_1^{!}\otimes f_2^{!} \text{ sur } L_1\otimes L_2
\end{align*}

\begin{align*}
f(x_1\otimes x_2)&=f_1(x_1)\otimes f_2(x_2)\\
f(u_1\otimes u_2)&=f_1^{!}(u_1)\otimes f_2^{!}(u_2)
\end{align*}

\[
f_1(x_1)\circledast u_2=f_1^{!}(\pi_1(x_1))\otimes u_2
\]

\[
f_1(x_1)=f_1(\pi\ \ldots
\]

\[
x_1-\pi_1(x_1)
\]

\[
E_1\otimes E_2\longrightarrow L_1\otimes L_2
\]

\[
\begin{array}{cc}
E_1 & E_2\\
{\scriptstyle f_1}\downarrow & \downarrow{\scriptstyle f_2}\\
L_1 & L_2\\
\lambda_1 & \lambda_2
\end{array}
\qquad
\begin{array}{cc}
f_1 & f_2\\
\lambda_1 & \lambda_2
\end{array}
\qquad
\frac{\lambda_1}{\lambda_2}\,f_2
\]

\[
\chi\, f_1(x_1)\otimes \overbrace{f_2(u_2)}^{\lambda_2 u_2}
\]

\[
\lambda_1\pi_1(x_1)=f_1(x_1)
\]

\[
T_1=2\cdot 1_1,\,-b_1\ =\ \begin{pmatrix}2&0\\-b_1&0\end{pmatrix}
\qquad
\bar T_1=\begin{pmatrix}0&0\\b_1&2\end{pmatrix}
\]

\begin{align*}
b_1 &: M_1\xrightarrow{\ \tau_1\ }E_1\xrightarrow{\ \pi_1\ }L_1\\
-b_1 &: M_1\xrightarrow{\ \varpi_1\ }E_1\xrightarrow{\ \tau_1\cdot\ }L_1
\end{align*}

\[
\left.\begin{aligned}
\tau'_1&=\tau_1+u_1\\
b'_1&=b_1+\chi u_1
\end{aligned}\ \right|\
\begin{aligned}
\tau'_2&=\tau_2+u_2\\
b'_2&=b_2+\chi u_2
\end{aligned}
\qquad u_i\in\mathrm{Hom}(M_i,L_i)
\]

\[
\beta_*(\xi)=b\in\mathrm{Hom}(M_1,L_1)\qquad \beta'(\eta)=-b
\]

\begin{align*}
x_1&\mapsto \pi_1(\tau_1^{*}(x_1))\\
x_1&\mapsto \pi_1(\tau_1(x_1)+u_1(x_1))=\pi_1(\tau_1(x_1))
\end{align*}

\begin{align*}
\sigma_1&=\mathrm{id}_1-\gamma\,\pi_1\\
\sigma_2&=\mathrm{id}_2-\gamma\,\pi_2
\end{align*}

\begin{align*}
\mathrm{id}_{12}+\sigma_1\otimes\sigma_2
&=\mathrm{id}_{12}+\mathrm{id}_{12}-\gamma(\pi'_1+\pi'_2)+\gamma^2\pi'_1\otimes\pi'_2\\
&=2\,\mathrm{id}_{12}-\gamma(\pi'_1+\pi'_2)+\gamma^2\pi'_1\otimes\pi'_2
\end{align*}

\[
\boxed{\gamma\chi=2}
\qquad
\begin{aligned}
\beta_1&=\beta_2=1-\gamma\chi\\
\alpha&=-\chi(1-\gamma\chi)
\end{aligned}
\qquad
\boxed{\begin{aligned}\beta&=-1\\ \alpha&=\chi\end{aligned}}
\]

\[
\sigma_i=\lambda 1_i+\mu\pi_i
\]

\begin{align*}
\mathrm{id}+\sigma_1\otimes\sigma_2
&=\underbrace{\mathrm{id}_{12}+\lambda^2\mathrm{id}_{12}}_{(1+\lambda^2)\,\mathrm{id}_{12}}
+\lambda\mu(\pi'_1+\pi'_2)+\mu^2\pi'_1\otimes\pi'_2
\end{align*}

\[
\boxed{\left\{\begin{aligned}
\lambda^2+1&=\chi\\
\lambda\mu&=-1\\
\mu^2&=\gamma
\end{aligned}\right.}
\]

\[
0\to L\to E\to M\to 0
\]

\[
0\to\mathrm{Hom}(M,L)\longrightarrow\mathcal{H}/\mathrm{id}\longrightarrow\frac{\mathcal{O}\times\mathcal{O}}{\mathcal{O}}\to 0
\qquad
\to\mathcal{H}\to\mathcal{O}\to
\]

\begin{align*}
\sigma(u)&=(\lambda(u)+\mu(u))\,\mathrm{id}_E-u\\
u+\sigma(u)&=(\lambda(u)+\mu(u))\,\mathrm{id}
\end{align*}

\[
\begin{array}{c}
\mathrm{Hom}(M\otimes M',\,L\otimes L')\\
\uparrow\\
\mathrm{Hom}(M,L)\otimes\mathrm{Hom}(M',L')
\end{array}
\]

\[
\bigl(x_1\otimes x_2+\bar x_1\otimes\bar x_2\bigr)
\quad
\bigl(x_1-\pi_1(x_1)\bigr)\bigl(x_2-\pi_2(x_2)\bigr)\bigr)
\]

\begin{align*}
\bar{\bar x}_1&=\bigl(x_1-\pi_1(x_1)\bigr)-\underbrace{\pi_1\bigl(x_1-\pi_1(x_1)\bigr)}_{-\pi_1(x_1)+\chi\pi_1(x_1)}=x_1\\
&\hphantom{=}\qquad\text{avec } \chi=2
\end{align*}

\[
\pi^2-\chi\pi=0\qquad \pi(\pi-\chi)=0 \qquad 0,\ \chi
\]

\[
\varpi_1(x_1)=2x_1-\pi_1(x_1)
\]

\begin{align*}
x_1-\varpi_1(x_1)
&=x_1-\bigl(2x_1-\pi_1(x_1)\bigr)=\pi_1(x_1)-x_1=-\bar x_1
\end{align*}

\[
\underbrace{(\chi-1)\,\mathrm{id}+(1-\pi_1)\otimes(1-\pi_2)}+(\gamma-1)\,\pi_1\otimes\pi_2
\]

\[
\underbrace{\beta_1\doteq\beta_2}_{\beta}=1-\gamma\chi
\qquad
\alpha=-\beta\chi=-\chi(1-\gamma\chi)
\]

\[
-\chi(1-\gamma\chi)\,\mathrm{id}+(1-\gamma\chi)(\pi'_1+\pi'_2)+\gamma\,\pi'_1\otimes\pi'_2
\qquad \gamma\in\mathbb{Z}
\]

\[
-\chi\underbrace{(1-\gamma\chi)}+2(1-\gamma\chi)\chi+\gamma\chi^2=\chi
\qquad
\text{\struck{$2\chi-\chi$}}\ \chi=\chi \quad \text{o.k.!}
\]

\begin{align*}
&-\chi\cdot(1-\gamma\chi)\,\mathrm{id}_{M_1\otimes M_2}
+(1-\gamma\chi)\bigl[(\tau_1\otimes\mathrm{id}_2)+\mathrm{id}_1\otimes\tau_2\bigr]\\
&\quad\text{\struck{$x_1\otimes x_2$}}+\gamma\,\tau_1\otimes\tau_2
\end{align*}

\[
\left\{\begin{aligned}
-\chi(1-\gamma\chi)&=\chi\\
1-\gamma\chi&=-1
\end{aligned}\right.
\qquad
1-\gamma\chi=-1
\qquad
\boxed{\gamma\chi=2}
\]

\[
\text{\struck{$2\,\mathrm{id}$}}\quad
2\,\mathrm{id}_{E_1\otimes E_2}-(\pi_{E_1}\otimes\mathrm{id}_{E_2}+\mathrm{id}_{E_1}\otimes\pi_{E_2})
+\pi_{E_1}\otimes\pi_{E_2}
\]

\begin{align*}
x_1\otimes x_2\longmapsto{}& 2x_1\otimes x_2-\pi_1(x_1)\otimes x_2-x_1\otimes\pi_2(x_2)\\
&+\pi_1(x_1)\otimes\pi_2(x_2)
\end{align*}

\begin{align*}
\bar x_1&=\tau_1\psi_1(x_1)-x_1=x_1-\pi_1(x_1)\\
\pi_1(x_1)&=2x_1-\tau_1\psi_1(x_1)
\end{align*}

\[
\text{\struck{$E_1\otimes E_2\to$}}\quad
L_1\otimes L_2+M_1\otimes L_2+L_1\otimes M_2+M_1\otimes M_2
\]

\[
\begin{array}{c|cccc}
 & L_1\otimes L_2 & L_1\otimes M_2 & M_1\otimes L_2 & M_1\otimes M_2\\
\hline
L_1\otimes L_2 & \chi & \mathrm{id}_{L_1}\otimes\tau_2 & \tau_1\otimes\mathrm{id}_{L_2} & ?\ (\tau_1\otimes\tau_2\,?)\\
L_1\otimes M_2 & 0 & 0 & 0 & \doteq\ \tau_1\otimes\mathrm{id}_{M_2}\\
M_1\otimes L_2 & 0 & 0 & 0 & \doteq\ \mathrm{id}_{M_1}\otimes\tau_2\\
M_1\otimes M_1 & 0 & 0 & 0 & \chi
\end{array}
\]

\[
E_1\otimes E_2\qquad E_1\otimes E_2
\]

\[
\pi'_1=\alpha_1\pi_1\ \Big|\ \varpi'_1=\varpi_1\psi_1\qquad \pi'_1+\varpi
\]

\[
\pi'_2=\alpha_2\pi_2\qquad \varpi'_2=\omega_2\psi_2
\]

\[
\boxed{\begin{aligned}
&-\chi(1-\delta\chi)\,x_1\otimes x_2+(1-\delta\chi)\bigl[x_1\otimes\pi_2(x_2)+\pi_1(x_1)\otimes x_2\bigr]\\
&\qquad+\delta\,\pi_1(x_1)\otimes\pi_2(x_2)
\end{aligned}}
\]

\begin{align*}
\alpha&=-2(1-2)=2\\
\beta&=\gamma=1-2=-1
\end{align*}

\[
2x_1\otimes x_2-\bigl[(x_1\otimes\pi_2(x_2)+\pi_1(x_1)\otimes x_2)\bigr]+\pi_1(x_1)\otimes\pi_2(x_2)
\]

\begin{align*}
&\text{\struck{$\pi_i(x_i)=2x_i-\tau_i\psi_i(x_i)$}}\\
\pi_i(x_i)&=2x_i-\tau_i\psi_i(x_i)=2x_i-(x_i+\bar x_i)=x_i-\bar x_i\\
\bar x_i&=\tau_i\psi_i(x_i)-x_i\\
x_i+\bar x_i&=\tau_i\psi_i(x_i)
\end{align*}

\begin{align*}
&2x_1\otimes x_2-\bigl[(x_1\otimes(x_2-\bar x_2))+(x_1-\bar x_1)\otimes x_2\bigr]\\
&\qquad+(x_1-\bar x_1)\otimes(x_2-\bar x_2)\\
&=x_1\otimes x_2+\bar x_1\otimes\bar x_2 \qquad\text{o.k.}
\end{align*}

\begin{align*}
\bar x_i&=x_i-\pi_i(x_i)=\Bigl\{\\
&x_1\otimes x_2+\text{\struck{$(x_i-\pi_i(x_i))$}}
\end{align*}

\[
-\chi(1-\delta\chi)\overset{?}{=}\chi\qquad -1+\delta\chi=1\qquad \boxed{\delta\chi=2}
\]

\[
(\alpha+\chi\gamma)\,u_1\otimes x_2+(\beta+\delta\chi)\,u_1\otimes\pi_2(x_2)
\]

\[
(\alpha+\chi\gamma)\,u_1\otimes x_2+(\beta+\delta\chi-1)\,u_1\otimes\pi_2(x_2)=0
\]

\[
(\alpha+\chi\gamma)+\chi(\beta+\delta\chi-1)=0
\]

\[
u_1\otimes\bigl[(\alpha+\chi\gamma)\,x_2+(\beta+\delta\chi-1)\,\pi_2(x_2)\bigr]=0
\]

\[
(\alpha+\chi\gamma)\,\mathrm{id}=0
\qquad
\alpha+\chi\gamma=0
\qquad
\alpha=-\chi\gamma
\]

\[
-\chi\beta\,x_1\otimes x_2+\beta\bigl(x_1\otimes\pi_2(x_2)+\pi_1(x_1)\otimes x_2\bigr)+\delta\,\pi_1(x_1)\otimes\pi_2(x_2)
\]

\[
\text{\struck{$\ldots\oplus(\beta\ldots$}}\quad(\beta+\delta\chi-1)\,\pi_2(x_2)=0
\]

\[
\boxed{\beta+\delta\chi=1}
\qquad
\begin{aligned}
\beta&=1-\delta\chi\\
\alpha&=-\chi\beta=\text{\struck{$-\chi+\chi^2$}}
\end{aligned}
\]

\[
\beta=\beta=1-\delta\chi\qquad \alpha=-\chi(1-\delta\chi)
\]

\[
\begin{gathered}
0\to L_1\to E_1\to L'_1\to 0\\
0\to L_2\to E_2\to L'_2\to 0
\end{gathered}
\]

\[
\longrightarrow\qquad 0\to L^{!}_1\otimes L_2\to E_1\wedge E_2\to L'_1\otimes L'_2\to 0
\]

\[
0\leftarrow L_1^{\vee}\otimes L_2^{\vee}\leftarrow E_1^{\vee}\wedge E_2^{\vee}\leftarrow L_1'^{\vee}\otimes L_2'^{\vee}\leftarrow 0
\]

\[
\mathrm{Hom}(L'_1,\mathcal{O})\otimes\mathrm{Hom}(L'_2,\mathcal{O})\simeq\mathrm{Hom}(L'_1\otimes L'_2,\mathcal{O})
\]

\[
\text{\struck{$f_1$}}\otimes f_2+\bar f_1\otimes\bar f_2,\qquad u_1\otimes u_2
\]

\[
\begin{array}{c}
A_1\otimes A_2 \curvearrowleft\\
\downarrow\quad A_1\otimes_0 A_2\\
L_1\otimes L_2\quad \pi_1\otimes\mathrm{id}\\
\phantom{L_1\otimes L_2}\quad\mathrm{id}\otimes\pi_2
\end{array}
\qquad
\begin{aligned}
&\bigl(\mathrm{Tr}_1(f_1)-f_1\bigr)\otimes\bigl(\mathrm{Tr}_2(f_2)-f_2\bigr)\\
&\bigl(\tau'_1(f_1)-f_1\bigr)\bigl(\pi_2\psi_2)-f_2\bigr)
\end{aligned}
\]

\[
\text{\struck{$f_1$}}\qquad \text{\struck{$f_2$}}\qquad f_1
\]

\[
\pi_1(x_1)-\ldots
\qquad
\text{\struck{$\sigma=\pi$}}\qquad
\sigma(x)=\pi(x)\cdot 1-x
\qquad
\sigma=\pi\otimes e-\mathrm{id}
\]

\[
\begin{aligned}
\pi_1(x_1)\qquad x_1\otimes x_2&\longmapsto \text{\struck{$\pi_1(x_1)\otimes x$}}\quad u_1\otimes\pi_2(x_2)\\
x_1\otimes x_2&\longmapsto \pi_1(x_1)\otimes u_2
\end{aligned}
\qquad
u_1\otimes\underline{\pi\sigma(x_2)}=\ldots
\]

\[
\boxed{x_1\otimes x_2+\pi'_1(x_1)\otimes\pi'_2(x_2)}
\]

\[
\left\{\begin{aligned}
&\pi_2(x)=\overbrace{x}^{(\chi-1)x}+\pi'_2(x)\\
&\text{si } x\in L_2,\ \ldots\\
&\pi'_2(x)=x
\end{aligned}\right.
\]

\[
x \longmapsto \pi'(x) = \pi(x) - (\chi - 1)\,x
\qquad\qquad 0 \to L \to E \to L' \to 0
\]

\[
\left\{
\begin{array}{l}
\pi'|L = \text{\struck{\ill{}}}\ \mathrm{id}_L \\
\pi'|E/L = L' \;\;)= -(\chi - 1)\,\mathrm{id}_{L'}
\end{array}
\right.
\iff \pi|L = \chi\,\mathrm{id}_L
\]

\[
\iff \pi : E \to L
\]

\[
x_1 \otimes x_2 + \underbrace{\pi_1'(x_1)}_{u_1} \otimes \pi_2'(x_2)
= u_1 \otimes \bigl(\underbrace{x_2 + \pi_2'(x_2)}_{\chi_2 + \pi_2(x_2) - (\chi - 1) x_2}\bigr)
\]

\[
= (2 - \chi)\,x_2 + \pi_2(x_2) \qquad ?
\]

\[
\alpha\, x_1 \otimes x_2 + \beta\, x_1 \otimes \pi_2(x_2) + \gamma\, \pi_1(x_1) \otimes x_2 + \delta\, \pi_1(x_1) \otimes \pi_2(x_2)
\]

\[
\pi_i' :\quad
\begin{array}{c|cc}
 & L_i & M_i \\ \hline
L_i & \chi & \tau_i \\
M_i & 0 & 0
\end{array}
\qquad\qquad
\varpi_i' :\quad
\begin{array}{c|cc}
 & L_i & M_i \\ \hline
L_i & 0 & -\tau_i \\
M_i & 0 & \chi
\end{array}
\]

\[
\begin{array}{c|ccc}
1 & 1 & 1 & 1 \\ \hline
\pi_1' & \chi & \chi & \tau_1 \otimes \mathrm{id}_{L_2} \\
\pi_2' & \chi & \mathrm{id}_{L_1} \otimes \tau_2 & \chi \\
\pi_1' \otimes \pi_2' & \chi^2 & \chi\, \mathrm{id}_{L_1} \otimes \tau_2 & \chi\, \tau_1 \otimes \mathrm{id}_{L_2}
\end{array}
\]

\[
\alpha + \beta_1 \pi_1' + \beta_2 \pi_2' + \gamma\, \pi_1' \otimes \pi_2'
\]

\[
\boxed{\alpha + (\beta_1 + \beta_2)\chi + \gamma \chi^2 = \chi}
\]

\[
\alpha + \beta_1 \chi + \underbrace{\beta_2\, \mathrm{id} \otimes \tau_2 + \gamma \chi\, \mathrm{id} \otimes \tau_2}_{(\beta_2 + \gamma\chi)\, \mathrm{id} \otimes \tau_2} = \mathrm{id}_{L_1} \otimes \tau_2
\]

\[
\boxed{\begin{array}{l}
\alpha + \beta_1 \chi = 0 \\
\beta_2 + \gamma \chi = 1
\end{array}}
\]

\[
(\pi_1, \varpi_1)\ (\pi_2, \varpi_2)
\left\{
\begin{array}{l}
\alpha_i \pi_i + \varpi_i \psi_i = \chi\, \mathrm{id}_{E_i} \\
\text{\struck{$\pi_1 \varpi_2 = \chi\, \mathrm{id}_{E_2}$}} \\
\pi_i \alpha_i = \chi\, \mathrm{id}_{L_i} \qquad \text{\struck{\ill{}}} = 0 \\
\psi_i \varpi_i = \chi\, \mathrm{id}_{M_i}
\end{array}
\right.
\]

\[
\left[\;
\begin{array}{c}
E_1 \otimes L_2 \searrow \\
L_1 \otimes L_2 \qquad E_1 \otimes_0 E_2 \quad \text{cocartésien} \\
L_1 \otimes E_2 \nearrow
\end{array}
\;\right]
\]

\[
u = u_1 \star u_2 - b_1 b_2 = 2 u_1 u_2 + b_2 u_1 + b_1 u_2 \in \mathcal{A}_1 \otimes \mathcal{A}_2
\]

\[
u^2 + bu + c = 0 \quad \text{dans } \mathcal{A}
\]

\[
b = b_1 b_2, \qquad c = b_1^2 c_2 + b_2^2 c_1 - 4 c_1 c_2
\]

\[
\delta = b^2 - 4c = \underbrace{\delta_1}_{(b_1^2 - 4c_1)} \underbrace{\delta_2}_{(b_2^2 - 4c_2)}
\]

\[
f \in \mathcal{A}_1 \otimes \mathcal{A}_2 \longmapsto f^{\sigma} \;\text{\struck{$(\sigma_1 \otimes \sigma_2 + \mathrm{id})(f)$}}
\]

\[
= \bigl(\underbrace{\sigma_1 \otimes \sigma_2}_{\sigma} + \mathrm{id}_{\mathcal{A}_1 \otimes \mathcal{A}_2}\bigr)(f) = f + \bar f
\]

\[
x \in E_1 \otimes E_2 \qquad x^{\sigma} = x + \bar x
\]

\[
0 \to L_1 \to E_1 \to L_1' \to 0, \qquad 0 \to L_2 \to E_2 \to L_2' \to 0
\]

\[
0 \to L \xrightarrow{\;i\;} E \xrightarrow{\;\varphi\;} L' \to 0
\]

\[
E \xrightarrow{\;\pi\;} L \quad \text{tel que } \pi(x) = \chi x \text{ si } x \in L
\]

\[
L' \xrightarrow{\;\varpi\;} E \quad \text{tel que } \varphi(\varpi(\lambda)) = \chi \lambda \text{ si } \lambda \in L'
\]

\[
\boxed{\pi = \chi\,\mathrm{id} - f}
\]

\[
0 \to L^{\vee} \hookrightarrow \mathcal{A} \to L^{\vee} \to 0
\]

\[
0 \to L \to E_n \to L' \to 0
\]

\[
0 \to L_1 \to E_1 \to L_1' \to 0 \qquad \text{\struck{$E_1 \otimes E_2$}} \qquad 0 \to L_2 \to E_2 \to L_2' \to 0
\]

\[
0 \to L_1 \to E_1 \to L_1' \to 0
\]

\[
0 \to L_2 \to E_2 \to L_2' \to 0 \qquad E_1 \simeq \mathcal{A}_1 \otimes \Delta_1
\]

\[
E_1 \wedge E_2 \qquad
\overset{\Delta_1^{-1}}{E_1^{\vee}} \wedge \overset{\Delta_2^{-1}}{E_2^{\vee}}
\simeq \bigl(\underbrace{E_1 \wedge E_2}_{\Delta_1 \Delta_2}\bigr) \otimes \Delta_1^{-1} \otimes \Delta_2^{-1}
\]

\[
\simeq (E_1 \wedge E_2)^{\vee}
\]

\[
\pi(x) = 2x - F(x)
\]

\[
\pi(x) - x = x - F(x)
\]

\[
\begin{array}{llcll}
\mathcal{A}_i & & & \mathcal{A} & u \qquad 1 \\[4pt]
E_i & x_i,\ e_i & & E & x_1 \star x_2 = x,\quad e = e_1 \otimes e_2 \\
 & \cap \quad \cap & & & \cap \\
 & P_i \quad L_i & & & P_3
\end{array}
\]

\[
\begin{array}{ll}
\langle u_i, x_i \rangle = \langle 1_i, e_i \rangle = 0 & \qquad \langle u, e \rangle = \langle 1, x \rangle = 1 \\
\langle u_i, e_i \rangle = \langle 1_i, x_i \rangle = 1 & \qquad \langle u, x \rangle = \langle 1, e \rangle = 0
\end{array}
\]

\[
\lambda = \langle \underbrace{u_1 \star u_2}_{u}, x \rangle, \qquad \mu = \langle u_1 \star u_2, e \rangle
\]

\[
\langle u_1 \star u_2, x_1 \otimes x_2 \rangle
= \langle \underbrace{u_1 \otimes u_2 + \bar u_1 \otimes \bar u_2}_{(u_1 \otimes u_2)^{\sigma}}, x_1 \otimes x_2 \rangle
\]

\[
= \langle 2 u_1 \otimes u_2 + b_1 u_2 + u_1 b_2 + b_1 b_2,\ x_1 \otimes x_2 \rangle
\]

\[
= b_1 b_2
\]

\[
\mu = \langle u_1 \star u_2, e \rangle
\]

\[
= 1 \qquad e \in T\,\mathcal{L} \subset T\!E
\]

\[
u_1 \star u_2 = b_1 b_2 + \text{\struck{\ill{}}}\ u
\]

\[
\boxed{u_1 \star u_2 - b_1 b_2 = u}
\]

\[
u_1 \star u_2 - \mathrm{Tr}_1(u_1)\,\mathrm{Tr}_2(u_2) = 2 \,\cdot
\]

\[
\boxed{2 u_1 \otimes u_2 + b_2 u_1 + b_1 u_2}
\]

\[
\langle \underbrace{f - f^{\sigma}}_{\mathrm{Tr}(f).1}, x \rangle = \mathrm{Tr}(f)\, \langle 1 \otimes \text{\uncertain{$1$}}, x \rangle
\]

\[
\langle f, x + \bar x \rangle = \langle f, \pi(x) \rangle = \varphi(f) . \pi(x)
\]

\[
\mathcal{A} \overset{\mathrm{def}}{=} (\mathcal{A}_1 \otimes \mathcal{A}_2) \oplus_{(\mathcal{A}_1 \otimes_0 \mathcal{A}_2)} \underline{O}
\]

\[
E \overset{\mathrm{def}}{=} (E_1 \otimes E_2) \oplus_{E_1 \otimes_0 E_2} L_1 \otimes L_2
\]

\[
\sigma = \sigma_1 \otimes \sigma_2 \quad \text{dans } \mathcal{A}_1 \otimes \mathcal{A}_2 \text{ et } E_1 \otimes E_2,
\]

\[
f \longmapsto f^{\sigma} = f + \sigma f, \quad f \in \mathcal{A}_1 \otimes \mathcal{A}_2, \qquad x^{\sigma} = x + \sigma x, \quad x \in E_1 \otimes E_2
\]

\[
\langle f, x^{\sigma} \rangle = \langle f^{\sigma}, x \rangle = \langle f, x^{\sigma} \rangle
\]

\[
\langle f_1 \otimes f_2, \underbrace{x_1 \otimes x_2}_{?} \rangle
= \bigl\langle f_1 \otimes f_2 + (\mathrm{Tr}_1(f_1) - f_1)(\mathrm{Tr}_2(f_2) - f_2),\ x_1 \otimes x_2 \bigr\rangle
\]

\[
= \bigl\langle f_1 \otimes f_2,\ (\pi_1(x_1)T_1 - x_1) \otimes (\pi_2(x_2)T_2 - x_2)
\]

\[
\qquad + x_1 \otimes x_2 \bigr\rangle
\]

\[
\begin{array}{ll}
\langle \lambda, u \rangle = 0 & \text{si } \lambda \in \underline{O},\ u \in L_1 \otimes L_2 \\[2pt]
\langle f, u \rangle = \langle \varphi(f), u \rangle = \varphi(f).u & \text{si } f \in \mathcal{A}_1 \otimes \mathcal{A}_2,\ u \in L_1 \otimes L_2 \\[2pt]
\langle \lambda, x \rangle = \lambda\, \pi(x) & \text{si } x \in E_1 \otimes E_2,\ \lambda \in \underline{O}
\end{array}
\]

\[
(f+\sigma(f))\,\varphi(x) \;=\; \langle f+\sigma(f),\, x\rangle
\]

\[
\varphi(f)\,(x+\sigma(x)) \;=\; \langle f,\, x+\sigma x\rangle
\]

\[
\langle (1+\sigma)f,\, x\rangle
\]

\[
\text{\struck{$A_1$}}\otimes A_2)\otimes(E_1\otimes E_2)\longrightarrow \underline{O}
\]

\[
\mapsto\;\bigl[(A_1\otimes_0 A_2)\otimes(E_1\otimes E_2)\bigr]\oplus
(A_1\otimes_0 A_2)\otimes(E_1\otimes_0 E_2)
\]

\[
\begin{array}{c|c}
f_1 & f_2\\
\varphi_1(f_1)\in L_1^{\vee} & \varphi_2(f_2)\\
\hline
x_1,\ T_1 & x_2,\ T_2\\
\varphi_1(x_1)\in k & \varphi_2(x_2)\\
\pi_1(x_1)\in L_1 & \pi_2(x_2)
\end{array}
\]

\[
\langle (1+\sigma)f,\, x\rangle =
\]

\[
\tfrac12\bigl[(f+\sigma f)\bigr]\bigl[
\]

\[
\langle f+\sigma f,\, x\rangle
\]

\[
A\simeq (A_1\otimes A_2)\oplus \cdots\;\big/\;(A_1\oplus_k A_2)
\]

\[
A_1\oplus_k A_2 \xrightarrow{\ \mathrm{Tr}_1,\ \mathrm{Tr}_2\ } k,
\qquad
A_1\oplus_k A_2 \xhookrightarrow{\ \text{can.}\ } A_1\otimes A_2
\]

\[
E\simeq (E_1\otimes E_2)\oplus(L_1\otimes L_2)\;\big/\;
(E_1\otimes L_2)+_{L_1\otimes L_2}(L_1\otimes E_2)
\]

\[
0\to k\to A\to L_1^{\vee}\otimes L_2^{\vee}\to 0
\]

\[
0\to L_1\otimes L_2\to E\to \underline{O}\to 0
\]

\[
\langle f_1\otimes f_2,\ \cdots\rangle
\;=\;
\text{\struck{$\langle\varphi_1(x_1),u_1\rangle\langle\varphi_2(x_2),u_2\rangle$}}
\;\varphi_1(f_1)\varphi_2(f_2)\cdot u
\]

\[
\langle\lambda,\ x_1\otimes x_2\rangle = \lambda\,\psi_1(f_1)\psi_2(f_2)
= \lambda\,\psi(x_1\otimes x_2)
\]

\[
\langle\lambda,\ u\rangle = 0
\]

\[
\langle f_1\otimes f_2,\ x_1\otimes x_2\rangle
\]

\[
\begin{array}{cccc}
1, & u_1, & u_2, & u_1u_2\\
\downarrow & \downarrow & \downarrow & \downarrow\\
2 & \cdot, & \cdot & 2u_1u_2=v
\end{array}
\qquad \sigma_1\sigma_2
\]

\[
v\longmapsto 2u_1u_2
\]

\[
-u_1^2=c_1\cdot 1,\qquad -u_2^2=c_2\cdot 1,\qquad u_1^2u_2^2=c_1c_2
\]

\[
x_1\otimes x_2\longmapsto x_1\otimes x_2+\overline{x}_1\otimes\overline{x}_2
\]

\[
E_1\otimes L_2 \ni x_1\otimes u_2
\]

\[
\pi_1(x_1)\otimes u_2=(2x_1-\overset{\psi_1(x_1)}{T_1})\otimes u_2
\]

\[
x_1\otimes u_2+(\psi_1(x_1)T_1-x_1)\otimes(-u_2)
\]

\[
=2x_1\otimes u_2-\psi_1(x_1)T_1\otimes u_2
\]

\[
(x_1+u_1)*(x_2+u_2)=(\sigma+\overline{\sigma})\bigl[(x_1+u_1)\otimes(x_2+u_2)\bigr]
\]

\[
=x_1*x_2+\pi_1(x_1)\otimes u_2+u_1\otimes\pi_2(x_2)+\underbrace{u_1*u_2}_{=\,2u_1\otimes u_2}
\]

\[
b_1\otimes u_2+u_2\otimes b_1+2u_1\otimes u_2
\]

\[
(x_1+u_1)*(x_2+u_2) = x_1*x_2+\underbrace{(b_1u_2+b_2u_1+2u_1u_2)}\ \in E_1\wedge E_2
\]

\[
A_1\otimes A_2\to (A_1\otimes A_2)\oplus(A_1\oplus_0 A_2)\to A
\]

\[
A_1\otimes A_2\xrightarrow{\ \text{can}\ }(A_1\otimes A_2)\oplus(A_1\oplus_0 A_2)
\xrightarrow{\ \text{somme}\ } A
\]

\[
\text{\struck{$A^{\vee}\simeq A_1\otimes A_2$}}\times\cdots\;\big/\;\mathrm{Im}(A_1\oplus_k A_2)
\]

\[
A^{\vee}\simeq(f,\overset{\lambda}{\cdots})\quad\ldots
\]

\[
f(x_1,1)=2g_1\qquad x_1\in A_1
\]

\[
f(1,x_2)=2g_2\qquad x_2\in A_2
\]

\[
f(x_1)=\lambda\,\mathrm{Tr}_1(x_1)
\]

\[
f(x_2)=\lambda\,\mathrm{Tr}_2(x_2)
\]

\[
0\to L_1\otimes L_2\to E\xrightarrow{\ \psi\ }\underline{O}\to 0
\]

\[
u_1\overset{\xi}{*}u_2=\underset{\sim}{\lambda}\,u+\underset{\sim}{\mu}
\qquad\text{trouver } \lambda,\mu
\]

\[
\mathrm{Tr}_{A/k}(u_1*u_2)=-\lambda\beta+2\mu
\]

\[
N_{A/k}(u_1*u_2)=\underline{\lambda^2}c+\underline{\mu^2}-\underline{\lambda\mu\beta}
\]

\[
\langle 1,x\rangle=1\qquad \langle 1,e\rangle=0
\]

\[
\langle u,e\rangle=1\qquad \langle u,x\rangle=0
\]

\[
\lambda=\text{\struck{$\langle u,e\rangle$}}\;\langle u_1*u_2,\ \overbrace{e_1\otimes e_2}^{e}\rangle
\]

\[
\mu=\langle u_1*u_2,\ \underbrace{x_1*x_2}_{x}\rangle
\]

\[
\text{\struck{$(x,y)\mapsto x\otimes\cdots$}}\qquad
{}^t(\mathrm{id}+\sigma_1\otimes\sigma_2)=\mathrm{id}+({}^t\sigma_1)\otimes({}^t\sigma_2)
\]

\[
\begin{cases}
\langle u,\ e_1\otimes e_1\rangle=1\\
\langle u,\ x_1*x_2\rangle=0
\end{cases}
\]

\[
u_1\qquad u_1^2+b_1u_1+c_1=0
\]

\[
u_2\qquad u_2^2+b_2u_2+c_2=0
\]

\[
u_1*u_2=u_1\otimes u_2+(u_1+b_1)\otimes(u_2+b_2)=2u_1u_2+b_2u_1+b_1u_2+\underline{\underline{b_1b_2}}
\]

\[
A_1\simeq\underline{O}\oplus L_1^{\vee}\qquad A_2=\underline{O}\oplus L_2^{\vee}
\]

\[
E_1\simeq\underline{O}+L_1,\qquad E_2\simeq\underline{O}+L_2
\]

\[
E_1\otimes E_2\to E_1\wedge E_2\to E_1\otimes E_2
\]

\[
x_1\otimes x_2\qquad x_1*x_2\qquad \mathrm{Tr}(x_1\otimes x_2)
\]

\[
A\simeq\underline{O}\oplus L_1^{\vee}\otimes L_2^{\vee}
\]

\[
b_1=\beta_1e_1\qquad c_1=\gamma_1e_1^{\otimes 2}
\]

\[
b_2=\beta_2e_2\qquad c_2=\gamma_2e_2^{\otimes 2}
\]

\[
A\simeq\underline{O}+e_1^{*}\ldots\qquad u_1=e_1^{*}\qquad u_2=e_2^{*}
\]

\[
u_1^2+\beta_1u_1+\gamma_1=0
\]

\[
u_2^2+\beta_2u_2+\gamma_2=0
\]

\[
x_1,\ e_1\ \longleftrightarrow\ u_1\in A_1\qquad u_1^2+\beta_1u_1+\gamma_1=0
\]

\[
\begin{cases}
e_1=\varphi(u_1)\\
\langle u_1,\ x_1\rangle=0\\
\varphi(x_1)=1
\end{cases}
\]

\[
(x_2,\ e_2)\ \longleftarrow\ u_2\in A_2\qquad u_2^2+\beta_2u_2+\gamma_2=0
\]

\[
(x_1*x_2,\ e_1\otimes e_2)\ \longleftrightarrow\ u\in A.
\]

\[
u^2+\beta u+\gamma=0
\]