\[
  \varphi_L \colon K^{\cdot}(A) \longrightarrow K^{\cdot}(X)
\]

\[
  \Phi_L(F) = \mathbb{R}pr_{2*}\bigl(\mathbb{L}pr_1^{*}(F) \otimes L\bigr)
\]

\[
  \Phi_L\bigl(F \overset{L}{*} G\bigr) \;\simeq\;
  \Phi_L(F) \overset{L}{\otimes} \Phi_L(G)
  \qquad (F, G \in \mathrm{Ob}\, D_{\mathrm{parf}}(A)).
\]

\[
  (*) \qquad F \overset{L}{*} G = \mathbb{R}\pi_{*}\bigl(F \overset{L}{\otimes}_S G\bigr)
  \quad \text{et}
\]

\begin{align*}
\Phi\bigl(F \overset{L}{*} G\bigr)
  &\underset{\text{déf}}{\simeq}
    \mathbb{R}pr_{2*}\bigl(\mathbb{L}pr_1^{*}(F \overset{L}{*} G) \otimes L\bigr)
  \underset{(*)}{=}
    \mathbb{R}pr_{2*}\bigl(\mathbb{L}pr_1^{*}(\pi_{*}(F \overset{L}{\otimes}_S G)) \otimes L\bigr) \\
  &\underset{\text{chgt. de base}}{\simeq}
    \mathbb{R}pr_{2*}\bigl(\mathbb{R}(\pi \times \mathrm{id}_B)_{*}(\mathbb{L}pr_{12}^{*}(F \overset{L}{\otimes}_S G)) \otimes L\bigr) \\
  &\underset{\text{formule de proj.}}{\simeq}
    \mathbb{R}pr_{2*}\bigl(\mathbb{R}(\pi \times \mathrm{id}_B)_{*}(\mathbb{L}pr_{12}^{*}(F \overset{L}{\otimes}_S G) \\
  &\hphantom{\underset{\text{formule de proj.}}{\simeq}\mathbb{R}pr_{2*}\bigl(}
    \otimes (\pi \times \mathrm{id}_B)^{*}(L))\bigr) \\
  &\underset{\text{transit.}}{\simeq}
    \mathbb{R}pr_{3*}\bigl(\mathbb{L}pr_{12}^{*}(F \overset{L}{\otimes}_S G) \otimes (\pi \times \mathrm{id}_B)^{*}(L)\bigr)
\end{align*}

\[
  (\pi \times \mathrm{id}_B)^{*}(L) \simeq pr_{13}^{*}(L)\, pr_{23}^{*}(L)
\]

\[
  \mathbb{L}pr_{12}^{*}(F \overset{L}{\otimes}_S G) \otimes (\pi \times \mathrm{id}_B)^{*}(L)
  \;\simeq\;
  \mathbb{L}f^{*}\bigl(pr^{\prime *}_{12}(F \overset{L}{\otimes}_S G) \otimes pr^{\prime *}_{13}(L) \otimes pr^{\prime *}_{24}(L)\bigr)
\]

\[
  pr^{\prime *}_{12}(F \overset{L}{\otimes}_S G) \otimes pr^{\prime *}_{13}(L) \otimes pr^{\prime *}_{24}(L)
  \;\simeq\;
  pr^{\prime *}_{13}(F \otimes L) \overset{L}{\otimes} pr^{\prime *}_{24}(G \otimes L)
\]

\begin{align*}
\Phi\bigl(F \overset{L}{*} G\bigr)
  &\simeq \mathbb{R}pr_{3*}\bigl(\mathbb{L}f^{*}(\quad)\bigr)
  \underset{\text{chgt. de base}}{\simeq}
    \mathbb{L}\Delta_B^{*}\Bigl(\mathbb{R}pr_{34*}\bigl(pr^{\prime *}_{13}(F \otimes_S L) \overset{L}{\otimes} pr^{\prime *}_{24}(G \otimes_S L)\bigr)\Bigr) \\
  &\underset{\text{Künneth}}{\simeq}
    \mathbb{L}\Delta_B^{*}\bigl(\mathbb{R}pr_{2*}(F \otimes L) \overset{L}{\otimes}_S \mathbb{R}pr_{2*}(G \otimes L)\bigr) \\
  &\underset{\text{déf}}{\simeq}
    \mathbb{L}\Delta_B^{*}\bigl(\Phi(F) \overset{L}{\otimes}_S \Phi(G)\bigr)
  \simeq \Phi(F) \overset{L}{\otimes} \Phi(G)
\end{align*}

\[
  K(A) \xrightarrow{\;\varphi_L\;} K(B) \xrightarrow{\;\varphi_{{}^{s}L}\;} K(A)
\]

\[
  D_{\mathrm{parf}}(A) \xrightarrow{\;\Phi_L\;} D_{\mathrm{parf}}(B)
  \xrightarrow{\;\Phi_{{}^{s}L}\;} D_{\mathrm{parf}}(A)
\]

\[
  \Phi_L\bigl(F \overset{L}{\otimes} G\bigr) \otimes \omega_A[n]
  \;\simeq\;
  \bigl[\Phi_L(F) \otimes \omega_A[n]\bigr] * \bigl[\Phi_L(G) \otimes \omega_A[n]\bigr]
\]

\[
  \Phi_L\bigl(F \overset{L}{\otimes} G\bigr)
  \;\simeq\;
  \bigl[\Phi_L(F) * \Phi_L(G)\bigr] \otimes \omega_A[n].
\]

\[
  M = \mathbb{R}pr_{13*}\bigl(pr_{12}^{*}(L)\, pr_{23}^{*}({}^{s}L)\bigr)
\]

\[
  \mathbb{R}pr_{2*}(N) \simeq \mathbb{R}pr_{2*}(\Psi \times \mathrm{id}_B)^{*}(W_B)
  \underset{\text{chgt de base}}{\simeq}
  \mathbb{L}\Psi^{*}\bigl(\mathbb{R}pr'_{2*}(W_B)\bigr).
\]

\[
  \mathbb{R}p_{*}(W) \simeq e_{*}\bigl(\struck{\ill{}}\; \tau_{B^{*}}[-n]\bigr)
\]

\[
  \mathbb{R}pr_{1*}(N) \simeq \mathbb{L}\Psi^{*}\bigl(\uncertain{e_{*}\tau_{B^{*}}}[-n]\bigr)
\]

\[
  \mathbb{R}pr_{1*}(N) \simeq \mathcal{O}_K[-n] \otimes_S \tau_{B^{*}}
\]

\[
  H^{i}(X, L) = 0 \quad \text{pour tout } i
\]

\[
  \mathbb{R}p_{*}(W) \longrightarrow \struck{\ill{}}\; e_{*}(\mathcal{O}_S) \otimes_{\mathcal{O}_S} \tau_{B^{*}}[-n]
\]

\[
  R^{n}p_{*}(W) \longrightarrow \struck{\otimes_S}\; \tau_{B^{*}}
\]

\[
  \struck{e_{B}}\; e^{*}\bigl(R^{n}p_{*}(W)\bigr) \longrightarrow \struck{\otimes_S}\; \tau_{B^{*}}
\]

\[
  R^{n}g_{*}(\mathcal{O}_B) \longrightarrow \struck{\otimes_S}\; \tau_{B^{*}}
\]

\[
  R^{n}g_{*}(\mathcal{O}_B) \simeq \hat\Lambda\, R^{1}g_{*}(\mathcal{O}_B) \simeq \hat\Lambda^{n}\, t_{B^{*}}
\]

\[
  \mathbb{R}p_{*}(W) \longrightarrow e_{*}(\mathcal{O}_S) \otimes \tau_{B^{*}}[-n]
\]

\[
  \varphi \colon \underline{\mathrm{Pic}}_{A/S} \longrightarrow \underline{\mathrm{Pic}}_{B/S}
\]

\[
  \underline{\mathrm{Pic}}^{\delta}_{A/S} \longrightarrow \underline{\mathrm{Pic}}^{-\delta'}_{B/S}
\]

\[
  \tilde\delta{}'\, \tilde\delta = -\chi(\delta)\, \mathrm{id}_A
\]

\[
  \varphi_L(\mathrm{cl}(L)) = \chi\; \struck{\ill{}}\; L'^{-1/\chi}
  \qquad \text{où } \chi = \chi(\delta) \in H^{0}(S, \mathbf{Z}).
\]

\[
  \mathrm{ch}\, \varphi_L(\uncertain{\mathrm{cl}(L)}) = \chi \exp\Bigl(-\frac{D'}{\chi}\Bigr)
  = \chi - D' + \frac{1}{2\chi} D'^{2} - \frac{1}{3!\,\chi^{2}} D'^{3} \ldots
  + (-1)^n \frac{1}{n!\,\chi^{n-1}} D'^{n},
\]

\begin{align*}
  c(\varphi_L(\uncertain{\mathrm{cl}(L)})) &= \Bigl(1 - \frac{D'}{\chi}\Bigr)^{\chi}
  = 1 - D' + \frac{\chi(\chi-1)}{2\chi} D'^{2} - \frac{\chi(\chi-1)(\chi-2)}{3!\,\chi^{2}} D'^{3} + \cdots \\
  &\qquad + (-1)^n \frac{\chi(\chi-1)\cdots(\chi-n+1)}{n!\,\chi^{n-1}} D'^{n}.
\end{align*}

\[
  \struck{\ill{}}\; \frac{\chi(\chi-1)\cdots(\chi-k+1)}{k!\,\chi^{k-2}}\, \delta'^{k}
\]

\[
  (*) \qquad \mathrm{ch}(x * y) = \mathrm{ch}(x) * \mathrm{ch}(y)\; \mathrm{Todd}(t_A)
\]

\[
  \Psi_D = \varphi_{\exp D} \colon \mathrm{Cw}^{\cdot}(A)_{\mathbf{Q}} \longrightarrow \mathrm{Cw}^{\cdot}(B)_{\mathbf{Q}}.
\]

\[
  \mathrm{ch}(\varphi_L(x)) = \Psi_D(\mathrm{ch}(x))\, T \qquad (T = \mathrm{Todd}\, t_A)
\]

\[
  \varphi_L(x * y) = \varphi_L(x)\, \varphi_L(y) \qquad \text{(prop 1)}
\]

\[
  \Psi_D(\mathrm{ch}(x * y))\, T = \Psi_D(\mathrm{ch}\, x)\, T\; \Psi_D(\mathrm{ch}\, y)\, T
\]

\[
  \Psi_D(\struck{\ill{}}\; \mathrm{ch}\, x * \mathrm{ch}\, y)\, T\, \struck{T^{2}}
  = \Psi_D(\mathrm{ch}\, x)\, \Psi_D(\mathrm{ch}\, y)\, T^{2}
\]

\[
  \Psi_D(a * b) = \Psi_D(a)\, \Psi_D(b)
\]

\[
  \varphi_L(xy) = \bigl(\varphi_L(x) * \varphi_L(y)\bigr)\; \struck{\ill{}}\; (-1)^n \omega_A
\]

\[
  \Psi_D(\mathrm{ch}(xy)) = \struck{\ill{}}\; \Psi_D(\mathrm{ch}\, x) * \Psi_D(\mathrm{ch}\, y)\bigr)\, T\, (-1)^n\, \struck{\exp}\; \mathrm{ch}\, \omega_A
\]

\[
  \Psi_D(ab) = \Psi_D(a) * \Psi_D(b)\, (-1)^n\, T\, \mathrm{ch}\, \omega_A
  \qquad
  \begin{cases}
    T = \mathrm{Todd}(t_A) \\
    \omega_A = \hat\Lambda\, t_A^{\vee}
  \end{cases}
\]

\[
  \Phi_L\bigl(\check F \otimes \omega_A[n]\bigr) \simeq \Phi_L(F)^{\vee}
\]

\[
  \Phi_L(F)^{\vee} \simeq \Phi_L\bigl(\check F \otimes \omega_A[n]\bigr)
\]

\[
  \varphi_L(x)^{\vee} = (-1)^n\, \omega_A\, \varphi_L(\check x)
\]

\[
  \Phi_L(u.F) \simeq u'.\Phi_L(F)
\]