\[
  f^{*} : Y_{\mathrm{ét}} \to X_{\mathrm{ét}}, \qquad
  \widetilde{Y}_{\mathrm{ét}} \to \widetilde{X}_{\mathrm{ét}}, \qquad
  f_{*} : \widetilde{X}_{\mathrm{ét}} \to \widetilde{Y}_{\mathrm{ét}}
\]

\[
  \xi \to X, \quad \text{d'où foncteur fibre} \quad
  u^{*} : \widetilde{X}_{\mathrm{ét}} \to \widetilde{\xi}_{\mathrm{ét}} \simeq (\mathrm{Ens})
\]

\[
  F_\xi = \varinjlim_{\substack{X' \text{ étale sur } X\\ \text{et pointé par } \xi}} F(X')
\]

\[
  \mathcal{O}_{X,\xi} = \varinjlim_{\substack{X' \text{ étale sur } X\\ \text{et } \xi\text{-pointé}}} \Gamma(X', \mathcal{O}_{X'})
\]

\[
  \overline{X}{}^{\xi} = \operatorname{Spec} \mathcal{O}_{X,\xi}, \qquad
  F_\xi = \Gamma(\overline{X}, F_{\overline{X}})
\]

\[
  H^i(X, F) \to H^i(X^{\mathrm{an}}, F^{\mathrm{an}})
\]

\[
  H^q(X, F) \to H^q(X^{\mathrm{an}}, F^{\mathrm{an}}), \qquad
  R^q f_{*}(F)^{\mathrm{an}} \to R^q f^{\mathrm{an}}_{*}(F^{\mathrm{an}})
\]

\[
  H^i(X \times_{Y} \mathcal{O}_{Y,y}, F \otimes \mathcal{O}_{Y,y})
  \simeq \varinjlim_{U \ni y} H^i((f^{\mathrm{an}})^{-1}(U), F^{\mathrm{an}})
\]

\[
  \Gamma_Y(F) \subset \Gamma(F), \qquad
  \underline{\Gamma}_Y(F) \subset F, \qquad
  \underline{\Gamma}_Y(F)(X') = \Gamma_{Y'}(X', F')
\]

\[
  H^i_Y(X, F) \simeq \operatorname{Ext}^i(X ; \mathbb{Z}_Y, F), \qquad
  \underline{H}^i_Y(F) \simeq \underline{\operatorname{Ext}}^i(\mathbb{Z}_Y, F)
  = R^{i-1} u_{*}(F|U) \quad \text{si } i \geq 2
\]

\[
  \underline{H}^i_Y((\mathbb{Z}/n\mathbb{Z})_X) =
  \begin{cases}
    0 & \text{si } i \neq 2p \\
    \simeq \mu_n^{\otimes(-p)} & \text{si } i = 2p
  \end{cases}
  \qquad n \text{ premier aux car.\ résiduelles de } Y
\]

\[
  \mu_n = \operatorname{Ker}(\mathbb{G}_m \xrightarrow{\;n\;} \mathbb{G}_m),
  \qquad
  \mu_n(X') = {}_n\Gamma(X', \mathcal{O}_{X'})^{*}
\]

\[
  X \xrightarrow{\;i\;} \overline{X} \xrightarrow{\;\bar{f}\;} Y
\]

\[
  R^q_{!} f_{*}(F) = (R^q \bar{f}_{*})(i_{!}(F))
\]

\[
  \operatorname{Tr}_f(F) : f_{!}(F) \to F
\]

\[
  R^q_{!} g'_{*}(F') \to R^q_{!} g_{*}(f_{!}(F'))
\]

\[
  R^q_{!} g'_{*}(f^{*}(F)) \to R^q_{!} g_{*}(F) \;]
\]

\[
  \varphi \in \Gamma\bigl(i_*(\mathbb{G}_{m,U})\bigr)
\]

\[
  0 \to \underline{H}^0_Y(\mathbb{G}_{m,X}) \to \mathbb{G}_{m,X} \to i_*(\mathbb{G}_{m,U})
  \xrightarrow{\ d\ } \underline{H}^1_Y(\mathbb{G}_{m,X}) \to 0
\]

\[
  \underline{H}^1_Y(\mathbb{G}_{m,X}) \xrightarrow{\ \sim\ } i_*(\mathbb{G}_{m,U})/\mathbb{G}_{m,X}
\]

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

\[
  \partial_n : \underline{H}^1_Y(\mathbb{G}_{m,X}) \longrightarrow \underline{H}^2_Y(\mu_{n,X})
\]

\[
  \gamma^{(n)}_{Y/X} = \partial_n(\beta_{Y/X}) \in \Gamma\bigl(X, \underline{H}^2_Y(\mu_{n,X})\bigr)
\]

\[
  Y = Y_1 \cap \dots \cap Y_d
\]

\[
  \gamma_{Y/X} = \gamma_{Y_1/X} \cdot \ldots \cdot \gamma_{Y_d/X}
  \in \Gamma\bigl(Y, \underline{H}^{2d}_Y(\mu_n^{\otimes d})\bigr)
\]

\[
  \underline{H}^i_Y\bigl((\mathbb{Z}/n\mathbb{Z})_X\bigr) = 0 \quad\text{pour } i < 2d
\]

\[
  \underline{H}^i_Y(\mu_{n,X}^{\otimes d}) = 0 \quad\text{pour } i < 2d .
\]

\[
  H^i_Y(X, \mu_{n,X}^{\otimes d}) = 0 \quad\text{si } i < 2d
\]

\[
  H^{2d}_Y(X, \mu_{n,X}^{\otimes d}) \simeq \Gamma\bigl(Y, \underline{H}^{2d}_Y(\mu_{n,X}^{\otimes d})\bigr)
\]

\[
  \gamma^{(n)}_{Y/X} \in H^{2d}_Y(X, \mu_{n,X}^{\otimes d})
\]

\[
  \gamma_{x/X} \in H^{2d}_{!}(X, \mu_{n,X}^{\otimes d}) .
\]

\[
  t : H^2(X, \mu_{n,X}) \xrightarrow{\ \sim\ } \mathbb{Z}/n\mathbb{Z} .
\]

\[
  t(\gamma_{x/X}) = 1 .
\]

\[
  T^{(n)}_{X/Y} \quad\text{ou simplement}\quad T_{X/Y}
\]

\[
  \mathrm{Tr}_f : R^{2d}_{!} f_{*}(T_{X/Y}) \longrightarrow A_Y
\]

\[
  \left\{
  \begin{aligned}
    \underline{H}^i_Y(T_{X/S}) &\simeq 0 \quad\text{si } i \neq 2d\\
    \underline{H}^{2d}_Y(T_{X/S}) &\simeq T_{Y/S}
  \end{aligned}
  \right.
\]

\[
  \Longrightarrow\quad
  \left\{
  \begin{aligned}
    R^{i}_{!} g_{*Y}(T_{X/S}) &= 0 \quad\text{si } i < 2d\\
    R^{2r}_{!} g_{*Y}(T_{X/S}) &\simeq R^{2r} g_{*Y}\bigl(\underline{H}^{2d}_Y(T_{X/S})\bigr)
  \end{aligned}
  \right.
\]

\[
  R^{2r} g_{*Y}\bigl(\underline{H}^{2d}_Y(T_{X/S})\bigr)
  \simeq R^{2r+2d}_{\substack{\text{supports propres}/S\\ \text{dans } Y}} f_{*}(T_{X/S})
\]

\[
  Lf^* : D(Y) \longrightarrow D(X) \quad\text{quand on a } f : X \to Y
\]

\[
  D^+(Y) \to D^+(X) \quad \text{morphismes plats}
\]

\[
  D^-(Y) \to D^-(X) \quad \text{à topos annelés}
\]

\[
  D^+(Y) \to D^+(X) \quad \text{définition directe via résolutions injectives}
\]

\[
  f^* : D^+(Y) \to D^+(X)
\]

\[
  Lf^* : D^-(Y) \to D^-(X),
\]

\[
  D^+(Y) \to D^+(X), \qquad D^b(Y) \to D^b(X) .
\]

\[
  \mathbb{R}\mathrm{Hom}(F^{\bullet}, G^{\bullet}) = \mathrm{Hom}^*\bigl(F^{\bullet}, C^{\bullet}(G^{\bullet})\bigr) .
\]

\[
  \mathbb{R}\underline{\mathrm{Hom}}(F^{\bullet}, G^{\bullet})
\]

\[
  \boxed{\ \mathbb{R}\mathrm{Hom}(F^{\bullet}, G^{\bullet})
  = \mathbb{R}\Gamma_X\, \mathbb{R}\underline{\mathrm{Hom}}(F^{\bullet}, G^{\bullet})\ }
\]

\[
  \left\{
  \begin{aligned}
    &\mathrm{Hom}(F^{\bullet}, C^{\bullet}) \text{ est aussi noté } \mathrm{Ext}^0(F^{\bullet}, C^{\bullet})\\
    &\mathrm{Hom}(F^{\bullet}, C^{\bullet}(i)) \simeq \mathrm{Hom}(F(-i), C^{\bullet}) = \mathrm{Ext}^i(F^{\bullet}, C^{\bullet})
  \end{aligned}
  \right.
\]

\[
  \mathrm{Ext}^*(F^{\bullet}; C^{\bullet}) \Longleftarrow
  E_2^{pq} = H^p\bigl(X, \underline{\mathrm{Ext}}^q(F^{\bullet}; C^{\bullet})\bigr)
\]

\[
  {}'E_2^{pq} = H^p\bigl(X, \mathrm{Ext}^q(F^*, C^*)\bigr)
\]

\[
\mathbb{R}_! f_* : D^+(X) \longrightarrow D^+(Y), \qquad
\mathcal{H}^i(\mathbb{R}_! f_*) = \mathbb{R}^i_! f_* .
\]

\[
\mathbb{R}_! f_* : D(X) \longrightarrow D(Y)
\]

\[
D^-(X) \longrightarrow D^-(Y), \quad D^+(X) \to D^+(Y), \quad
D^b(X) \longrightarrow D^b(Y).
\]

\[
\text{\struck{$\mathbb{R} f'_*(h^*(F^\cdot)) \longleftarrow$}}
\qquad
\boxed{\,g^*(\mathbb{R} f_*(F^\cdot)) \longrightarrow \mathbb{R} f'_*(h^*(F^\cdot))\,}
\]

\[
(*_i) \qquad g^* \mathbb{R}^i f_*(F) \longrightarrow \mathbb{R}^i f'_*(h^*(F)).
\]

\[
H^i_!(X, F)^\vee \simeq H^{-i}(X, DF).
\]

\[
H^i_!(X, F)^\vee \simeq H^{2d-i}(X, \underline{\mathrm{Hom}}(F, \mu_{n,X}^{\otimes d})).
\]

\[
\underline{\mathrm{Hom}}(R^i_! f_*(F), (\mathbb{Z}/n\mathbb{Z})_X) \simeq R^{-i} f_*(D(F))
\]

\[
\underline{\mathrm{Hom}}(R^i_! f_*(F), (\mathbb{Z}/n\mathbb{Z})_X)
\simeq R^{2d-i} f_*(\underline{\mathrm{Hom}}(F, \mu_{n,X}^{\otimes d})).
\]

\[
\mathbb{R} i_! = i_! : D^+(Y) \longrightarrow D^+(X)
\]

\[
i^! : C(Y) \longrightarrow C(X)
\]

\[
i^!(F) = \text{\struck{$\mathbb{R}$}}\; i'^* \underline{\Gamma}_Y(j^*(F)).
\]

\[
\mathrm{Hom}_Y(G, i^!(F)) \simeq \mathrm{Hom}_X(i_!(G), F)
\]

\[
i_! = (j i')_! = j_! i'_! , \qquad i^! = i'^! j^! = i'^! j^* .
\]

\[
\mathbb{R} i^! : D^+(X) \longrightarrow D^+(Y).
\]

\[
(\mathbb{R}_! i_*)(\mathbb{R} i^!) \simeq \mathbb{R} \underline{\Gamma}_Y
\]

\[
\underline{H}^q(\mathbb{R} i^!(F)) \simeq i^*(\underline{H}^q_Y(F)).
\]

\[
\boxed{\,\mathrm{Hom}(G^\cdot ; \mathbb{R} i^!(F^\cdot)) \simeq
\mathrm{Hom}(\mathbb{R}_! i_*(G^\cdot), F^\cdot)\,}
\]

\begin{align*}
\mathrm{Hom}(G^\cdot ; \mathbb{R} i^!(F^\cdot))
&= \mathrm{Hom}(G^\cdot ; i^!(C^\cdot(F^\cdot))) \\
&= H^0(\mathrm{Hom}^\bullet(G^\cdot ; i^!(C^\cdot(F^\cdot))))
 = H^0(\mathrm{Hom}(i_!(G^\cdot), C^\cdot(F^\cdot))) \\
&\text{\struck{$= \mathrm{Hom}(i_!(G^\cdot), \ldots)$}} \\
&= \mathrm{Hom}(\mathbb{R} i_!(G^\cdot), C^\cdot(F^\cdot))
\end{align*}

\[
\mathrm{Tr}_i(F^\cdot) : \mathbb{R}_! i_* \mathbb{R} i^!(F^\cdot) \longrightarrow F^\cdot
\]

\[
\mathbb{R}_! f_*\bigl(\mathbb{R}\underline{\mathrm{Hom}}(A_Y, \mathbb{R}^! f(G^\cdot))\bigr)
\simeq \mathbb{R}\underline{\mathrm{Hom}}(\mathbb{R}_! g(A_Y), G^\cdot)
\]

\[
g_*\bigl(\mathbb{R} i^!(\mathbb{R}^! f(G^\cdot))\bigr) \xrightarrow{\ \sim\ } G^\cdot ,
\]

\[
\begin{cases}
\underline{H}^i_Y(f^*(G)) = 0 & \text{si } i \neq 2d \\
\underline{H}^{2d}_Y(f^*(G)) = g^*(G) \otimes T^{-1}_{Y/X} &
\end{cases}
\]

\[
g^*(\mathbb{R} i^!) \longrightarrow (\mathbb{R} i'^!) f^*
\]

\[
(\mathbb{R}^! g^*)(\mathbb{R} i^!) \longrightarrow (\mathbb{R} i'^!)(\mathbb{R}^! f)
\]

\[
  \mathbb{R}f^! : D^+(Y) \longrightarrow D^+(X)
\]

\[
  \mathbb{R}f^!(G) = (\mathbb{R}i^!)(\mathbb{R}f'^!)
\]

\[
  \mathrm{Hom}(F^\bullet, \mathbb{R}f^!(G^\bullet)) \simeq \mathrm{Hom}(\mathbb{R}_!f_*F^\bullet, G^\bullet)
\]

\[
  \bigl(F^\bullet \in \mathrm{Ob}\, D^b(X),\ G^\bullet \in \mathrm{Ob}\, D^+(Y)\bigr).
\]

\[
  X''' = X' \times_Y X'' ,
\]

\[
  (*) \qquad \mathbb{R}i^! \simeq \text{\struck{$\mathbb{R}g$}}\ \mathbb{R}(j\sigma)^!\, \mathbb{R}^! g .
\]

\[
  \mathbb{R}(j\sigma)^! \simeq \mathbb{R}\sigma^!\, \mathbb{R}j^!
\]

\[
  (j\sigma)^! \simeq \sigma^!\, j^!
\]

\[
  \mathbb{R}i^! \simeq \mathbb{R}\sigma^!\,(\mathbb{R}j^!\, \mathbb{R}^! g)
\]

\[
  \begin{cases}
    (\mathbb{R}^! g')(\mathbb{R}i^!) \simeq (\mathbb{R}j^!)(\mathbb{R}^! g) & (a)\\
    \text{\struck{$\mathbb{R}\,\mathrm{id}_{D^+(Y)}$}}\ \mathbb{R}^!\,\mathrm{id}_{Y'} \simeq (\mathbb{R}\sigma^!)(\mathbb{R}^! g') & (b),
  \end{cases}
\]

\[
  \mathbb{R}i^! = \text{\struck{$\mathbb{R}i^!$}}\,(\mathbb{R}^!\,\mathrm{id}_{Y'})\,\mathbb{R}i^!
  \simeq \text{\struck{$\mathbb{R}i^!$}}\ \mathbb{R}\sigma^!\, \mathbb{R}^! g'\, \mathbb{R}i^!
\]

\[
  \simeq \mathbb{R}\sigma^!\,\mathbb{R}j^!\,\mathbb{R}^! g\ \text{\struck{$\mathbb{R}i^!$}}
\]

\[
  \Bigl(\text{N.B.}\quad \underline{\Omega}^d_{X'/Y'} \simeq \underline{\Omega}^d_{X/Y} \otimes_{\mathcal{O}_X} \mathcal{O}_{X'}\Bigr)
\]

\[
  g'^*(\mathbb{R}i^!) \simeq (\mathbb{R}j^!)\, g^* .
\]

\[
  \mathrm{Tr}_f : (\mathbb{R}_!f)(\mathbb{R}^!f) \longrightarrow \mathrm{id}_{D^+(Y)}
\]

\[
  \mathbb{R}i_!\bigl(\mathbb{R}i^!(T_{X/S})\bigr) \simeq \mathbb{R}i_!\bigl(T_{Y/S}[2d]\bigr)
  \longrightarrow \mathbb{R}_!f_*(T_{X/S})
\]

\[
  \mathbb{R}_!i_*\bigl(T_{Y/S}[-2d]\bigr) \longrightarrow \mathbb{R}_!f_*(T_{X/S})
\]

\[
  \mathbb{R}_!g_*(T_{Y/S})[-2d] \longrightarrow \mathbb{R}_!f_*(T_{X/S})
\]

\[
  R^{\,i-2d}_!\, g_*(T_{Y/S}) \longrightarrow R^{\,i}_!\, f_*(T_{X/S})
\]

\[
  R^{2(r-d)}_!\, g_*(T_{Y/S}) \longrightarrow T_{X/S}
\]

\[
  \text{\struck{$\mathbb{R}^! g^*(G) \simeq \mathbb{R}i^!\,\mathbb{R}f^!(G)$}}
\]

\[
  \mathbb{R}g^!(G) \simeq \mathbb{R}i^!\,\mathbb{R}f^!(G)
\]

\[
  \underline{H}^i_Y(f^*(G)) =
  \begin{cases}
    0 & i \neq 2d\\
    g^*(G) \otimes T^{-1}_{Y/X} & i = 2d
  \end{cases}
\]

\[
  \mathbb{R}g^!(G^\bullet) \simeq \mathbb{R}i^!\bigl(\mathbb{R}f^!(G^\bullet)\bigr)
\]

\[
  \text{\struck{$\mathbb{R}_!g_*(\mathbb{R}g^!(G^\bullet)) \to \mathbb{R}f$}}
\]

\[
  \mathbb{R}_!i_*\bigl(\mathbb{R}g^!(G^\bullet)\bigr) \longrightarrow \mathbb{R}f^!(G^\bullet)
\]

\[
  \mathbb{R}^!(gf) \simeq \mathbb{R}^!f\ \mathbb{R}^!g
\]

\[
  G^\bullet = \mathbb{L}g^*(F^\bullet)
\]

\[
  G^\bullet \simeq g^!(F^\bullet)\ \text{\struck{$[-2d]$}}\otimes T_{Y/Z}\,[-2d]
\]

\[
  f^!(G^\bullet) = \bigl[f^!\, g^!(F^\bullet)\bigr] \otimes T^{-1}_{Y/Z}\,[-2d]
\]

\[
  f^!\, g^!(F^\bullet) \simeq h^!(F^\bullet) = \mathbb{L}h^*(F^\bullet) \otimes T_{X/Z}\,[2d']
\]

\[
  f^!(G^\bullet) \simeq \mathbb{L}h^*(F^\bullet) \otimes T_{X/Y}\,[2d_{X/Y}]
\]

\[
  T_{X/Y} \simeq T_{X/Z} \otimes T^{-1}_{Y/Z} , \qquad
  d_{X/Y} = \dim_{X/Z} - \dim_{Y/Z}
\]

\[
\mathbb{R}_!h\bigl(\mathbb{L}h^{*}F^{\cdot}\otimes T_{X/Y}[2d_{X/Y}]\bigr)\longrightarrow\mathbb{R}_!g\bigl(\mathbb{L}g^{*}(F^{\cdot})\bigr)
\]

\[
\mathbb{R}_!^{\,i+2d_{X/Y}}h\bigl(\mathbb{L}h^{*}(F^{\cdot})\bigr)\otimes T_{X/Y}\longrightarrow\mathbb{R}_!^{\,i}g\bigl(\mathbb{L}g^{*}(F^{\cdot})\bigr)
\]

\[
\mathbb{H}^{i+2d_{X/Y}}\bigl(X,\mathbb{L}h^{*}(F^{\cdot})\bigr)\otimes T_{X/Y}\longrightarrow\mathbb{H}^{i}\bigl(Y,\mathbb{L}g^{*}(F^{\cdot})\bigr)
\]

\[
(*)\qquad \mathbb{R}_!f\,\mathbb{R}^!f(G^{\cdot})\longrightarrow G^{\cdot}
\]

\[
\begin{cases}
\mathbb{R}^{i}_!f\bigl(\mathbb{R}^!f(G^{\cdot})\bigr)\longrightarrow\mathcal{H}^{i}(G^{\cdot})\\
\mathbb{H}^{i}_{\Phi(f)}\bigl(X,\mathbb{R}^!f(G^{\cdot})\bigr)\longrightarrow\mathbb{H}^{i}(Y,G^{\cdot})
\end{cases}
\]

\[
\mathbb{H}^{i}\bigl(X,\mathbb{R}^!f(G^{\cdot})\bigr)\longrightarrow\mathbb{H}^{i}(Y,G^{\cdot})
\]

\[
\text{d'où}\quad
\begin{cases}
\mathbb{R}_!h\bigl(\mathbb{R}^!f(G^{\cdot})\bigr)\longrightarrow\mathbb{R}_!g(G^{\cdot})\\
\mathbb{H}^{i}_{\Phi(h)}\bigl(X,\mathbb{R}^!f(G^{\cdot})\bigr)\longrightarrow\mathbb{H}^{i}_{\Phi(g)}(Y,G^{\cdot})
\end{cases}
\]

\[
\text{Trans.}*\ \begin{cases}
\mathbb{R}(gf)_{*}\simeq\mathbb{R}g_{*}\,\mathbb{R}f_{*}\\
\mathbb{L}(gf)^{*}\simeq\mathbb{L}f^{*}\,\mathbb{L}g^{*}
\end{cases}
\qquad
\text{Trans.}!\ \begin{cases}
\mathbb{R}_!(gf)\simeq\mathbb{R}_!g\,(\mathbb{R}_!f)\\
\mathbb{R}^!(gf)\simeq(\mathbb{R}^!f)(\mathbb{R}^!g)
\end{cases}
\]

\[
\boxed{\operatorname{Hom}\bigl(\mathbb{R}_!f(F_{\cdot}),G^{\cdot}\bigr)\simeq\operatorname{Hom}\bigl(F_{\cdot},\mathbb{R}^!f(G^{\cdot})\bigr)}
\]

\[
[F_{\cdot}\in\operatorname{Ob}D^{b}(X),\ G^{\cdot}\in\operatorname{Ob}D^{+}(Y)]
\]

\[
\boxed{\operatorname{Hom}\bigl(\mathbb{L}f^{*}(G_{\cdot}),F^{\cdot}\bigr)\simeq\operatorname{Hom}\bigl(G_{\cdot},\mathbb{R}f_{*}(F^{\cdot})\bigr)}
\]

\[
[F^{\cdot}\in\operatorname{Ob}D^{+}(X),\ G^{\cdot}\in\operatorname{Ob}D^{b}(Y)]
\]

\begin{align*}
\operatorname{Hom}\bigl(G^{\cdot},\mathbb{R}f_{*}(F^{\cdot})\bigr)
&\simeq H^{0}\Bigl(\operatorname{Hom}^{\cdot}\bigl(G^{\cdot},f_{*}(C^{\cdot}(F^{\cdot}))\bigr)\Bigr)\\
&\simeq H^{0}\Bigl(\operatorname{Hom}^{\cdot}\bigl(f^{*}G^{\cdot},C^{\cdot}(F^{\cdot})\bigr)\Bigr)\\
&\simeq\operatorname{Hom}\bigl(\mathbb{L}f^{*}G^{\cdot},F\bigr)
\end{align*}

\[
\boxed{\mathbb{R}\mathcal{H}om\bigl(\mathbb{R}_!f(F_{\cdot}),G^{\cdot}\bigr)\simeq\mathbb{R}f_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(\mathbb{R}^!f(F_{\cdot}),G^{\cdot}\bigr)\Bigr)}
\]

\[
\boxed{\mathbb{R}\mathcal{H}om\bigl(G_{\cdot},\mathbb{R}f_{*}(F^{\cdot})\bigr)\simeq\mathbb{R}f_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(\mathbb{L}f^{*}(G_{\cdot}),F^{\cdot}\bigr)\Bigr)}
\]

\[
\boxed{G_{\cdot}\overset{\mathbb{L}}{\otimes}\mathbb{R}f_{*}(F_{\cdot})\simeq\mathbb{R}f_{*}\bigl(G_{\cdot}\overset{\mathbb{L}}{\otimes}F_{\cdot}\bigr)}
\]

\begin{align*}
\operatorname{Tr}_{f}(G^{\cdot})&\colon\ \mathbb{R}_!f\,\mathbb{R}^!f(G^{\cdot})\longrightarrow G^{\cdot}\\
\tau_{f}(G^{\cdot})&\colon\ G_{\cdot}\longrightarrow\mathbb{R}f_{*}\bigl(\mathbb{L}f^{*}(G_{\cdot})\bigr)
\end{align*}

\[
\boxed{\mathbb{L}g^{*}\,\mathbb{R}_!f\xrightarrow{\ \sim\ }\mathbb{R}_!f'\,\mathbb{L}g'^{*}}
\]

\[
\begin{cases}
\mathbb{L}g^{*}\,\mathbb{R}f_{*}\xrightarrow{\ \sim\ }\mathbb{R}f'_{*}\,\mathbb{L}g'^{*} & \text{si $f$ propre [ou $g$ lisse]}\\
\mathbb{R}^!f'\,\mathbb{L}g^{*}\xleftarrow{\ \sim\ }\mathbb{L}g'^{*}\,\mathbb{R}^!f & \text{si $f$ ou $g$ lisse}
\end{cases}
\]

\[
\text{\struck{$\mathbb{R}^!g\,\mathbb{R}_!f\Rightarrow\mathbb{R}_!f'\,\mathbb{R}^!g'$}}
\]

\[
\boxed{\mathbb{R}^!g\,\mathbb{R}f_{*}\simeq\mathbb{R}f'_{*}\,\mathbb{R}^!g'}
\]

\[
\boxed{\mathbb{R}^!f\bigl(\mathbb{R}\mathcal{H}om(F_{\cdot},G^{\cdot})\bigr)\simeq\mathbb{R}\mathcal{H}om\bigl(\mathbb{L}f^{*}(F_{\cdot}),\mathbb{R}^!f(G^{\cdot})\bigr)}
\]

\[
\boxed{\mathbb{L}f^{*}\bigl(F_{\cdot}\overset{\mathbb{L}}{\otimes}G_{\cdot}\bigr)\simeq\mathbb{L}f^{*}(F_{\cdot})\overset{\mathbb{L}}{\otimes}\mathbb{L}f^{*}(G_{\cdot})}
\]

\[
\mathbb{L}f^{*}\bigl(\mathbb{R}\mathcal{H}om(F_{\cdot},G^{\cdot})\bigr)\simeq\mathbb{R}\mathcal{H}om\bigl(\mathbb{L}f^{*}F_{\cdot},\mathbb{L}f^{*}G^{\cdot}\bigr)
\]

\begin{align*}
\mathbb{R}\mathcal{H}om\bigl(\mathbb{R}i_!(F'),G^{\cdot}\bigr)&\simeq\mathbb{R}i_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(F',\mathbb{R}i^!(G^{\cdot})\bigr)\Bigr)
&&\text{(dualité pour $i$)}\\
\mathbb{L}f^{*}(\ \cdots\ )&\simeq\mathbb{L}f^{*}\,\mathbb{R}i_{*}(\ \cdots\ )\\
&\simeq\mathbb{R}j_{*}\,\mathbb{L}f'^{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(F',\mathbb{R}i^!(G^{\cdot})\bigr)\Bigr)
&&\text{(chang.\ de base lisse)}\\
&\simeq\mathbb{R}j_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(f'^{*}(F'),f'^{*}(\mathbb{R}i^!(G^{\cdot}))\bigr)\Bigr)\\
&\simeq\mathbb{R}j_{*}\Bigl(\mathbb{R}\mathcal{H}om\bigl(F'_{X'},\mathbb{R}j^!G^{\cdot}_{X}\bigr)\Bigr)\\
&\simeq\mathbb{R}\mathcal{H}om\bigl(j_!(F'_{X'}),G^{\cdot}_{X}\bigr)
&&\text{(dualité pour $j$)}\\
&\simeq\mathbb{R}\mathcal{H}om\bigl(f^{*}(F_{X}),f^{*}(G)\bigr)
\end{align*}

\begin{align*}
\mathbb{R}^!f\bigl(\mathbb{R}\mathcal{H}om(F_{\cdot},G^{\cdot})\bigr)&\simeq f^!\,\mathcal{H}om^{\cdot}\bigl(F_{\cdot},C^{\cdot}(G^{\cdot})\bigr)\\
&\simeq\mathcal{H}om^{\cdot}\bigl(f^{*}F_{\cdot},f^!C^{\cdot}(G^{\cdot})\bigr)\simeq\mathbb{R}\mathcal{H}om\bigl(\mathbb{L}f^{*}(F_{\cdot}),\mathbb{R}f^!(G^{\cdot})\bigr)
\end{align*}

\[
\begin{cases}
\mathbb{R}f_{*}\,D_{X}\simeq D_{Y}\,\mathbb{R}_!f\\
\mathbb{R}^!f\,D_{Y}\simeq D_{X}\,\mathbb{L}f^{*}
\end{cases}
\]

\[
  \mathbb{L}f^{*}(M_{\bullet} \overset{\mathbb{L}}{\otimes} L_{\bullet})
  \simeq \mathbb{L}f^{*}(M_{\bullet}) \overset{\mathbb{L}}{\otimes} \mathbb{L}f^{*}(L_{\bullet})
\]

\[
  \mathbb{R}\Gamma_{X}\, L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet} .
\]

\begin{align*}
      \mathbb{R}\mathcal{H}om^{\bullet}(L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet}, K^{\bullet})
        &\simeq \mathbb{R}\mathcal{H}om^{\bullet}(L_{\bullet}, \mathbb{R}\mathcal{H}om^{\bullet}(M_{\bullet}, K^{\bullet})) \\
        &\simeq \mathbb{R}\mathcal{H}om^{\bullet}(M_{\bullet}, \mathbb{R}\mathcal{H}om^{\bullet}(L_{\bullet}, K^{\bullet}))
    \end{align*}

\begin{align*}
      \mathbb{R}\mathrm{Hom}^{\bullet}(L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet}, K^{\bullet})
        &\simeq \mathbb{R}\mathrm{Hom}^{\bullet}(L_{\bullet}, \mathbb{R}\mathcal{H}om^{\bullet}(M_{\bullet}, K^{\bullet})) \\
        &\simeq \mathbb{R}\mathrm{Hom}^{\bullet}(M_{\bullet}, \mathbb{R}\mathcal{H}om^{\bullet}(L_{\bullet}, K^{\bullet}))
    \end{align*}

\begin{align*}
      \mathrm{Hom}(L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet}, K^{\bullet})
        &\simeq \mathrm{Hom}(L_{\bullet}, \mathbb{R}\mathcal{H}om^{\bullet}(M_{\bullet}, K^{\bullet})) \\
        &\simeq \mathrm{Hom}(M_{\bullet}, \mathbb{R}\mathcal{H}om^{\bullet}(L_{\bullet}, K^{\bullet}))
    \end{align*}

\[
  \mathbb{L}f^{*}(L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet})
  \simeq \mathbb{L}f^{*}(L_{\bullet}) \overset{\mathbb{L}}{\otimes} \mathbb{L}f^{*}(M_{\bullet}) .
\]

\[
  \mathbb{L}(gf)^{*} \simeq \mathbb{L}f^{*}\, \mathbb{L}g^{*}
\]

\[
  L_{\bullet} \overset{\mathbb{L}}{\otimes}_{Y} M_{\bullet}
  \simeq L_{\bullet} \overset{\mathbb{L}}{\otimes} \mathbb{L}f^{*}(M_{\bullet}) .
\]

\[
  \mathrm{Hom}(\mathbb{L}f^{*}(L_{\bullet}), K^{\bullet})
  \xrightarrow{\ \sim\ } \mathrm{Hom}(L_{\bullet}, \mathbb{R}f_{*}(K^{\bullet}))
\]

\[
  \mathrm{Hom}^{\bullet}(f^{*}L_{\bullet}, K^{\bullet}) \simeq \mathrm{Hom}^{\bullet}(L_{\bullet}, f_{*}(K^{\bullet})) \qquad \text{OK.}
\]

\[
  \mathbb{R}f_{*}\, \mathbb{R}\mathcal{H}om^{\bullet}(\mathbb{L}f^{*}(L_{\bullet}), K^{\bullet})
  \simeq \mathbb{R}\mathcal{H}om^{\bullet}(L_{\bullet}, \mathbb{R}f_{*}(K^{\bullet}))
\]

\[
  \mathbb{R}\mathrm{Hom}^{\bullet}(\mathbb{L}f^{*}(L_{\bullet}), K^{\bullet})
  \simeq \mathbb{R}\mathrm{Hom}^{\bullet}(L_{\bullet}, \mathbb{R}f_{*}(K^{\bullet}))
\]

\begin{align*}
  \mathbb{R}\mathrm{Hom}^{\bullet}(f^{*}L_{\bullet}, K^{\bullet})
  &= \mathbb{R}\mathrm{Hom}^{\bullet}(A_{X'}, K^{\bullet})
   = \mathbb{R}\Gamma_{X'}(\mathbb{L}j'^{*}(K^{\bullet})) \\
  &\simeq \mathbb{R}\Gamma_{Y'}(\mathbb{R}f'_{*}\, \mathbb{L}j'^{*}(K^{\bullet})) \\
  &\simeq \mathbb{R}\mathrm{Hom}^{\bullet}(A_{Y'}, \mathbb{L}i^{*}(\mathbb{R}f_{*}(K^{\bullet}))) \\
  &= \mathbb{R}\mathrm{Hom}^{\bullet}(L_{\bullet}, \mathbb{R}f_{*}(K^{\bullet}))
\end{align*}

\begin{align*}
  \mathbb{R}_{!}f_{*} :\ & D(X) \to D(Y) \\
  & D^{+}(X) \to D^{+}(Y) \\
  & D^{-}(X) \to D^{-}(Y) \\
  & D^{b}(X) \to D^{b}(Y)
  && \text{si $Y$ qu.\ cpt, $f$ compactifiable}
\end{align*}

\begin{align*}
  \mathbb{R}f_{*} :\ & D(X) \to D(Y) \\
  & D^{+}(X) \to D^{+}(Y) \\
  & D^{-}(X) \to D^{-}(Y) \\
  & D^{b}(X) \to D^{b}(Y)
  && \text{si $Y$ de t.f.\ sur un corps}\\
  &&& \text{ou sur $\mathrm{Spec}\, \mathbb{Z}$, et $f$ de t.f.}
\end{align*}

\begin{align*}
  \mathbb{L}f^{*} :\ & D(\text{\struck{$X$}}\, Y) \to D(X) \\
  & D^{-}(Y) \to D^{-}(X) \\
  & D^{+}(Y) \to D^{+}(X)
  \qquad D^{b}(Y) \to D^{b}(X)
\end{align*}

\begin{align*}
  \mathbb{R}^{!}f :\ & D(\text{\struck{$X$}}\, Y) \to D(X) \\
  & D^{+}(Y) \to D^{+}(X) \\
  & D^{-}(Y) \to D^{-}(X) \\
  & D^{b}(Y) \to D^{b}(X)
  && \text{si $f$ compactifiable de t.f., $Y$ de t.f.}\\
  &&& \text{sur un corps, ou sur $\mathrm{Spec}\, \mathbb{Z}$.}
\end{align*}

\[
  \mathbb{R}_{!}f(L_{\bullet}) \overset{\mathbb{L}}{\otimes} M_{\bullet}
  \simeq \mathbb{R}_{!}f(L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet})
\]

\[
  \mathbb{R}f_{*}(L_{\bullet}) \overset{\mathbb{L}}{\otimes} M_{\bullet}
  \simeq \mathbb{R}f_{*}(L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet})
\]

\[
  f_{*}(L_{\bullet}) \otimes M_{\bullet} \longrightarrow f_{*}(C^{\bullet}(L_{\bullet} \otimes M_{\bullet}))
\]

\[
  f_{*}(L_{\bullet}) \otimes M_{\bullet} \text{\struck{$\otimes$}} \longrightarrow f_{*}(L_{\bullet} \otimes M_{\bullet})
\]

\[
  M_{\bullet} \longrightarrow \mathcal{H}om^{\bullet}(f_{*}(L_{\bullet}), f_{*}(L_{\bullet} \otimes M_{\bullet}))
\]

\[
  f_{*}(L_{\bullet} \otimes M_{\bullet}) \longrightarrow f_{*}(C^{\bullet}(L_{\bullet} \otimes M_{\bullet}))
\]

\[
  i_{!}(Q) \otimes P \simeq i_{!}(Q \otimes i^{*}P)
\]

\begin{align*}
  i_{!}(A_{Y'}) \otimes \mathbb{R}_{!}f(L_{\bullet})
  &\simeq \mathbb{R}i_{!}(\mathbb{L}i^{*}(\mathbb{R}_{!}f(L_{\bullet}))) \\
  &\simeq \mathbb{R}i_{!}\, \mathbb{R}_{!}f'(\mathbb{L}j^{*}(L_{\bullet})) \\
  &\simeq \mathbb{R}_{!}f(\mathbb{R}_{!}j(\mathbb{L}j^{*}(L_{\bullet}))) \\
  &\simeq \mathbb{R}_{!}f(\mathbb{L}f^{*}i_{!}(A_{Y'}) \otimes \mathbb{L}j^{*}(L_{\bullet}))
\end{align*}

\begin{align*}
  &\mathbb{R}_{!}h_{*}(\mathbb{L}g'^{*}(L_{\bullet}) \overset{\mathbb{L}}{\otimes} \mathbb{L}f'^{*}(M_{\bullet})) \\
  &\qquad \simeq \mathbb{R}_{!}f_{*}(L_{\bullet}) \overset{\mathbb{L}}{\otimes} \mathbb{R}_{!}g_{*}(M_{\bullet})
\end{align*}

\[
  \mathbb{R}^{n}_{!}h(g'^{*}(F) \otimes f'^{*}(G))
  \simeq \sum_{i+j=n} R^{i}_{!}f(F) \otimes R^{j}_{!}g(G) .
\]

\begin{align*}
  (*) \qquad &\mathbb{R}(f \times g)_{*}(L_{\bullet} \overset{\mathbb{L}}{\boxtimes} M_{\bullet}) \\
  &\qquad \xleftarrow{\ \sim\ } \mathbb{R}f_{*}(L_{\bullet}) \overset{\mathbb{L}}{\boxtimes} \mathbb{R}g_{*}(M_{\bullet})
\end{align*}

\begin{align*}
  \mathbb{R}(f \times g)_{*}(L_{\bullet} \overset{\mathbb{L}}{\boxtimes} M_{\bullet})
  &\simeq \mathbb{R}g_{*}\big[\mathbb{R}f'_{*}(\mathbb{L}g'^{*}(L_{\bullet})) \overset{\mathbb{L}}{\otimes} M_{\bullet}\big] \\
  &\simeq \mathbb{R}f_{*}(L_{\bullet}) \overset{\mathbb{L}}{\otimes} \mathbb{R}g_{*}(M_{\bullet})
\end{align*}

\[
  M_{\bullet} = A_{U,X} ,
\]

\[
  \mathbb{R}f_{*}(L_{\bullet}) \overset{\mathbb{L}}{\otimes} M_{\bullet}
  \longrightarrow \mathbb{R}f_{*}(L_{\bullet} \overset{\mathbb{L}}{\otimes} M_{\bullet})
\]

\[
  \sum (-1)^i \operatorname{Tr} f^{(i)} = \text{nb des pts fixes de } f
\]

\[
  \operatorname{Tr}_* u \in A
\]

\[
  \operatorname{Tr}_* u = \sum (-1)^i \operatorname{Tr} u_i ,
\]

\[
  \check{L}_\bullet \simeq \mathbf{R}\mathcal{H}om(L_\bullet, A),
\]

\[
  L_\bullet \to \check{\check{L}}_\bullet
\]

\[
  \bigl(L_\bullet \overset{\mathbf{L}}{\otimes} L'_\bullet\bigr)^{\vee}
  \simeq \check{L}_\bullet \overset{\mathbf{L}}{\otimes} \check{L}'_\bullet .
\]

\[
  \mathbf{R}\mathcal{H}om^\bullet(L_\bullet, M_\bullet) \simeq
  \check{L}_\bullet \overset{\mathbf{L}}{\otimes} M_\bullet ,
\]

\[
  \mathbf{R}\mathcal{H}om^\bullet(M_\bullet, L_\bullet) \simeq
  \check{M}_\bullet \overset{\mathbf{L}}{\otimes} L_\bullet ,
\]

\[
  u \in H^0\bigl(\mathbf{R}\mathcal{H}om^\bullet(L_\bullet, M_\bullet)\bigr)
  = \operatorname{Hom}_{D(A)}(L_\bullet, M_\bullet) ,
\]

\[
  v \in H^0\bigl(\mathbf{R}\mathcal{H}om(M_\bullet, L_\bullet)\bigr)
  = \operatorname{Hom}_{D(A)}(M_\bullet, L_\bullet) ,
\]

\[
  \langle u, v \rangle = H^0(A) = A .
\]

\[
  \langle u, v \rangle = \langle v, u \rangle
  = \sum (-1)^i \operatorname{Tr}(vu)^i = \sum (-1)^i \operatorname{Tr}(uv)^i
\]

\[
  \sum (-1)^i \operatorname{Tr} u^i = \langle u, \mathrm{id}_{L_\bullet}
  \rangle = \langle \mathrm{id}_{L_\bullet}, u \rangle ,
\]

\[
  \mathcal{T}or_i(F, L_\bullet) = 0
\]

\[
  \mathcal{T}or_i(F, L_\bullet) = 0 \quad \text{pour } i \notin [N, M] .
\]

\[
R_!f(L_\bullet)\overset{L}{\otimes} L'_\bullet \;\simeq\; R_!f\bigl(L_\bullet \overset{L}{\otimes} Lf^*(L'_\bullet)\bigr) \qquad 2.
\]

\[
\struck{\text{Ext}}\;\mathrm{Hom}^{\circ}\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) \simeq R\Gamma_{X\times_S Y}\,\mathbb{R}\mathcal{H}om^{\bullet}\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr)
\]

\[
\begin{aligned}
&\wr\;\; \mathrm{Hom}^{\circ}\bigl(\mathrm{pr}_{2!}\,\mathrm{pr}_1^*(F_\bullet), G_\bullet\bigr)\\
&\wr\;\; \mathrm{Hom}^{\circ}\bigl(g^*f_!(F_\bullet), G_\bullet\bigr)\\
&\wr\;\; \mathrm{Hom}^{\circ}\bigl(f_!(F_\bullet), g_*(G_\bullet)\bigr)
\end{aligned}
\]

\[
\begin{aligned}
\mathrm{Hom}^{\circ}\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) &\simeq \mathrm{Hom}^{\circ}\bigl(f_!(F_\bullet), g_*(G_\bullet)\bigr)\\
\mathrm{Hom}^{\circ}\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr) &\simeq \mathrm{Hom}^{\circ}\bigl(g_!(G_\bullet), f_*(F_\bullet)\bigr)
\end{aligned}
\]

\[
\text{\struck{par}}\quad \mathbb{R}\mathcal{H}om^{\bullet}\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) \times \mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr) \longrightarrow h^!(A_S)
\]

\[
\mathcal{H}^{0}\bigl(\mathbb{R}\mathcal{H}om(\mathcal{L}_\bullet, \mathcal{L}_\bullet)\bigr) \longrightarrow A_X
\]

\[
\varphi \longmapsto \sum (-1)^i\, \mathrm{Tr}\, \varphi_i
\]

\[
\begin{aligned}
\mathbb{R}\mathcal{H}om(\mathcal{L}_\bullet, \mathcal{M}_\bullet) \times \mathbb{R}\mathcal{H}om(\mathcal{M}_\bullet, \mathcal{L}_\bullet) &\xrightarrow{\;\varphi_{\mathcal{L}_\bullet,\mathcal{M}_\bullet}\;} \mathbb{R}\mathcal{H}om(\mathcal{L}_\bullet, \mathcal{L}_\bullet) & (u,v)&\mapsto vu\\
\mathbb{R}\mathcal{H}om(\mathcal{M}_\bullet, \mathcal{L}_\bullet) \times \mathbb{R}\mathcal{H}om(\mathcal{L}_\bullet, \mathcal{L}_\bullet) &\longrightarrow \mathbb{R}\mathcal{H}om(\mathcal{M}_\bullet, \mathcal{M}_\bullet) & (u,v)&\mapsto uv
\end{aligned}
\]

\[
\tau_{\mathcal{L}_\bullet}\,\varphi_{\mathcal{L}_\bullet,\mathcal{M}_\bullet} = \tau_{\mathcal{M}_\bullet}\,\varphi_{\mathcal{M}_\bullet,\mathcal{L}_\bullet}^{\,\mathrm{sym}}
\]

\[
\mathbb{R}\mathcal{H}om(\mathcal{L}_\bullet, \mathcal{M}_\bullet) \times \mathbb{R}\mathcal{H}om(\mathcal{M}_\bullet, \mathcal{L}_\bullet) \longrightarrow A_X
\]

\[
P_\bullet \simeq \mathbb{R}\mathcal{H}om(Q_\bullet, A_X), \qquad Q_\bullet \simeq \mathbb{R}\mathcal{H}om(P_\bullet, A_X^\bullet)
\]

\[
\langle u^{(i)}, v^{(-i)}\rangle = \sum \mathrm{Tr}\,(uv)^{\varepsilon}_i\,(-1)^i
\]

\[
\begin{aligned}
&\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) \times \mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr)\\
&\qquad\longrightarrow K^\bullet_{X\times_S Y} = h^!(A_S)
\end{aligned}
\]

\[
\mathrm{pr}_2^!(G_\bullet) \longrightarrow D_Z\,\mathrm{pr}_2^*(D_Y G_\bullet)
\]

\[
\begin{aligned}
\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) &\longrightarrow \mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), D_Z\,\mathrm{pr}_2^*(D_Y G_\bullet)\bigr)\\
&\;\wr\;\; D_Z\bigl(\mathrm{pr}_1^*(F_\bullet)\overset{L}{\otimes}\mathrm{pr}_2^*(D_Y G_\bullet)\bigr)
\end{aligned}
\]

\[
D_Z\bigl(\mathrm{pr}_1^*(F_\bullet)\overset{L}{\otimes}\mathrm{pr}_2^*(D_Y(G_\bullet))\bigr) \longrightarrow D_Z\,\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_2^*G_\bullet, \mathrm{pr}_1^!(F_\bullet)\bigr)
\]

\[
\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr) \longrightarrow \mathrm{pr}_1^*(F_\bullet)\overset{L}{\otimes}\mathrm{pr}_2^*(D_Y(G_\bullet))
\]

\[
D_Z\bigl(\mathrm{pr}_2^*(G_\bullet)\overset{L}{\otimes}\mathrm{pr}_1^*(D_X F_\bullet)\bigr)
\]

\[
D_Z\bigl(\mathrm{pr}_2^*(G_\bullet)\overset{L}{\otimes}\mathrm{pr}_1^* D_X(F_\bullet)\bigr) \longrightarrow \mathrm{pr}_1^*(F_\bullet)\overset{L}{\otimes}\mathrm{pr}_2^*(D_Y(G_\bullet))
\]

\[
(\ast)\quad \mathrm{pr}_1^*(F_\bullet)\overset{L}{\otimes}\mathrm{pr}_2^* D_Y G_\bullet \overset{L}{\otimes} \mathrm{pr}_2^* G_\bullet \overset{L}{\otimes} \mathrm{pr}_1^* D_X(F_\bullet) \longrightarrow K^\bullet_Z
\]

\[
\begin{aligned}
\mathbb{R}\mathcal{H}om(\mathrm{pr}_1^*F_\bullet, \mathrm{pr}_2^!G_\bullet) &\to D_Z\bigl(\mathrm{pr}_1^*(F_\bullet)\overset{L}{\otimes}\mathrm{pr}_2^*(D_Y G_\bullet)\bigr)\\
\mathbb{R}\mathcal{H}om(\mathrm{pr}_2^*G_\bullet, \mathrm{pr}_1^!F_\bullet) &\to D_Z\bigl(\mathrm{pr}_2^*(G_\bullet)\overset{L}{\otimes}\mathrm{pr}_1^*(D_X F_\bullet)\bigr)
\end{aligned}
\]

\[
\begin{aligned}
\mathrm{pr}_1^*(F_\bullet)\otimes\mathrm{pr}_1^*(D_X F_\bullet) &\longrightarrow \mathrm{pr}_1^*(R^\bullet_X)\\
\mathrm{pr}_2^*(G_\bullet)\otimes\mathrm{pr}_2^* D_Y G_\bullet &\longrightarrow \mathrm{pr}_2^*(R^\bullet_Y)
\end{aligned}
\]

\[
\mathrm{pr}_1^*(R^\bullet_X)\otimes\mathrm{pr}_2^*(R^\bullet_Y)
\]

\[
(\ast\ast)\quad \mathrm{pr}_1^*(F_\bullet)\otimes\mathrm{pr}_2^*(D_Y G_\bullet) \longrightarrow D_Z\bigl(\mathrm{pr}_2^* G_\bullet\otimes\mathrm{pr}_1^* D_X(F_\bullet)\bigr)
\]

\[
(1)\quad R'_X \overset{L}{\boxtimes} R'_Y \xrightarrow{\;\sim\;} R'_{X\times Y}
\]

\[
(2)\quad F_\bullet \overset{L}{\boxtimes} D_Y G_\bullet \xrightarrow{\;\sim\;} D_Z\bigl(D_X F_\bullet \overset{L}{\boxtimes} G_\bullet\bigr)
\]

\[
\mathbb{R}\mathcal{H}om\bigl(F_\bullet \overset{L}{\boxtimes} G_\bullet,\, P_\bullet \overset{L}{\boxtimes} Q_\bullet\bigr) \xleftarrow{\;\simeq\;} \mathbb{R}\mathcal{H}om(F_\bullet, P_\bullet) \overset{L}{\boxtimes} \mathbb{R}\mathcal{H}om(G_\bullet, Q_\bullet)
\]

\[
R(f\times g)^!\bigl(P_\bullet \overset{L}{\boxtimes} Q_\bullet\bigr) \simeq Rf^!(P_\bullet) \overset{L}{\boxtimes} Rg^!(Q_\bullet)
\]

\[
(\varphi\times\psi)^!\;\text{\struck{$\varphi^!$}}\;\bigl(P_\bullet \boxtimes Q_\bullet\bigr) \simeq \varphi^!(P_\bullet)\boxtimes\psi^!(Q_\bullet)
\]

\[
\mathbb{R}\mathcal{H}om\bigl(F_\bullet\boxtimes G_\bullet,\, P_\bullet\boxtimes Q_\bullet\bigr) \simeq \mathbb{R}\mathcal{H}om(F_\bullet, P_\bullet)\overset{L}{\boxtimes}\mathbb{R}\mathcal{H}om(G_\bullet, Q_\bullet)
\]

\[
\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) \times \mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr) \longrightarrow K^\bullet_{X\times_S Y}
\]

\[
\begin{aligned}
\xi^\circ &\in \mathbb{R}^0\mathrm{Hom}\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) = \mathbb{R}^0\Gamma_{X\times_S Y}\,\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr)\\
\eta^\circ &\in \mathbb{R}^0\mathrm{Hom}\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr) = \mathbb{R}^0\Gamma_{X\times_S Y}\,\mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr)
\end{aligned}
\]

\[
\xi^\circ\cup\eta^\circ \in \mathbb{R}^0\Gamma_{X\times_S Y}\bigl(K^\bullet_{X\times_S Y}\bigr) \xrightarrow{\;\mathrm{Tr}_h\;} A.
\]

\[
\boxed{\;\mathrm{Tr}(\xi^\circ\cup\eta^\circ) = \langle \xi^\circ, \eta^\circ\rangle\;}
\qquad \Bigl(= \sum_i \mathrm{Tr}\bigl((\xi^\circ)^{(i)}(\eta^\circ)^{(-i)}\bigr)(-1)^i\Bigr)
\]

\[
\boxed{\;\mathrm{Tr}\bigl(\xi^\circ\cup\delta^\bullet_{X,F}\bigr) = \sum (-1)^i\,\mathrm{Tr}\,[\xi^\circ]^i\;}
\]

\[
u : T \longrightarrow X\times_S Y
\]

\[
u_1 : T\to X, \qquad u_2 : \text{\struck{$T$}}\;T\to Y
\]

\[
\mathbb{R}u_*\bigl(\mathbb{R}\mathcal{H}om(u_1^*(F_\bullet), u_2^!(G_\bullet))\bigr) \longrightarrow \mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr)
\]

\[
\alpha^\circ \in \mathbb{R}\mathrm{Hom}^\circ\bigl(u_1^*(F_\bullet), u_2^!(G_\bullet)\bigr) = \mathbb{R}^0\Gamma_T\bigl(\mathbb{R}\mathcal{H}om(u_1^*(F_\bullet), u_2^!(G_\bullet))\bigr)
\]

\[
\overline{\alpha^\circ} \in \mathrm{Corr}(X,F;\, Y,G)
\]

\[
\overline{\beta^\circ} \in \mathrm{Corr}(Y,G;\, X,F).
\]

\[
  R^{0}\Gamma_{\overline{T}}\bigl(R\mathcal{H}om(\mathrm{pr}_{1}^{*}(F), \mathrm{pr}_{2}^{!}(G))\bigr)
  = R^{0}\Gamma_{X \times_{S} Y}\bigl(R\underline{\Gamma}_{\overline{T}}\, R\mathcal{H}om( \ , \ )\bigr)
\]

\[
  R^{0}\Gamma_{\overline{T'}}\bigl(R\mathcal{H}om(\mathrm{pr}_{2}^{*}(G), \mathrm{pr}_{1}^{!}(F))\bigr)
  = R^{0}\Gamma_{X \times_{S} Y}\bigl(R\underline{\Gamma}_{\overline{T'}}, R\mathcal{H}om \ldots\bigr)
\]

\[
  \overline{\alpha^{0}} \cup \overline{\beta^{0}}
  \in R^{0}\Gamma_{\overline{T} \cap \overline{T'}}\, \underline{K}^{\bullet}_{X \times_{S} Y}
  \simeq R^{0}\Gamma_{\overline{T} \cap \overline{T'}}\, K^{\bullet}_{\overline{T} \times_{S} \overline{T'}}
\]

\[
  \langle \xi^{0}, \eta^{0}\rangle = \sum_{\alpha} \langle \xi^{0}, \eta^{0}\rangle_{S_{\alpha}} .
\]

\[
  \operatorname{Hom}(F_{\bullet}, R^{!}f(G^{\bullet}))
  \xleftarrow{\ \sim\ } \operatorname{Hom}(R_{!}f(F_{\bullet}), G^{\bullet})
\]

\[
  F_{\bullet} \xrightarrow{\ \sim\ } D_{X} D_{X}(F_{\bullet})
\]

\[
  D_{X}(F_{\bullet}) = R\mathcal{H}om(F_{\bullet}, A_{X}),
  \qquad F_{\bullet} \in \operatorname{Ob} D^{b}(X)
\]

\[
  D_{X} \
  \begin{cases}
    D_{c}(X)^{\circ} \xrightarrow{\ \approx\ } D_{c}(X) \\
    D_{c}^{-}(X)^{\circ} \xrightarrow{\ \approx\ } D_{c}^{+}(X) \\
    D_{c}^{+}(X)^{\circ} \xrightarrow{\ \approx\ } D_{c}^{-}(X) \\
    D_{c}^{b}(X) \xrightarrow{\ \approx\ } D_{c}^{b}(X)
  \end{cases}
\]

\[
  R\mathcal{H}om(P_{\bullet}, D_{X}(F_{\bullet}))
  \simeq R\mathcal{H}om(P_{\bullet}, R\mathcal{H}om(F_{\bullet}, R^{\bullet}_{X}))
  \simeq R\mathcal{H}om(P_{\bullet} \overset{L}{\otimes} F_{\bullet}, R^{\bullet}_{X})
\]

\[
  \operatorname{Ext}^{i}(\cdot\,, G) = 0 \quad \text{pour } i > n, \qquad
  \operatorname{Ext}^{i}(F, G) = 0 \quad \text{pour } i > 2n .
\]

\[
  \mathcal{E}xt^{p}(F, G)
\]

\[
  \operatorname{Ext}^{n}(i_{!}(F), G) \simeq \operatorname{Ext}^{n}(F, Ri^{!}(G))
\]

\[
  \operatorname{Ext}^{*}(F, Ri^{!}(G)) \Longleftarrow E_{2}^{pq}
  = \operatorname{Ext}^{p}(F, R^{q}i^{!}(G))
\]

\[
  \operatorname{Ext}^{p}(F, \underline{R^{q}i^{!}G})
\]

\[
  \mathcal{E}xt^{0}, \mathcal{E}xt^{1}\ \mathcal{E}xt^{2} \,/\,
  \mathcal{E}xt^{3}\ \mathcal{E}xt^{4} \,/ \quad \cdots \quad /\,
  \mathcal{E}xt^{2d-1}, \mathcal{E}xt^{2d}
\]

\[
  H^{p}(X, \mathcal{E}xt^{q}) \qquad
  \text{\struck{$H^{q}$}}\ \underline{H}^{n}(X, \mathcal{E}xt^{0}),\ H^{n-1}\mathcal{E}xt^{1},
\]

\[
  n \leq 2d \qquad
  n-1 \leq 2(d-1) \qquad
  n-3 \leq 2(d-2)
\]

\[
  \text{\struck{$n-1 \leq 2d-1$}} \qquad
  n-2 \leq 2(d-1) \qquad
  n-4 \leq 2(d-2) \qquad
  \underline{2d-1}
\]

\[
  \text{\struck{$\operatorname{Ext}^{*}(F, G) \simeq$}}
\]

\[
  R\mathcal{H}om(F, \text{\struck{$D$}}G) \simeq R\mathcal{H}om(F \otimes DG, K_{X})
\]

\[
  \operatorname{Ext}^{n}(X; F, G) \Longleftarrow E_{2}^{pq}
  = \operatorname{Ext}^{p}(X; \mathcal{E}xt^{q}(X; G, A_{X}), A_{X})
\]

\[
  \operatorname{Ext}^{*}(X; F, A_{X}) \Longleftarrow H^{p}(X, \mathcal{E}xt^{q}(F, A_{X}))
  \qquad 2d
\]

\[
  \mathcal{E}xt^{i}(F, G) \qquad \text{\struck{$F$}}\ DG
\]

\[
  R\mathcal{H}om(F, G) \simeq R\mathcal{H}om(F, DDG)
  \simeq R\mathcal{H}om(F \overset{L}{\otimes} DG, R^{\bullet}_{X})
\]

\[
  F \otimes DG \qquad -2d \longrightarrow 0 \qquad -2d \longrightarrow 0
\]

\[
  D(F \overset{L}{\otimes} DG \qquad \mathcal{E}xt^{i}(F, G)
\]

\[
  \mathcal{E}xt^{*}(F, G) \Longleftarrow E_{2}^{pq}
  = \mathcal{E}xt^{p}(F \otimes \mathcal{E}xt^{q}(G, A_{X}), A_{X})
\]

\[
  \text{\struck{$\mathcal{E}xt^{*}$}}\ \mathcal{E}xt^{2d}(F, G)
  \simeq \mathcal{E}xt^{2d}(F \otimes \mathcal{H}om(G, A_{X}), A_{X})
\]

\[
  \cdots \to R^{i}\Gamma_{x}(F) \to R^{i}\Gamma_{X}(F) \to R^{i}\Gamma_{U}(F)
  \to R^{i+1}\Gamma_{x}(F) \to \cdots
\]

\[
  R^{-i}\Gamma_{X}(DF) \leftarrow R^{-i}\Gamma_{x}DF \leftarrow R^{-i-1}\Gamma_{U}(DF)
  \leftarrow R^{-i-1}\Gamma_{x}DF \leftarrow \cdots
\]

\[
  R\mathcal{H}om(F, G) \qquad D(F \otimes DG) \qquad
  R\mathcal{H}om(G, DF) \simeq D(G \overset{L}{\otimes} F)
\]

\[
  R\mathcal{H}om(F, D(DG)) \qquad
  D(F \otimes DG) \simeq DR\mathcal{H}om(F, G)
\]

\[
  D(\mathcal{H}^{i}_{x}(F)) \longrightarrow \text{\struck{$D$}}\ \mathcal{H}^{-i}_{x}(DF)_{x}
  \qquad F \to DDF
\]

\[
  \text{\struck{$\mathcal{H}^{i}_{x}(F) \to$}} \qquad
  \mathcal{H}^{i}(F)_{x} \longrightarrow D_{x}\,\underline{\mathcal{H}^{-i}_{x}(DF)}
  \qquad D_{x} \text{\struck{$\mathcal{H}$}}\, D_{x}(\mathcal{H}^{i}(F)_{x})
\]

\[
  \Updownarrow
\]

\[
  \Updownarrow
\]

\[
  DF \overset{L}{\otimes} G \simeq DR\mathcal{H}om(G, F),
\]

\[
  \text{\struck{$DF \otimes DDG \simeq D(F \otimes D$}}
  \qquad [P, Q]
\]