Cote n° 32 · pages 4–47
· 61 displayed formulas · Corrections (références, rajouts) [dont SGA 4] : tapuscrit (s.d.), notes manuscrites (s.d.).
Inventory dating : [vers 1963-1977]
Édition de démonstration
\[H^i(LT(K)) = 0 \ \text{si}\ i \geqslant nd .\]
LaTeX source
\[
H^i(LT(K)) = 0 \ \text{si}\ i \geqslant nd .
\]\[N T N^{-1} K \quad
\begin{cases}
\text{si } 0 \leqslant p \leqslant q, & K \text{ dans } [p, nq]\\
\text{si } p \leqslant q \leqslant 0, & K \text{ dans } [np, q]
\end{cases}
\quad \text{comme complexes.}\]
LaTeX source
\[
N T N^{-1} K \quad
\begin{cases}
\text{si } 0 \leqslant p \leqslant q, & K \text{ dans } [p, nq]\\
\text{si } p \leqslant q \leqslant 0, & K \text{ dans } [np, q]
\end{cases}
\quad \text{comme complexes.}
\]\[\text{(4.2)}\qquad \add{\mathrm{Tr}_f \text{ ou }} f_{*}\colon\;
Rf_{!}(F_X) \longrightarrow F_Y(d'-d)[2(d'-d)] ,\]
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\[
\text{(4.2)}\qquad \add{\mathrm{Tr}_f \text{ ou }} f_{*}\colon\;
Rf_{!}(F_X) \longrightarrow F_Y(d'-d)[2(d'-d)] ,
\]\[\text{(4.2.1)}\qquad H^i(f_{*}) \text{ ou } f_{*}\colon\;
R^i f_{!}(F_X) \longrightarrow
\underline{H}^{i\supplied{+2(d'-d)}}(F_Y)(d'-d)\,\struck{[2(d'-d)]} ,\]
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\[
\text{(4.2.1)}\qquad H^i(f_{*}) \text{ ou } f_{*}\colon\;
R^i f_{!}(F_X) \longrightarrow
\underline{H}^{i\supplied{+2(d'-d)}}(F_Y)(d'-d)\,\struck{[2(d'-d)]} ,
\]\[\text{(4.2.2)}\qquad Rq_{!}(f_{*}) \text{ ou } f_{*}\colon\;
Rp_{!}(F_X) \longrightarrow Rq_{!}(F_Y)(d'-d)[2(d'-d)] ,\]
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\[
\text{(4.2.2)}\qquad Rq_{!}(f_{*}) \text{ ou } f_{*}\colon\;
Rp_{!}(F_X) \longrightarrow Rq_{!}(F_Y)(d'-d)[2(d'-d)] ,
\]\[g\colon Y \longrightarrow Z\]
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\[ g\colon Y \longrightarrow Z \]
\[\text{(4.2.3)}\qquad R(gf)_{!}(F_X) \longrightarrow
Rg_{!}(F_Y)(d'-d)[2(d'-d)] ,\]
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\[
\text{(4.2.3)}\qquad R(gf)_{!}(F_X) \longrightarrow
Rg_{!}(F_Y)(d'-d)[2(d'-d)] ,
\]\[\text{(4.2.4)}\qquad Rg_{*}Rf_{!}(F_X) \overset{\mathrm{dfn}}{=}
R(gf)_{c(f)}(F_X) \longrightarrow Rg_{*}(F_Y)(d'-d)[2(d'-d)]\]
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\[
\text{(4.2.4)}\qquad Rg_{*}Rf_{!}(F_X) \overset{\mathrm{dfn}}{=}
R(gf)_{c(f)}(F_X) \longrightarrow Rg_{*}(F_Y)(d'-d)[2(d'-d)]
\]\[\text{(4.2.5)}\qquad R\Gamma_X(Rf_{!}(F_X)) \overset{\mathrm{dfn}}{=}
R\Gamma_{X,\,c(f)}(F_X) \longrightarrow R\Gamma_Y(F_Y(d'-d))[2(d'-d)] .\]
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\[
\text{(4.2.5)}\qquad R\Gamma_X(Rf_{!}(F_X)) \overset{\mathrm{dfn}}{=}
R\Gamma_{X,\,c(f)}(F_X) \longrightarrow R\Gamma_Y(F_Y(d'-d))[2(d'-d)] .
\]\[\text{(4.2.6)}\qquad Rf_{*}(F_X) \longrightarrow F_Y(d'-d)[2(d'-d)] ,\]
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\[
\text{(4.2.6)}\qquad Rf_{*}(F_X) \longrightarrow F_Y(d'-d)[2(d'-d)] ,
\]\[\begin{aligned}
&\text{(4.2.7)}\qquad R(gf)_{*}(F_X) \longrightarrow Rg_{*}(F_Y)(d'-d)[2(d'-d)] ,\\
&\text{(4.2.8)}\qquad R\Gamma_X(F_X) \longrightarrow R\Gamma_Y(F_Y(d'-d))[2(d'-d)] .
\end{aligned}\]
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\[
\begin{aligned}
&\text{(4.2.7)}\qquad R(gf)_{*}(F_X) \longrightarrow Rg_{*}(F_Y)(d'-d)[2(d'-d)] ,\\
&\text{(4.2.8)}\qquad R\Gamma_X(F_X) \longrightarrow R\Gamma_Y(F_Y(d'-d))[2(d'-d)] .
\end{aligned}
\]\[\text{(4.3)}\qquad \boxed{\,Rf_{!}(f^{*}(G)) \longrightarrow
G(d'-d)[2(d'-d)]\,} ,\]
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\[
\text{(4.3)}\qquad \boxed{\,Rf_{!}(f^{*}(G)) \longrightarrow
G(d'-d)[2(d'-d)]\,} ,
\]\[\text{(4.3 bis)}\qquad Rf_{!}(f^{*}G(d-d')) \longrightarrow G[2(d'-d)] .\]
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\[
\text{(4.3 bis)}\qquad Rf_{!}(f^{*}G(d-d')) \longrightarrow G[2(d'-d)] .
\]\[\text{(4.4)}\qquad Rf_{!}(A_X(d-d')[2(d-d')]) \longrightarrow A_Y .\]
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\[
\text{(4.4)}\qquad Rf_{!}(A_X(d-d')[2(d-d')]) \longrightarrow A_Y .
\]\[\textstyle\sum d_i n_i = N\]
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\[ \textstyle\sum d_i n_i = N \]
\[\Bigl(\sum d_i\, f_{i*} u_i^{*}\Bigr) f^{*} = \sum d_i\, f_{i*} u_i^{*} f^{*}
= \sum d_i\, (f_{i*} f_i^{*}) = \sum d_i\, n_i\, \mathrm{id}\]
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\[
\Bigl(\sum d_i\, f_{i*} u_i^{*}\Bigr) f^{*} = \sum d_i\, f_{i*} u_i^{*} f^{*}
= \sum d_i\, (f_{i*} f_i^{*}) = \sum d_i\, n_i\, \mathrm{id}
\]\[\mathrm{Tr}_f(1_X) = N\, 1_Y\]
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\[
\mathrm{Tr}_f(1_X) = N\, 1_Y
\]\[\text{(4.4)}\qquad f^{*}(G)(d-d')[2(d-d')] \longrightarrow Rf^{!}G ,\]
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\[
\text{(4.4)}\qquad f^{*}(G)(d-d')[2(d-d')] \longrightarrow Rf^{!}G ,
\]\[\text{(4.5)}\qquad \varphi_f\colon A_X \longrightarrow
Rf^{!}(A_Y)(d'-d)\ 2(d'-d)\add{)} ,\quad
\add{\text{i.e. } \varphi_f \in H^{2(d-d')}(Y, Rf^{!}(A_Y)(d-d'))}\]
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\[
\text{(4.5)}\qquad \varphi_f\colon A_X \longrightarrow
Rf^{!}(A_Y)(d'-d)\ 2(d'-d)\add{)} ,\quad
\add{\text{i.e. } \varphi_f \in H^{2(d-d')}(Y, Rf^{!}(A_Y)(d-d'))}
\]\[\text{(4.6)}\qquad
\begin{aligned}
&\struck{\varphi_f \in H^0(Y, \underline{H}^0 Rf^{!}(A_Y)(d'-d)\ 2(d'-d))}\\
&\add{\varphi_f} \in H^0(Y, \underline{R}^{2(d'-d)} f^{!}(A_Y))(d'-d)) ,
\end{aligned}\]
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\[
\text{(4.6)}\qquad
\begin{aligned}
&\struck{\varphi_f \in H^0(Y, \underline{H}^0 Rf^{!}(A_Y)(d'-d)\ 2(d'-d))}\\
&\add{\varphi_f} \in H^0(Y, \underline{R}^{2(d'-d)} f^{!}(A_Y))(d'-d)) ,
\end{aligned}
\]\[\text{(4.7)}\qquad R^i f^{!}(A_Y) = 0 \quad \text{pour } i \neq 2(d-d') ,\]
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\[
\text{(4.7)}\qquad R^i f^{!}(A_Y) = 0 \quad \text{pour } i \neq 2(d-d') ,
\]\[\text{(4.8)}\qquad \add{\gamma_{X/Y}} \in H^{2(d-d')}_{\add{X}}(Y, A_Y(d-d')) .\]
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\[
\text{(4.8)}\qquad \add{\gamma_{X/Y}} \in H^{2(d-d')}_{\add{X}}(Y, A_Y(d-d')) .
\]\[\text{(4.9)}\qquad \add{\gamma_{X/Y}} \in H^{2(d-d')}(Y, A_Y(d-d'))\]
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\[
\text{(4.9)}\qquad \add{\gamma_{X/Y}} \in H^{2(d-d')}(Y, A_Y(d-d'))
\]\[K(K_0, \varphi) = p^{*}(K_0) + p^{*}(\varphi)[\mathcal{O}(1)]\]
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\[
K(K_0, \varphi) = p^{*}(K_0) + p^{*}(\varphi)[\mathcal{O}(1)]
\]\[\Gamma(S, R^1 p_{*} G) = \frac{H^1(P(E), G)}{H^1(S, G)}\]
LaTeX source
\[
\Gamma(S, R^1 p_{*} G) = \frac{H^1(P(E), G)}{H^1(S, G)}
\]\[0 \to H^1(S, G) \to H^1(P(E), G) \xrightarrow{\ \theta\ } \Gamma(S, R^1 p_{*} G) \to 0\]
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\[
0 \to H^1(S, G) \to H^1(P(E), G) \xrightarrow{\ \theta\ } \Gamma(S, R^1 p_{*} G) \to 0
\]\[\struck{H^1(S_0}\]
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\[
\struck{H^1(S_0}
\]\[H^1(X_{\struck{s}}, f^{*} \underline{\operatorname{Hom}}(L, I)) = 0,\]
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\[
H^1(X_{\struck{s}}, f^{*} \underline{\operatorname{Hom}}(L, I)) = 0,
\]\[0 \to G \to G_1 \to G_2 \to 0\]
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\[ 0 \to G \to G_1 \to G_2 \to 0 \]
\[0 \to R^1 p_{*} G \to \underline{\operatorname{Hom}}(\mathbb{G}_m, G_1) \to
\underline{\operatorname{Hom}}(\mathbb{G}_m, G_2)\]
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\[
0 \to R^1 p_{*} G \to \underline{\operatorname{Hom}}(\mathbb{G}_m, G_1) \to
\underline{\operatorname{Hom}}(\mathbb{G}_m, G_2)
\]\[0 \to \mathbb{G}_m^n \to G_0 \to H \to 0\]
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\[
0 \to \mathbb{G}_m^n \to G_0 \to H \to 0
\]\[g^{*} Rf_{!} \xrightarrow{\ \sim\ } Rf'_{!}\, g'^{*}\]
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\[
g^{*} Rf_{!} \xrightarrow{\ \sim\ } Rf'_{!}\, g'^{*}
\]\[(3.1.6.3)\qquad Rg'_{*}\, Rf'^{!} \xrightarrow{\ \sim\ } Rf^{!}\, Rg_{*}\]
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\[
(3.1.6.3)\qquad Rg'_{*}\, Rf'^{!} \xrightarrow{\ \sim\ } Rf^{!}\, Rg_{*}
\]\[(3.1.6.4)\qquad
\begin{array}{ll}
g^{*} Rf_{*} \longrightarrow Rf'_{*}\, g'^{*} & \qquad g^{*} Rf_{!} \xrightarrow{\ \sim\ } Rf'_{!}\, g'^{*} \\[1ex]
Rg^{!} Rf_{*} \xrightarrow{\ \sim\ } Rf'_{*}\, Rg'^{!} & \qquad Rg^{!} Rf_{!} \longleftarrow Rf'_{!}\, Rg'^{!}
\end{array}\]
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\[
(3.1.6.4)\qquad
\begin{array}{ll}
g^{*} Rf_{*} \longrightarrow Rf'_{*}\, g'^{*} & \qquad g^{*} Rf_{!} \xrightarrow{\ \sim\ } Rf'_{!}\, g'^{*} \\[1ex]
Rg^{!} Rf_{*} \xrightarrow{\ \sim\ } Rf'_{*}\, Rg'^{!} & \qquad Rg^{!} Rf_{!} \longleftarrow Rf'_{!}\, Rg'^{!}
\end{array}
\]\[(3.1.7.1)\qquad f^{\bullet}_{!}\underline{F} \otimes_{\mathcal{B}} \underline{G} \longrightarrow f^{\bullet}_{!}(\underline{F} \otimes_{\mathcal{B}} f^{*}\underline{G}).\]
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\[
(3.1.7.1)\qquad f^{\bullet}_{!}\underline{F} \otimes_{\mathcal{B}} \underline{G} \longrightarrow f^{\bullet}_{!}(\underline{F} \otimes_{\mathcal{B}} f^{*}\underline{G}).
\]\[(3.1.7.2)\qquad \underline{\operatorname{Hom}}_{\mathcal{A}}(f^{*}\underline{G}, f^{!}_{\bullet}\underline{H}) \longrightarrow f^{!}_{\bullet}\underline{\operatorname{Hom}}_{\mathcal{A}}(\underline{G}, \underline{H}).\]
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\[
(3.1.7.2)\qquad \underline{\operatorname{Hom}}_{\mathcal{A}}(f^{*}\underline{G}, f^{!}_{\bullet}\underline{H}) \longrightarrow f^{!}_{\bullet}\underline{\operatorname{Hom}}_{\mathcal{A}}(\underline{G}, \underline{H}).
\]\[(3.1.7.3)\qquad R\underline{\operatorname{Hom}}_{\mathcal{A}}(f^{*}L, Rf^{!}M) \xrightarrow{\ \sim\ } Rf^{!} R\underline{\operatorname{Hom}}_{\mathcal{A}}(L, M),\]
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\[
(3.1.7.3)\qquad R\underline{\operatorname{Hom}}_{\mathcal{A}}(f^{*}L, Rf^{!}M) \xrightarrow{\ \sim\ } Rf^{!} R\underline{\operatorname{Hom}}_{\mathcal{A}}(L, M),
\]\[R\underline{\operatorname{Hom}}_{\mathcal{A}}(f^{*}L, Rf^{!}M) \xrightarrow{\ \sim\ } Rf^{!} R\underline{\operatorname{Hom}}_{\mathcal{A}}(L, M).\]
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\[
R\underline{\operatorname{Hom}}_{\mathcal{A}}(f^{*}L, Rf^{!}M) \xrightarrow{\ \sim\ } Rf^{!} R\underline{\operatorname{Hom}}_{\mathcal{A}}(L, M).
\]\[\struck{E}\ \operatorname{Mod}(S, \mathcal{A}) \to \operatorname{Mod}(S, \mathcal{B}) : \underline{F} \mapsto \underline{\operatorname{Hom}}_{\mathcal{A}}(L, \underline{F}),\]
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\[
\struck{E}\ \operatorname{Mod}(S, \mathcal{A}) \to \operatorname{Mod}(S, \mathcal{B}) : \underline{F} \mapsto \underline{\operatorname{Hom}}_{\mathcal{A}}(L, \underline{F}),
\]\[K(U) = \bar f_{*}\, \tau^{*}_{\leqslant d}\, C_{\ell}^{*} \mathcal{A}_U
= f_!^{\bullet}(\mathcal{A}_U) .\]
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\[
K(U) = \bar f_{*}\, \tau^{*}_{\leqslant d}\, C_{\ell}^{*} \mathcal{A}_U
= f_!^{\bullet}(\mathcal{A}_U) .
\]\[Rf_!\, \mathcal{A}_U \simeq \mathcal{A} \overset{\mathbb{L}}{\otimes} Rf_!(\mathbb{Z}/n)_U .\]
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\[
Rf_!\, \mathcal{A}_U \simeq \mathcal{A} \overset{\mathbb{L}}{\otimes} Rf_!(\mathbb{Z}/n)_U .
\]\[\underline{F}/\ell \underline{F} \xrightarrow{\ \sim\ } \ell^{i}\underline{F}/\ell^{i+1}\underline{F} ,\]
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\[
\underline{F}/\ell \underline{F} \xrightarrow{\ \sim\ } \ell^{i}\underline{F}/\ell^{i+1}\underline{F} ,
\]\[\struck{f_!^{\bullet}(\mathbb{Z}/n)_U \xrightarrow{\ \sim\ }}\]
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\[
\struck{f_!^{\bullet}(\mathbb{Z}/n)_U \xrightarrow{\ \sim\ }}
\]\[f_!^{\bullet}(\mathbb{Z}/n)_U \otimes \mathcal{A} \xrightarrow{\ \sim\ } f_!^{\bullet}(\mathcal{A}_U)\]
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\[
f_!^{\bullet}(\mathbb{Z}/n)_U \otimes \mathcal{A} \xrightarrow{\ \sim\ } f_!^{\bullet}(\mathcal{A}_U)
\]\[U \longmapsto f_!^{\bullet}((\mathbb{Z}/n)_U)\]
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\[
U \longmapsto f_!^{\bullet}((\mathbb{Z}/n)_U)
\]\[U \longmapsto \add{K_0(U) \overset{\mathrm{df}}{=}}\; L_{\bullet}(f_!^{\bullet}(\mathbb{Z}/n)_U)\; \struck{= K_0(U)} .\]
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\[
U \longmapsto \add{K_0(U) \overset{\mathrm{df}}{=}}\; L_{\bullet}(f_!^{\bullet}(\mathbb{Z}/n)_U)\; \struck{= K_0(U)} .
\]\[K_0^{!}(\underline{G}) \xrightarrow{\ \sim\ } RK_0^{!}(\underline{G})\]
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\[
K_0^{!}(\underline{G}) \xrightarrow{\ \sim\ } RK_0^{!}(\underline{G})
\]\[(3.1.15.1) \qquad K(W) = \bar f_{V *}\, \tau_{\leqslant 2d}\, C^{*}_{\ell, \bar X_V}(\mathcal{A}_U)\]
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\[
(3.1.15.1) \qquad K(W) = \bar f_{V *}\, \tau_{\leqslant 2d}\, C^{*}_{\ell, \bar X_V}(\mathcal{A}_U)
\]\[\struck{j_! K(W)}\]
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\[
\struck{j_! K(W)}
\]\[j_!\, K(W) \xrightarrow{\ \sim\ } \bar f_{*}\, \tau_{\leqslant 2d}\, C^{*}_{\ell, \bar X}(\mathcal{A}_U)\]
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\[
j_!\, K(W) \xrightarrow{\ \sim\ } \bar f_{*}\, \tau_{\leqslant 2d}\, C^{*}_{\ell, \bar X}(\mathcal{A}_U)
\]\[(3.1.16.1) \qquad \mathrm{Hom}(Rf_!\, K, L) \simeq \mathrm{Hom}(K, Rf^! L)\]
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\[
(3.1.16.1) \qquad \mathrm{Hom}(Rf_!\, K, L) \simeq \mathrm{Hom}(K, Rf^! L)
\]\[\mathrm{Hom}^{\bullet}(f_!^{\bullet} K, L) \simeq \mathrm{Hom}^{\bullet}(K, f^{!}_{\bullet} L) .\]
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\[
\mathrm{Hom}^{\bullet}(f_!^{\bullet} K, L) \simeq \mathrm{Hom}^{\bullet}(K, f^{!}_{\bullet} L) .
\]\[\mathrm{Hom}^{*}_{\uncertain{/V}}(K, f^{!}_{\bullet} L)
= \mathrm{Hom}^{\bullet}(j'_! j'^{*} K, f^{!}_{\bullet} L)
\xrightarrow{\ \sim\ } \mathrm{Hom}^{\bullet}(f_!^{\bullet}(j'_! j'^{*} K), L) \longrightarrow\]
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\[
\mathrm{Hom}^{*}_{\uncertain{/V}}(K, f^{!}_{\bullet} L)
= \mathrm{Hom}^{\bullet}(j'_! j'^{*} K, f^{!}_{\bullet} L)
\xrightarrow{\ \sim\ } \mathrm{Hom}^{\bullet}(f_!^{\bullet}(j'_! j'^{*} K), L) \longrightarrow
\]\[\struck{\longrightarrow \mathrm{Hom}\ \mathrm{Hom}}\]
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\[
\struck{\longrightarrow \mathrm{Hom}\ \mathrm{Hom}}
\]\[\longrightarrow \mathrm{Hom}(j_! j^{*}(f_!^{\bullet} K), L) = \mathrm{Hom}_{V}(f_!^{\bullet} K, L)\]
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\[
\longrightarrow \mathrm{Hom}(j_! j^{*}(f_!^{\bullet} K), L) = \mathrm{Hom}_{V}(f_!^{\bullet} K, L)
\]\[(3.1.16.2) \qquad Rf_{*}\, R\underline{\mathrm{Hom}}(K, Rf^! L) \xrightarrow{\ \sim\ } R\underline{\mathrm{Hom}}(Rf_!\, K, L)\]
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\[
(3.1.16.2) \qquad Rf_{*}\, R\underline{\mathrm{Hom}}(K, Rf^! L) \xrightarrow{\ \sim\ } R\underline{\mathrm{Hom}}(Rf_!\, K, L)
\]\[\underline{H}^{-i}(Rf^! \mathbb{Z}/n\mathbb{Z})_x \simeq \varinjlim_{U(x)}
\mathrm{Hom}\bigl(\struck{\ill{}}\, H^{i}_{c}(U, \mathbb{Z}/n\mathbb{Z}), \mathbb{Z}/n\mathbb{Z}\bigr) ,\]
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\[
\underline{H}^{-i}(Rf^! \mathbb{Z}/n\mathbb{Z})_x \simeq \varinjlim_{U(x)}
\mathrm{Hom}\bigl(\struck{\ill{}}\, H^{i}_{c}(U, \mathbb{Z}/n\mathbb{Z}), \mathbb{Z}/n\mathbb{Z}\bigr) ,
\]\[H^{i}_{x}(X, \mathbb{Z}/n\mathbb{Z}) \longrightarrow \struck{\ill{}}\;
\underset{U(x)}{\underset{\longleftarrow}{\text{“lim”}}}\; H^{i}_{c}(U, \mathbb{Z}/n\mathbb{Z})\]
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\[
H^{i}_{x}(X, \mathbb{Z}/n\mathbb{Z}) \longrightarrow \struck{\ill{}}\;
\underset{U(x)}{\underset{\longleftarrow}{\text{“lim”}}}\; H^{i}_{c}(U, \mathbb{Z}/n\mathbb{Z})
\]\[\underline{H}^{-i}(Rf^! \mathbb{Z}/n\mathbb{Z})_x \xrightarrow{\ \sim\ } H^{i}_{x}(X, \mathbb{Z}/n\mathbb{Z})^{\vee}\]
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\[
\underline{H}^{-i}(Rf^! \mathbb{Z}/n\mathbb{Z})_x \xrightarrow{\ \sim\ } H^{i}_{x}(X, \mathbb{Z}/n\mathbb{Z})^{\vee}
\]\[(3.1.19.1) \qquad R^{i} f^{!} L = \underline{H}^{i}_{X}(L)\]
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\[
(3.1.19.1) \qquad R^{i} f^{!} L = \underline{H}^{i}_{X}(L)
\]\[f^{!} \simeq (p_1 \Delta)^{*} f^{!} \simeq \Delta^{*}(p_1^{*} f^{!})
\underset{\text{chgt de base par lisse}}{\simeq} \Delta^{*}(p_2^{!} f^{*})
\underset{\text{pureté relative ??}}{\simeq} \Delta^{!} p_2^{!} f^{*}[2d](d)
\simeq f^{\uncertain{*}}[2d](d)\]
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\[
f^{!} \simeq (p_1 \Delta)^{*} f^{!} \simeq \Delta^{*}(p_1^{*} f^{!})
\underset{\text{chgt de base par lisse}}{\simeq} \Delta^{*}(p_2^{!} f^{*})
\underset{\text{pureté relative ??}}{\simeq} \Delta^{!} p_2^{!} f^{*}[2d](d)
\simeq f^{\uncertain{*}}[2d](d)
\]