\[
  M = \coprod_{i \in \mathbb{Z}} M_{i}
\]

\[
  M = \coprod_{i} M'_{i}\Big(\frac{n-i}{2}\Big) \qquad
  M'_{i} \in \mathcal{M}^{+i}_{\text{pur}}(k)
\]

\[
  \operatorname{Filt}^{j} H^{i}(\overline{X}, \mathbb{Q}_{\ell}) =
  \varinjlim_{U} \operatorname{Ker}\big(H^{i}(\overline{X}, \mathbb{Q}_{\ell})
  \to H^{i}(\overline{U}, \mathbb{Q}_{\ell})\big)
\]

\[
  \operatorname{Filt}^{j} H^{i}(\overline{X}, \mathbb{Q}_{\ell}) =
  \text{partie de } T^{\ell}(h^{i}(X))
\]

\[
  \check{M} \otimes N \xrightarrow{\ \sim\ } \underline{\mathrm{Hom}}(M, N)
\]

\[
  (\mathcal{C}, F) \simeq (\mathrm{Modf}(G), \text{oubli}).
\]

\[
  \text{groupes affines sur } K \longrightarrow
  \begin{array}{l}
    \otimes\text{-catégories avec foncteur fibre donné, avec} \\ \otimes\text{-foncteurs modulo isom., compatibles avec } F
  \end{array}
\]

\[
  F' = P \times^{G} F
\]

\[
  F \mid \mathcal{M}^{+0}(S) \simeq F_{\xi}
\]

\[
  \pi_{1}\text{-}\mathbb{Q}\text{-modules continus de rang fini sur }
  \mathbb{Q} \longrightarrow \text{motifs sur } S
\]

\[
  G = \varprojlim G_{\alpha} \longrightarrow \pi_{1}(S, \xi)
\]

\[
  (*) \qquad \boxed{\varprojlim_{\alpha} G_{\alpha}/G_{\alpha}^{0}
  \xrightarrow{\ \sim\ } \pi_{1}(S, \xi)}
\]

\[
  F \otimes_{K} \mathbb{Q}_{\ell} \xrightarrow{\ \sim\ } T^{\ell}_{\xi}
\]

\[
  \pi_{1}(S, \xi) \to \operatorname{Aut} T^{\ell}_{\xi}
\]

\[
  \boxed{\rho_{\ell} : \pi_{1}(S, \xi) \longrightarrow G(\mathbb{Q}_{\ell})}
\]

\[
  G \longrightarrow \pi_{1}(S, \xi)
\]

\[
  \eta_{\ell} : G(\mathbb{Q}_{\ell}) \longrightarrow \pi_{1}(S, \xi) .
\]

\[
  \boxed{\eta_{\ell} \rho_{\ell} = \mathrm{id}_{\pi_{1}(S, \xi)}}
\]

\[
  \mathrm{Frob}_{k} \in \operatorname{Aut} \underline{\mathrm{id}}_{\mathcal{M}(k)}
  \longrightarrow \operatorname{Aut} \mathrm{id}_{\mathcal{C}}
  \quad [\simeq \mathfrak{Z}(\mathbb{Q}) \text{, où } \mathfrak{Z}
  \text{ est le centre de } G]
\]

\[
  \boxed{\mathrm{Frob}_{k,F} \in \mathfrak{Z}(\mathbb{Q}) \subset G(K)} .
\]

\[
  \mathrm{Frob}_{k} \in \pi_{1}(k, \xi) \simeq \hat{\mathbb{Z}}
\]

\[
  \rho_{\ell} : \hat{\mathbb{Z}} \longrightarrow G(\mathbb{Q}_{\ell})
\]

\[
  \mathrm{Frob}_{k,F} \in G(\mathbb{Q})
\]

\[
  G \longrightarrow \pi_{1}(S, \xi)
\]

\[
  \pi_{1}(S, \xi) \longrightarrow G(\mathbb{Q}_{\ell}) .
\]

\[
  T = D_{K}(M) \longrightarrow G
\]

\[
  y : \mathbb{G}_{m,K} \longrightarrow G
\]

\[
  i : \mathbb{G}_{m,K}^{2} \longrightarrow G
\]

\[
  i_{1}, i_{2} : \mathbb{G}_{m} \longrightarrow G
\]

\[
  i_{1}(\lambda) i_{2}(\lambda) = y(\lambda)
\]

\[
  i_{1}, i_{2} : \mathbb{G}_{m} \to G^{\mathrm{Betti}} \otimes_{\mathbb{Q}}
  \mathbb{C}
\]

\[
  i_{1}(\lambda) i_{2}(\lambda) = y(\lambda)
\]

\[
  \overline{i_{1}(\lambda)} = i_{2}(\lambda)
\]

\[
  \varepsilon : G \longrightarrow \mathbb{G}_{m,K}
\]

\[
  \boxed{\varepsilon(y(\lambda)) = \lambda^{2}}
\]

\[
  F' = \mathrm{Gr}(F)
\]

\[
  F \simeq F'
\]

\[
  i_{1}, i_{2} : \mathbb{G}_{m,\mathbb{C}} \longrightarrow \Gamma_{\mathbb{C}}
\]

\[
  i(\lambda, \mu) \text{ est } \lambda^{p}\mu^{q} \text{ sur } V^{pq}
\]

\[
  f^{*} : \mathcal{M}^{+}(Y) \to \mathcal{M}^{+}(X),
\]

\[
  Lf^{*} : \mathrm{D}^{b}(\mathcal{M}^{+}(Y)) \to \mathrm{D}^{b}(\mathcal{M}^{+}(X)),
\]

\[
  Rf_{*} : \mathrm{D}^{b}(\mathcal{M}^{+}(X)) \to \mathrm{D}^{b}(\mathcal{M}^{+}(Y))
\]

\[
  Rf_{*}(M \otimes Lf^{*}(N)) \simeq Rf_{*}(M) \otimes N .
\]

\[
  \mathcal{M}^{+}(X) \simeq \varinjlim \mathcal{M}^{+}(X_{i}) .
\]

\[
  Rf_{*}(M) = \varinjlim v_{i}^{*}(Rf_{i*}(M_{i}))
\]

\[
  T_{\ell} = T_{\ell}^{(X)} : \mathcal{M}^{+}(X) \to \mathcal{M}_{\ell}(X)
\]

\[
  T_{\infty} = T_{\infty}^{(X)} : \mathrm{D}^{b}(\mathcal{M}(X)) \to \mathcal{M}_{\infty}(X)
\]

\[
  \mathbb{Q}_{X}(-1) \quad\text{ou}\quad \mathbf{1}_{X}(-1) \in \operatorname{Ob} \mathcal{M}^{+}(X)
\]

\[
  T_{\ell}(\mathbb{Q}(-1)) \simeq \mathbb{Q}_{\ell}(-1)
  = \underbrace{T_{\ell}(\mathbb{G}_{m})^{-1}}_{\text{Module de Tate inverse de } \mathbb{G}_{m}}
\]

\[
  R^{2d} f_{*}(\mathbf{1}_{X}) \simeq \mathbb{Z}_{S}(-d)
\]

\[
  R^{2d} f_{!}(\mathbf{1}_{X}) \simeq \mathbb{Z}_{S}(-d)
\]

\[
  Ri^{!}(\mathbf{1}_{X}) \simeq \mathbb{Q}_{Y}(-d)
\]

\[
  \mathcal{M}^{+} \xrightarrow{\otimes \mathbb{Q}(-1)} \mathcal{M}^{+}
  \xrightarrow{\otimes \mathbb{Q}(-1)} \mathcal{M}^{+} \to \cdots
\]

\[
  \mathrm{Hom}(P \otimes Q, R) \simeq \mathrm{Hom}(P, \underline{\mathrm{Hom}}(Q, R))
  \simeq \mathrm{Hom}(Q, \underline{\mathrm{Hom}}(P, R))
\]

\[
  R\underline{\mathrm{Hom}}(P, Q) \qquad P, Q \in \operatorname{Ob} \mathrm{D}^{b}\mathcal{M}(X)
\]

\[
  \begin{array}{c}
    T_{\ell}(M) \text{ faisceau constant tordu} \\
    \Updownarrow \\
    T_{\ell'}(M) \text{ faisceau constant tordu}
  \end{array}
\]

\[
  T_{\ell}(u) \in \operatorname{End} T_{\ell}(M), \qquad
  T_{\ell'}(u) \in \operatorname{End}(T_{\ell'}(M))
\]

\[
  \boxed{\operatorname{Tr} T_{\ell}(u) = \operatorname{Tr} T_{\ell'}(u) \in \mathbb{Q}}
\]

\[
  \boxed{P(T_{\ell}(u), t) = P(T_{\ell'}(u), t) \in \mathbb{Q}[t]}
\]

\[
  u \in \operatorname{Hom}(\mathbf{1}_{X}, \underline{\mathrm{Hom}}(M, M))
  = \operatorname{Hom}(\mathbf{1}_{X}, M^{\vee} \otimes M)
\]

\[
  M^{\vee} \otimes M \to \mathbf{1}_{X}
\]

\[
  \operatorname{Tr} T_{\ell}(u) = T_{\ell}(c(u)) \in \mathbb{Q}_{\ell} .
\]

\[
  \boxed{X \text{ connexe} \Longrightarrow \operatorname{Hom}(\mathbf{1}_{X}, \mathbf{1}_{X}) = \mathbb{Q}\, \mathrm{id}_{\mathbf{1}_{X}}}
\]

\[
  \mathcal{M}^{+0}(X) \subset \mathcal{M}^{+1}(X) \subset \cdots \subset \mathcal{M}^{+i}(X) \subset \cdots
\]

\[
  R^{j}f_{!} : \mathcal{M}^{+i}(X) \to \mathcal{M}^{+i+j}(Y) ;
\]

\[
  R^{i}f_{!}(\mathbb{Z} \cdot \mathbf{1}_{X})
\]

\[
  \left\{
  \begin{array}{l}
    \mathcal{M}^{+i}(X) \otimes \mathcal{M}^{+k}(X) \subset \mathcal{M}^{+i+k}(X) \\
    \mathbb{Z}(-1) \in \operatorname{Ob} \mathcal{M}^{+2}(X)
  \end{array}
  \right.
\]

\[
  \mathcal{M}^{+i}(X) \otimes \mathbb{Z}(-j) \subset \mathcal{M}^{+i+2j}(X)
  \qquad \text{pour } j \geqslant 0
\]

\[
  M \in \operatorname{Ob} \mathcal{M}^{+i}(X) \Longleftrightarrow
  M(-j) \in \operatorname{Ob} \mathcal{M}^{+i+2j}(X)
\]

\[
  M \in \operatorname{Ob} \mathcal{M}^{i}(X) \Longleftrightarrow
  M(-j) \in \operatorname{Ob} \mathcal{M}^{i+2j}(X)
\]

\[
  \mathcal{M}^{+i}(X) = \mathcal{M}^{i}(X) \cap \mathcal{M}^{+}(X)
\]

\[
  \mathcal{M}^{+i}(X) = \{0\} \struck{\mathcal{M}^{+}(X)} \quad \text{si } i < 0
\]

\[
  \begin{array}{rcl}
    \otimes \mathbb{Q}(-1) & : & \mathcal{M}^{i}(X) \overset{\text{équiv.}}{\simeq} \mathcal{M}^{i+2}(X) \\
    \otimes \mathbb{Q}(j) & : & \mathcal{M}^{i}(X) \overset{\text{équiv.}}{\simeq} \mathcal{M}^{i-2j}(X)
  \end{array}
\]

\[
  \begin{array}{l}
    \mathbb{Q}(-1) \otimes \mathcal{M}^{+i}(X) \hookrightarrow \mathcal{M}^{+(i+2)}(X) \qquad \text{plus gén.} \\
    \mathbb{Q}(-j) \otimes \mathcal{M}^{+i}(X) \hookrightarrow \mathcal{M}^{+(i+2j)}(X)
  \end{array}
\]

\[
  \mathcal{M}^{i}(X) \text{ épaisse dans } \mathcal{M}^{j}(X) \qquad (i \leqslant j).
\]

\[
  \mathcal{G}_{i}(X) \simeq \mathcal{G}_{i+2j}(X) ,
\]

\[
  \mathcal{G}^{+}_{i}(X) \subsetneq \mathcal{G}_{i}(X) \simeq \mathcal{G}_{0}(X)
\]

\[
  D_{i}(\mathcal{M}^{+}(X)) = \struck{\mathcal{M}^{i}} \text{ sous-catégorie pleine}
\]

\[
  \struck{\ill{}} \; M(-j), \quad \text{avec } M \in \operatorname{Ob} \mathcal{M}^{+i}(X), \; j \in \mathbb{Z},\ j \geqslant 0
\]

\[
  \begin{array}{l}
    R^{j}f_{!}(D_{i}(\mathcal{M}^{+}(X))) \subset D_{i+j}(\mathcal{M}^{+}(Y)) \\
    R^{j}f_{*}(D_{i}(\mathcal{M}^{+}(X))) \subset D_{i+j}(\mathcal{M}^{+}(Y)) \quad \text{?}
  \end{array}
\]

\[
  M \in \operatorname{Ob} D_{i}(\mathcal{M}^{+}(X)) \Longrightarrow
  M(-j) \in \operatorname{Ob} D_{i}(\mathcal{M}^{+}(X))
\]

\[
  M \in \operatorname{Ob} D_{i}(\mathcal{M}(X)) \Longleftrightarrow
  M(-j) \in \operatorname{Ob} D_{i}(\mathcal{M}(X))
\]

\[
  D_{i}(\mathcal{M}(X)) \otimes D_{j}(\mathcal{M}(X)) \subset D_{i+j}(\mathcal{M}(X))
\]

\[
  \underline{\underline{\mathrm{Ext}}}^{i}(D_{j}(\mathcal{M}(X)), D_{k}(\mathcal{M}(X))) \subset D_{j+k}(\mathcal{M}(X))
\]

\[
  M \text{ est tordu} \Longleftrightarrow M(j) \text{ est tordu.}
\]

\[
  M(X) = M^{+}(X)_{\xi} = M^{+}(X)\bigl[\tfrac{1}{\xi}\bigr] .
\]

\[
  \Sigma(X) = \varinjlim_{\text{par } e \mapsto \xi e} \Sigma^{+}(X) ,
\]

\[
  \Sigma^{+}_{(i)}(X) \quad \text{et} \quad \Sigma_{i}(X),
\]

\[
  M^{+}_{i}(X) = \mathbb{Z}^{\Sigma^{+}_{(i)}(X)}
\]

\[
  \Sigma^{+}_{i}(X) = \Sigma^{+}_{(i)}(X) - \Sigma^{+}_{(i-1)}(X)
\]

\[
  \Sigma^{+}_{i}(X) \cdot \Sigma^{+}_{j}(X) \subset \mathbb{Z}^{\Sigma^{+}_{i+j}(X)}
\]

\[
  M_{0}(X) \simeq \mathbb{Z}^{\Sigma_{0}(X)}
\]

\[
  \Sigma_{0}(X) = \varinjlim_{\text{par } x \mapsto \xi x} \Sigma^{+}_{i}(X)
\]

\[
  \begin{array}{l}
    M^{\text{pair}}(X) = M_{0}(X)[\xi] \qquad (\xi \text{ de degré } 1) \\
    M^{\text{impair}}(X) = \underline{M}_{1}(X) \otimes_{M_{0}(X)} M_{0}(X)[\xi]
  \end{array}
\]

\[
  M_{1}(X) \otimes_{M_{0}(X)} M_{1}(X) \to M_{0}(X)
\]

\[
  \mathbb{Z}^{D_{i}(\Sigma^{+}(X))}
\]

\[
  \begin{aligned}
    D_{i}(\Sigma^{+}(X)) &= \coprod_{j \geqslant 0} \xi^{j}\, \Sigma^{+}_{(i)}(X) \\
    &= \coprod_{\substack{j \geqslant 0 \\ 0 \leqslant k \leqslant i}} \xi^{j}\, \Sigma^{+}_{k}(X)
  \end{aligned}
\]

\[
  \struck{\Sigma}\; \sigma_{i}(X) = \Sigma^{+}_{i}(X) - \bigcup_{\alpha \geqslant 0} \xi^{\alpha}\, \Sigma^{+}_{i-2\alpha}(X)
\]

\[
  \Sigma_{i}(X) = \sigma_{i}(X) \cup \xi\, \sigma_{i-2}(X) \cup \cdots \cup
  \left\{
  \begin{array}{l}
    \xi^{i/2}\, \sigma_{0}(X) \\
    \xi^{\frac{i-1}{2}}\, \sigma_{1}(X)
  \end{array}
  \right.
\]

\[
  D_{i}(\Sigma^{+}(X)) \cap \Sigma^{+}_{j}(X) =
  \begin{cases}
    \Sigma^{+}_{j}(X) & \text{si } j \leqslant i \\
    \xi^{j-i}\, \sigma_{i}(X) \cup \xi^{j-i+1}\, \sigma_{i-1}(X) \cup \cdots \cup \xi^{j}\, \sigma_{0}(X) & \text{si } j > i
  \end{cases}
\]

\[
  \begin{aligned}
    D_{i}(M^{+}(X)) / D_{i-1}(M^{+}(X)) &\simeq \mathbb{Z}^{D_{i}(\Sigma^{+}(X)) - D_{i-1}(\Sigma^{+}(X))} \\
    &\simeq \mathbb{Z}^{\sigma_{i}(X) + \xi \sigma_{i-2}(X) + \xi^{2} \sigma_{i-4}(X) + \cdots}
  \end{aligned}
\]

\[
  \sigma_{i}(X) \times \sigma_{j}(X) \longrightarrow
  \mathbb{Z}^{\left(\coprod_{k \leqslant i+j-2} \sigma_{k}(X)\right)}
\]

\[
  \struck{\mathbb{Z}}\mathbb{Q}(-i)\, [\mathcal{M}^{+}_{i}(X)]^{\vee} \subset \mathcal{M}^{+}_{i}(X)
\]

\[
  H^{*}(X, \mathbb{Q}_{\ell}) \Longleftarrow E_{1}^{pq} = \coprod_{x \in X[p]} H^{q-p}(x, \mathbb{Q}_{\ell})(-p)
\]

\[
  \begin{cases}
    X[p] = \{x \in X \mid \dim \mathcal{O}_{X,x} = p\} \\
    H^{*}(x, \mathbb{Z}_{\ell}(-p)) = \varinjlim_{U} H^{*}(U, \mathbb{Q}_{\ell}(-p))
  \end{cases}
\]

\[
  \sigma : L(K) \longrightarrow M^{+}(K)
\]

\[
  \mathrm{cl}(X) \longmapsto \sum (-1)^{i}\, \mathrm{cl}\, H^{i}_{!}(X, \mathbb{Q}(0))
\]

\[
  L_{i}(K) \longrightarrow D_{i}(M^{+}_{\uncertain{\varepsilon}}(K))
\]

\[
  L_{i}(K) \to M^{+}_{(i)}(K) + \zeta M^{+}_{(i-1)}(K) + \zeta^{2} M^{+}_{(i-2)}(K) + \cdots
\]

\[
  L_{(i)}(K) \longrightarrow \underbrace{M^{+}_{(i)}(K) + \zeta M^{+}_{(i-1)}(K) + \zeta^{2} M^{+}_{(i-2)}(K) + \cdots}_{=\ D'_{i}(M^{+}(K))}
\]

\[
  \operatorname{gr}_{n}(L(K)) \longrightarrow \operatorname{gr}'_{n}(M^{+}(K)) \simeq \mathbb{Z}^{(\sigma_{n} \cup \zeta\sigma_{n-1} \cup \zeta^{2}\sigma_{n-2} \cup \cdots)}
\]

\[
  \gamma_{n}(X) + \zeta\gamma_{n-1}(X) + \zeta^{2}\gamma_{n-2}(X) + \cdots
\]

\[
  \gamma_{j}(X) \in \mathbb{Z}^{\sigma_{j}(X)} \qquad 0 \leq j \leq n
\]

\[
  \varinjlim_{U \text{ ouvert dense}} H^{j}(U, \mathbb{Q}(0))
\]

\[
  H^{i}(X) \longrightarrow H^{i}(U)
\]

\[
  \mathcal{M}^{+}_{(i)}(X) \big/ \bigl[\mathcal{M}^{+}_{(i-1)}(X) + \zeta\,\mathcal{M}^{+}_{(i-2)}(X)\bigr],
\]

\[
  \underline{\Gamma}^{i}(X) \in \operatorname{Ob} \mathcal{M}^{+}_{i\,\mathrm{bipur}}(K),
\]

\[
  C'[t_{1}, \ldots, t_{N}] \longrightarrow C \quad ??
\]

\[
  \begin{cases}
    H^{i}_{!}(X, \mathbb{Z}(0)) \in \operatorname{Ob} \mathcal{M}^{+}_{i}(K) \ \text{et} \in \operatorname{Ob} \mathcal{M}^{+}_{n}(K) \quad \forall i \quad \text{mieux :} \\
    H^{n+k}_{!}(X, \mathbb{Z}(0)) \in \operatorname{Ob} \bigl[\zeta^{k} \mathcal{M}^{+}_{n-k}(K)\bigr]
  \end{cases}
\]

\[
  H^{i}(X, \mathbb{Z}(0)) \in \operatorname{Ob} \underline{D}_{i}(\mathcal{M}^{+}(K)), \ \text{et} \in \operatorname{Ob} D_{n}(\mathcal{M}^{+}(K)) \quad \forall i
\]

\[
  \begin{aligned}
    H^{i}(X, \mathbb{Z}(0)) &\in \operatorname{Ob} \bigl[\mathcal{M}^{+}_{i} + \zeta \mathcal{M}^{+}_{i-1} + \zeta^{2} \mathcal{M}^{+}_{i-2} + \cdots\bigr] && \text{si } i \leq n \\
    &\in \operatorname{Ob} \zeta^{i-n} \bigl[\mathcal{M}^{+}_{2n-i} + \zeta \mathcal{M}^{+}_{2n-i-1} + \zeta^{2} \mathcal{M}^{+}_{2n-i-2} + \cdots\bigr] && \text{si } i \geq n
  \end{aligned}
\]

\[
  \begin{aligned}
    \mathrm{cl}\, H^{i}(X, \mathbb{Z}(0)) &\in \sum_{i \leq \alpha \leq \operatorname{Inf}(2i, 2n)} \operatorname{Gr}_{i}(\mathcal{M}^{+}(K)) && \text{plus gén. :} \\
    \mathrm{cl}\, H^{i}(X, M) &\in \sum_{\rho + i \leq \alpha \leq \rho + \operatorname{Inf}(2i, 2n)} \operatorname{Gr}_{i} && \text{si }M\text{ pure de poids }\rho\text{}
  \end{aligned}
\]

\[
  L_{i}(K)/L_{i-1}(K) \longrightarrow D_{i}\mathcal{M}^{+}(K) / D_{i-1}\mathcal{M}^{+}(K)
\]

\[
  \boxed{L_{i}(K)/L_{i-1}(K) \longrightarrow \mathbb{Z}^{\sigma_{i}(K)}}
\]

\[
  \mathcal{M}^{+}(X) \longrightarrow \mathcal{M}^{+}_{\ell}(X)
\]

\[
  \mathrm{cl}(E) \in \mathbf{M}_{i}(X) \Longleftrightarrow
  \begin{array}{l}
    \text{pour tout pt fermé }x\text{ de }X\text{,} \\
    \text{les valeurs propres de Frobenius} \\
    \text{dans }T_{\ell}(E)(x)\text{ sont de val.\ abs.\ }N(x)^{i/2}\text{}
  \end{array}
\]

\[
  \mathrm{cl}(E) \in \mathcal{M}^{+}(X) \Longleftrightarrow
  \begin{array}{l}
    \forall x \text{ pt fermé de } X, \text{ le polynôme caractéristique} \\
    \text{de }\mathrm{Frob}_{x}^{-1}\text{ dans }T_{\ell}(E)(x)\text{ est à coefficients \textit{entiers}} \\
    \text{[i.e.\ les valeurs propres de Frobenius sont des \textit{entiers} algébriques]}
  \end{array}
\]

\[
  \boxed{E \in \operatorname{Ob} D_{i}\mathcal{M}(X) \text{ i.e.\ } \mathrm{cl}(E) \in D_{i}\mathcal{M}(X) \Longleftrightarrow S_{i}}
\]

\[
  E \in \operatorname{Ob} D_{0}(\mathcal{M}(X)) \Longleftrightarrow
  \begin{array}{l}
    E\text{ est de degré \textit{pair} }\alpha = 2\beta \\
    \text{et }E\zeta^{-\beta} = F\text{ (qui est de degré }0\text{) est }\in \operatorname{Ob} \mathcal{M}^{+}(X)\text{}
  \end{array}
\]

\[
  H^{2i}(\overline{X}, \mathbb{Q}_{\ell}(i))^{\pi} \xleftarrow{\ \approx\ } \mathcal{A}^{i}(X/K) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell}
\]

\[
  \begin{cases}
    H^{2i}(\overline{X}, \mathbb{Q}_{\ell}(i))^{\mathfrak{G}_{\ell}(\overline{K}/K)} \xleftarrow{\ \sim\ } \mathcal{A}^{i}(\overline{X}) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell} \\
    \mathcal{U}^{i}(X) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell} \text{ et l'algèbre enveloppante de } \mathfrak{G}_{\ell}(\overline{K}/K) \text{ dans } \operatorname{End}_{\mathbb{Q}_{\ell}}(H^{i}(\overline{X}, \mathbb{Q}_{\ell}(0))) \\
    \qquad \text{sont commutant l'une de l'autre.}
  \end{cases}
\]

\[
  E_{1}^{pq} = \coprod_{x \in X(p)} H^{n-2p}(x, \mathbb{Q})_{\ell}(-p)
\]

\[
  \zeta^{p}\bigl[\mathcal{M}^{+}_{n-2p} + \zeta \mathcal{M}^{+}_{n-2p+1} + \cdots\bigr]
\]

\[
  \begin{gathered}
    H^{i}_{Y}(X) \to H^{i}(X) \to H^{i}(U) \to H^{i+1}_{Y}(X) \\
    H^{i}_{!}(Y) \leftarrow H^{i}_{!}(X) \leftarrow H^{i}_{!}(U) \leftarrow H^{i-1}_{!}(Y)
  \end{gathered}
\]

\[
  H^{i}_{!}(Y) \in \zeta^{i-m} \mathcal{M}^{+}_{2m-i} \quad (i \geq 2m) \qquad
  \zeta^{i-m} \mathcal{M}^{+}_{2n-i-2} = \zeta^{i-m} \mathcal{M}^{+}_{j-2}
\]

\[
  H^{i-1}_{!}(Y) \in \zeta^{i-m-1} \mathcal{M}^{+}_{2m-(i-1)-2} = \zeta^{i-m-1} \mathcal{M}^{+}_{j-1}
\]

\[
  \mathfrak{G}_{\ell}(\overline{K}/K) \longrightarrow \operatorname{End}(M_{\ell}) \simeq \check{M}_{\ell} \otimes M_{\ell}
\]

\[
  H^{i}(X, M) = \bigl(H^{i}(X, M)\bigr)_{i} \,;
\]

\[
  \mathbb{R}\Gamma_{X} : D^{b}(\mathcal{M}^{+}(X)) \longrightarrow D^{b}(\mathrm{Ab})
\]

\[
  \mathbb{R}\Gamma_{X} = \mathbb{R}\Gamma_{Y}\, \mathbb{R}f_{*}
\]

\[
  H^{*}(X, M) \Longleftarrow H^{p}(Y, R^{q}f_{*}(M))
\]

\[
  H^{0}(X, M) = \operatorname{Hom}(\mathbb{Q}_{X}(0), M)
\]

\[
  \mathbb{E}\mathrm{xt}^{i}(X ; M, N) = \mathbb{H}^{i}(X, \mathbb{R}\underline{\mathrm{Hom}}(M, N)) .
\]

\[
  \operatorname{Hom}(M, N) \simeq \mathbb{E}\mathrm{xt}^{0}(X, M, N)
\]

\[
  \mathbb{H}^{*}(X, M^{\bullet}) \Longleftarrow H^{p}(X, H^{q}(M^{\bullet}))
\]

\[
  \mathbb{E}\mathrm{xt}^{*}(X, M, N) \Longleftarrow H^{p}(X, \underline{\mathrm{Ext}}^{q}(M, N))
\]

\[
  H^{i}(X, M) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell} \longrightarrow H^{i}(X, T_{\ell}(M))
\]

\[
  \mathrm{Ext}^{i}(X ; M, N) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell} \longrightarrow \mathbb{E}\mathrm{xt}^{i}(X ; T_{\ell}(M), T_{\ell}(N))
\]

\[
  H^{i}(k, M) = 0 \quad \text{pour } i > 0, \quad \forall \text{ motif } M .
\]

\[
  a^{i}(X) \otimes_{\mathbb{Z}} \mathbb{Q} \xrightarrow[\;\sim\;]{\text{ét.\ pr.\ } \mathbb{Q}_{\ell}} \Gamma_{k}\bigl(R^{i}f_{*}(\mathbb{Q}_{X}(i))\bigr)
\]

\[
  \boxed{\; a^{i}(X) \otimes_{\mathbb{Z}} \mathbb{Q} \simeq \operatorname{Im}\bigl[ H^{2i}(X, \mathbb{Q}(i)) \longrightarrow H^{2i}(X, \mathbb{Q}_{\ell}(i)) \bigr] \;}
\]

\[
  H^{i}(X, \mathbb{Q}(j)) =
  \begin{cases}
    0 & \text{si } i \neq 2j, 2j+1 \\
    a^{j}(X) \otimes_{\mathbb{Z}} \mathbb{Q} & \text{si } i = 2j \quad \text{(canonique)} \\
    a^{j}(X) \otimes_{\mathbb{Z}} \mathbb{Q} & \text{si } i = 2j+1 \quad \text{(\emph{pas} canonique)}
  \end{cases}
\]

\[
  H^{j}(X, \mathbb{Q}(i)) \longrightarrow H^{j}(X, \mathbb{Q}_{\ell}(i))
\]

\[
  \boxed{\;
  \begin{aligned}
    &H^{i}(X, M) \ \text{dual de}\ \mathrm{Ext}_{!}^{2n+1-i}(X ; M, \mathbb{Q}(n)) \\
    &H^{i}_{!}(X, M) \ \text{dual de}\ \mathrm{Ext}^{2n+1-i}(X ; M, \mathbb{Q}(n))
  \end{aligned}
  \;}
\]

\[
  H^{i}(\mathbb{F}_{p}, M) \quad \text{et} \quad H^{1-i}(\mathbb{F}_{p}, M^{*})
\]

\[
  \bigl( M^{*} = \underline{\mathrm{Hom}}(M, \mathbb{Q}(0)) \bigr)
\]

\[
  H^{1}(\mathbb{F}_{p}, \mathbb{Q}(0)) \simeq \mathbb{Q}
\]

\[
  H^{i}(\pi, V) \quad \text{et} \quad H^{1-i}(\pi, V^{*}) \quad \text{en dualité}
\]

\[
  \longrightarrow H^{i-1}(\overline{X}, \mathbb{Q}_{\ell}(j))_{\pi} \longrightarrow H^{i}(X, \mathbb{Q}_{\ell}(j)) \longrightarrow H^{i}(\overline{X}, \mathbb{Q}_{\ell}(j))^{\pi} \longrightarrow 0
\]

\[
  \begin{aligned}
    &\text{si } i \neq 2j, 2j+1, && H^{i}(X, \mathbb{Q}_{\ell}(j)) = 0 \\
    &\text{si } i = 2j, && H^{2j}(X, \mathbb{Q}_{\ell}(j)) = H^{2j}(\overline{X}, \mathbb{Q}_{\ell}(j))^{\pi} \underset{\text{conj.\ de Tate}}{\simeq} a^{j}(X/k) \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell} \\
    &\text{si } i = 2j+1, && H^{2j+1}(X, \mathbb{Q}_{\ell}(j)) = H^{2j}(\overline{X}, \mathbb{Q}_{\ell}(j))_{\pi} \underset{\text{Tate} + \text{action semi-simple}}{\simeq} a^{j}(X/k) \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell}
  \end{aligned}
\]

\[
  \varinjlim H^{i}(X_{k_{n}}, \mathbb{Q}_{\ell}(j))
\]

\[
  \varinjlim H^{i}(X_{k_{n}}, \mathbb{Q}(j)) =
  \begin{cases}
    0 & \text{si } i \neq 2j, 2j+1 \\
    a^{j}(\overline{X}) \otimes_{\mathbb{Z}} \mathbb{Q} & \text{si } i = 2j \\
    \struck{\ill{}} & \text{si } i = 2j+1, \ \text{mais pas canoniquement isomorphe à} \ldots
  \end{cases}
\]

\[
  ?\ \ H^{*}(S, M_{\ell}) \overset{?}{\Longleftarrow} H^{p}(\mathbb{F}_{p}, R^{q}f_{*}(M_{\ell}))
\]

\[
  0 \longrightarrow \underbrace{H^{i-1}(\overline{S}, \overline{M}_{\ell})_{\pi}}_{\substack{\text{poids entre} \\ \rho + i - 1 \text{ et } \rho + \mathrm{Inf}(2i-2, 2n)}} \longrightarrow H^{i}(S, M_{\ell}) \longrightarrow \underbrace{H^{i}(\overline{S}, \overline{M}_{\ell})^{\pi}}_{\substack{\text{poids entre} \\ \rho + i \text{ et } \rho + \mathrm{Inf}(2i, 2n)}} \longrightarrow 0
\]

\[
  \rho + i - 1 \leq 0 \leq \rho + 2i - 2 \quad \text{ou} \quad \rho + i \leq 0 \leq \rho + 2i
\]

\[
  2 - 2i \leq \rho \leq 1 - i \quad \text{ou} \quad -2i \leq \rho \leq -i
\]

\[
  \begin{aligned}
    -2i \leq \rho &\leq -i + 1 && (\text{si } i \geq 2) \\
    -2 \leq \rho &\leq 1 && (\text{si } i = 1) \\
    \rho &= 0 && (\text{si } i = 0)
  \end{aligned}
\]

\[
  A_{K}(K) = A(S)
\]

\[
  \mathbb{R}^{1}f_{*}(\mathbb{Q}_{A}(1)) \simeq T_{\mathrm{mot}}(A)
\]

\[
  A(S) \otimes_{\mathbb{Z}} \mathbb{Q} \simeq H^{1}(S, T_{\mathrm{mot}}(A))
\]

\[
  0 \longrightarrow {}_{\ell^{\nu}}A \longrightarrow A \longrightarrow A \longrightarrow 0
\]

\[
  \longrightarrow H^{0}(S, A)_{\ell^{\nu}} \longrightarrow H^{1}(S, {}_{\ell^{\nu}}A) \longrightarrow H^{1}(S, A)_{\ell^{\nu}} \longrightarrow 0
\]

\[
  0 \longrightarrow \underbrace{H^{0}(S, A)}_{\text{de type fini sur } \mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell} \longrightarrow \underset{\substack{\wr\wr \\ H^{1}(S, T_{\mathrm{mot}}(A)) \otimes_{\mathbb{Q}} \mathbb{Q}_{\ell}}}{H^{1}(S, T_{\ell}(A))_{\mathbb{Q}_{\ell}}} \longrightarrow T_{\ell}(H^{1}(S, A))_{\mathbb{Q}_{\ell}} \longrightarrow 0
\]

\[
  \boxed{\; H^{1}(S, A)(\ell) \ \underline{\text{fini}} \ \text{si } \ell \text{ “premier à } S \text{”} \quad (S \text{ de type fini sur } \operatorname{Spec} \mathbb{Z}) \;}
\]

\[
  \longrightarrow H^{1}(\mathbb{F}_{p}, R^{0}f_{*}A) \longrightarrow H^{1}(S, A) \longrightarrow H^{0}(\mathbb{F}_{p}, \mathbb{R}^{1}f_{*}(A)) \longrightarrow \struck{\ill{}}
\]

\[
  H^{0}(\mathbb{F}_{p}, \mathbb{R}^{1}f_{*}(A)) = H^{1}(\overline{S}, \overline{A})^{\pi}
\]

\[
  \longrightarrow \underset{\substack{\wr\wr \\ M \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}}}{H^{0}(\overline{S}, \overline{A}) \otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}} \longrightarrow H^{1}(\overline{S}, {}_{\ell^{\infty}}\overline{A}) \longrightarrow H^{1}(\overline{S}, \overline{A})(\ell) \longrightarrow 0
\]

\[
  0 \longrightarrow \underbrace{A^{\mathrm{tr}}}_{\ill{}\ \text{de la V.A. } A \text{ sur } S} \longrightarrow \mathbb{R}^{0}f_{*}(A) \longrightarrow \underbrace{M}_{\substack{\pi\text{-module,} \\ \text{de type fini sur } \mathbb{Z}}} \longrightarrow 0
\]

\[
  \underset{\substack{\| \\ 0 \\ \text{par Lang}}}{H^{1}(\mathbb{F}_{p}, A^{\mathrm{tr}})} \longrightarrow H^{1}(\mathbb{F}_{p}, \mathbb{R}^{0}f_{*}(M)) \longrightarrow \underset{\substack{\| \\ (M \otimes \mathbb{Q}/\mathbb{Z})^{\pi}}}{H^{1}(\mathbb{F}_{p}, M)} \longrightarrow 0
\]

\[
  0 \longrightarrow \underset{\substack{\wr \\ \text{gr.\ fini}}}{H^{1}(\mathbb{F}_{p}, \mathbb{R}^{0}f_{*}(A))} \longrightarrow H^{1}(S, A) \longrightarrow \underset{\substack{\| \\ H^{1}(\overline{S}, \overline{A})^{\pi}}}{H^{0}(\mathbb{F}_{p}, \mathbb{R}^{1}f_{*}(A))} \longrightarrow 0
\]

\[
  0 \longrightarrow \widetilde{A}^{\mathrm{tr}} \longrightarrow \mathbb{R}^{0}f_{*}(A) \longrightarrow M \longrightarrow 0 ,
\]

\[
  H^{2}(S, T_{\ell}(A)) \simeq H^{1}(\overline{S}, T_{\ell}(\overline{A}))_{\pi} = \bigl[ T_{\ell}(\overline{J}) \otimes T_{\ell}(\overline{J}) \bigr]_{\pi}
\]

\[
  \simeq \operatorname{End}(J) \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell} \quad \bigl( \simeq (\text{classes de corr.\ divisorielles sur } S \times S) \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell} \bigr)
\]

\[
  H^{1}(S, A)(\ell) \xrightarrow{\;\sim\;} H^{1}(\overline{S}, \overline{A})^{\pi}(\ell)
\]

\[
  \boxed{\; H^{1}(\overline{S}, \overline{A})(\ell)^{\pi} = \text{groupe fini} \;}
\]

\[
  H^{2}(S, \mathbb{G}_{m}) \longrightarrow H^{2}(X, \mathbb{G}_{m})
\]

\[
  H^{1}(S, T_{\ell}(A)) \simeq H^{1}(\overline{S}, T_{\ell}(\overline{A}))^{\pi}
\]

\[
  \operatorname{rang}_{\mathbb{Q}_{\ell}} \bigl( H^{1}(\overline{S}, T_{\ell}(\overline{A})) \bigr)^{\pi} \mathrel{\underline{\simeq}} \operatorname{rang}_{\mathbb{Z}} A(K)
\]

\[
  H^{i}(C, A) = 0 \quad \text{si } i \geqslant 2
\]

\[
  H^{1}(C, A) = 0
\]

\[
  H^{1}(U, A) \simeq H^{2}_{x}(C, A)
\]

\[
  H^{2}_{x}(C, A) \simeq H^{1}(K_{x}, A)
\]

\[
\begin{array}{lll}
  H^{0}(X, \mathbb{Q}_{\ell}(1)) & = 0 & \\[1ex]
  H^{1}(X, \mathbb{Q}_{\ell}(1)) & = H^{0}(X, \mathcal{O}_{X}^{*}) \otimes_{\mathbb{Z}} \mathbb{Q}_{\ell} & \\[1ex]
  H^{2}(X, \mathbb{Q}_{\ell}(1)) & \text{extension de } T_{\ell}(H^{2}(X, \mathbb{G}_{m})) & \text{par } H^{1}(X, \mathcal{O}_{X}^{*}) \otimes \mathbb{Q}_{\ell}
\end{array}
\]

\[
  H^{3}(X, \mathbb{Q}_{\ell}(1)) \simeq T_{\ell}(H^{3}(X, \mathbb{G}_{m}))
  \ \text{est ce qu'il peut}
\]

\[
  H^{i}(X, \mathbb{Q}_{\ell}(1)) = 0 \quad \text{si } i \geqslant 4 \ ??
\]

\[
  \Vert
\]

\[
  T_{\ell}(H^{i}(X, \mathbb{G}_{m}))
\]

\[
  M \times M \longrightarrow \mathbb{Q}(-p)
\]

\[
  \Gamma_{k}(\operatorname{Hom}(M \otimes M, \mathbb{Q}(-p))) = \mathcal{L}(M)
\]

\[
  \mathcal{L}^{s}(M) \otimes \mathcal{L}^{s}(N) \longrightarrow \mathcal{L}^{s}(M \otimes N)
\]

\[
  \mathcal{L}^{s}(M' \times M'') \simeq \mathcal{L}^{s}(M') \times \mathcal{L}^{s}(M'') \times \mathcal{B}(M', M'')
\]

\[
  \mathcal{B}(M', M'') = \operatorname{Hom}(M' \otimes M'', \mathbb{Q}(p)) .
\]

\[
  N = M \oplus M^{\perp} ,
\]

\[
  \mathcal{L}^{s}(M) \simeq \mathcal{L}^{s}(M(n))
\]

\[
  \Gamma_{k} M \times \Gamma_{k} M \longrightarrow \Gamma_{k} \mathbb{Q}(0) = \mathbb{Q}
\]

\[
  \mathcal{L}^{s}(M) \simeq \bigoplus_{\alpha} \mathcal{L}^{s}(M_{\alpha})
\]

\[
  I \otimes_{A} M
\]

\[
  P(t) = \det(t \operatorname{id} - f)
\]

\[
  Q(t) = t^{d} P(\tfrac{1}{t}) = \det(1 - tf)
\]

\[
  (V, f) \rightsquigarrow \frac{1}{\det(\operatorname{id}_{V} - f t)} = L(V, f)
\]

\[
  R(\mathbb{N}, k) \longrightarrow 1 + k[[t]]^{+}
\]

\[
  \sum_{\substack{P \text{ pl.} \\ \text{unitaire}}} c_{P} (V_{P}, f_{P})
\]

\[
  L = \prod_{Q} Q^{c_{P}}
\]

\[
  1 + k[[t]]^{+} \longrightarrow k[[t]]
\]

\[
  \varphi(t) \rightsquigarrow \frac{\varphi'(t)}{\varphi(t)}
\]

\[
  \Lambda(V, f)(t) = \frac{L(V, f)'(t)}{L(V, f)(t)} = [\log L(V, f)]' = \frac{1}{t} \sum_{1}^{\infty} (\operatorname{Tr} f^{n})\, t^{n}
\]

\[
  \Lambda : R(\mathbb{N}, k) \longrightarrow k[[t]]
\]

\[
  \chi(F) = \sum (-1)^{i} \operatorname{cl} H^{i}_{!}(\overline{X}, \overline{F}) \quad \text{de } R(\mathbb{N}, \mathbb{Q}_{\ell})
\]

\[
  \sum (-1)^{i} \underline{h}^{i}(F) , \qquad \underline{h}^{i}(F) \in R(\mathbb{N}, \mathbb{Q}_{\ell}) ,
\]

\[
  \underline{h}^{i}(F) = H^{i}(\overline{X}, \overline{F})
\]

\[
  H^{0}(X, \operatorname{Hom}(F, \mathbb{Q}_{\ell}(n))) = H^{0}(X, \check{F})(n) ,
\]

\[
  H^{1}(X, \operatorname{Hom}(F, \mathbb{Q}_{\ell}(n))) ,
\]

\[
  L(X, F) \ \text{modulo} \ L(f^{-1}(Y'), F) ,
\]

\[
  \rho_{X} : C^{i}(X) \longrightarrow H^{2i}_{\ell}(X)
\]

\[
  H^{j}_{\ell}(X) \times H^{2n-j}_{\ell}(X) \longrightarrow H^{2n}_{\ell}(X) \xrightarrow[\ \sim\ ]{\ \kappa_{X}\ } \mathbb{Q}_{\ell} \qquad (j = 0, \ldots, 2n)
\]

\[
  \kappa_{X}(\alpha \cdot \rho_{X}(Z)) = \kappa_{Z}(\operatorname{res}_{Z}(\alpha)) \quad \text{for all } \alpha \in H^{2n-2i}_{\ell}(X) .
\]

\[
  \rho_{X} : C^{*}(X) \longrightarrow H^{*}_{\ell}(X) .
\]

\[
  \left\{
  \begin{array}{l}
    X = \mathbb{P}^{2} - T' \\
    Y = T - T' \\
    U = \mathbb{P}^{2} - T
  \end{array}
  \right.
\]

\[
\begin{aligned}
H^{*}_{\ell}(X \times Z) &\simeq H^{*}_{\ell}(X) \otimes_{\mathbb{Q}_{\ell}} H^{*}_{\ell}(Z)
&& \text{(Kunneth formula)} \\
&\simeq \check{H}^{*}_{\ell}(X) \otimes_{\mathbb{Q}_{\ell}} H^{*}_{\ell}(Z)
&& \text{(Poincaré duality)} \\
&\simeq \operatorname{Hom}_{\mathbb{Q}_{\ell}}\bigl(H^{*}_{\ell}(X), H^{*}_{\ell}(Z)\bigr) .
\end{aligned}
\]

\[
\begin{array}{c|ccc}
\text{poids} & 0 & 1 & 2 \\
\hline
\text{niveau} & 0 & 1 & 0
\end{array}
\]

\[
\begin{array}{c|ccccc}
\text{poids} & 0 & 1 & 2 & 3 & 4 \\
\hline
\text{niveau} & 0 & 1 & 2 & 1 & 0
\end{array}
\]

\[
X = X' - Y, \qquad Y = \bigcup_{i} Y_{i},
\]

\[
H^{i-1}(X') \to H^{i-1}(Y) \to H^{i}_{!}(X) \to H^{i}(X') \to H^{i}(Y) \to
H^{i+1}_{!}(X) \to \cdots
\]

\[
H^{*}(Y) \Longleftarrow E_{2}^{pq} = H^{p}\bigl(i_{0} \ldots i_{p} \mapsto
H^{q}(Y_{i_{0}} \cap \cdots \cap Y_{i_{p}})\bigr)
\]

\[
\leqslant \struck{\ill{}}\; 2(n-p-1) - q \;=\; \struck{\operatorname{Inf}(q,}
(2n-i-2) - p \qquad (i = p+q),
\]

\[
\begin{array}{c|c|c|c|c|c|c}
 & E_{2}^{i-1,0} & E_{2}^{i-2,1} & \cdots & E_{2}^{1,i-2} &
 E_{2}^{0,i-1}/\operatorname{Im} H^{i-1}(X') &
 \operatorname{Ker}\bigl(H^{i}(X') \to E_{2}^{0,i}\bigr) \\
\hline
\text{poids} & 0 & 1 & & i-2 & i-1 & i
\end{array}
\]

\[
\begin{array}{c|c|c|c|c|c|c}
 & 0 \cdots 0 & \scriptstyle E_{2}^{2n-i-1,2(i-n)} & \scriptstyle E_{2}^{2n-i-2,2(i-n)+1} & \cdots
 & \scriptstyle E_{2}^{0,i-1}/\operatorname{Im} H^{i-1}(X') &
 \scriptstyle \operatorname{Ker}(H^{i}(X') \to E_{2}^{0,i}) \\
\hline
\text{poids} & \times & 2(i-n) & 2(i-n)+1 & \cdots & i-1 & i \\
\hline
\text{niveau} & \times & 0 & 1 & \cdots & 2n-i-1 & 2n-i
\end{array}
\]

\[
H^{*}(X) \Longleftarrow E_{2}^{pq} = H^{p}\bigl(\sigma \mapsto
H^{q}(U_{\sigma})\bigr) .
\]

\[
\to H^{i}_{!}(U) \to H^{i}_{!}(X) \to H^{i}_{!}(Y) \to
\]

\[
H^{*}(X) \Longleftarrow E_{2}^{pq} = H^{p}\bigl(s \mapsto
H^{q}((X'/X)^{s})\bigr)
\]

\[
F = R^{j}f_{*}(\mathbb{Q}_{\ell,Y}) .
\]

\[
H^{*}(Y) \Longleftarrow E_{2}^{pq} = H^{p}\bigl(X, R^{q}f_{*}(Y)\bigr) .
\]

\[
\begin{array}{c|c|c|c|c}
\text{poids} & 2(i-n) & 2(i-n)+1 & \cdots & (i+j) \\
\hline
\text{niveau} & \leqslant 0 & \leqslant 1 & \cdots & \leqslant 2n-i+j
\end{array}
\]

\[
R^{*}_{!}(fg)(\mathbb{Q}_{\ell}) \Longleftarrow E_{2}^{pq} =
R^{p}_{!}f\bigl(R^{q}_{!}g(\mathbb{Q}_{\ell,X})\bigr)
\]

\[
X_{0} \overset{i}{\hookrightarrow} X \overset{f}{\hookrightarrow} Y, \qquad
i_{!}(\mathbb{Q}_{\ell,X_{0}}) \to Ri_{*}(\mathbb{Q}_{\ell,X_{0}}) .
\]

\[
R^{*}f_{*}(\mathbb{Q}_{\ell,X}) \Longleftarrow E_{2}^{pq} =
R^{p}h_{*}\bigl(R^{q}g_{*}(\mathbb{Q}_{\ell})\bigr),
\]

\[
R^{q}g_{*}(\mathbb{Q}_{\ell}) = \bigwedge^{q} \sum \mathbb{Q}_{\ell,Z_{i}} =
\sum_{i_{1} < \cdots < i_{q}} \mathbb{Q}_{\ell,Z_{i_{1}} \cap \cdots \cap Z_{i_{q}}}(-q)
\]

\[
\begin{array}{c|ccc}
\text{poids} & 0 & 1 \;\cdots & 2i \\
\hline
\text{niveau} & 0 & 1 \;\cdots & 2i
\end{array}
\]

\[
\begin{array}{l|c|c|c|c|c|c}
E_{2}^{i,0} & \text{poids} & 0 & 1 & \cdots & i & \\
 & \text{niveau} \leqslant & 0 & 1 & \cdots & i & \\
\hline
E_{2}^{i-1,1} & \text{poids} & 2 & 3 & \cdots & i & i+1 \\
 & \text{niveau} & 0 & 1 & \cdots & i-2 & i-1 \\
\hline
E_{2}^{i-2,2} & \text{poids} & 4 & 5 & \cdots & i \;\; i+1 & i+2 \\
 & \text{niveau} & 0 & 1 & \cdots & i-4 \;\; i-3 & i-2 \\
\hline
E_{2}^{0,i} & \text{poids} & 2i & & & & \\
 & \text{niveau} & 0 & & & &
\end{array}
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c}
\text{poids} & 0 & 1 & \cdots & i-1 & i & i+1 & \cdots & 2i \\
\hline
\text{niveau} & 0 & 1 & \cdots & i-1 & i & i-1 & \cdots & 0
\end{array}
\]

\[
\begin{array}{l|c|c|c|c|c}
E_{2}^{i,0} & 2(i-n+1) & 2(i-n+1)+1 & \cdots & & i \\
 & 0 & 1 & 2 \;\cdots & & 2(n-1)-i \\
\hline
E_{2}^{i-1,1} & 2(i-n)+2 & 2(i-n)+3 & \cdots & i & i+1 \\
 & 0 & 1 & \cdots & 2n-i-2 & 2n-i-1 \\
\hline
E_{2}^{i-2,2} & 2(i-n)+4 & 2(i-n)+5 & \cdots & i \;\; i+1 & i+2 \\
 & 0 & 1 & \cdots & 2n-i-2 \;\; 2n-i-3 & 2n-i-2 \\
\hline
E_{2}^{i-3,3} & 2(i-n)+6 & 2(i-n)+7 & \cdots & i \;\; i+1 \;\; i+2 & i+3 \\
 & 0 & 1 & \cdots & 2n-i-6 \;\; \ldots & 2n-i-3 \\
\hline
E_{2}^{0,i} & 2i & & & & \\
 & 0 & & & &
\end{array}
\]

\[
q = 2n - i, \quad p = 2(i-n) : \qquad
E_{2}^{2(i-n),\,2n-i} = R^{2(i-n)}f_{Z\,*}(\mathbb{Q}_{\ell}\ldots, \qquad
2(i-n) + 2(2n-i) = 2n .
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c}
\text{poids} & 2(i-n)+2 & 2(i-n)+3 & 2(i-n)+4 & \cdots & i+1 & i+2 & \cdots & 2n \\
\hline
\text{niveau} & 0 & 1 & 2 & \cdots & 2n-i-1 & 2n-i-2 & \cdots & 0
\end{array}
\]

\[
E_{2}^{2(i-n),\,2n-i} = \sum R^{2(i-n)}f_{Z_{\alpha_{1}} \cap Z_{\alpha_{2}}
\cap \cdots \cap Z_{\alpha_{2n-i}}\,*}(\mathbb{Q}_{\ell})\bigl(-(2n-i)\bigr),
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c|l}
i & 0 & 1 & \cdots & n-1 & n & n+1 & \cdots & 2n-1 & \\
\hline
\text{niveau} & 0 & 1 & \cdots & n-1 & n-1 & n-2 & \cdots & 0 & \leqslant n-1 \\
\hline
\text{poids} \leqslant & 0 & 2 & \cdots & 2n-2 & 2n & 2n & 2n & 2n & \leqslant 2n
\end{array}
\]

\[
\boxed{\begin{array}{l}
\text{poids} \leqslant 2\operatorname{Inf}(i, n) \\
\text{niveau} \leqslant \operatorname{Inf}(i, n-1)
\end{array}}
\]

\[
\text{poids} \leqslant 2\operatorname{Inf}(i, n) + j \;? \qquad
\text{niveau} \leqslant \operatorname{Inf}(i, n-1) + j .
\]

\[
n = d_{y}(f) = \operatorname*{Sup}_{x \in X} \bigl[\deg \operatorname{tr}
k(x) : k(f(x)) + \dim \overline{f(x)}_{y}\bigr] ,
\]

\[
\boxed{\begin{array}{l}
\text{poids} \leqslant 2\operatorname{Inf}(i, n) + j \\
\text{niveau} \leqslant \operatorname{Inf}(i, n) + j
\end{array}}\ ?
\]

\[
R^{\bullet}(f\varphi)_{*}(\mathbb{Q}_{\ell X'}) \Longleftarrow E_{2}^{pq} = R^{p}f_{*}(R^{q}\varphi_{*}(\mathbb{Q}_{\ell}))
\]

\[
R^{i}f_{*}(F) = R^{i}f_{*}(R^{j}\varphi_{*}(\mathbb{Q}_{\ell X'}))
\]

\[
R^{n}(f\varphi)_{*}(\mathbb{Q}_{\ell X'}) \Longleftarrow E_{2}^{pq} = R^{p}g_{*}(R^{q}f'_{*}(\mathbb{Q}_{\ell X'}))
\]

\[
R^{n}g_{*}(\mathbb{Q}_{\ell}),
\]

\[
R^{i}_{!}f_{U}(F|U) \to R^{i}_{!}f_{*}(F) \to R^{i}_{!}f_{Z*}(F|Z)
\]

\[
0 \to G \to R^{m}g_{*}(\mathbb{Q}_{\ell}) \to H \to 0
\]

\[
0 \to I \to G \to F \to 0
\]

\[
R^{p}_{!}f_{*}(R^{q}g_{*}(\mathbb{Q}_{\ell})) \qquad p, q \geq 0
\]

\[
R^{0}_{!}f(R^{q}g_{*}(\mathbb{Q}_{\ell})) = 0,
\]

\[
0 \to R^{2}_{!}f_{*}(R^{n-2}g_{*}(\mathbb{Q}_{\ell})) \to R^{n}_{!}(gf)_{*}(\mathbb{Q}_{\ell}) \to R^{1}_{!}f_{*}(R^{n-1}g_{*}(\mathbb{Q}_{\ell})) \to 0
\]

\[
R^{n}_{!}(gf)_{!}(\mathbb{Q}_{\ell})
\]

\[
V = \hat{V} - Z, \qquad \hat{V} \text{ projectif et lisse sur } k,
\]

\[
R^{n-1}_{!}h_{Z}(\mathbb{Q}_{\ell}) \to R^{n}_{!}h(\mathbb{Q}_{\ell}) \to R^{n}_{!}\hat{h}(\mathbb{Q}_{\ell})
\]

\[
F \to \mathbf{R}j_{*}F \to Q^{*},
\]

\[
\mathbf{R}^{*}\uncertain{f_{*}}(F) \Longleftarrow \mathbf{R}^{p}f_{*}(R^{q}i'_{*}(F))
\]

\[
\mathbf{R}^{*}(if)_{*} \Longleftarrow \mathbf{R}^{p}i_{*}(\mathbf{R}^{q}f_{*}(\mathbb{Q}_{\ell}))
\]

\[
E_{2}^{p2} \simeq \mathbf{R}^{p}i_{*}(\mathbf{R}^{2}f_{*}(\mathbb{Q}_{\ell})) \simeq \mathbb{Q}_{\ell}(-1) \otimes E_{2}^{p0}
\]

\[
= \bigl\{ M, \{E_{\ell}\}, \{\alpha_{\ell}\} \bigr\}
\]

\[
\bigl(M, \{E_{\ell}\}, \{\alpha_{\ell}\}\bigr) \to \bigl(M', \{E'_{\ell}\}, \{\alpha'_{\ell}\}\bigr)
\]

\[
\mathfrak{P}(A) = (M, E_{\ell}, u)
\]

\[
\begin{cases}
M = R^{1}(f_{A^{0}})_{*}(\mathbb{Q}_{A^{0}}) \\
E_{\ell} = \underline{\mathrm{Hom}}({}_{\ell^{\infty}}A, \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}) \quad \text{extension de } \underline{\mathrm{Hom}}(A/A^{0}, \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}) \text{ par } R^{1}f_{A^{0}*}(\mathbb{Z}_{\ell A^{0}}) \\
u_{\ell} \text{ provenant de l'hom.\ induit } E_{\ell} \to R^{1}f_{A^{0}*}(\mathbb{Z}_{\ell A^{0}})
\end{cases}
\]

\[
  H \colon \mathcal{V}^{\circ} \longrightarrow \mathcal{C}
\]

\[
  \underline{h} \colon \mathcal{V}^{\circ} \longrightarrow
  \mathcal{M}(\mathcal{V}) = \mathcal{M}
\]

\[
  T \longmapsto T \circ \underline{h} \colon\quad
  \operatorname{Fonc\,exacts}(\mathcal{M}, \mathcal{C}) \longrightarrow
  \operatorname{Fonc\,add}(\mathcal{V}^{\circ}, \mathcal{C})
\]

\[
  \operatorname{Fonc\,exacts}(\mathcal{M}, \mathcal{C}) \longrightarrow
  \operatorname{Fonc\,add}(\mathcal{V}^{\circ}, \mathcal{C})
\]

\[
  \underline{h}^{\#} \colon \mathcal{V} \longrightarrow \mathcal{M}^{\#}
\]

\[
  \operatorname{Fonc\,add}(\mathcal{M}, \mathcal{C})
  \xrightarrow{\ \varphi\ } \operatorname{Fonc\,add}(\mathcal{V}, \mathcal{C})
\]

\[
  \psi \colon \operatorname{Fonc\,add}(\mathcal{V}, \mathcal{C})
  \longrightarrow \operatorname{Fonc\,add}(\mathcal{M}, \mathcal{C})
\]

\[
  \psi(H)(X,\pi) = \operatorname{Im} H(\pi)
\]

\[
  \varphi\psi(H) \simeq H, \qquad \psi\varphi(T) \simeq T .
\]

\[
  (\pi\pi' = \pi'\pi = 0,\quad \pi + \pi' = 1,\quad \pi^{2} = \pi,\quad
  \pi'^{2} = \pi')
\]

\[
\left\{
\begin{array}{l}
  vu = \text{projecteur } \pi \\
  uv = \text{projecteur } \varpi
\end{array}
\right.
\]

\[
    A \times B \longrightarrow B \times C, \qquad
    \operatorname{inj}_{1} \circ \operatorname{pr}_{2} .
  \]

\[
\left\{
\begin{array}{l}
  T \text{ fidèle} \\
  T(M) = 0 \Longrightarrow M = 0 \\
  T \text{ conservatif} \\
  T \colon \operatorname{End}(M,M) \to \operatorname{End}(T(M),T(M))
  \text{ injectif}
\end{array}
\right.
\]

\[
\begin{array}{l}
  \mathcal{V}_{0} \quad \text{catégorie des schémas projectifs lisses sur } k \\
  \quad\downarrow \\
  \mathcal{V} \quad \text{catégorie ayant mêmes objets que } \mathcal{V}_{0},
  \text{ mais}
\end{array}
\]

\[
  \operatorname{Fl}(X,Y) = \bigoplus_{i,j} C^{m_{j}}(X_{i} \times Y_{j})
\]

\[
  \mathcal{V}^{\circ} \xrightarrow{\ \underline{h}\ } \mathcal{M}
\]

\[
  H^{*}(\ ,\mathbf{Q}_{\ell}) \colon \mathcal{V}^{\circ} \longrightarrow
  \operatorname{Mod}_{\mathrm{t.f.}}(\mathbf{Q}_{\ell})
\]

\[
  T_{\ell} \colon \mathcal{M} \longrightarrow
  \operatorname{Mod}_{\mathrm{t.f.}}(\mathbf{Q}_{\ell})
\]

\[
  X, Y \rightsquigarrow X \times Y
\]

\[
  \pi_{i} \in \operatorname{End}\operatorname{id}_{\mathcal{V}}, \quad
  \text{avec } \pi_{i}^{2} = \pi_{i},\ \pi_{i}\pi_{j} = \pi_{j}\pi_{i} = 0
  \text{ si } i \neq j,
\]

\[
  \pi_{n}(X \boxtimes Y) = \sum_{i+j=n} \pi_{i}(X) \boxtimes \pi_{j}(Y) ;
\]

\[
  \operatorname{Hom}(\Lambda(n), \Lambda(n)) \simeq
  \operatorname{Hom}(\Lambda(0), \Lambda(0)) \simeq \mathbf{Q}
\]

\[
  M(i) \qquad i \in \mathbf{Z},\ M \in \operatorname{Ob}\mathcal{M},
\]

\[
  \operatorname{Hom}(M(i), N(j)) = \operatorname{Hom}(M(i+r), N(j+r))
\]

\[
  \operatorname{Ps}\varinjlim_{k} \mathcal{M}^{i+kp}, \qquad
  \text{où } \mathcal{M}^{i+kp} \longrightarrow \mathcal{M}^{i+(k+1)p}
  \text{ est le foncteur pl.\ fid.\ } M \rightsquigarrow M(1).
\]

\[
  \mathcal{M}^{i} \simeq \mathcal{P}^{i} \times \mathcal{P}^{i-p}(1) \times
  \mathcal{P}^{i-2p}(2) \times \cdots ,
\]

\[
  \mathcal{Q}^{i,j,k} \colon \mathcal{P}^{i} \times \mathcal{P}^{j}
  \longrightarrow \mathcal{P}^{i+j-kp}
  \qquad
  \mathcal{Q}^{i,j,k}(M,N) = K_{i,k}(M \otimes N)(k) .
\]

\[
  \operatorname{Hom}(P, \underline{\operatorname{Hom}}(M,N)) \simeq
  \operatorname{Hom}(P \otimes M, N)
\]

\[
  M^{*} = \underline{\operatorname{Hom}}(M, \Lambda(-i)) \qquad
  \deg M = i
\]

\[
  \underline{\operatorname{Hom}}(M, N(-i)) \simeq M^{*} \otimes N
\]

\[
  B(M,N) \simeq \operatorname{Hom}(M \otimes N, \Lambda(-i)) \simeq
  \operatorname{Hom}(M, N^{*}) \simeq \operatorname{Hom}(N, M^{*})
\]

\[
\begin{aligned}
  (u \pm v)' &= u' \pm v' \\
  (uv)' &= v'u' \\
  1' &= 1 \\
  (u')' &= u
\end{aligned}
\]

\[
  \widetilde{\alpha}_{1} = \widetilde{\alpha} \circ A, \qquad
  \widetilde{\beta}_{1} = \widetilde{\beta} \circ B \qquad
  (A \in \operatorname{End}(M),\ B \in \operatorname{End}(N))
\]

\[
  u'' = \widetilde{\alpha}_{1}^{-1} \circ u^{*} \circ \widetilde{\beta}_{1}
  = A^{-1}\widetilde{\alpha}^{-1} u^{*} \widetilde{\beta} B \quad \text{i.e.}
\]

\[
  u'' = A^{-1} u' B
\]

\[
    \boxed{\operatorname{Tr}\bigl(s_{u}^{(i)} u^{(i)}\bigr) \geq 0,
    \ \text{égalité ssi } u^{(i)} = 0}
  \]

\[
  \operatorname{Tr}\,(\mu u \lambda')'(\mu u \lambda')
  = \operatorname{Tr} \lambda u' \mu' \mu u \lambda'
  = \operatorname{Tr} \lambda' \lambda u' \mu' \mu u
\]

\[
  B(M(-1), M(-1)) \simeq B(M, M)
\]

\[
  1 \in B(\Lambda(-1), \Lambda(-1))
\]