\[
(\mathrm{A}^f_n) \quad
\left.
\begin{array}{l}
  H^i(U, F) \text{ de dim finie si } i \leq n \\
  \Updownarrow \\
  F \text{ de prof} > n+1 \text{ sur } U \text{ au voisinage de } Z \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ pt isolé de } \operatorname{Supp} E^{r-i-1} \cup \lbrace z \rbrace \text{ si } i \leq n \\
  \Updownarrow \\
  \alpha_i \text{ injectif pour } \underline{m \text{ grand}} \text{ si } i \leq n+1 \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  R^i g_*(F) \text{ coh.\ pour } i \leq n \iff \underline{H}^i_z(F) \text{ de dim finie pour } z \in Z,\ i \leq n+1 \\
  \Updownarrow \\
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S \text{ pour } i \leq n
\end{array}
\right.
\]

\begin{align*}
  S &\simeq \operatorname{Pent}_\omega(E) && \text{struct.\ pent.\ comp.\ avec } \omega \\
  A &\simeq \operatorname{Car}(E) && \text{partitions du type } (2, 2, 1) \\
    & && \simeq \text{structures carrées sur une partie à 4 él.\ de } E \\
  F &\simeq \operatorname{Tr}_3(E) && \text{triangles} \subset E \\
  \vec{A} &\simeq \operatorname{Pent}^{s}_\omega(E) && \omega\text{-structures pentagonales à somm.\ marqué} \\
  A^{\uparrow} &\simeq \operatorname{Tr}_3(E) && \text{triangles pointés} \\
  A^{\wedge} &\simeq \operatorname{Pent}^{a}_\omega(E) \simeq \operatorname{Pent}^{s}_\omega(E)
    && \text{structures pol.\ à arête marquée} \\
  A^{\uparrow\to} &\simeq \operatorname{Rep}_\omega(E) && \text{ordres totaux sur } E \text{ comp.\ avec } \omega \\
  \tilde{S} &\simeq \overrightarrow{\operatorname{Pent}}_\omega(E) && \text{permutations circulaires comp.\ avec } \omega \\
  \tilde{A} &\simeq \overrightarrow{\operatorname{Car}}(E) && \text{carrés orientés} \subset E \\
  \tilde{F} &\simeq \overrightarrow{\operatorname{Tr}}_3(E) && \text{triangles orientés} \subset E
\end{align*}

\[
\begin{array}{rll}
180 & \left\lbrace \begin{array}{l}
  60 = 3\times 20 = 2\times 30 \\
  120 = 4\times 30 = 2\times 60
  \end{array}\right. &
  \begin{array}{l} f_{B,s} \\ f_{\vec{Q},\alpha} \end{array} \\[1ex]
420 & \left\lbrace \begin{array}{l}
  60 = 1\times 60 = 1\times 60 \\
  120 = 1\times 120 = 1\times 120 \\
  60 = 1\times 60 = 5\times 12 \\
  60 = 1\times 60 = 3\times 20 \\
  120 = 1\times 120 = 6\times 20
  \end{array}\right. &
  \begin{array}{l} \tau_{\pi,s} \\ \tau^{0}_{r} \\ \gamma_{\pi,s} \\
  \gamma'_{\pi,s} \\ \gamma^{0}_{r} \end{array} \\[1ex]
1020 & \left\lbrace \begin{array}{l}
  120 = 2\times 60 = 1\times 120 \\
  120 = 2\times 60 = 2\times 60 \\
  120 = 1\times 120 = 2\times 60 \\
  120 = 1\times 120 = 2\times 60 \\
  3\times 120 = 3\times 40 = 3\times 40 \\
  120 = 2\times 60 = 2\times 60 \\
  60 = 1\times 60 = 2\times 30
  \end{array}\right. &
  \begin{array}{l} \tau_{\pi,\rho} \\ \gamma_{\pi,\rho} \\ \tau'_{r} \\
  \gamma^{1}_{r} \\ \gamma^{1\,\prime}_{r} \\ \gamma^{2}_{r} \\
  \gamma^{2\,\prime}_{r} \end{array}
\end{array}
\]

\[
(19)\quad\left\lbrace
\begin{array}{l}
\hat X=X^{*}\amalg_{\dot X=\coprod\dot X_\beta}\underbrace{\mathcal C\bigl(p=\textstyle\coprod p_\beta\bigr)}_{\coprod_\beta\mathcal C(p_\beta)},\\[10pt]
T_\beta=\mathcal C(p_\beta)\\[4pt]
X_\beta\ \text{pour}\ \beta\in I_0\ \text{clair}\\[4pt]
X_i\ \text{pour}\ i\in I^{*}\ \text{est donné par}\\[4pt]
\qquad\bar X_i=X^{*}_i\cup\bigcup_{\beta\in I_0,\ \beta\le i}\bigl(\mathbf{I}.\dot X_{i\beta}\cup X_\beta\bigr),\qquad \dot X_{i\beta}\overset{\text{déf}}{=}X^{*}_i\cap\dot X_\beta\\[10pt]
B_\alpha=B^{*}_\alpha\cup\bigcup_{\beta\in I_0}\mathbf{I}.\dot B_\alpha,\qquad \dot B_\alpha\overset{\text{déf}}{=}B^{*}_\alpha\cap\dot X_\beta\\[10pt]
\mathcal E_\alpha\ \text{défini par}\ \mathcal E^{*}_\alpha\ \text{et les}\ (B_{\beta,\alpha},\mathcal E_{\beta\alpha})
\end{array}\right.
\]

\[
(\mathrm{B}_n) \quad
\left.
\begin{array}{l}
  \alpha_i \text{ bijectif pour } m \text{ grand},\ i \leq n, \\
  \quad \text{et } \alpha_i \text{ injectif pour } m \text{ grand},\ i = n+1 \\
  \Updownarrow \\
  \operatorname{Supp} E^{r-i} \subset Z \text{ pour } i \leq n,\ \operatorname{Supp} E^{r-n-1} \ldots Z \\
  \Downarrow \\
  \operatorname{prof}(F_x) > n \text{ si } x \text{ fermé dans } U, \text{ et} \\
  \operatorname{prof}(F_x) > n+1 \text{ si de plus } x \in V \cap U,\ V \text{ voisinage ouvert convenable de } Z \\
  \Updownarrow \\
  H^i(U, F(-m)) = 0 \text{ pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S \text{ pour } i \leq n
\end{array}
\right.
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c}
 & \mu_p & W_{11} & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \mathbb{G}_m & \mathbb{G}_a & Ab & \mathbb{Z} \\ \hline
{}_p\mathbb{G}_m = \mu_p & \mathbb{Z}/p & 0 & 0\ \ast & \ast\ 0 & ? & 0 & [a] & 0\ \ast \\
{}_F\mathbb{G}_a = W_{11} & 0 & \mathbb{G}_a^2\ (?) & 0\ \ast & \ast\ 0 & 0 & (?) & [b] & 0\ \ast \\
\mathbb{Z}/\ell\mathbb{Z} & 0\ \times & 0\ \times & \mathbb{Z}/\ell\mathbb{Z} & 0 & \emptyset\ 0 & \emptyset\ 0 & \emptyset\ 0 & \emptyset\ \mathbb{Z}/\ell \\
\mathbb{Z}/p\mathbb{Z} & 0\ \ast & 0\ \times & 0 & \mathbb{Z}/p\mathbb{Z} & \emptyset\ 0 & \mathbb{G}_a & \emptyset\ 0 & \emptyset\ \mathbb{Z}/p\mathbb{Z} \\
\mathbb{G}_m & \mathbb{Z}/p\mathbb{Z} & 0 & [c] & 0 & 0 & 0 & [d] & \\
\mathbb{G}_a & 0 & (?) & 0\ 0 & (?) & 0 & (?) & [e] & \\
Ab & [f] & \longleftarrow & \longrightarrow & & (Ab)' & H^1(Ab, \underline{O}) & & \\
\mathbb{Z} & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast & 0\ \ast
\end{array}
\]

\[
  \left\lbrace
  \begin{array}{l}
    L_{E_\eta}\Bigl(\dfrac{1}{pt}\Bigr) = \prod_x \lambda_x(E)\,
      (-pt)^{\chi^{\natural}(E_\eta)}\, \delta^{\natural}(E_\eta)^{-1}\, L_{\check E_\eta}(t) \\[10pt]
    \chi^{\natural}(E_\eta) = \chi(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
    \delta^{\natural}(E_\eta) = \delta(E_\eta^{\natural}) \qquad \text{à déterminer} \ldots \\[4pt]
    \lambda_x(E) = L_{\mu_x(E)}(t) = L_{\alpha_x(E_\eta)^{\vee}}(t) / L_{\alpha_x(\check E_\eta)}(p^{-1} t)
      = (-t)^{\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
      \dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(\check E_\eta)}(t)}
  \end{array}
  \right.
\]

\[
\begin{aligned}
(y_2)\,\lambda_2 &= \lambda_1 (y_0 + y_1) - \lambda_0 y_{n-1}
 = \lambda_0 (-y_{n-1}) + \lambda_1 (y_0 + y_1) \qquad
 (= \lambda_4 y_3 - \lambda_5 y_5) \\
(y_2 y_3)\,\lambda_3 &= \underbrace{(y_2 \lambda_2)}\, y_1
 - y_2 (\lambda_0 y_{n-1} - \lambda_1 y_1) \\
 &= \lambda_0 \bigl( -y_{n-1} (y_1 + y_2) \bigr)
 + \lambda_1 y_1 [y_0 + y_1 + y_2] \\
(y_2 y_3 y_4)\,\lambda_4 &= (y_2 y_3 \lambda_3)\, y_3
 - y_2 y_3 (\lambda_0 y_{n-1} - \lambda_1 y_1) \\
 &= \lambda_0 \bigl( -y_2 y_{n-1} (y_1 + y_2 + y_3) \bigr)
 + \lambda_1 y_1 y_2 (y_0 + y_1 + y_2 + y_3) \\
(y_2 y_3 y_4 y_5)\,\lambda_5 &= \underbrace{(y_2 y_3 y_4 \lambda_4)}\, y_3
 - y_2 y_3 y_4 (\lambda_0 y_{n-1} - \lambda_1 y_1)
\end{aligned}
\]

\[
(20)\quad
\begin{array}{ll}
(a) & \text{Les } X_\beta \text{ disjoints}\\
(b) & \text{Chaque } \dot X_\beta \text{ transverse : } (B^{*}_\alpha,\mathcal E^{*}_\alpha)\ \Longrightarrow\ \text{le syst. induit dans } (\dot X_\beta,\ldots)\ (\dot B^{\alpha},\dot{\mathcal E}^{\alpha})\\
(c) & (\dot X_\beta,\underbrace{\dot B_{\beta\alpha},\dot{\mathcal E}_{\beta\alpha}}_{\text{induit sur }\dot X_\beta\text{ par }(B_\alpha,\mathcal E_\alpha)})\longrightarrow(X_\beta,B_{\beta,\alpha},\mathcal E_{\beta,\alpha})\ \text{est transverse}\\
(d) & \text{Les } X^{*}_i \text{ transverses aux } \bigl((\dot X_\beta,\mathcal E_\beta)_{\beta\in I_0},(B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}\bigr)
\end{array}
\]

\begin{gather*}
  f_1 \dots f_n \text{ régulier} \Longleftrightarrow f_1 \dots f_n \text{ quasi-régulier} \Longleftrightarrow \\
  A/I[T_1 \dots T_n] \otimes_{A/I} M/I \to \operatorname{gr}_I(A) \otimes_{\operatorname{gr}^0_I(A)} \operatorname{gr}^0_I(M) \to \operatorname{gr}_I(M) \text{ bijectif} \\
  \Longleftrightarrow \operatorname{gr}_I(A) \otimes_{A/I} M/I \to \operatorname{gr}_I(M) \text{ bijectif et} \\
  A/I[T_1 \dots T_n] \otimes A/\operatorname{Ann}(M/I) \to \operatorname{gr}_I(A) \otimes A/\operatorname{Ann}(M/I) \text{ bijectif.} \\
  \Longleftrightarrow \operatorname{gr}_I(A) \otimes M/I \xrightarrow{\sim} \operatorname{gr}_I(M),
  \text{ et modulo } \operatorname{Ann}(M/I),\ I/I^2 \text{ libre de rg } n \text{ et}
  \operatorname{Sym}_{A/I}(I/I^2) \simeq \operatorname{gr}_I(A).
\end{gather*}

\[
  (1)\quad
  \left\lbrace
  \begin{array}{l}
    \boxed{L_{DE}(t) = (-t)^{-\chi(E)}\, \delta(E)\, L_E(t^{-1})} \\[6pt]
    \chi(E) = \operatorname{rang} R f_{*} E = \sum (-1)^i \operatorname{rg} H^i(\bar X, \bar E) \\[6pt]
    \delta(E) = \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^i}
      = \varepsilon(E)\, p^{\sum_i (-1)^i \frac{i + \rho}{2} b_i}
  \end{array}
  \right.
\]

\[
\begin{array}{c|c|c|c|c|c|c|c|c|c}
 & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mathbb{G}_m & \mathbb{G}_a & \mathbb{Z} & \mathbb{V} & A \\ \hline
\mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/\ell\mathbb{Z} & 0 & 0 & 0 & \mu_\ell & 0 & 0 & 0 & {}_\ell A \\
\mathbb{Z}/p\mathbb{Z} & 0 & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mu_p & \mathbb{G}_a & 0 & \mathbb{V} & {}_p A \\
\alpha_p & 0 & 0 & \mathbb{G}_a & \alpha_p & \alpha_p & \mathbb{G}_a & 0 & (\mathrm{gros}) & ? \\
\mu_p & 0 & 0 & 0 & \mathbb{Z}/p\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & 0 & 0 & 0 & ? \\
\mathbb{G}_m & 0 & 0 & 0 & 0 & \mathbb{Z} & 0 & 0 & 0 & 0 \\
\mathbb{G}_a & 0 & 0 & (\mathrm{gros}) & \mathbb{V} & \mathbb{V} & ? & 0 & (\mathbb{G}_a ?) & \mathrm{gros} \\
\mathbb{Z} & \mathbb{Z}/\ell\mathbb{Z} & \mathbb{Z}/p\mathbb{Z} & \alpha_p & \mu_p & \mathbb{G}_m & \mathbb{G}_a & \mathbb{Z} & \mathbb{V} & A \\
\mathbb{V} & 0 & 0 & \mathbb{G}_a & \mathbb{G}_a & \mathbb{G}_a & \mathbb{G}_a ? & 0 & (\mathrm{Imm}) & \underline{V}(\omega_A) \\
A & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ? & \mathbb{Z}^n
\end{array}
\]

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

\[
\begin{gathered}
\ldots\ \tfrac{6}{9} > \tfrac{11}{17}, \qquad \tfrac{2}{3} > \tfrac{11}{17}, \qquad
\tfrac{5}{12} > \tfrac{7}{17}, \qquad \tfrac{1}{4} < \tfrac{?}{11} < \tfrac{2}{7}, \qquad
\tfrac{2}{7} < \tfrac{5}{17}, \\
\tfrac{4}{17} < \tfrac{1}{4}, \qquad \tfrac{1}{9} < \tfrac{2}{17}, \qquad
\tfrac{1}{6} < \tfrac{3}{17}, \qquad \tfrac{7}{16} < \tfrac{12}{17}, \qquad
\tfrac{5}{7} \mathrel{?} \tfrac{13}{18}, \\
\tfrac{4}{13} > \tfrac{7}{23}, \qquad {<}\ \tfrac{51}{13}\,(?), \qquad
\tfrac{3}{10} > \tfrac{7}{24}, \qquad \tfrac{2}{7} < \tfrac{7}{24}, \qquad
\tfrac{1}{3} > \tfrac{7}{23}, \qquad \tfrac{2}{7} < \tfrac{1}{3}, \\
\tfrac{7}{9} < \tfrac{7}{8}\,(?), \qquad \tfrac{4}{5} < \tfrac{14}{17}\,(?), \qquad
\tfrac{11}{17} > \tfrac{13}{23}, \qquad \tfrac{?}{13} > \tfrac{15}{23}, \qquad
\tfrac{10}{14} < \tfrac{13}{18}, \\
\tfrac{13}{17} < \tfrac{10}{13}, \qquad \tfrac{3}{5} < \tfrac{14}{23}, \qquad
\tfrac{3}{4} < \tfrac{13}{17}
\end{gathered}
\]

\[
\begin{aligned}
&\operatorname{Hom}(\Psi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } \mathrm{id})
\end{cases} \\
&\operatorname{Hom}(\Phi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } u_i)
\end{cases} \\
&\operatorname{Hom}(\Psi^{i}, \Phi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j})) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \ (i \neq 0) \\
k & \text{si } j = i+1 \ (\text{base } \partial_i) \\
k & \text{si } j = i = 0 \ (\text{base } \mathrm{id})
\end{cases}
\end{aligned}
\]

\begin{align*}
  &\pi_{1}^{\ell}(\underline{\mathrm{Alb}}^{0}_{X}) \xleftarrow[\ \sim\ ]{(t)} \pi_{1}^{\ell}(X)^{\mathrm{ab}}(\ell)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours surjectif ?}] \\
  &\pi_{1}(\underline{\mathrm{Pic}}^{00}_{X}) \simeq H^{1}(\underline{\mathrm{Alb}}^{0}_{X})(1)
  \xrightarrow[\ \sim\ ]{t} H^{1}(X)(1)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours injectif}] \\
  &H^{1}(\underline{\mathrm{Alb}}^{0}_{X}) \xrightarrow[\ \sim\ ]{t} H^{1}(X)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours injectif}] \\
  &H^{1}(\underline{\mathrm{Pic}}^{00}_{X}) \simeq \pi_{1}(\underline{\mathrm{Alb}}_{X})(-1)
  \xleftarrow[\ \sim\ ]{(t)} H^{2n-1}(X)(n-1)
  && \text{toujours}
\end{align*}

\[
(\mathrm{A}'_i) \quad
\left.
\begin{array}{l}
  H^i(U, F) \text{ de dim finie sur } k \\
  \Updownarrow \\
  H^{i+1}_z(F_z) \text{ de dim finie [i.e.\ de long.\ finie] pour tt } z \in Z \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ pt isolé de } \operatorname{Supp} E^{r-i-1} \cup \lbrace z \rbrace \\
  \Updownarrow \\
  \alpha_{i+1} \text{ injectif pour } m \text{ grand} \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand}
\end{array}
\right.
\]

\[
  \left\{
  \begin{array}{ll}
    (\mathrm{vii}) \not\Rightarrow (\mathrm{vi}) & = \text{si } \psi
      \text{ birationnel} : \mathrm{Sl}(2, k), \text{ car. } 2 \\[3pt]
    (\mathrm{x}) \not\Rightarrow (\mathrm{v}) & [\mathrm{x} \not\Rightarrow
      (\mathrm{i}),\ (\mathrm{x}) \not\Rightarrow (\mathrm{xiii}),\
      (\mathrm{vi}) \Rightarrow (\mathrm{v})] \\
      & \text{Cas d'un système de } G \text{ semi-simple} \ldots \\[3pt]
    (\mathrm{i}) \not\Rightarrow (\mathrm{x}) & [(\mathrm{v}) \not\Rightarrow
      (\mathrm{x}),\ (\mathrm{vi}) \not\Rightarrow (\mathrm{x}),\
      (\mathrm{i}) \not\Rightarrow (\mathrm{xiii})] \\
      & GP(2, k), \text{ car. } 2
  \end{array}
  \right.
\]

\[ (\mathrm{M}11) \qquad \begin{cases}
\text{a) } 0 \neq 1 \text{ sont } \in R_0 \text{ i.e.\ } \lbrace 0 \rbrace,
\lbrace 1 \rbrace \in \mathcal{M}_1 \\
\text{b) } \forall A \in \mathcal{M}_2, \text{ les fonctions } (x,y) \mapsto
x - y, \ (x,y) \mapsto xy \\ \qquad \text{sur } A \text{ sont modérées, i.e.\ les} \\
\qquad \lbrace (x, y, x-y) \mid (x,y) \in A \rbrace \text{ et } \lbrace (x,
y, xy) \mid (x,y) \in A \rbrace \\ \qquad \text{sont modérées dans } R^3 \\
\text{c) } \forall A \subset R^*,\ A \in \mathcal{M}_1,\ \exists B \in
\mathcal{M}_1 \text{ avec } A^{-1} \subset B
\end{cases} \]

\[
\begin{aligned}
  0 &\to \underset{\substack{\parallel \\ \mathbb{Z}^{*} = \{+1, -1\}}}{H^{0}(\overline{X}, \mathbb{G}_{m})}
  \to H^{0}(\overline{U}, \mathbb{G}_{m}) \to \mathbb{Z}^{S}
  \to \underset{\substack{\parallel \\ 0}}{H^{1}(\overline{X}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0\,!}}{H^{1}(\overline{U}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0}}{H^{2}_{S}(\overline{X}, \mathbb{G}_{m})} \\
  0 &\to \underset{\substack{\parallel \\ 0}}{H^{2}(\overline{X}, \mathbb{G}_{m})}
  \to H^{2}(\overline{U}, \mathbb{G}_{m})
  \to \underset{\substack{\parallel \\ (\mathbb{Q}/\mathbb{Z})^{S}}}{H^{1}(S, \mathbb{Q}/\mathbb{Z})}
  \to \underset{\substack{\parallel \\ \mathbb{Q}/\mathbb{Z}}}{H^{3}(\overline{X}, \mathbb{G}_{m})}
  \to \underset{\substack{\parallel \\ 0\,?}}{H^{3}(\overline{U}, \mathbb{G}_{m})} \to 0
\end{aligned}
\]

\[
\begin{array}{ll}
s_0 = (0, 0) & u_0 = (1, 0) \\
s_1 = (1, 0) & u_1 = (x - 1, y) \\
{[s_4 = (0, f_0)]} & u_2 = (x' - x, y' - y) \\
s_2 = (x, y) & u_3 = (-x', f_0 - y') \\
s_3 = (x', y') & u_4 = (0, -f_0)
\end{array}
\qquad
\begin{aligned}
f_1 &= y \\
f_2 &= (x - 1)(y' - y) - y(x' - x) \\
 &= (xy' - yx') - (y' - y) \\
f_3 &= (x' - x)(f_0 - y') + x'(y' - y) \\
 &= (xy' - yx') + f_0 (x' - x) \\
f_4 &= f_0 x' \\
f_0 &= f_0
\end{aligned}
\]

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

\[
\left.
\begin{array}{l}
  \alpha_i \text{ bijectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \operatorname{Supp} E^{r-i} \subset Z \text{ pour } i \leq n \\
  \Updownarrow \\
  \operatorname{prof}(F_x) > n \text{ pour tt } x \text{ fermé dans } U \\
  \Updownarrow \\
  H^i(U, F(-m)) \leftarrow H^i(X, F(-m)) \text{ pour } m \text{ grand},\ i < n \text{ et} \\
  H^n(U, F(-m)) \to H^{n+1}_Z(X, F(-m)) \text{ injectif}
\end{array}
\right.
\]

\[
\left\{
\begin{aligned}
&\text{a)}\ \sigma_0^{2} = \tau_0^{2} = 1, \quad \tau_0\sigma_0 =
\sigma_0\tau_0 \quad (\text{i.e.\ } (\underbrace{\sigma_0\tau_0}_{\tilde\sigma_0})^{2} = 1)\\
&\text{b)}\ \rho^{3} = 1\\
&\underbrace{\sigma_0\rho\sigma_0^{-1}\rho^{-1}}_{\uncertain{\sigma_1}}
= \underbrace{\rho^{2}\tau_0\rho^{-2}}_{\tau_\infty}\,
\underbrace{\rho\tau_0\rho^{-1}}_{\tau_1}\,\rho
\quad \text{i.e.} \quad \sigma_0\rho\sigma_0^{-1}\rho^{-1} =
\rho^{-1}\tau_0\rho\tau_0\\
&\rho\sigma_0\rho^{-1}\sigma_0 = \tau_0\rho^{-1}\tau_0\rho
\quad \text{\uncertain{soit}} \quad \struck{\operatorname{int}(\rho)\sigma_0}\
[\rho, \sigma_0] = [\tau_0, \rho^{-1}]\\
&\text{\uncertain{soit}} \quad [\rho, \sigma_0][\rho^{-1}, \tau_0] = 1
\end{aligned}
\right.
\]

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

\[
  \left.
  \begin{array}{l}
    \text{Géom.\ de drapeaux } D \\
    + \text{« fonction dimension »} \\
    \delta : \Phi \longrightarrow I \\
    \quad \text{\scriptsize él.\ min.}
  \end{array}
  \right|
  \Longleftrightarrow
  \left|
  \begin{array}{l}
    \text{foncteur } D' : \mathfrak{P}_f^*(I)^\circ \to (\mathrm{Ens}) \\
    \text{\emph{NB} Posant } I_0 = \lbrace i \in I \mid D'(\lbrace i
    \rbrace) \neq \emptyset \rbrace \\
    \text{on a sur } I_0 \text{ une structure } \dots
  \end{array}
  \right.
\]

\[
\begin{array}{lcl}
\widetilde{(\mathrm{Aff}_k)} \simeq \underline{\operatorname{Hom}}^{\sim}(\mathrm{Alg}_k, \mathrm{Ens}) & & \\
\widetilde{(\mathrm{Sch})_k} \xrightarrow{\ \mathrm{restr.}\ } \widetilde{\mathrm{Aff}_k} \simeq \underline{\operatorname{Hom}}^{\sim}(\mathrm{Alg}_k, \mathrm{Ens}) & & \text{(faisceaux pour la top. « de Zariski »)} \\
\qquad \downarrow \text{inc.} \qquad\qquad \downarrow \text{inc.} & & \\
\widehat{(\mathrm{Sch})_k} \xrightarrow{\ \mathrm{restr.}\ } \widehat{\mathrm{Aff}_k} \simeq \underline{\operatorname{Hom}}(\mathrm{Alg}_k, \mathrm{Ens}) & & \text{(pas plein\uncertain{fid.})} \\
\widehat{(\mathrm{Aff}_k)} \simeq \underline{\operatorname{Hom}}(\mathrm{Alg}_k, \mathrm{Ens}) & &
\end{array}
\]

\[
  (10) \qquad
  \begin{cases}
    X_{\Delta_r} = \bigl\{ (x_0, \ldots, x_r) \in X_{\Delta_0}^{\,r+1}
      \ \big|\ x_0 \leq \cdots x_r \bigr\} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \bigl\{ (x, i_0, \ldots, i_r)
      \in X \times I^{\,r+1} \ \big| \\
    \qquad\qquad x \in X_{i_0},\ i_0 \leq \cdots \leq i_r \bigr\} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \coprod_{i_{*} \in I(\Delta_r)} X_{i_{*}} \\[2pt]
    \phantom{X_{\Delta_r}} \simeq \coprod_{i_0 \in I} X_{i_0} \times I_i(\Delta_{r-1})
  \end{cases}
\]

\[
  \begin{cases}
    c^{i}(1) = 0 & i \geqslant 1, \\[3pt]
    c^{n}(x+y) = \displaystyle\sum_{i+j=n} c^{i}(x)\,c^{j}(y)
      & x, y \in K,\ n \geqslant 1, \\[8pt]
    c^{n}(x \ast y) = Q_{n}\bigl(c^{1}(x), \ldots, c^{n}(x);\;
      c^{1}(y), \ldots, c^{n}(y)\bigr)
      & x, y \in I(K),\ n \geqslant 1, \\[5pt]
    c^{n}(E_{s}) = 0 & n > n_{s},
  \end{cases}
\]

\[
  \left\lbrace
  \begin{array}{l}
    A(t) = L^{\infty}_E\Bigl(\dfrac{1}{p^{n+\rho}\, t}\Bigr)^{-1} \\[8pt]
    \chi(E) = \sum (-1)^i \operatorname{rang} H^i_!(\bar X, \bar F) \\[6pt]
    \delta(E) = \prod_i \bigl(\det f_{H^i_!(\bar X, \bar F)}\bigr)^{(-1)^i}
  \end{array}
  \right.
\]

\[
\begin{array}{ccc}
\mathrm{Hom}_{A^{\wedge}}\bigl(a, \Gamma''_M(X)\bigr) & \simeq &
\mathrm{Hom}_{A^{\wedge}}\bigl(a, p_{B*}\, \underline{\mathrm{Hom}}(p_A^{*}(M), X)\bigr)\\
\| & & \wr\\
\Gamma''_M(X)(a) & &
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a), \underline{\mathrm{Hom}}(p_A^{*}(M), X)\bigr)\\
\| & & \wr\\
\mathrm{Hom}_{B^{\wedge}}\bigl(M, X''(a)\bigr) & &
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a) \times p_A^{*}(M), X\bigr)\\
\wr & &\\
p_{A*}\bigl(\underline{\mathrm{Hom}}(p_B^{*}(a), X)\bigr) & &\\
\| & &\\
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_A^{*}(M), \underline{\mathrm{Hom}}(p_B^{*}(a), X)\bigr) & &\\
\| & &\\
\mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_A^{*}(M) \times p_B^{*}(a), X\bigr) & &
\end{array}
\]

\[
\begin{aligned}
  H^{1}(\pi, \mathbb{Z}_{\ell}(n)) &\underset{\text{non canoniquement}}{\simeq}
  \mathbb{Z}_{\ell}/(1-q^{n})\mathbb{Z}_{\ell} \\
  &= \begin{cases}
    \mathbb{Z}_{\ell} & \text{si } n = 0 \\
    0 & \text{si } \ell \nmid (1-q^{n}) \ \text{i.e. } n \not\equiv 0 \ (w_{\ell}(q)) \\
    \mathbb{Z}/\ell^{v_{\ell}(1-q^{n})} & \text{si } \ell \mid 1-q^{n},\ n \neq 0 \ \text{i.e. } n \equiv 0 \ (w_{\ell}(q))
  \end{cases}
\end{aligned}
\]

\[
  (23) \qquad
  \begin{cases}
    \text{a) } X_{\Delta_1} \text{ est un sous-espace top. fermé} \\
    \qquad \text{de } X_{\Delta_0} \times X_{\Delta_0}
      \text{ (avec la top. induite)} \\[2pt]
    \text{b) } \forall (i,j) \in I \times I,\
      \Gamma_{ij} \underset{\text{déf}}{=}
      X_{\Delta_1} \cap (X_i \times X_j) \\
    \qquad \text{est ou bien vide, ou bien le graphe} \\
    \qquad \text{d'une immersion fermée } X_i \hookrightarrow X_j .
  \end{cases}
\]

\begin{align*}
  (\widetilde{v \circ u})(x)
  &\overset{\text{déf.\ de }\sim}{=} \struck{\ill{}}\; \beta''_{*}\bigl[(v \circ u)\, \beta^{*}(x)\bigr] \\
  &\overset{\text{déf.\ de }v \circ u}{=} \beta''_{*}\bigl[\underbrace{p_{31*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\bigr) \beta^{*}(x)}_{=\ \text{formule de proj.}}\bigr] \\
  &= \beta''_{*}\, p_{31*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\, \underbrace{p_{31}^{*} \beta^{*}(x)}_{p_{1}^{*}(x)}\bigr) \\
  &\overset{\text{transitivité}}{=} p_{3*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\, p_{1}^{*}(x)\bigr) \\
  &\struck{\overset{\text{projection}}{=} p_{3*}\bigl(p_{21}^{*}(u)\, p_{32}^{*}(v)\bigr) x}
\end{align*}

\[
    \left\{
    \begin{array}{l}
      \dim \mathfrak{g}_\alpha = 1 \\
      \exists\,! \; x_\alpha : \mathbf{G}_a \longrightarrow Z_\chi \subset G,
        \quad x_\alpha(\mathbf{G}_a) = U_\alpha, \;
        \mathrm{Lie}(U_\alpha) = \mathfrak{g}_\alpha \\
      \phantom{\exists\,! \;} t\,x_\alpha(\lambda)\,t^{-1} =
        x_\alpha\bigl(\alpha(t)\lambda\bigr) \\
      \rho_\alpha : \mathbf{G}_m \longrightarrow T, \quad
        f\bigl(x_\alpha(\lambda)\bigr) = \rho_\alpha(1+\lambda), \quad
        \langle \rho_\alpha, \alpha \rangle = 1
    \end{array}
    \right.
  \]

\[
\begin{array}{lll}
  u(e_{0}) = (z \mapsto -z) = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}
    & & \text{pts fixes } 0, \infty \\[1ex]
  u(e_{1}) = (z \mapsto 1/z) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
    & & \text{pts fixes } \pm 1 \\[1ex]
  u(e_{0}+e_{1}) = (z \mapsto -1/z) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}
    & & \text{pts fixes } \pm i
\end{array}
\]

\[
  (8) \qquad
  \begin{array}{l}
    \text{Catégorie des ens. } E \text{ de} \\
    \text{cardinal } 6, \text{ munis d'une} \\
    \text{partition } (P_i)_{i \in T} \text{ du type } (3,3), \\
    \text{indexée par } T, \text{ et d'une} \\
    \struck{\text{famille}}\ \text{paire de bijections} \\
    \qquad D_i^{*} \simeq \omega_{P_i}
  \end{array}
  \quad \xrightarrow{\ \approx\ } \quad
  \begin{array}{l}
    \text{Catégorie analogue} \\
    \text{définie en termes} \\
    \text{de } T' \text{ et de} \\
    \quad (D_i^{*})_{i \in T'}
  \end{array}
\]

\[ (\mathrm{M}6) \qquad \begin{cases}
\forall\, Z, X \in \mathcal{M}_n,\ Z \subset X,\ \exists\,
(f_\alpha)_{\alpha \in A} \text{ famille finie de fonctions modérées } X \to
R \\
\text{et des } (e_\alpha)_{\alpha \in A},\ e_\alpha \in R_0, \text{ telles
que } Z = \bigcap f_\alpha^{-1}(e_\alpha) . \\
\text{De plus OPS que les } f_\alpha \text{ séparent les points de } X
\smallsetminus Z
\end{cases} \]

\[
  \left\lbrace
  \begin{array}{l}
    \chi(E) = \sum (-1)^i \operatorname{rang} R^i f_{*}(E) \\[6pt]
    \delta(E) = p^{+(\rho+n)\chi(E)}\, \delta\bigl(R f_{*}(E)\bigr)^{-1}
      = p^{(n+\rho)\chi(E)} \prod_i \bigl(\det f_{H^i(\bar X, \bar F)}\bigr)^{(-1)^{i+1}}
  \end{array}
  \right.
\]

\[
(18)\quad\left\lbrace
\begin{array}{l}
\overbrace{X^{*},\ (\dot X_\beta)_{\beta\in I_0},\ (B^{*}_\alpha,\mathcal E^{*}_\alpha)_{\alpha\in\Lambda}}^{\text{syst. de cylindres-bords transversaux}},\ (X^{*}_i)_{i\in I^{*}}\ ;\ (X_\beta)_{\beta\in I},\ (B_{\beta,\alpha},\mathcal E_{\beta,\alpha})_{\alpha\in\Lambda,\ \beta\in I_0},\\[6pt]
(\dot X_\beta\xrightarrow{\ p_\beta\ }X_\beta)_{\beta\in I_0}
\end{array}\right\rbrace
\]

\[
\left\lbrace
\begin{array}{l}
\mathcal{A}^{0} = \prod_{x \in X} \hat{\mathcal{O}}_{x}
\quad \text{\uncertain{d'où} \uncertain{des}}\ \pi_{x} \in \mathcal{A}_0 \quad (x \in X) \\
\qquad \text{(\uncertain{comme} \uncertain{A-algèbre} \uncertain{topologique})} \\[4pt]
\Theta \in \mathcal{A}^{1}, \quad \text{avec}\ \underline{\Theta^{2} = 0}, \quad
\underline{\pi_{x'} \Theta \pi_{x} = 0\ \text{si}\ x'\ \text{\uncertain{ne} \uncertain{précède} \uncertain{pas}}\ x}
\end{array}
\right.
\]

\[
\begin{array}{lll}
s_0 = (0, 0) & & \\
s_1 = (1, 0) & & \\
s_2 = (1, 1) & & (y_2 = 1) \\
s_3 = (\beta_1, \beta_1) & \beta_1 \neq 1, 0 & \\
s_4 = (\beta_1, y_4) & y_4 \neq \beta_1 & \\
s_5 = (\beta_2 \beta_1, \beta_2 y_4) & \beta_2 \neq 1, 0 & \\
s_6 = (\beta_2 \beta_1, y_6) & y_6 \neq \beta_2 y_4 & \\
s_7 = (\beta_3 \beta_2 \beta_1, \beta_3 y_6) & \beta_3 \neq 1, 0 & \\
s_8 = (\beta_3 \beta_2 \beta_1, y_8) & y_8 \neq \beta_3 y_6 & \\
{[s_9 = s_0]} & [\beta_4 = 0] &
\end{array}
\qquad
\begin{array}{l}
\beta_1\ \beta_2\ \beta_3 \\
y_4\ y_6\ y_8
\end{array}
\]

\[
  \begin{aligned}
    D(E_\eta^{\natural}) &= \bigl(R i_{*}(\check E_U)\bigr)(1)
      + \sum_{x \in Y} j_{x*}\bigl(\check E_\eta^{\natural(x)}\bigr) \\
    &= R i_{!}(\check E_U)(1) + \sum_{x \in Y} j_{x*}\Bigl[\check E_\eta^{\natural(x)}
      + j_x^{*} R i_{*}(\check E_U)(1)\Bigr] \\
    &= \check E_\eta^{\natural}(1) + \sum_{x \in Y} j_{x*}\bigl(\mu_x(E)\bigr)
  \end{aligned}
\]

\[
  \left\lbrace
  \begin{array}{l}
    q = p^{1+\rho} \\[4pt]
    \lambda_x(E_\eta) = L_{\mu_x(E_\eta)}(t)
      = (-t)^{-\chi_x(E_\eta)}\, \delta_x(E_\eta)\,
      \dfrac{L_{\alpha_x(E_\eta)}(t^{-1})}{L_{\alpha_x(E_\eta)}(p^{-\rho} t)} \\[10pt]
    \chi_x(E_\eta) = \chi\bigl(\alpha_x(E_\eta)\bigr) \\[4pt]
    \delta_x(E_\eta) = \delta\bigl(\alpha_x(E_\eta)\bigr)
  \end{array}
  \right.
\]

\begin{align*}
&\sum_{\substack{\text{perm. circ.}\\ \text{de } X,Y,Z}}
\Big[ -\langle \delta\omega, [X,Y] \wedge Z \rangle
+ \theta_{[X,Y]} \langle \omega, Z \rangle
- \theta_Z \langle \omega, [X,Y] \rangle \Big] \\
&= \sum \Big[ -\langle i_Z \delta\omega, [X,Y] \rangle
+ (\theta_X\theta_Y - \theta_Y\theta_X) \langle \omega, Z \rangle \\
&\qquad + \theta_Z \langle \delta\omega, X \wedge Y \rangle
- \theta_Z\theta_X \langle \omega, Y \rangle
+ \theta_Z\theta_Y \langle \omega, X \rangle \Big] \\
&= \sum -\langle i_Z \delta\omega, [X,Y] \rangle
+ \theta_Z \langle \delta\omega, X \wedge Y \rangle \\
&= \sum \langle \delta i_Z \delta\omega, X \wedge Y \rangle
- \theta_X i_Y i_Z \delta\omega + \theta_Y i_X i_Z \delta\omega
+ \theta_Z \langle \delta\omega, X \wedge Y \rangle \\
&= \sum \big( i_X i_Y \delta i_Z + \theta_Z i_X i_Y
- \theta_X i_Y i_Z + \theta_Y i_X i_Z \big) . (\delta\omega)
\end{align*}

\[
  \left\{
  \begin{array}{ll}
    H^i(E) = 0 & \text{si}\ i \neq 0, 2\nu - 2, 4\nu - 1 \\[4pt]
    H^{2(\nu-1)}(E) \simeq
      \left\{ \begin{array}{ll} 0, & \ell \neq 2 \\ \mathbf{Z}/2\mathbf{Z} & \text{si}\ \ell = 2 \end{array} \right. \\[10pt]
    H^{4\nu-1}(E) \simeq \mu(-2\nu)
  \end{array}
  \right.
  \qquad \boxed{n + 1 = 2\nu} \ \ n = 2\nu - 1
\]

\begin{gather}
  (\Lambda^{p} u) \wedge (\Lambda^{q} v) = \frac{1}{(p+q)!} \sum_{\substack{i_1 < \cdots < i_p \\ j_1 < \cdots < j_q}} \rho_{i_1} \cdots \rho_{i_p}\, \sigma_{j_1} \cdots \sigma_{j_q}\; \times \\
  \overline{b_{i_1} \wedge \cdots \wedge b_{i_p} \wedge d_{j_1} \wedge \cdots \wedge d_{j_q}} \otimes \bigl(a_{i_1} \wedge \cdots \wedge a_{i_p} \wedge c_{j_1} \wedge \cdots \wedge c_{j_q}\bigr)
\end{gather}

\[
  \left\{
  \begin{array}{l}
    \rho_{\mathrm{aff}} \geqslant \rho'_{\mathrm{aff}} \qquad
      d \geqslant d' \\[2pt]
    (5)\ \rho_{\mathrm{n}} \geqslant \rho'_{\mathrm{n}}, \quad
    (6)\ \rho_{\mathrm{u}} \geqslant \rho'_{\mathrm{u}}, \quad
    (7)\ \rho_{\mathrm{na}} \geqslant \rho'_{\mathrm{na}} \\[4pt]
    (1)\ \rho_{\mathrm{ab}} \leqslant \rho'_{\mathrm{ab}}, \quad
    (2)\ \rho_{\mathrm{ab}} + \rho_{\mathrm{r}} \leqslant \rho'_{\mathrm{ab}}
      + \rho'_{\mathrm{r}}, \quad
    (3)\ \rho_{\mathrm{s}} \leqslant \rho'_{\mathrm{s}}
  \end{array}
  \right.
\]

\[
  \boxed{
  \left\lbrace
  \begin{array}{l}
    D(E_\eta^{\natural}) = \check E_\eta^{\natural}(1)
      + \displaystyle\sum_{x \in X^{(0)}} j_{x*}\bigl(\mu_x(E)\bigr) \\[10pt]
    \mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(\check E_\eta)(1) \qquad
    \alpha_x(E_\eta) = E_\eta^{\natural(x)} - E_\eta(x)
  \end{array}
  \right.}
\]

\[
(48)\quad\left\lbrace
\begin{array}{l}
X^{(1)}=X\setminus\mathring T_{(0)}\\[4pt]
X^{(1)}_{(d)}=X^{(1)}\cap X_{(d)}\qquad(=\emptyset\ \text{si}\ d\le0)\\[4pt]
X^{(1)}_i=X^{(1)}\cap X_i\qquad\text{de sorte que}\quad X^{(1)}_{(d)}=\bigcup_{\substack{i\in I\\ \delta(i)\le d}}X^{(1)}\cap X_i
\end{array}\right.
\]

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

\[
(*) \quad
\left.
\begin{array}{l}
  \alpha_i \text{ injectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \exists \text{ voisinage ouvert } V \text{ de } Z, \text{ tel que } F \text{ soit de profondeur } \ldots \text{ sur } U \cap V \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand},\ i < n
\end{array}
\right.
\]

\[
  (10) \qquad
  \left\{
  \begin{array}{l}
    T \simeq \operatorname{diag} Q
    \quad \text{et} \quad D_i^{*} \simeq d_i \quad \text{pour } i \in T \\[4pt]
    S_Q = \bigcup_{i \in T} D_i^{*} = \coprod_{i \in T} D_i^{*} \\[4pt]
    A_Q = \prod_{i \in T} D_i^{*} \simeq V^{*} \smallsetminus \bigcup_{i \in T} D_i^{*}
    = \bigcup_{i \in T'} D_i^{*}
  \end{array}
  \right.
\]

\[
\begin{aligned}
P_{a}(X) &\simeq \operatorname{Ker}\bigl(\mathcal{A}^{i}_{a}(X)
\xrightarrow{L^{n-2i}} \mathcal{A}^{n-i}_{a}(X)\bigr)
&& \text{partie « primitive » de la « coh.\ algébrique »} \\
I_{r}(X) &= \operatorname{Ker}\bigl(\mathcal{A}^{i}_{r}(X)^{a}
\xrightarrow{L^{n-2i+1}} \mathcal{A}^{n-i+1}_{r}(X)^{a}\bigr)
&& \text{partie « primitive » de la V.A.\ intermédiaire}
\end{aligned}
\]

\[
  \begin{array}{l}
  x \preceq y \ \text{ssi} \\
  x \in F_\alpha,\ y \in F_\beta
  \end{array}
  \quad
  \left\lbrace
  \begin{array}{l}
  \text{ou bien } x = y \ (\text{donc } \alpha = \beta) \\
  \text{ou bien } \alpha < \beta \text{ et } \exists\, z_\alpha = x \in
  F_\alpha,\ z_{\alpha+1} \in F_{\alpha+1}, \dots, z_{\beta-1} \in
  F_{\beta-1}, \\
  \qquad z_\beta = y \in F_\beta \text{ tels que } \forall\, \alpha
  \leqslant i < \beta,\ z_i \text{ et } z_{i+1} \text{ soient incidents}
  \end{array}
  \right.
\]

\begin{align*}
  \mathrm{Bil\,cont}_k(\check{P}, \check{Q}; \check{R})
  &\simeq \mathrm{Bil\,cont}_k\bigl(\check{P}, \check{Q}; \varprojlim_{\gamma} \check{R}_\gamma\bigr) \\
  &\simeq \varprojlim_{\gamma} \bigl[\mathrm{Bil\,cont}_k(\check{P}, \check{Q}; \check{R}_\gamma)\bigr] \\
  &\simeq \varprojlim_{\gamma} \Bigl[\varinjlim_{\alpha,\beta} \underbrace{\mathrm{Bil}_k(\check{P}_\alpha, \check{Q}_\beta; \check{R}_\gamma)}_{\mathrm{Hom}_k(\check{P}_\alpha \otimes \check{Q}_\beta,\, \check{R}_\gamma)}\Bigr] \\
  &\simeq \varprojlim_{\gamma} \varinjlim_{\alpha,\beta} \mathrm{Hom}_k(R_\gamma, P_\alpha \otimes Q_\beta) \\
  &\simeq \varprojlim_{\gamma} \mathrm{Hom}_k(R_\gamma, P \otimes Q) \\
  &\simeq \mathrm{Hom}_k(R, P \otimes Q) \qquad !\ ]
\end{align*}

\[
\left.
\begin{array}{l}
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S \text{ pour } i \leq n \\
  \Updownarrow \\
  R^i g_*(F) \text{ coh.\ pour } i \leq n, \text{ i.e.\ } \underline{H}^i_Z(F) \text{ coh.\ pour } i \leq n+1, \\
  \quad \text{i.e.\ } H^i_z(F_z) \text{ de dim finie pour } i \leq n+1,\ z \in Z
\end{array}
\right.
\]

\[
p\text{-}\mathrm{Grf}(k) \xrightarrow{\;\approx\;} \left\lbrace
\begin{array}{l}
\text{catégorie des modules de longueur} \\
\text{finie } M \text{ sur } W, \text{ munis de} \\
F_M, V_M : M \to M \text{ satisfaisant} \\
\quad F\lambda = \lambda^{\sigma} F,\ \lambda V = V \lambda^{\sigma} \\
\quad FV = p\cdot\mathrm{id},\ FV = p\cdot\mathrm{id}
\end{array}
\right.
\qquad G \longmapsto D^{*}(G)
\]

\[
  (25) \qquad
  \begin{cases}
    X_m^{*} = \coprod\limits_{i \in I_m} X_i^{*} \\[10pt]
    \mathcal{V}_{m,m+1} = \coprod\limits_{\substack{(i,j) \in I \times I \\
      i \leq j ,\ d(i)=m ,\ d(j)=m+1}} \mathcal{V}_{i,j} \\[16pt]
    \mathcal{V}^{*}_{m,m+1} = \coprod\limits_{i,j \text{ comme dessus}}
      \mathcal{V}^{*}_{i,j}
  \end{cases}
\]

\[
  \begin{array}{ll}
    (1)\ \rho_{\mathrm{ab}} & (5)\ \rho_{\mathrm{n}} \\
    (2)\ \rho_{\mathrm{r}} + \rho_{\mathrm{ab}} = \rho_{\mathrm{rig}} &
      (6)\ \rho_{\mathrm{u}} \\
    (3)\ \rho_{\mathrm{ss}} & (7)\ \rho_{\mathrm{naff}} \\
    (4)\ d_{\mathrm{ss}} & (8)\ d_{\mathrm{aff}} \\
     & (9)\ d_{\mathrm{rad}} \\
     & (10)\ d_{\mathrm{radaff}}
  \end{array}
\]

\begin{gather*}
e_1 e_2 = e_1^2 = 0 \\
(x e_1 + y e_2)(x' e_1 + y' e_2) = y y' \\
(e_1 + e_2)(e_1 + e_2) \qquad e_1 + e_2 - \ldots + e_n \qquad
\Sigma x_i^2 \\
e_1\ \ \underbrace{e_2\ e_3 - \ldots}\ \ e_1 - e_2 \\
e_1,\ e_1 - e_2,\ e_1 - e_3,\ \ldots,\ e_1 - e_n \\
e'_i e'_j = 0 \quad i, j \neq 1 \\
a_{11} x_1 x'_1 + \sum_{1 \leq i < j} a_{ij} (x_i x'_j + x_j x'_i)
\qquad (e_1 - e_i)(e_1 - e_j) = 1 \\
a_{11} x_1 x'_1
\end{gather*}

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

\[
(3)\ \begin{cases}
H^{1}(K, A) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(B, \mathbb{Q}/\mathbb{Z}) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(B'^{\circ}, \mathbb{Q}/\mathbb{Z}) \times \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(V, \mathbb{Q}/\mathbb{Z}) \\
H^{1}(K, B) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(A, \mathbb{Q}/\mathbb{Z}) \simeq \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(A'^{\circ}, \mathbb{Q}/\mathbb{Z}) \times \underline{\mathrm{Ext}}^{1}_{k\text{-}\mathrm{gr}}(U, \mathbb{Q}/\mathbb{Z})
\end{cases}
\]

\[
  (12) \qquad
  \begin{cases}
    R_{j,k} = \bigcup_{\ell \in I^{(k)'}} X_\ell \\[2pt]
    S_{j,k} = \bigcup_{\ell \in I^{(k)''}} X_\ell
  \end{cases}
  \qquad
  \begin{aligned}
    &\text{Donc } R_{j,k} \cup S_{j,k} = \dot{X}_k \\
    &(\text{avec } R_{j,k} \cap S_{j,k} = \dot{X}_i) .
  \end{aligned}
\]

\[
\begin{array}{ll}
\mathbf{P}_1 \times \mathbf{P}_1 & \mathrm{PGL}_2 \times \mathrm{PGL}_2 \\[2pt]
\mathbf{P}_2 & \mathrm{PGL}_3 \\[2pt]
F_n \quad n \geq 2 & \bigl(\mathrm{GL}(2)\cdot \Gamma(\mathcal{O}(n))\bigr)/\mu_n
\end{array}
\qquad
\begin{array}{l}
(F_0 = \mathbf{P}_1 \times \mathbf{P}_1) \\[2pt]
(F_1 = \mathbf{P}_2 \text{ éclaté})
\end{array}
\]

\[
\begin{array}{ll}
u_0 \wedge u_1 = 1\,.\,e & \mu'_1 = 1 \\
u_1 \wedge u_2 = -\alpha\, e & \mu'_2 = -\alpha \\
u_2 \wedge u_3 = (\beta - \alpha)\, e & \mu'_3 = \beta - \alpha \\
u_3 \wedge u_0 = (1 + \beta)\, e & \mu'_0 = 1 + \beta
\end{array}
\qquad
\begin{aligned}
u_2 \wedge u_3 &= -u_2 \wedge u_0 - u_2 \wedge u_1 \\
 &= \beta \underbrace{u_0 \wedge u_1}_{e} - \alpha\, e \\
u_3 \wedge u_0 &= -u_1 \wedge u_0 - u_2 \wedge u_0 \\
 &= e + \beta\, e
\end{aligned}
\]

\[
\begin{aligned}
  \operatorname{Ext}^{\bullet}_{Z' \cap U' = T'}(X'; F'; G')
  &= H^{\bullet}\, \struck{\ill{}}\; \underline{\operatorname{Hom}}_{Z' \cap U'\,
  T'}(F'; C(G'))\\
  &= H^{\bullet}\, \struck{\ill{}}\; \underline{\operatorname{Hom}}_{Z \cap U\,
  T}(F, p_{U*} C(G'))\\
  &= H^{\bullet}\, \underline{\operatorname{Hom}}_{Z \cap U\, T}(F,
  C(p_{U*} G'))\\
  &= \operatorname{Ext}^{\bullet}_{Z \cap U\, T}\Bigl(F, \coprod_{n \in
  \mathbf{Z}} G(n)\Bigr)
\end{aligned}
\]

\[
\left\lbrace
\begin{array}{l}
\mathrm{Hom}(R_{X/Y}, P^{1}) \simeq \mathrm{Hom}_{Y\text{-gr}}(\mathrm{Pic}^{1}_{\infty}, P) \\
\underline{H}^{0}(R_{X/Y}, G) \xleftarrow{\ \sim\ } \mathrm{Hom}_{Y\text{-gr}}(\mathrm{Pic}_{\infty}, G) \\
\underline{H}^{1}(R_{X/Y}, G) \xleftarrow{\ \sim\ } \mathrm{Ext}^{1}_{Y\text{-gr}}(\mathrm{Pic}_{\infty}, G) \quad (\text{et } \ill{}\ \ill{}) \\
\underline{H}^{i}(R_{X/Y}, G) \xleftarrow{\ \sim\ } \mathrm{Ext}^{i}_{Y\text{-gr}}(\mathrm{Pic}_{\infty}, G)
\end{array}\right.
\]

\[
  (7) \qquad
  \begin{cases}
    \text{a) } X_{\Delta_1} \xrightarrow{\ \sigma\ } X_{\Delta_0}
      \text{ est un revêtement étale} \\
    \qquad (\text{sur chaque pièce } X_i)
      \text{ — en fait constant,} \\
    \qquad \text{de fibre égale à } I_i = \{ j \in I \mid j \geq i \}, \\
    \qquad \text{donc rev. fini si les } I_i \text{ sont finis} \\
    \text{b) } X_{\Delta_1} \xrightarrow{\ b\ } X_{\Delta_0}
      \text{ est une immersion locale.}
  \end{cases}
\]

\[
\left\lbrace
\begin{array}{l}
A_0^n(\lambda_1, \ldots, \lambda_{n-2})\, u_0 + B_0^n(\lambda_1, \ldots, \lambda_{n-2})\, u_1
 + C_0^n(\lambda_1, \ldots, \lambda_{n-2})\, u_2 = 0 \\
A_1^n(\lambda_0, \ldots, \lambda_{n-1})\, u_0 + B_1^n(\lambda_0, \ldots, \lambda_{n-1})\, u_1
 + C_1^n(\lambda_0, \ldots, \lambda_{n-1})\, u_2 = 0 \\
A_2^n(\quad)\, u_0 + B_2^n(\quad)\, u_1 + C_2^n(\quad)\, u_2 = 0 \\
A_3^n(\quad)\, u_0 + B_3^n(\quad)\, u_1 + C_3^n(\quad)\, u_2 = 0
\end{array}
\right.
\]

\[
(14) \qquad
\left\lbrace
\begin{array}{l}
\bigwedge_{i \in I} \Omega_i \simeq \bigwedge_{i \in I} \Phi_i \\[1ex]
\underbrace{\Bigl(\bigwedge_{i \in I} \Omega_i\Bigr) \wedge \operatorname{or} I}_{\text{orientations de l'oct.\ comb.\ défini par } (\Omega_i)_{i \in I}}
\simeq
\underbrace{\Bigl(\bigwedge_{i \in I} \Phi_i\Bigr) \wedge \operatorname{or} I}_{\text{orientations de l'oct.\ comb.\ défini par } (\Phi_i)_{i \in I}}
\end{array}
\right.
\]

\[
  \left\lbrace
  \begin{array}{ll}
    R_n(X_n) \simeq E_n^\circ(X_n) & G_{n-1}\text{-ens} \\
    R_{n-1}(X_n) \simeq E_n^\circ(X_n)/\sigma_n & G_{n-2}\text{-ens} \\
    R_{n-2}(X_n) \simeq E_n^\circ(X_n)/(\sigma_n, \sigma_{n-1}) &
    G_{n-3}\text{-ens} \\
    \vdots & \\
    R_1(X_n) \simeq E_n(X_n)/(\sigma_n, \ldots, \sigma_2) & \\
    R_0(X_n) \simeq E_n(X_n)/(\sigma_n, \ldots, \sigma_2, \sigma_1) &
  \end{array}
  \right.
\]

\[
\begin{aligned}
&X_f = \bigcup X_{f_i} \\
&\text{i.e. } V(f) = \bigcap V(f_i) \\
&\text{i.e. } \widetilde{fA} = \widetilde{\textstyle\sum f_i A} \\
&\text{i.e. } \begin{cases} \forall i \ f_i \prec f \\ \exists n \in \mathbb{N} \text{ et } (h_i)_{i \in I} \text{ avec } f^n = \sum f_i h_i \end{cases}
\end{aligned}
\]

\[
  \begin{cases}
    k[[x]][Y] / (Y^{p} - x^{q}) \\
    k[[x]][Y] / (Y^{2} + P(x)\,Y + Q(x)) \\
    k[x_{1}, \ldots, x_{n}] / (x_{1}^{p_{1}}, x_{2}^{p_{2}}, \ldots, x_{n}^{p_{n}}) \qquad \text{p. ex. } k[x]/(x^{p}) \\
    k[x, y] / \mathfrak{m}^{2} \\
    k[x, y] / (x^{3}, y^{2}, x^{2}y) \\
    k[x, y] / (x^{3}, y^{2}, xy)
  \end{cases}
\]

\[
  \begin{array}{lll|l}
    0 & 1 & 2 & \text{drapeaux minimaux (de long.\ 0)} \\
    S & A & F & \text{ou « facettes »} \\[4pt]
    (0,1) & (1,2) & (0,2) & \text{drapeaux de long.\ 1} \\
    R \subset S \times A & R' \subset A \times F & R'' \subset S \times F & \\[4pt]
    (0,1,2) & & & \text{repères, ou drapeaux} \\
    \operatorname{Rep} \subset S \times A \times F & & & \text{maximaux}
  \end{array}
\]

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

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

\[
\begin{array}{ccccc}
  \mu_n^{T} & \subset & \mathcal{U} & & \\
  \cup & & & & \\
  \mu_n^{*T} & \subset & \mathcal{U} & & \zeta + \bar\zeta \\
  \Big\downarrow{\scriptstyle \text{étale rang } 2} & & \Big\downarrow{\scriptstyle \operatorname{Tr}_{K/\mathbb{Q}}} & & \\
  \vartheta_n^{T} = \mathcal{V}_n & \subset & \mathcal{O} & & \\
  \Big\downarrow{\scriptstyle \text{étale rang } \uncertain{\varphi(n)/2}} & & & & \\
  S & & & &
\end{array}
\]

\[
  \pi^i(r,u) = (r,u) \quad \forall\, (r,u) \in R \times \mathrm{Rep}(I)
  \iff
  \begin{cases}
    i \equiv 0\ (2), \text{ i.e. } i = 2n \\[2pt]
    \bigl(\varphi(r,u(\ell-1))\,\varphi(r,u(\ell))\bigr)^n = 1
    \quad \forall\, r, u
  \end{cases}
\]

\[
  (16) \qquad
  \begin{cases}
    \text{a) si } \alpha : \Delta_{r'} \to \Delta_r
      \text{ est tel que } \alpha(0) = 0, \text{ alors} \\
    \qquad X_{\Delta_r} \xrightarrow{\ \alpha^{*}\ } X_{\Delta_{r'}}
      \text{ est un morphisme de rev. étale} \\
    \qquad \text{(en particulier } X_{\Delta_r} \to X_{\Delta_0}, \\
    \qquad\quad (x_0, \ldots, x_r) \mapsto x_0
      \text{ ou } (x, i_0, \ldots, i_r) \mapsto x, \\
    \qquad\quad \text{est un morphisme de rev. étale).}
  \end{cases}
\]

\[
  \left.
  \begin{array}{ll}
    H^{*}(X_0) : & 1, \xi_0, \ldots, \xi_0^{\nu-1}, [\xi_0^{\nu}, \lambda'_{\nu}],
      \tfrac{1}{2}\xi_0^{\nu+1}, \ldots, \tfrac{1}{2}\xi_0^{2\nu-1} \\[4pt]
    H^{*}(\overline{X}_1) : & 1, \xi_1, \ldots, \xi_1^{\nu-1}, \tfrac{1}{2}\xi_1^{\nu},
      \tfrac{1}{2}\xi_1^{\nu+1}, \ldots, \tfrac{1}{2}\xi_1^{2\nu-1}
  \end{array}
  \right]
  \quad
  \begin{array}{l}
    \xi_0 \mapsto \xi_1 \\
    \lambda'_{\nu} \mapsto 0
  \end{array}
\]

\[
  \left\lbrace
  \begin{array}{l}
    \text{géométries de drapeaux} \\
    \text{(ens.\ ordonnés ainsi)}
  \end{array}
  \right.
  \Longleftrightarrow
  \begin{array}{l}
    \text{Couples } (\Phi, D) \text{ d'un ens.\ } \Phi \text{ et d'un
    foncteur} \\
    \mathfrak{P}_f^*(\Phi)^\circ \longrightarrow (\mathrm{Ens}) \\
    \text{tels que } \mathrm{card}\, D(\lbrace i \rbrace) = 1\ \forall\, i
    \in \Phi
  \end{array}
\]

\begin{align*}
  \Gamma^{p^{\nu}}(x+y) &= \Gamma^{p^{\nu}}x + \Gamma^{p^{\nu}}y
    + \sum_{\substack{\alpha, \beta \geqslant 1 \\ \alpha + \beta = p^{\nu}}} \struck{\ill{}}\ \Gamma^{\alpha}(x)\,\Gamma^{\beta}(y)
  \qquad \underbrace{\phantom{xxxxxxxxxxxxxx}}_{\text{expression en les } \Gamma^{p^{i}}(x),\ \Gamma^{p^{j}}(y),\ i, j < \nu} \\
  \Gamma^{p^{\nu}}(xy) &= x^{p^{\nu}}\, \Gamma^{p^{\nu}}(y) \\
  \Gamma^{p^{m}}(x)\,\Gamma^{p^{n}}(x) &= \frac{(p^{m} + p^{n})!}{p^{m}!\; p^{n}!}\; \Gamma^{p^{m}+p^{n}}(x) \qquad \ldots
\end{align*}

\[
\begin{array}{lcl}
\operatorname{diag} g \in \mathcal{M} & \Longleftrightarrow & \forall S' \to S, \text{ les sections de } Y' \text{ sur } S' \text{ sont } \in \mathcal{M} \\
g : Y \to S & \Longleftrightarrow & \forall (u, v) : X \rightrightarrows Y \text{ sur } S,\ \operatorname{Ker}(u, v) \to X \text{ est } \in \mathcal{M} \\
 & \Longleftrightarrow & \forall f : X \to Y,\ \Gamma_f : X \to X \times_S Y \text{ est } \in \mathcal{M} \\
 & \Longleftrightarrow & gf \in \mathcal{M} \Rightarrow f \in \mathcal{M}
\end{array}
\]

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

\[
\begin{aligned}
  \varphi(x^{2}) &= \varphi(x) + x\varphi(x) \\
  \varphi(x^{3}) &= \varphi(x) + x\varphi(x) + x^{2}\varphi(x) \\
  \varphi(x^{n}) &= (1 + x + \cdots + x^{n-1})\varphi(x) \\
  \text{or } \varphi(x^{-1}x) &= \varphi(x^{-1}) + x^{-1}\varphi(x) \\
  \varphi(x^{-1}) &= -x^{-1}\varphi(x) \\
  \varphi(x^{-n}) &= -(1 + x^{-1} + \cdots + x^{-(n-1)})x^{-1}\varphi(x)
  = -\frac{1 - x^{-n}}{1 - x^{-1}}\, x^{-1}\varphi(x) = \frac{1 - x^{-n}}{1 - x}
\end{aligned}
\]

\[
  \begin{array}{c|cccc|l}
    q & 2 & 3 & 4 & 5 & \\
    \hline
    & 1 & 2 & 3 & 4 & Gl(1, \mathbf{F}_q) \\
    & 2 & 6 & 12 & 20 & \mathrm{Aff}(1, \mathbf{F}_q) \\
    & 6 & 24 & 60 & 120 & Sl(2, \mathbf{F}_q) \\
    & 6 & 24 & 60 & 120 & GP(1, \mathbf{F}_q) \\
    & 6 & 48 & 180 & 480 & Gl(2, \mathbf{F}_q) \\
    & 24 & 9\cdot 48 & 16 \cdot 180 & 25 \cdot 480 & \mathrm{Aff}(2,
    \mathbf{F}_q)
  \end{array}
\]

\[
  (6) \qquad \left\lbrace
  \begin{array}{l}
    (x, y, z) \longmapsto \begin{pmatrix} z & x \\ y & -z \end{pmatrix} \\[4pt]
    E_0 = \underline{O}^3_{S_0} \xrightarrow{\ \sim\ } \widetilde{\mathfrak{g}}_0 = \mathfrak{sl}(2)_{S_0}
  \end{array}\right.
\]

\[
\begin{aligned}
(A \times B)^{\wedge} &\overset{\text{déf}}{=}
\underline{\mathrm{Hom}}\bigl((A \times B)^{\circ}, (\mathrm{Ens})\bigr)
\simeq \underline{\mathrm{Hom}}\bigl(A^{\circ} \times B^{\circ}, (\mathrm{Ens})\bigr)\\
&\simeq \underline{\mathrm{Hom}}(A^{\circ}, B^{\wedge})
\simeq \underline{\mathrm{Hom}}_{!}(A^{\wedge\circ}, B^{\wedge})\\
&\simeq \underline{\mathrm{Hom}}(B^{\circ}, A^{\wedge})
\simeq \underline{\mathrm{Hom}}_{!}(B^{\wedge\circ}, A^{\wedge})\\
&\simeq \underline{\mathrm{Hom}}_{!!}\bigl((A^{\wedge})^{\circ} \times (B^{\wedge})^{\circ}, \mathrm{Ens}\bigr)
\simeq \underline{\mathrm{Hom}}_{!}\bigl((A^{\wedge} \times B^{\wedge})^{\circ}, \mathrm{Ens}\bigr)
\end{aligned}
\]

\[
\left.
\begin{array}{l}
  \alpha_i \text{ injectif pour } m \text{ grand} \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ isolé dans } (\operatorname{Supp} E^{r-i} \cup \lbrace z \rbrace) \\
  \Updownarrow \\
  H^{i-1}(X, F(-m)) \to H^{i-1}(U, F(-m)) \text{ surjectif pour } m \text{ grand}
\end{array}
\right.
\]

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

\[
\boxed{
\begin{array}{l}
\operatorname{cd}_{\ell}\bigl(k(x)\bigr) \leqslant n \\[2pt]
\forall\, Y \ \uncertain{\text{réunion}} \text{ dans } X, \quad
\operatorname{cd}_{\ell}(Y) < n \\[2pt]
\forall\, y \in X,\ y \ne x, \quad
\operatorname{cd}_{\ell}(\bar{y}) + \operatorname{cd}_{\ell}
\operatorname{Frac} \underline{O}_{X,\bar y} < n
\end{array}}
\]

\[
  \boxed{
  \begin{array}{l}
    D(E_\eta^{\natural}) = E_\eta^{\natural}(\rho+1)
      + \displaystyle\sum_{x \in X^{(0)}} j_{x*}\bigl(\mu_x(E)\bigr) \\[10pt]
    \mu_x(E) = \alpha_x(E_\eta)^{\vee} - \alpha_x(E_\eta)(\rho+1)
  \end{array}}
\]

\[
\begin{aligned}
\underline{\Sigma}=\Bigl(&\Sigma(0)=\mathrm{Ob}(\underline{\Sigma}),\ \Sigma(1)=\mathrm{Fl}(\underline{\Sigma}),\ \Sigma(1)\overset{\underline s}{\underset{\underline b}{\rightrightarrows}}\Sigma(0),\\
&(\Sigma(1),\underline b)\times_{\Sigma(0)}(\Sigma(1),\underline s)\overset{\text{déf}}{\Longrightarrow}\Sigma(2)\xrightarrow[\text{compos.\ des flèches}]{\text{compl.}}\Sigma(1)\Bigr),
\end{aligned}
\]

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

\[
  \begin{array}{l}
  = \delta'_{\rho}\, t^{\chi'_{\rho}} L'_{\rho}(t) \cdots
    \delta'_{\rho+n-1}\, t^{\chi'_{\rho+n-1}} L'_{\rho+n-1}(t) \\[4pt]
  \qquad \delta_{\rho+n}\, t^{\chi_{\rho}} L_{\rho+n}(t)\,
    \delta_{\rho+n-1}\, t^{\chi_{\rho+n-1}} L_{\rho+n-1}(pt) \cdots
    \delta_{\rho}\, t^{\chi_{\rho}} L_{\rho}(p^{n}t) \\[4pt]
  = \delta\, t^{\chi} L'_{\rho}(t) \cdots L'_{\rho+n-1}(t)\,
    L_{\rho+n}(t)\, L_{\rho+n-1}(pt) \cdots L_{\rho}(p^{n}t)
  \end{array}
\]

\[
\begin{aligned}
(15)&\quad (\operatorname{int}\tilde\rho)(p, \sigma, g_0, g_1, g_\infty) =
(p, \dot\rho\sigma\dot\rho^{-1}, \rho(g_\infty), \rho(g_0), \rho(g_1))\\
(16)&\quad \operatorname{int}(\tilde\sigma_0)(p, \sigma, g_0, g_1, g_\infty) =
(p, \dot\sigma_0\sigma\dot\sigma_0^{-1}, \sigma_0(g_0), \sigma_0(g_\infty),
\sigma_0(g_1))
\end{aligned}
\]

\[
\begin{array}{l}
F(S) \to F(S') \rightrightarrows F(S'') \\
O(S) \to O(S') \rightrightarrows O(S'') \\
LF_{rc}(S) \to LF_{rc}(S') \rightrightarrows LF_{rc}(S'') \\
{}[LF(S) \to LF \rightrightarrows LF(S'')\ ?]
\end{array}
\quad
\left(\begin{array}{l}
\text{où } F = \text{parties fermées} \\
O = \text{parties ouvertes} \\
LF \text{ parties loc. fermées \uncertain{rétrocompactes}} \\
S'' = S' \times_S S'
\end{array}\right) \text{ est \emph{exact}}
\]

\[
\left\lbrace
\begin{array}{l}
q_{\mathcal{U}}(y) . 1_C = \eta_{\mathcal{U}}(y) . 1_{C'} \\[4pt]
x, y \in C,\ u \in \mathcal{U},\ \Delta_{\mathcal{U}} u = \sum v_i \otimes w_i
\;\Longrightarrow\;
p_{\mathcal{U}}(u)(xy) = \sum \bigl(p_{\mathcal{U}}(v_i) x\bigr)\bigl(p_{\mathcal{U}}(w_i) y\bigr)
\end{array}
\right.
\]

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

\[
  \begin{array}{rcl}
  H^{2n}(A) \simeq {\textstyle\bigwedge^{2n}} H^{1}(A) & \xrightarrow[\ \sim\ ]{\eta_{A}} & \mathbb{Q}_{\ell}(-n) \\[2pt]
  H^{2n}(A^{*}) \simeq {\textstyle\bigwedge^{2n}} H^{1}(A^{*}) & \xrightarrow[\ \sim\ ]{\eta_{A^{*}}} & \mathbb{Q}_{\ell}(-n) \\[2pt]
  \wr & & \wr\ \text{(can.)} \\
  \mathbb{Q}_{\ell}(2n) & & \mathbb{Q}_{\ell}(-2n)
  \end{array}
\]

\[
\begin{cases}
\boxed{d\omega = 0} & \text{exprime que la connexion } \nabla
\text{ définie par } \omega \text{ est à courbure nulle} \\
\boxed{F_S^{*}(\omega) - p\omega = d\Phi} & \text{exprime que } F_M
\text{ est compatible à la connexion} \\
\boxed{\Phi_0 = 0} & \text{exprimant que } (F_M)_0 \text{ respecte }
\operatorname{Fil}_0 .
\end{cases}
\]

\[
\begin{array}{ccc}
\text{idèles entiers} & & \\
\prod G(\hat{\mathcal{O}}_x) \longleftarrow 0 & \qquad & G(C) = H^{0}(C, G) \qquad 0 \\
\downarrow \qquad\quad \downarrow & & \\
\bigl(\prod_{\text{local}} G(\hat{K}_x)\bigr)/G(K_\xi) \longleftarrow 0 & & H^{1}(C, G) \qquad 0 \\
\text{classes d'idèles} & & \\
H_{\mathrm{II}}\,DD(G) & & H_{\mathrm{I}} H_{\mathrm{II}}\,DD(G)
\end{array}
\]

\[
\begin{cases}
U = U_K \setminus B = \operatorname{Int}(U_K) \\
V = X \setminus \overline{U_K} = (X \setminus U_K) \setminus C = \operatorname{Int}(X_1),
\qquad X_1 = X \setminus U_K \\
U = \complement\overline{V}, \quad V = \complement\overline{U}, \quad
\dot{U} = \dot{V} = \overline{U} \cap \overline{V} = \complement(U \cup V) = B \cup C \\
\struck{U \cap K = U \subset K,\ V \cap K = V_K}\quad V \subset X_1 \subset \overline{V}
\end{cases}
\]

\[
\begin{array}{ccc}
 & \check{\ell}^{G^{*}}_{\bullet} & \\
{\scriptstyle ?}\swarrow & & \nwarrow \\
\ell^{G}_{\bullet}[-1] & \longrightarrow & \Delta^{*}(G)
\end{array}
\qquad \text{et} \qquad
\begin{array}{ccc}
 & \mathcal{D}/V\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M & \\
{\scriptstyle 1}\swarrow & & \nwarrow \\
\mathcal{D}/F\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M & \longrightarrow & \mathcal{D}/p\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M
\end{array}
\]

\[
\mathcal{E} \longrightarrow \mathbb{Z}/12\mathbb{Z} \simeq (\mathrm{SL}(2,\mathbb{Z}))_{\mathrm{ab}}
\qquad
\begin{aligned}
\rho &\longmapsto 10 \bmod 12 & \sigma\rho = \varepsilon_0 &\longmapsto 1 \bmod 12 \\
\sigma &\longmapsto 3 \bmod 12 & (\sigma\rho)^2\omega = \rho_0 &\longmapsto 8 \bmod 12 \\
\omega &\longmapsto 6 \bmod 12 & &
\end{aligned}
\]

\[
  \boxed{\begin{array}{ccccccc}
    P_{i}^{+\circ} & \subset & \mathcal{A}^{i} & \xrightarrow{\ \alpha_{i}\ } &
    \mathcal{A}_{d-i} & \supset & P_{d-i}^{+} \\
    \Vert & & & & & & \\
    P^{i}_{+} & & & & & & \\[1ex]
    P_{d-i}^{+\circ} & \subset & \mathcal{A}^{2d-i} &
    \xrightarrow{\ \alpha_{d-i}\ } & \mathcal{A}_{i} & \supset & P_{i}^{+}
    \\[1ex]
    & & \mathrm{int}\bigl(P_{i}^{+\circ}\bigr) & \longrightarrow &
    P_{d-i}^{+} & &
  \end{array}}
\]

\[
\left\lbrace
\begin{aligned}
E(U; F, \mu_{\infty})^{n} &\simeq E\bigl(k, \underline{H}_{U}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-1} \\
E(X; F, \mu_{\infty})^{n} &\simeq E\bigl(k, \underline{H}_{X}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-2} \\
E_{x}(X; F, \mu_{\infty})^{n} &\simeq E\bigl(k, \underline{H}_{x}(F), \mathbb{Q}/\mathbb{Z}\bigr)^{n-2}
\end{aligned}
\right.
\]

\begin{align*}
  \det\begin{pmatrix}
    X_1 & \cdots & X_n \\
    X_1^{p} & \cdots & X_n^{p} \\
    \vdots & & \vdots \\
    X_1^{p^{n-1}} & \cdots & X_n^{p^{n-1}}
  \end{pmatrix}
  &= \prod_{(\alpha_1, \dots, \alpha_{n-1}) \in \mathbb{F}_p^{n-1}}\Bigl(X_n + \sum_1^{n-1}\alpha_i X_i\Bigr) \\
  &\qquad \prod_{(\alpha_1, \dots, \alpha_{n-2}) \in \mathbb{F}_p^{n-2}}\Bigl(X_{n-1} + \sum_1^{n-2}\alpha_i X_i\Bigr)\cdots
\end{align*}

\[
\begin{cases}
  \|x(u,v;\alpha) - u\|^2 \underset{\text{par raison de sym.}}{=} \|x(u,v';\alpha) - u\|^2 \\ \qquad = (\cos\alpha - 1)^2 + \sin^2\alpha = 2(1 - \cos\alpha) = 4 \sin^2 \frac{\alpha}{2} \\[1ex]
  \|x(u,v,\alpha) - x(u,v',\alpha)\|^2 = \sin^2\alpha \, \|v - v'\|^2 \\ \qquad = \sin^2\alpha \; 4 \sin^2 \frac{\theta}{2} = \Bigl(4 \sin^2 \frac{\alpha}{2} \cos^2 \frac{\alpha}{2}\Bigr)\Bigl(4 \sin^2 \frac{\theta}{2}\Bigr)
\end{cases}
\]

\[
\begin{aligned}
(8.5.11.1) \qquad \dim_{k} H^{n-1}(X, \underline{O})_{\mathrm{ss}} &=
\dim_{\mathbf{F}_{p}} H^{n-1}(\overline{X}, \mathbf{Z}/p\mathbf{Z}) \\
&\geq
\dim\bigl((H^{n-1}(X)/\mathrm{Tors}^{n-1}(X)) \otimes k\bigr)_{\mathrm{ss}}
\end{aligned}
\]

\[
\left\lbrace
\begin{array}{l}
  H^i(U, F) \simeq H^i(\hat{X}, \hat{F}) \quad \text{est} \quad
  \left\lbrace
  \begin{array}{l}
    \text{isom si } i < n \\
    \text{mono si } i = n
  \end{array}
  \right. \\[1ex]
  H^i(\hat{X}, \hat{F}) \simeq \varprojlim H^i(X_m, F_m) \text{ pour } i \leq n
\end{array}
\right.
\]

\[
\begin{aligned}
\underset{X(a, b)}{\underset{\|}{\mathrm{Hom}_{A^{\wedge}}\bigl(a, \Gamma''_b(X)\bigr)}}
&\simeq \mathrm{Hom}_{A^{\wedge}}\bigl(a, p_{B*}\, \underline{\mathrm{Hom}}(p_A^{*}(b), X)\bigr)
&&\text{fonctoriel en } a, X, b\\
&\simeq \mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a), \underline{\mathrm{Hom}}(p_A^{*}(b), X)\bigr)\\
&\simeq \mathrm{Hom}_{(A \times B)^{\wedge}}\bigl(p_B^{*}(a) \times p_A^{*}(b), X\bigr)
&&\text{OK}
\end{aligned}
\]

\[
\left\lbrace
\begin{array}{l}
\mathrm{Ext}_{\mathcal{O}_S}(\mathcal{O}_X, \mathcal{J}) \simeq \mathrm{Ext}^1_{\mathcal{O}_X}(L^{X/S}_{\bullet}, \mathcal{J}) \\[2pt]
\text{autom.\ d'une extension} \simeq \mathrm{Ext}^0_{\mathcal{O}_X}(L^{X/S}_{\bullet}, \mathcal{J}) = \mathrm{Hom}(\Omega^1_{X/S}, \mathcal{J})
\end{array}
\right.
\]

\[
\begin{aligned}
t^3(\lambda_0) &= \lambda_0 \\
t^3(\lambda_1) &= (\lambda_1 \lambda_2 \lambda_1^{-1})\, \lambda_1\, (\lambda_1 \lambda_2^{-1} \lambda_1^{-1})
= \lambda_1 \lambda_2 \lambda_1 \lambda_2^{-1} \lambda_1^{-1} \\
&= \operatorname{int}(\lambda_1 \lambda_2)\, \lambda_1 = \operatorname{int}(\lambda_0^{-1})\, \lambda_1 \\
t^3(\lambda_2) &= t(\lambda_1) = \lambda_1 \lambda_2 \lambda_1^{-1} = \struck{\operatorname{int}(\lambda_1)} \\
&= (\lambda_1 \lambda_2)\, \lambda_2\, (\lambda_2^{-1} \lambda_1^{-1})
= \operatorname{int}(\lambda_1 \lambda_2)\, \lambda_2\ \operatorname{int}\ill{}
\end{aligned}
\]

\[
\begin{array}{ccc}
\mathcal{C} \hookrightarrow \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{C}^{\circ}, \mathit{Ab}_k^{\bullet}) & \xrightarrow{\ \sim\ } & \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{C}, \mathit{Ab}_k^{\circ})^{\circ} \\
\downarrow{\scriptstyle \wr} & & \downarrow{\scriptstyle \wr} \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{A}^{\circ}, \mathit{Ab}_k^{\bullet}) & \xrightarrow{\ \sim\ } & \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{A}, \mathit{Ab}_k^{\circ})^{\circ} \\
\downarrow & & \downarrow{\scriptstyle \wr} \\
\lbrace \mathit{Ab}_k^{*} \rbrace^{\circ} & & (\mathit{Ab}_k^{\circ *})^{\circ}
\end{array}
\]

\[
  \left|
  \begin{array}{l}
    \widetilde{J} = \widetilde{J}' \neq \emptyset \Rightarrow
    \varepsilon_{R,J}\, \varepsilon_{R',J'} = 0 \\[1ex]
    \widetilde{J} \neq \widetilde{J}' \text{ ou } \widetilde{J} = \widetilde{J}' = \emptyset
    \quad \text{alors} \quad
    \varepsilon_{R,J}\, \varepsilon_{R',J'} =
    \Bigl(\prod_{i \in \widetilde{J} \cup \widetilde{J}'} y_{i}\Bigr)
    \prod_{\substack{A'' \in R'' \\ A'' \neq \widetilde{J} \\ A'' \neq \widetilde{J}'}}
    \varepsilon_{R_{A''}}\, \varepsilon_{R'_{A''}}
  \end{array}
  \right.
\]

\[
\begin{array}{ccc}
H^{i}_{Z}(F) & \times & H^{-i}_{a}(\hat{F}|Z) \\
\downarrow & & \uparrow \\
H^{i}_{Y}(F) & \times & H^{-i}_{a}(\hat{F}|Y) \\
\downarrow & & \uparrow \\
H^{i}_{Y-Z}(F) & \times & H^{-i-1}\bigl(Y - a, \mathbb{R}k_{!}(\hat{F}|Y - Z)\bigr) \\
\downarrow & & \uparrow \\
H^{i+1}_{Z}(F) & \times & H^{-i-1}_{a}(\hat{F}|Z) \\
\downarrow & & \uparrow \\
H^{i+1}_{Y}(F) & \times & H^{-i-1}_{a}(\hat{F}|Y)
\end{array}
\]

\[
  \left\{
  \begin{array}{l}
    \xi^0, \ldots, \xi^{\nu-1}, [\xi^{\nu}, \lambda_{\nu}], \xi_{\nu+1}, \ldots, \xi_{2\nu} \\
    \xi^0, \ldots, \xi^{\nu}, \xi_{\nu+1}, \ldots, \xi_{2\nu+1}
  \end{array}
  \right.
  \qquad
  \left[
  \begin{array}{l}
    2\xi_i = \xi^i \\
    \xi^i \xi_j = \xi_{i+j} \\
    \xi_i \xi_j = 0 \ \text{pr raisons de degré} \\
    \lambda_{\nu}^2 = 2\xi_{2\nu} = \xi^{2\nu} \\
    \lambda_{\nu} \xi = 0 \\
    \lambda_{\nu} \xi_{\alpha} = 0 \quad \alpha \geqslant \nu+1 \ \text{raisons de degré}
  \end{array}
  \right.
\]

\[
(74)\qquad \dot\Sigma_J=X_J\cap\big(\text{bord de }\underbrace{T_{\overline J\smallsetminus J}}_{\bigcup_{i\in\overline J\smallsetminus J}T_i}\big)
=\bigcup_{\substack{i<j\\ i,j\in J}}\mathrm{Im}(\Sigma_{ij}\to\Sigma_j)
=\varinjlim_{\substack{d\in\mathrm{Drap}\\ d \text{ ayant au moins 2 termes}}}\Sigma_d
\]

\[
\left\{
\begin{aligned}
  & \mathbb{H}^{0}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}) = \mathbb{Z} \ (\text{mod } p\text{-groupes ?}), \quad \mathbb{H}^{1}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}) = \mathbb{Z} \ (\text{mod } p\text{-groupes ?}) \\
  & \mathbb{H}^{0}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}(n)) = 0 \ (\text{id}), \quad \mathbb{H}^{1}(\mathbb{F}_{q}, \underset{\sim}{\mathbb{Z}}(n)) = \mathbb{Z}/(1-q^{n})\mathbb{Z} \ (\text{mod } p\text{-groupes ?})
\end{aligned}
\right.
\]

\begin{gather}
\underline{S} = \operatorname{gr}_{\mathcal{J}} \underline{O}_{X'} =
\coprod_{k \in \underline{N}} \mathcal{J}^k / \mathcal{J}^{k+1} \tag{2.5} \\
\operatorname{gr}_{\mathcal{J}}(\underline{F}) = \coprod_{k \in
\underline{N}} \mathcal{J}^k \underline{F} / \mathcal{J}^{k+1}
\underline{F} \tag{2.6} \\
\underline{K}^i = R^i f_{*}(\operatorname{gr}_{\mathcal{J}}(\underline{F}))
= \coprod_{k \in \underline{N}} R^i f_{*}(\mathcal{J}^k \underline{F} /
\mathcal{J}^{k+1} \underline{F}), \tag{2.7}
\end{gather}

\[
\begin{array}{c}
e \in E \\
\cup \\
M(e) = M(E_e) \subset E_0 = M(E) \\
\big\downarrow{\scriptstyle \varphi} \\
I \ \text{ordonné}
\end{array}
\qquad
\begin{cases}
|E| = X \supset |E_e| \\
X_i = \bigcup_{\varphi(M(e)) \leq i} |E_e|
\end{cases}
\qquad
E_e = \{x \in E \mid x \leq e\}
\]

\[
\boxed{
\begin{aligned}
&\lambda_0 \bigl[ -x_{i-1} x_i y_{-1} (y_1 + \cdots + y_{i-1})
 + y_{i-1} y_i x_{-1} (x_1 + \cdots + x_{i-1}) \bigr] \\
&\quad + \lambda_1 \bigl[ x_{i-1} x_i y_1 (y_0 + \cdots + y_{i-1})
 - y_{i-1} y_i x_1 (x_0 + \cdots + x_{i-1}) \bigr] = 0
\end{aligned}}
\qquad i = 2, \ldots, n-1
\]

\[
  \begin{array}{ccccccccc}
    1 & \to & ST(X) & \to & \mathcal{A}(X)/S\mathcal{A}^\circ(X) & \to &
    \mathcal{A}(\partial X) & \to & 1 \\
    & & \uparrow & & \uparrow & & \uparrow\!\wr & & \\
    1 & \to & \widetilde{ST}(X) & \to & \widetilde{\mathcal{A}}(X)/
    S\mathcal{A}^\circ(X) & \to & \mathcal{A}(\widetilde{\partial X}) &
    \to & 1 \\
    & & \uparrow & & \uparrow & & \uparrow & & \\
    1 & \to & \mathbf{Z}^I & \Rightarrow & \mathbf{Z}^I & \to & 1 & \to &
    1
  \end{array}
\]

\[
  \begin{array}{rcl}
    C \otimes T \subset C & \Longleftrightarrow & TF \text{ stable by } \underline{\mathrm{Hom}}(T, ?) \\
    TC \otimes T \subset TC & \Longleftrightarrow & F \text{ stable by } \underline{\mathrm{Hom}}(T, ?) \\
    C \boxtimes C \subset C & \Longleftrightarrow & \bigl( i \in C,\ f \in TF \Rightarrow \alpha(i,f) \in TF \bigr) \\
    \ \boxtimes TC \subset TC & \Longleftrightarrow & \bigl( i \in TC,\ f \in F \Rightarrow \alpha(i,f) \in TF \bigr) \\
    TC \boxtimes C \subset TC & \Longleftrightarrow & \bigl( i \in C,\ f \in F \Rightarrow \alpha(i,f) \in F \bigr)
  \end{array}
\]

\[
  \begin{cases}
    \Sigma_{\alpha+1}(f) = \Sigma_1(\Sigma_\alpha(f)), \quad
    X_{\alpha+1}(f) = \text{source } \Sigma_{\alpha+1}(f), \\
    i_{\alpha+1}(f) = \text{composition }
    X \xrightarrow{i_\alpha(f)} X_\alpha(f)
    \xrightarrow{i_1(\Sigma_\alpha(f))} X_{\alpha+1}(f) \\[1ex]
    \Sigma_\alpha(f) = \varinjlim_{\alpha' < \alpha} \Sigma_{\alpha'}(f)
    \quad \text{if } \alpha \text{ is a limiting ordinal.}
  \end{cases}
\]

\[
  \begin{aligned}
  H^{i}_{!}(W - T, F) &\simeq H^{i}_{!}(\mathring{P}, F) \times H^{i}_{!}(\mathring{Q}, F) \\
  &\simeq H^{n-i}(\mathring{P}, \check{F} \otimes \mathcal{T}_W)^\vee \times
     H^{n-i}(\mathring{Q}, \check{F} \otimes \mathcal{T}_W)^\vee \\
  &\simeq H^{n-i}(P, \check{F} \otimes \mathcal{T}_W)^\vee \times
     H^{n-i}(Q, \check{F} \otimes \mathcal{T}_W)^\vee
  \end{aligned}
\]

\[
  \left\{\begin{aligned}
    &H^{i}(V_\eta) \to H^{i-1}(V_s) \ \text{un \emph{isom} si } i \neq 0, 2n \\
    &\emph{et}\ \left\{\begin{aligned}
      &H^{0}(V_\eta) \simeq \Lambda \\
      &H^{2n}(V_\eta) \to H^{2n-1}(V_s) \ \text{inj et conoyau} \simeq \Lambda \\
    \end{aligned}\right. \qquad (\text{quand } n \geq 1)
  \end{aligned}\right.
\]

\[
  \text{(4)}\qquad \left\{
  \begin{array}{l}
    \Phi_A(t) = \displaystyle\int_0^t \varphi_A(s)\, ds \\[1ex]
    \Psi_A(t) = \displaystyle\int_0^t \log\varphi_A(s)\, ds \\[1ex]
    \Delta_A = \Delta_f = \exp\Psi_f \qquad \Delta_A(t) = \exp\displaystyle\int_0^t \log\varphi_{|A|}(s)\, ds
  \end{array}\right.
\]

\[
  \begin{array}{ccccccc}
    Q^0 & Q^1 & Q^2 & Q^3 & Q^4 & Q^5 & Q^6 \\[4pt]
    [\lambda_0, \xi^0] & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 \\
     & \tfrac12 \xi & [\lambda_1, \xi] & \xi & \xi & \xi & \xi \\
     & & \tfrac12 \xi^2 & \tfrac12 \xi^2 & [\lambda_2, \xi^2] & \xi^2 & \xi^2 \\
     & & & \tfrac12 \xi^3 & \tfrac12 \xi^3 & \tfrac12 \xi^3 & [\lambda_3, \xi^3] \\
     & & & & \tfrac12 \xi^4 & \tfrac12 \xi^4 & \tfrac12 \xi^4 \\
     & & & & & \tfrac12 \xi^5 & \tfrac12 \xi^5 \\
     & & & & & & \tfrac12 \xi^6
  \end{array}
\]

\[
  \begin{aligned}
  \lambda (b_0 \otimes \cdots \otimes b_n) &= (\lambda b_0) \otimes b_1 \cdots \otimes b_n \\
  &= (\lambda_0 \otimes 1) \otimes b_2 \cdots \otimes b_n
  && \text{pour } \lambda_0 \in I \text{ conv.} \\
  &= 1_B \otimes \bigl[\lambda_1 (b_2 \otimes \cdots \otimes b_n)\bigr] = \\
  &= 1_B \otimes \mu (1_B \otimes \cdots \otimes 1_B)
  && \mu \in I \text{ conv.} \\
  &= \mu (1_B \otimes \cdots \otimes 1_B)
  \end{aligned}
\]

\begin{align*}
  \Gamma^{n}(x+y) &= \Gamma^{n}x + \Gamma^{n}y + \sum_{\substack{p, q \geqslant 1 \\ p+q = n}} \Gamma^{p}(x)\,\Gamma^{q}(y) \qquad (x, y \in I)
    \quad [\Gamma^{1}x = x \text{ par déf.}] \\
  \Gamma^{n}(xy) &= x^{n}\,\Gamma^{n}(y) \qquad x \in I,\ y \in A \\
  \struck{\Gamma^{m}(\Gamma^{n}(x))} &\struck{= \tfrac{(mn)!}{(n!)^{m}\, m!}\, \Gamma^{mn}(x)} \qquad x \in I \\
  \Gamma^{m}(x)\,\Gamma^{n}(x) &= \frac{(m+n)!}{m!\, n!}\; \Gamma^{m+n}(x)
\end{align*}

\[
\begin{aligned}
  H^{0}(\overline{U}, \mathbb{G}_{m}) &= \mathbb{Z}/2 + \mathbb{Z}^{S} \\
  H^{1}(\overline{U}, \mathbb{G}_{m}) &= 0 \\
  H^{2}(\overline{U}, \mathbb{G}_{m}) &= \operatorname{Ker}\bigl((\mathbb{Q}/\mathbb{Z})^{S} \to \mathbb{Q}/\mathbb{Z}\bigr) \\
  H^{i}(\overline{U}, \mathbb{G}_{m}) &= 0 \quad \text{si } i \geq 3
\end{aligned}
\]

\[
  \left\{\begin{aligned}
    H^{i}(W_0) &\xrightarrow{\ \sim\ } H^{2n-i-1}(M)^\vee && \text{si } i \geq n+2
      && (\text{épim si } i = n+1) \quad (\text{donc } 2n-i-1 \leq n-3) \\
    H^{i}(M) &\xrightarrow{\ \sim\ } H^{i}(W_0) && \text{si } i \leq n-2
      && (\text{mono si } i = n-1)
  \end{aligned}\right.
\]

\begin{align*}
\tilde\rho_{n-1}\tilde\rho_{n-2}\cdots\tilde\rho_0
&= \underbrace{\tilde\rho^{\,n-1}\tilde\rho_0\tilde\rho^{-(n-1)}}\,
\underbrace{\tilde\rho^{\,n-2}\tilde\rho_0\tilde\rho^{-(n-2)}} \cdots
(\tilde\rho\tilde\rho_0\tilde\rho^{-1})\,\tilde\rho_0 \\
&= \tilde\rho^{\,n-1}\tilde\rho_0\tilde\rho^{-1}\tilde\rho_0\tilde\rho^{-1} \cdots
\tilde\rho^{-1}\tilde\rho_0\tilde\rho^{-1}\tilde\rho_0
\qquad (n \text{ facteurs } \tilde\rho_0) \\
&= \tilde\rho^{\,n}\,\underbrace{(\tilde\rho^{-1}\tilde\rho_0)\cdots(\tilde\rho^{-1}\tilde\rho_0)}_{n \text{ facteurs}}
= \underbrace{\tilde\rho^{\,n}}_{\tilde\omega_0}(\tilde\rho^{-1}\tilde\rho_0)^n
\end{align*}

\[
  \mathrm{II}\quad \left\{
  \begin{array}{l}
    \mathcal{G}' = \text{groupe des autom.\ du schéma absolu } \mathfrak{X}
      \text{ qui invarie } D \text{ et } J \\
    G' = \text{s-gp de } \mathcal{G}' \text{ formé des }
      \overline{\mathbb{Q}}\text{-autom.} \\
    T(\overline{\mathbb{Q}}) = \varprojlim_n \mu_n(\overline{\mathbb{Q}}) \\
    \Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})
  \end{array}\right.
\]

\[
  \left\{
  \begin{array}{l}
    \forall t \in \mathcal{T} = \mathrm{Ob}\,\Lambda, \text{ la famille des }
      \pi_{t,t'} \subset \pi \ (t' \in \mathcal{T} \text{ variable}) \text{ forme une}\\
    \text{pseudo-partition de } \pi, \text{ i.e. } \forall \lambda \in \pi,\
      \exists \text{ un unique } t' \text{ tel que } \lambda \in \pi_{t,t'},\\
    \text{i.e. une unique flèche de } \Lambda \text{ au-dessus de } \lambda
      \text{ qui soit d'origine } t.
  \end{array}
  \right.
\]

\[
  \begin{array}{ccc}
    \boxed{P_{i}^{+\circ} \cdot P_{d-i}^{+\circ} \geqslant 0 \ ?} & &
    \\[1ex]
    \Big\Updownarrow & \Longleftarrow & P_{i}^{+\circ}\, P_{j}^{+\circ}
    \subset P_{i+j}^{+\circ} \\[1ex]
    \boxed{\alpha_{i}(P_{i}^{+\circ}) \subset P_{d-i}^{+}} & & \\[1ex]
    \boxed{\alpha_{d-i}(P_{d-i}^{+\circ}) \subset P_{i}^{+}} & &
  \end{array}
\]

\[
  (28) \qquad
  \begin{cases}
    \dot{X}_{\Delta_r} = \text{image inverse de } \dot{X}_{\Delta_0}
      \text{ par } \sigma_r : X_{\Delta_r} \to X_{\Delta_0} \\
    \qquad (\text{fermé de } X_{\Delta_r}) \\[2pt]
    X^{*}_{\Delta_r} = X_{\Delta_r} \smallsetminus \dot{X}_{\Delta_r}
      = \text{image inverse de } X^{*}_{\Delta_0} \text{ par } \sigma_r \\
    \qquad (\text{ouvert de } X_{\Delta_r}) .
  \end{cases}
\]

\[
  \left.
  \begin{array}{ll}
    H^{*}(X_0) : & 1, \xi_0, \ldots, \xi_0^{\nu-1}, \xi_0^{\nu},
      \tfrac{1}{2}\xi_0^{\nu+1}, \ldots, \tfrac{1}{2}\xi_0^{2\nu} \\[4pt]
    H^{*}(\overline{X}_1) : & 1, \xi_1, \ldots, \xi_1^{\nu-1}, [\xi_1^{\nu}, \lambda_{\nu}],
      \tfrac{1}{2}\xi_1^{\nu+1}, \ldots, \tfrac{1}{2}\xi_1^{2\nu}
  \end{array}
  \right]
  \quad \xi_0 \mapsto \xi_1
\]

\[
\begin{aligned}
x_1 x_2 y_1 \underbrace{(y_0 + y_1)}_{-y_2 - y_3}
 - y_1 y_2 x_1 \underbrace{(x_0 + x_1)}_{-x_2 - x_3} &= \\
= -x_1 x_2 y_1 (y_2 + y_3) + y_1 y_2 x_1 (x_2 + x_3) & \\
= -x_1 x_2 y_1 y_3 + y_1 y_2 x_1 x_3 &= -x_1 y_1 (x_2 y_3 - y_2 x_3)
\end{aligned}
\qquad\qquad \boxed{\lambda_0 + \lambda_1 = 0}
\]

\[
\begin{aligned}
F(\Psi^i_{\bullet}) \to \operatorname{Hom}^{\bullet}_k(\Psi^{\bullet}_i, F(\mathbb{C}^{\bullet})) &\simeq Z^i(F(\mathbb{C}^{\bullet})) \\
&\simeq \operatorname{Ker}(F(\mathbb{C}^i_{\bullet}) \to F(\mathbb{C}^{i+1}_{\bullet})) \\
&\simeq F(\underbrace{\operatorname{Ker}(\mathbb{C}^i \to \mathbb{C}^{i+1})}_{\Sigma\text{-Ker}\,!}) \simeq F(\Psi^i_{\bullet})
\end{aligned}
\]

\begin{align}
  u \circ f_a(x) &= c_0\bigl(x^{p^n} + a_{n-1}x^{p^{n-1}} + a_{n-2}x^{p^{n-2}} + \dots + a_1 x^p + a_0 x\bigr) \notag \\
   &\quad {} + c_1\bigl(x^{p^{n+1}} + a_{n-1}^{p} x^{p^{n}} + \dots + a_0^{p} x^{p}\bigr) \notag \\
   &\quad {} + c_2\bigl(x^{p^{n+2}} + a_{n-1}^{p^2} x^{p^{n+1}} + \dots + a_0^{p^2} x^{p^2}\bigr) \notag \\
   &\quad \dots \notag \\
   &\quad {} + c_N\bigl(x^{p^{n+N}} + a_{n-1}^{p^N} x^{p^{n+N-1}} + \dots + a_0^{p^N} x^{p^N}\bigr) \notag
\end{align}

\[
\begin{array}{ll}
i = 0 & H^0(X, D(G)) = D(G) \\
i = 1 & H^1(X, D(G)) = \operatorname{Hom}(G, \underline{\operatorname{Pic}}^{(1)}) \\
i = 2 & H^2(X, D(G)) \neq \mathcal{E}xt^1(G, \underline{\operatorname{Pic}}^{(1)}_{X/Y})
+ \operatorname{Hom}(G, \underline{\operatorname{Pic}}^{(2)}_{X})
\end{array}
\]

\[
  \begin{aligned}
    \xi_E\Bigl(\frac{1}{qt}\Bigr)
      &= \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)/2} L_E\Bigl(\frac{1}{qt}\Bigr)
       = \Bigl(\frac{1}{qt}\Bigr)^{\chi(E)/2} A(t)\, \delta(E)\, (-t)^{\chi(E)} L_E(t) \\
      &= q^{-\chi(E)/2} A(t)\, \delta(E)\, (-1)^{\chi(E)}\; t^{\chi(E)/2} L_E(t)
  \end{aligned}
\]

\[
\mathrm{Polg}_n(E)(S') = \Bigl\lbrace s_0, s_1, \ldots, s_{n-1}, d_0, d_1,
\ldots, d_{n-1} \Bigm|
\begin{array}{l}
s_i \in E(S') \simeq \Gamma(E_{S'}/S') \\
d_i \in \mathrm{Dr}(E_{S'}) \quad (0 \leqslant i \leqslant n-1)
\end{array} \Bigr.
\]

\[
\left\lbrace
\begin{array}{l}
\rho_f(\vec{a}, \omega_s) = (\vec{b}, \omega_t) \\
\vec{b} = \rho_{\omega_t}^{-1}(-\vec{a})
\end{array}
\right.
\qquad
\begin{array}{l}
\text{où } \omega_t = \varphi_{\vec{a}}(\omega_s) \in \underline{\omega}(A_t) \\
\text{et où } b \in \vec{A}_t \ (t = \operatorname{ex}(\vec{a})) \\
\text{est donné par}
\end{array}
\]

\[
  \begin{array}{ccl}
    X & = & \bigl( X', X'',\ X' \xrightarrow{u'} \varphi'(X'') ,\
      X'' \xrightarrow{u''} \varphi''(X') \bigr) \\[4pt]
    \downarrow \beta' & & \\[4pt]
    g' f'(X) & = & \bigl( X',\ \varphi''(X'),\
      X' \xrightarrow{\lambda'(X')} \varphi'\varphi''(X') ,\
      \varphi''(X') \xrightarrow{\mathrm{id}} \varphi''(X') \bigr)
  \end{array}
  \tag{3.20}
\]

\[
  (16) \qquad
  \begin{cases}
    [[e'_1 \wedge e'_2], e'_3] = (ap' - qr')e'_1 + (p'r - bq)e'_2 + (p'q' - pq)e'_3 \\
    [[e'_2 \wedge e'_3], e'_1] = (q'r' - qr)e'_1 + (bq' - rp')e'_2 + (q'p - cr)e'_3 \\
    [[e'_3, e'_1], e'_2] = (r'q - ap)e'_1 + (r'p' - rp)e'_2 + (cr' - pq')e'_3
  \end{cases}
\]

\[
\begin{aligned}
X(L, M) &= \underline{\mathrm{Hom}}_{A \times B}(L \boxtimes M, X)
 = \Gamma_{A \times B}\, \underline{\mathrm{Hom}}(L \boxtimes M, X)\\
{}^{g}X(L) &= p_{A*}\, \underline{\mathrm{Hom}}_{A \times B}({}^{g}L, X)
 = \underline{\mathrm{Hom}}_B({}^{g}L, X)\\
{}^{d}X(M) &= \underbrace{p_{B*}\, \underline{\mathrm{Hom}}_{A \times B}({}^{d}M, X)}_{X^{M}}
 = \underline{\mathrm{Hom}}_A({}^{d}M, X)
\end{aligned}
\]

\[
  \begin{bmatrix}
    L_1 = \mathfrak{Z}(G) \\
    D_1 = D(G) \\
    L_0 = G/\mathfrak{Z}(G) \\
    D_0 = G/D(G)
  \end{bmatrix}
  \qquad
  \begin{cases}
    D_1^{D_0} = \Pi_1 \\
    D_{1\,D_0} = \{e\}
  \end{cases}
  \qquad
  \begin{array}{l}
    \Pi_1 = L_1 \cap D_1 \\
    \Pi_0 = G/L_1 D_1
  \end{array}
\]

\begin{align*}
  F_0 &= \text{ens.\ des él.\ minimaux de } F_* \\
  F_1 &= \text{\quad''\quad''\quad''\quad} F_* - F_0 \\
  &\ \ \vdots \\
  F_i &= \text{\quad''\quad''\quad''\quad} F_* - F_0 - \dots - F_{i-1}
  \qquad (i \leqslant n) \\
  I_\alpha &\subset F_\alpha \times F_{\alpha+1}, \quad I_\alpha = \lbrace
  (x,y) \mid x \in F_\alpha,\ y \in F_{\alpha+1},\ x \preceq y \rbrace
\end{align*}

\[
\begin{aligned}
&\text{i.e. } V(f) \supset V(g) \\
&\text{i.e. } \widetilde{fA} \subset \widetilde{gA} \\
&\text{i.e. } f \in \widetilde{gA} \\
&\text{i.e. } \begin{cases} \exists n \in \mathbb{N}, \\ h \in A \end{cases} f^n = gh \qquad \Big| \text{ écrivons } f \prec g
\end{aligned}
\]

\[
  \begin{aligned}
    \delta\bigl(\mu_x(E)\bigr) &= \delta\bigl(\alpha_x(E_\eta)\bigr)^{-1}
      \Bigl[\delta\bigl(\alpha_x(E_\eta)\bigr)\, p^{-\chi_x(E_\eta)(\rho+1)}\Bigr]^{-1} \\
    &= \delta_x(E_\eta)^{-2}\, q^{+\chi_x(E_\eta)}
  \end{aligned}
\]

\[
\begin{aligned}
\sigma_0\sigma_1(l_0) &= \sigma_0(l_\infty) = l_1 = \rho(l_0) \\
\sigma_0\sigma_1(l_1) &= \sigma_0(l_\infty^{-1} l_0^{-1}) = \sigma_0(l_\infty)^{-1}\sigma_0(l_0)^{-1}
= l_1^{-1}\,(l_1^{-1} l_\infty^{-1})^{-1} = l_1^{-1}\, l_\infty\, l_1 \\
\sigma_0\sigma_1(l_\infty) &= (l_1^{-1} l_\infty l_1)^{-1}\, l_1^{-1}
= l_1^{-1} l_\infty^{-1} \struck{l_1} = l_1^{-1}\,(l_\infty^{-1} l_1^{-1})\, l_1 = l_1^{-1}\, l_0\, l_1
\end{aligned}
\]

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

\[
  \left\{
  \begin{array}{l}
    H^i(E) = 0 \quad \text{si}\ i \neq (0), 2\nu - 1, 2\nu + 1, 4\nu + 1 \\
    H^0(E) = \mu(0) \\
    H^{2\nu-1}(E) \simeq \mu(-\nu+1) \\
    H^{2\nu+1}(E) \simeq \mu(-\nu-1) \\
    H^{\uncertain{4\nu+1}}(E) \simeq \mu(-2\nu-1)
  \end{array}
  \right.
  \qquad \boxed{n + 1 = 2\nu + 1} \ \ n = 2\nu
\]

\[
\left(\lambda,\ \begin{pmatrix} 1 & \alpha & \beta \\ 0 & \lambda^2 & \gamma \\ 0 & 0 & \lambda^3 \end{pmatrix}\right)
\qquad
\begin{array}{l}
\lambda\in\Gamma(\mathbf{G}_{m,S}/S)=\Gamma(S,\underline{O}_S^*)\\
\alpha,\beta,\gamma\in\Gamma(S,\underline{O}_S)
\end{array}
\]

\[
\left\{
\begin{array}{l}
V_1, V_2, V_3 \\
\Gamma^{\circ}_1, \Gamma^{\circ}_2, \Gamma^{\circ}_3 \\
\partial
\end{array}
\right.
\quad
\left\{
\begin{array}{l}
U_1 = V_2 \cup V_3 \cup \Gamma^{\circ}_1 \\
U_2 = V_3 \cup V_1 \cup \Gamma^{\circ}_2 \\
U_3 = V_1 \cup V_2 \cup \Gamma^{\circ}_3
\end{array}
\right.
\qquad
\begin{array}{l}
V_3 = U_1 \cap U_2 \\
V_2 = U_3 \cap U_1 \\
V_1 = U_1 \cap U_2
\end{array}
\quad
\left\{
\begin{array}{l}
\Gamma_1 = \Gamma^{\circ}_2 \cup \Gamma^{\circ}_3 \cup \partial \\
\Gamma_2 = \Gamma^{\circ}_3 \cup \Gamma^{\circ}_1 \cup \partial \\
\Gamma_3 = \Gamma^{\circ}_1 \cup \Gamma^{\circ}_2 \cup \partial
\end{array}
\right.
\]

\[
\begin{aligned}
  H^{1}(\overline{U}, \mu_{\ell^{n}}) &\simeq (\mathbb{Z}/\ell^{n}\mathbb{Z})^{S} \\
  H^{2}(\overline{U}, \mu_{\ell^{n}}) &\simeq \operatorname{Ker}\bigl((\mathbb{Z}/\ell^{n}\mathbb{Z})^{S} \to (\mathbb{Z}/\ell^{n}\mathbb{Z})\bigr) \\
  H^{i}(\overline{U}, \mu_{\ell^{n}}) &= 0 \quad \text{si } i \geq 3
\end{aligned}
\]

\begin{align*}
  \underline{\underline{\mathrm{Prépic}}}_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Prépic}}_{X'/S'}\bigr)
     = \mathrm{Prépic}(X'/S')
     \qquad = H^1(X', \mathcal{O}_{X'}^{*}) \\
  \underline{\underline{\mathrm{Prépic}}}'_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Prépic}}'_{X'/S'}\bigr)
     = \mathrm{Prépic}'(X'/S')
     \qquad = \Gamma\bigl(S', \underline{H}^{1}_{X'/S'}(\underline{\mathcal{O}}^{*}_{X'})\bigr) \\
  \underline{\underline{\mathrm{Pic}}}_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Pic}}_{X'/S'}\bigr)
     = \mathrm{Pic}(X'/S')
     \qquad = \widetilde{\underline{\underline{\mathrm{Prépic}}}}_{X/S}(S')
\end{align*}

\[
0 \to
\begin{array}{c}
\text{s.-groupe de } \struck{\ill{}}\ E(K) \\
\text{des éléments qui sont} \\
\equiv 1\ (\underline{m})
\end{array}
\longrightarrow K^{*} \longrightarrow \underbrace{\struck{\ill{}}\ \mathcal{D}_{\underline{m}}(K) \times \prod_{\mathfrak{p}\mid\underline{m}} \struck{\ill{}}}\ \underbrace{K_{\mathfrak{p}}/V_{\mathfrak{p},\underline{m}}}_{(*)} \longrightarrow C_{\underline{m}}(K) \to 0
\]

\[
  \left\lbrace
  \begin{array}{l}
    P_{f \otimes f'}(t) = P_f(t) * P_{f'}(t) \\[4pt]
    L_{f \otimes f'}(t) = \dfrac{1}{L_f(t) * L_{f'}(t)}
  \end{array}
  \right.
  \qquad
  L_{f_1 \otimes \cdots \otimes f_{\nu}}(t)
    = \Bigl(\prod L_{f_i}(t)\Bigr)^{(-1)^{\nu+1}}
\]

\[
\begin{aligned}
\lambda_3 \lambda_5 u_5 &= \lambda_3 \bigl[ -\lambda_1 u_0
 + (\lambda_2 + \lambda_3)\, u_2 \bigr]
 + \lambda_4 (-\lambda_1 u_0 + \lambda_2 u_1 + \lambda_2 u_2)
 - \lambda_3^2 u_2 \quad \cdots \\
 &= \bigl( -\lambda_1 (\lambda_3 + \lambda_4) \bigr) u_0
 + \lambda_2 \lambda_4 u_1 + \lambda_2 (\lambda_3 + \lambda_4)\, u_2
\end{aligned}
\]

\begin{align*}
\lambda_{i+1} u_{i+1} &= \lambda_i (u_i + u_{i-1}) - \lambda_{i-1} u_{i-2} \\
 &= \frac{1}{\lambda_3 \cdots \lambda_{i-2}}
    \bigl[ P_i u_0 + Q_i u_1 + R_i u_2 \bigr]
  + \frac{\lambda_i \lambda_{i-2}}{\lambda_3 \cdots \lambda_{i-1}}
    \bigl[ P_{i-1} u_0 + Q_{i-1} u_1 + R_{i-1} u_2 \bigr] \\
 &\quad - \frac{\lambda_{i-1} \lambda_{i-3}}{\lambda_3 \lambda_4 \cdots \lambda_{i-2}}
    \bigl[ P_{i-2} u_0 + Q_{i-2} u_1 + R_{i-2} u_2 \bigr]
\end{align*}

\[
  \begin{aligned}
    &\mathrm{Cas}_{E'}(H, H) = 1 + 1 + 0 = 2 \\
    &\varphi'(H, H) = 1 \\
    &\mathrm{Cas}_{E}(\tilde{H}, \tilde{H}) = \mathrm{Cas}_{E'}(2H, 2H) = 4\, \mathrm{Cas}_{E'}(H, H) = 8 \\
    &\qquad (\ldots = 4 + 4 + 0) \\
    &\varphi(\tilde{H}, \tilde{H}) = 2 \\
    &\qquad \Delta = -2
  \end{aligned}
\]

\[
  \begin{gathered}
  H^{i}_{Q}(W) \to H^{i}(W) \to H^{i}(W - Q) \simeq H^{i}(P) \\
  H^{i}_{Q}(W) \simeq H^{i}_{!}(W - P) = H^{i}_{!}(Q - \partial Q)
  \simeq \bigl(H^{n-i}(Q - \partial Q)\bigr)^\vee = H^{n-i}(Q)^\vee \\
  H^{i}_{Q}(W) \times H^{n-i}(Q)
  \end{gathered}
\]

\[
\left\lbrace
\begin{array}{ll}
\varepsilon_{F'}(a') = -\varepsilon_F(a) & \text{si } F, F' \text{ sont les deux faces incidentes à } a \\
\varepsilon_F(a) = \varepsilon_F(b) & \text{si } a \text{ et } b \text{ sont adjacentes sur } F, \text{ le sommet commun étant d'ordre } 2 \\
\varepsilon_{a_1}(F_1) = \varepsilon_{a_2}(F_2) & \text{si } F_1 \text{ et } F_2 \text{ sont adjacentes le long d'une arête}
\end{array}
\right.
\]

\[
\begin{array}{ll}
\struck{A\,B\,C\,D\,E\,F\,G} & \\
A\,B\,C\,D\,E\,F\,G \quad \struck{A\,B\,C\,E\,D\,G\,F} & 1' \\
A\,B\,C\,E\,D\,G\,F & 2' \\
A\,B\,C\,E\,G\,D\,F & 3' \\
A\,B\,C\,G\,E\,D\,F & 4' \\
A\,B\,G\,C\,E\,D\,F & 5' \\
A\,G\,B\,C\,E\,D\,F & 6' \\
(A\,G)\,C\,B\,E\,D\,F & 7' \\
(A\,G)\,C\,E\,B\,D\,F & 8' \\
(A\,G)\,E\,C\,D\,B\,F & 9' \\
G\,A\,E\,D\,C\,F\,B &
\end{array}
\]

\[
\left\{
\begin{aligned}
&\sigma_0^{2} = \sigma_1^{2} = \sigma_\infty^{2} = \tau_0^{2}
= [\tau_1^{2} = \tau_\infty^{2}] = 1\\
&\rho^{3} = 1\\
&\rho\tau_0\rho^{-1} = \tau_1, \quad \rho\tau_1\rho^{-1} = \tau_\infty,
\quad \rho\tau_\infty\rho^{-1} = \tau_0\\
&\tau_0\sigma_0 = \sigma_0\tau_0, \quad \tau_1\sigma_1 = \sigma_1\tau_1,
\quad \tau_\infty\sigma_\infty = \sigma_\infty\tau_\infty\\
&\sigma_0\sigma_1 = \tau_\infty\tau_1\rho, \quad \sigma_1\sigma_\infty =
\tau_0\tau_\infty\rho, \quad \sigma_\infty\sigma_0 = \tau_1\tau_0\rho
\end{aligned}
\right.
\]

\[
\begin{cases}
  (C_0, \delta_0) = (C, \delta), \\
  (C_{i+1}, \delta_{i+1}) = K(C_i, \delta_i), \quad
    \tau_i : C_i \to C_{i+1}, \quad \delta_{i+1} = \tau_i(\delta_i), \\
  C_i = \varinjlim_{j < i,\ (\Delta)} C_j \ \text{ si $i$ ordinal limite,} \quad
    \delta_i = \textstyle\bigcup_{j<i} \mathrm{Im.\ de\ } \delta_j \ \text{dans } C_i .
\end{cases}
\]

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

\[
  \begin{aligned}
  &\underbrace{H^{2n}(W)}_{=0} \to H^{2n}(W - W_0) \xrightarrow{\ \mathrm{inj}\ }
  \underbrace{H^{2n-1}(W_0)}_{\simeq \Lambda} \xrightarrow{\ \mathrm{surj}\ }
  \underbrace{H^{2n+1}(W)}_{=\Lambda} \\
  &\qquad \to \underbrace{H^{2n+1}(W - W_0)}_{=0}
  \to \underbrace{H^{2n}(W_0)}_{=0} \to 0
  \end{aligned}
\]

\[
\begin{array}{cccccccccccccc}
0 \to & ({}_F M)^{(p)} & \to & ({}_p M)^{(p)} & \xrightarrow{F} & {}_V M & \to & M_F & \xrightarrow{V} & (M_p)^{(p)} & \to & (M_V)^{(p)} & \to 0 \\
 & \wr\| & & \wr\| & & \wr\| & & \wr\| & & \wr\| & & \wr\| & \\
0 \to & \underline{n}_G & \to & D^{*}(G_p)^{\vee} & \to & t_{G^{*}} & \to & \omega_G & \to & D^{*}(G_p) & \to & \nu_{G^{*}} & \to 0 ,
\end{array}
\]

\[
  \left\lbrace
  \begin{array}{l}
    L^{*}_{E_\eta}(t) = L^{*}_{E'_\eta}(t)\, L^{*}_{E''_\eta}(t)\, L_Q(t)^{-1}
      = L^{*}_{E'_\eta}(t)\, L^{*}_{E''_\eta}(t) \prod_x P_{Q_x}(t) \\[8pt]
    Q_x = \operatorname{Coker}(E_x \to E''_x)
  \end{array}
  \right.
\]

\[
  \begin{array}{lll}
    R_n(X_n) \overset{\text{déf}}{=} E_n(X_n) & G_{n-1}\text{-ens} &
    \sigma_0 \cdots \sigma_{n-1} \\
    R_{n-1}(X_n) = E_{n-1}(X_{n-1}) & G_{n-2}\text{-ens} & \sigma_0 \cdots
    \sigma_{n-2} \\
    \cdots & & \\
    R_1(X_n) = E_1(X_1) & G_0\text{-ens} & \sigma_0 \\
    R_0(X_n) = E_0(X_0) \overset{\text{déf}}{=} X_0 & G_{-1}\text{-ens} &
    \text{i.e.\ ensemble}
  \end{array}
\]

\[
  \left.
  \begin{array}{l}
    \alpha_1^+ \ \cdots\ \alpha_n^+ \in \mathbf{N} \\
    \alpha_1^- \ \cdots\ \alpha_n^- \in \mathbf{N}
  \end{array}
  \right| \quad
  \begin{array}{l}
    \alpha_i^+ \ (\alpha_i^-) = \text{nb des cycles de long.\ } i \\
    \text{de } u_I \text{ au-dessus desquels il y a} \\
    \text{un cycle de long.\ } i \ (\text{resp.\ } 2i)
  \end{array}
\]

\[
\begin{aligned}
  \operatorname{Ext}^{\bullet}_{T, \mathcal{O}_X}(X, F, G(n))
  &\Longleftarrow H^p\bigl(U, \underline{\operatorname{Ext}}^q_{T,
  \mathcal{O}_U}(F, G(n))\bigr)\\
  \operatorname{Ext}_{T', \mathcal{O}_{X'}}(\overline{X'}; \overline{F'},
  \overline{G'})
  &\Longleftarrow H^p\bigl(U', \underline{\operatorname{Ext}}_{T',
  \mathcal{O}_{U'}}(\overline{F'}, \overline{G'})\bigr)
\end{aligned}
\]

\[
  \left\lbrace
  \begin{array}{l}
    \dfrac{\delta^{\natural}(E_\eta)^2}{q^{\chi^{\natural}(E)}}
      = \displaystyle\prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}} \\[14pt]
    \qquad \parallel \\[2pt]
    \varepsilon^{\natural}(E_\eta)^2
      = \displaystyle\prod_x \frac{\delta_x(E_\eta)^2}{q^{\chi_x(E_\eta)}}
  \end{array}
  \right.
\]

\[
\begin{array}{ccccccccc}
0 & \to & \underline{t}_G^{\vee} & \to & H(G) & \to & \underline{t}_{G^*} & \to & 0 \\
 & & \| & & {\scriptstyle H(V) = V_0}\downarrow\;\uparrow{\scriptstyle H(F) = F_0} & & & & \\
 & & \omega_G & & & & & & \\
0 & \to & \underline{t}_G^{\vee(p)} & \to & H(G)^{(p)} & \to & \underline{t}_{G^*}^{(p)} & \to & 0 \\
 & & \| & & & & & & \\
 & & \omega_G^{(p)} & & & & & & \\
 & & \dim g & & \dim g + g^* & & \dim g^* & &
\end{array}
\]

\[
  \begin{array}{ccccc}
    H^1(G, I \otimes \check{M}) & \to & H^1(G, C \otimes \check{M}) & \to & H^2(G, K^{*} \otimes \check{M}) \\
    \wr & & \wr & & \wr \\
    \sum_{\mathfrak{p}} \widehat{H}^{-1}(G, D_{\mathfrak{p}} \otimes \check{M}) & \to & \widehat{H}^{-1}(G, \check{M})
    & \to & \widehat{H}^{0}(G, \check{M} \otimes Y)
  \end{array}
\]

\[
\begin{aligned}
\operatorname{Bil}_{k!!}(P^{k}, Q^{k}; M)
 &\simeq \operatorname{Hom}_{k!}\bigl(P^{k},
    \underbrace{\operatorname{Hom}_{k!}(Q^{k}, M)}_{\textstyle
    \operatorname{Hom}_k(Q, M)}\bigr) \\
 &\simeq \operatorname{Hom}_k\bigl(P, \operatorname{Hom}_k(Q, M)\bigr) \\
 &\simeq \operatorname{Hom}_k(P \otimes_k Q, M)
    \overset{\sim}{\longleftarrow}
    \operatorname{Hom}_{k!}\bigl((P \otimes_k Q)^{k}, M\bigr)
\end{aligned}
\]

\[
\begin{cases}
u(l_0) = l_0^{p} \\
u(l_1) = u\rho(l_0) = \mathrm{int}(g)\,\rho\, \underbrace{u(l_0)}_{l_0^{p}} = \mathrm{int}(g)\, l_1^{p} \\
u(l_\infty) = u\rho(l_1) = \mathrm{int}(g)\, \rho\, \underbrace{u(l_1)} = \mathrm{int}(g)\, \underbrace{\rho\, \mathrm{int}(g)\, \rho^{-1}}_{\mathrm{int}(\rho(g))}\, \underbrace{(\rho\, l_1^{p})}_{l_\infty^{p}} = \struck{\mathrm{int}(g\rho(g))\ill{}}
\end{cases}
\]

\[
\begin{aligned}
&p\sigma(w''_1) + \Bigl(\sigma(w) F + \sum_{2}^{\infty} \sigma(w'_{i-1}) F^{i}\Bigr)
+ \Bigl(\sum_{1}^{\infty} p\,\sigma(w''_{j+1}) V^j\Bigr) \\
&p\sigma^{-1}(w'_1) + \Bigl(\sum_{1}^{\infty} p\,\sigma^{-1}(w'_{i+1}) F^i\Bigr)
+ \Bigl(\sigma^{-1}(w) V + \sum_{1}^{\infty} \sigma^{-1}(w''_j) V^{j+1}\Bigr)
\end{aligned}
\]

\[
  G_i(j)=G_0(j)\cup\left\{x\in C\ \middle|\ \begin{array}{l}
    x\in C(j) \text{ et } R(x)\cap G_{i-1}(j)\neq\emptyset\\
    x\in(\overline{C}(j)\smallsetminus\overline{C}_0(j)) \text{ et } R(x)\subset G_{i-1}(j)
  \end{array}\right\}
\]

\[
\left\{
\begin{aligned}
  \underline{E}_{U/k}(G, {}_{\infty}\mu) &\simeq \underline{E}_k(N^i_{U/k}\,G,
  \mathbf{Q}/\mathbf{Z})\\
  \underline{E}_{S/k}(G, {}_{\infty}\mu) &\simeq
  \underline{E}_k(N^i_{Y/k}(G), \mathbf{Q}/\mathbf{Z})\\
  \underline{E}_{Y/k}(G, {}_{\infty}\mu) &\simeq E_k(N^i_{S/k}(G),
  \mathbf{Q}/\mathbf{Z}).
\end{aligned}
\right.
\]

\[
\begin{array}{lll}
H^{0}_{A}(G) = G(K) & & \text{si } A = K \text{ de dim.\ } 0 \\
H^{1}_{A}(G) = G(K)/G(A) & & \text{si } A \text{ de dim.\ } 1,\ \text{de corps des fractions } K \\
H^{i}_{A}(G) = H^{i-1}(\operatorname{Spec}(A) - \mathfrak{m}_{A}, G) & & \text{si } A \text{ de dim.\ } \geq 2
\end{array}
\]

\[
  (72) \qquad
  \begin{cases}
    \rho_E(X) = \tilde{\rho}_E(\tilde{X}) : & \tilde{X} \mapsto 0 ,\ \tilde{Y} \mapsto \tilde{H} ,\ \tilde{H} \mapsto -2\tilde{X} \\
    \rho_E(Y) = \tilde{\rho}_E(\tilde{Y}) : & \tilde{X} \mapsto -\tilde{H} ,\ \tilde{Y} \mapsto 0 ,\ \tilde{H} \mapsto 2\tilde{Y} \\
    \rho_E(H)\ (= \tfrac{1}{2}\tilde{\rho}_{\tilde{H}}\ \text{sic}) : & \tilde{X} \mapsto \tilde{X} ,\ \tilde{Y} \mapsto -\tilde{Y} ,\ \tilde{H} \mapsto 0
  \end{cases}
\]

\[
  \mathrm{Hom}((u_1, v_1, \alpha_1), (u_2, v_2, \alpha_2)) =
  \left\{ (\lambda, \mu) \;\middle|\;
  \begin{array}{l}
    \lambda \in \mathbb{Z}^I,\ \mu \in G' , \\
    \boxed{v_2 = \operatorname{int}(\mu) \circ v_1} , \\
    \alpha_2(i)\, {\ell'_{u(i)}}^{\lambda_i} = \mu\, \alpha_1(i)
  \end{array}
  \right\}
\]

\[
\begin{array}{ccccccccc}
1 & \to & \mathbb{Z} & \to & \widetilde\Pi^{D}_{03}(4) & \to & \Pi^{D}_{0,3} \times \mathbb{Z}/4\mathbb{Z} & \to & 1 \\
 & & \parallel & & \parallel & & \updownarrow & & \\
1 & \to & \mathbb{Z} & \to & \widetilde{\mathcal{T}}_{11} & \Longrightarrow & \mathcal{T}_{1,1} = \Pi^{D}_{0,3} \times_{\mathbb{Z}/12\mathbb{Z}} \mathbb{Z}/4\mathbb{Z} & &
\end{array}
\]

\[
\left\lbrace
\begin{array}{l}
u . 1_C = \eta(u) . 1_C \\[4pt]
x, y \in C,\ u \in \mathcal{U},\ \Delta_{\mathcal{U}} u = \sum v_i \otimes w_i
\;\Longrightarrow\; u(xy) = \sum (v_i x)(w_i y)
\end{array}
\right.
\]

\[
\begin{aligned}
(1-a)(1-b) + (1-b)(a+1) &= 2(1-b) = f_1 \\
(a+1)(b+1) + (a+1)(1-b) &= 2(a+1) = f_2 \\
(a+1)(b+1) + (b+1)(1-a) &= 2(b+1) = f_3 \\
(a-1)(b-1) - (b+1)(a-1) &= 2(1-a) = f_0
\end{aligned}
\]

\[
  (R, J) \quad
  \left\{
  \begin{array}{l}
    R \subset \mathfrak{P}(I) \text{ tel que } \struck{\ill{}}
    \left\{
    \begin{array}{l}
      \text{a)}\ \bigcup_{A \in R} A = I \\
      \text{b)}\ A, A' \in R,\ A \neq A' \Rightarrow A \cap A' = \emptyset \\
      \text{c)}\ \emptyset \in R
    \end{array}
    \right. \\[1ex]
    J \in R
  \end{array}
  \right.
\]

\[ (4.4)_{i} \qquad \begin{cases} \exists\, v_{i} \in \mathcal{E}^{(2n-2i)},\ w_{i} \in \mathcal{E}^{-(2n-2i)} \ \text{tels que l'on ait} \\[2pt] (w_{i} v_{i} - 1)\, \pi_{i} = 0 , \qquad (v_{i} w_{i} - 1)\, \pi_{2n-i} = 0 \end{cases} \]

\[
\begin{aligned}
X(a, b) &= \Gamma_{A \times B}\,
\underline{\mathrm{Hom}}\bigl(p_B^{*}(a) \times p_A^{*}(b), X\bigr)\\
&= \Gamma_{A \times B}\bigl(\underline{\mathrm{Hom}}_{A \times B}(a \boxtimes b, X)\bigr)
\end{aligned}
\qquad
\begin{cases}
= \Gamma_A\, p_{B*}\, \underline{\mathrm{Hom}} \\
= \Gamma_B\, p_{A*}\, \underline{\mathrm{Hom}}
\end{cases}
\]

\[
  \left\{
  \begin{array}{l}
    \Delta_t(A) = \displaystyle\sup_{\operatorname{Tr} E < t} \struck{\det}\ \Delta_E\, EAE \\[2ex]
    \Delta_t(1 + |A|) = \displaystyle\sup_{\substack{\operatorname{Tr}|B| < t\\ \|B\| \leq 1}} \struck{\det}\ \Delta_E\bigl(1_E + BA\uncertain{B}\bigr)
  \end{array}\right.
\]

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

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

\[
\begin{array}{ccc}
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \mathcal{B}) & \underset{\sigma}{\overset{\rho}{\rightleftarrows}} & \mathcal{B}^{\bullet} \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \widehat{\mathcal{B}}) & & \\
\downarrow & \nwarrow{\scriptstyle \sigma} & \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \mathcal{B}) & \xrightarrow{\ \rho\ } & \mathcal{B}^{\bullet}
\end{array}
\qquad \text{$\mathcal{B}$ cat.\ additive avec $\Sigma$ classe de suites exactes et $\Sigma$-noyaux}
\]