\[
\mathrm{Hom}(M \otimes M, L) \simeq \mathrm{End}(M)
\]

\[
\operatorname{Tr} u\, u^{\mathrm{ad}(\varphi,\psi)} > 0 \text{ pour } \forall\, u \neq 0
\]

\[
\Updownarrow
\]

\[
\operatorname{Tr} u\, u^{*} \alpha > 0 \text{ pour } \forall\, u \neq 0
\qquad (= \operatorname{Tr} u^{*} \alpha u)
\]

\[
\operatorname{Tr} u\, u^{\mathrm{ad}(\psi,\psi)} > 0 \text{ pour } u \neq 0,
\quad \text{i.e.}\quad
\operatorname{Tr} u\, \alpha^{-1} u^{*} \alpha > 0 \text{ pour } u \neq 0.
\]

\[
\varphi_0(J_M x, y) = \varphi_0(x, J_{\overline{M}}\, y) = -\varphi_0(x, J_M y),
\qquad
\varphi_0(J_M x, J_M y) = \varphi_0(x, y).
\]

\[
\varphi_0(x,y) = \psi_0(x,y) + \psi_0(J_M x, J_M y)
\]

\[
\varphi_0(J_M x, J_M y) = \psi_0(J_M x, J_M y) + \psi_0(-x, -y) = \varphi_0(x,y).
\]

\[
M_m = \Bigl(\bigotimes_{1 \leqslant i \leqslant r} \bigl(\otimes^{m_i} M_i\bigr)\Bigr)(m_{r+1}),
\qquad
\varphi_m = \Bigl(\bigotimes_{1 \leqslant i \leqslant r} \bigl(\otimes^{m_i} \varphi_i\bigr)\Bigr)(m_{r+1}),
\]

\[
\operatorname{Tr} u^{*(\varphi_m, \varphi_{m'})} u \geqslant 0.
\]

\[
G_m \xrightarrow{\;i\;} G \xrightarrow{\;\varepsilon\;} G_m,
\]

\[
u^{*(\varphi^{C},\psi^{C})} = C_M^{-1}\, u^{*(\varphi,\psi)}\, C_N,
\quad\text{donc}\quad
u^{*(\varphi,\psi)} = \underbrace{C_M\, u^{*(\varphi^{C},\psi^{C})}\, C_N^{-1}}_{u'},
\]

\[
\operatorname{Tr} u\, u^{*(\varphi,\psi)}
= \operatorname{Tr} u\, C_M\, u'\, C_N^{-1}
= \operatorname{Tr} C_N^{-1} u\, C_M\, u'
= \operatorname{Tr} u\, u' \geqslant 0.
\]

\[
g C g^{-1} = \varepsilon(g)\, C,
\qquad
\varphi^{C}(gx, gy) = \varphi(gx, Cgy) = \varepsilon(g)\, \varphi(gx, gCy)
= \varepsilon(g)\, \varphi^{C}(x, y).
\]

\[
\varphi^{C'}(x, y) = \varphi(x, gCg^{-1} y) = \varphi(g^{-1}x, Cg^{-1}y)
= \varphi^{C}(g^{-1}x, g^{-1}y).
\]

\[
g C g^{-1} C'^{-1} \in Z'(\mathbb{R})
\]

\[
C' = z\, g C g^{-1}, \qquad g \in G'(\mathbb{R})^{\circ},\ z \in Z'(\mathbb{R}).
\]

\[
\varphi^{C'} = \varphi^{zC} = -\varphi^{C} < 0,
\]

\[
u^{*(\varphi^{z}, \psi^{z})} = z_M^{-1}\, u^{*(\varphi,\psi)}\, z_N
= u^{*(\varphi,\psi)}.
\]

\[
{}^{z}(P(M)) = \{ {}^{z}\varphi \mid \varphi \in P(M) \}
\qquad (M \in \mathrm{Ob}\,\mathcal{M}^{i},\ i \in \mathbb{Z})
\]

\[
\mathrm{End}(M) \simeq \struck{\ill{}} \mathcal{Q}(M)
\]

\[
M \underset{\tilde{\psi}}{\overset{\tilde{\varphi}}{\rightrightarrows}} M^{*} = \check{M}(p),
\qquad
\varphi(x,y) = \struck{\ill{}} (y, \tilde{\varphi}(x))
\]

\[
\mathcal{Q}(M) = B(M, M ; \mathbb{Q}(p)) \simeq \mathrm{Hom}(M, M^{*})
\qquad \text{où } M^{*} = \check{M}(p)
\]

\[
\mathrm{End}(M) \xrightarrow[\;\sim\;]{\;\Theta_{\varphi}\;} \struck{\ill{}}\ \mathcal{Q}(M) \simeq \mathrm{Hom}(M, M^{*})
\]

\[
\Theta_{\varphi}(u) = \varphi \circ (u \otimes 1)
\qquad\text{donc}\qquad
\widetilde{\Theta_{\varphi}(u)} = \tilde{\varphi} \circ u
\]

\[
\Theta_{\varphi}^{-1}(\psi) = \tilde{\varphi}^{-1} \tilde{\psi}
\qquad
M \underset{\tilde{\psi}}{\overset{\tilde{\varphi}}{\rightrightarrows}} M^{*}
\]

\[
u^{\varphi} = \struck{\ill{}}\ \tilde{\varphi}^{-1}\bigl[{}^{s}[\varphi \circ (u \otimes 1)]\bigr]
= \tilde{\varphi}^{-1}\bigl(\varphi^{s} \circ (1 \otimes u)\bigr)
\]

\[
\struck{\varphi(ux, y) = \varphi(x, u^{\varphi} y)}
\]

\[
\varphi(u^{\varphi} x, y) = (-1)^{\delta} \varphi(u\struck{x}, \struck{x})
\]

\[
(-1)^{\delta}\, \Theta_{\varphi}(u)(x,y) = \Theta_{\varphi}(u)'(y,x)
\]

\[
\struck{(a)}\quad \varphi(ux, y) = \varphi(x, u^{\varphi} y)
\]

\[
\struck{(a)}\quad \varphi(ux, y) = \struck{\ill{}}\ \varphi(x, u^{\varphi} y)
\]

\[
(u+v)^{\varphi} = u^{\varphi} + v^{\varphi}
\]

\[
(\lambda u)^{\varphi} = \lambda u^{\varphi} \quad (\lambda \in \mathbb{Q})
\]

\[
(u^{\varphi})^{\varphi} = u
\]

\[
(uv)^{\varphi} = \struck{\ill{}}\ v^{\varphi} u^{\varphi}
\]

\[
\psi(x,y) \struck{\ill{}} = \varphi(ax, y)
\]

\[
u^{\psi} = \struck{\ill{}}\ a^{-1} u^{\varphi} a \qquad (a \text{ étant inversible}).
\]

\[
\mathrm{Tr}\, v^{\varphi} u = \mathrm{Tr}\, u^{\varphi} v
\]

\[
u \mapsto u^{\psi} = a^{-1} u^{\varphi} a
\]

\[
\mathrm{Tr}\, u^{\psi} u \;\bigl(= \mathrm{Tr}\, (a^{-1}ua)^{\varphi}
(a^{-1}ua)\bigr) \geqslant 0
\]

\[
\mathrm{Tr}\, u^{\varphi} u \geqslant 0 \qquad \forall u
\]

\[
\varphi : \mathcal{A}(X) \otimes_{\mathcal{O}_{S}} \mathcal{A}(Y)
\longrightarrow \mathcal{A}(X \times_{S} Y)
\]

\[
g'_{*}(\mathcal{O}_{Y'}) = v^{*} g_{*}(\mathcal{O}_{Y})
= \mathcal{A}(Y) \otimes_{\mathcal{O}_{S}} \mathcal{O}_{S'} .
\]

\[
p'_{*}(\mathcal{O}_{X \times_{S} Y}) = g'^{*} p_{*}(\mathcal{O}_{X})
= g'^{*}(\mathcal{O}_{S'}) = \mathcal{O}_{Y'} ,
\]

\[
\mathcal{A}(X \times_{S} Y) = v_{*} g'_{*} p'_{*}(\mathcal{O}_{X \times_{S} Y})
= v_{*} g'_{*}(\mathcal{O}_{Y'})
= v_{*}(\mathcal{A}(Y) \otimes_{\mathcal{O}_{S}} \mathcal{O}_{S'})
= \mathcal{A}(Y) \otimes_{\mathcal{O}_{S}} \mathcal{A}(X)
\]

\[
(\mathcal{A}(\eta) \otimes \mathrm{id}_{\mathrm{S}\mathcal{F}}) \circ \rho :
\mathcal{E} \longrightarrow
\mathcal{O}_{S} \otimes_{\mathcal{O}_{S}} \mathrm{S}\mathcal{F}
= \mathrm{S}\mathcal{F}
\]

\[
\sum \Delta(\varphi_{j}) \otimes b_{ji}
= \sum \varphi_{j} \otimes \Delta(b_{ji}) ,
\]

\[
A/I \to (A/I) \otimes_{k} (A/I)
= ((A/I) \otimes_{k} A) / ((A/I) \otimes_{k} I)
\]

\[
A/I \longrightarrow (A/I') \otimes_{k} (A/I)
= ((A/I') \otimes_{k} A) / ((A/I') \otimes_{k} I)
\]

\[
H^{2}(k, G) \simeq \operatorname{Hom}(\Gamma, \mathrm{Br}(k)) ,
\]

\[
\add{\xi}\, u : \Gamma \longrightarrow \mathrm{Br}(k) .
\]

\[
K(\mathcal{M}) \subset \underline{\mathbb{Z}}[\Gamma] ,
\]

\[
\sum n_{i} e^{i} \in K(\mathcal{M}) \iff n_{i} \equiv 0 \ (d(i))
\quad \text{pour tout } i \in \Gamma
\]

\[
\Sigma(\mathcal{M}) \simeq \Sigma(\mathcal{M}')/\pi .
\]

\[
\sigma \longmapsto (Z_{\sigma}, \xi_{\sigma} \in \mathrm{Br}(Z_{\sigma}))
\]

\[
\sigma \longmapsto n_{i}(\sigma, i) ,
\]

\[
\sigma\tau = \sum_{\rho} c^{\rho}_{\sigma,\tau}\, \rho
\qquad (c^{\rho}_{\sigma,\tau} \in \underline{\mathbb{N}}),
\]

\[
\sigma \longmapsto (Z_{\sigma}, \xi_{\sigma}) , \qquad
\xi_{\sigma} \in \mathrm{Br}(Z_{\sigma})
\]

\[
\text{rang} \;:\; \mathcal{M} \longrightarrow \underline{\mathbb{Z}}
\]

\[
K^{\mathrm{eff}} = \text{sous-groupe de } K(\mathcal{M})
\text{ engendré par les } c\ell(M),\ M \in \mathrm{Ob} ,
\]

\[
= \underline{\mathbb{N}}^{(\Sigma)}
\]

\begin{align*}
\operatorname{det}_{M} f &= \operatorname{N}_{Z/k}(\operatorname{Nr}_{K/Z}(f))^{n'/d} , \qquad (n'/d = n/dr) \\
\operatorname{Tr}_{M} f &= \operatorname{Tr}_{Z/k}(\operatorname{Trr}_{K/Z}(f))\, \frac{n'}{d} \\
\operatorname{Pol}_{M}(f,t) &= \operatorname{N}_{Z/k}(\operatorname{Polr}_{K/Z}(f,t))^{n'/d}
\end{align*}

\[
G \longrightarrow \underline{\operatorname{Aut}}(M_{\sigma}) = \underline{K}^{*}_{\sigma}
\]

\[
u_{\sigma} : G \longrightarrow \underline{Z}^{*}_{\sigma}
\;\add{\simeq}\; \prod\nolimits_{Z/k} \underline{G}_{m} ,
\]

\[
u_{\sigma}(\xi) = \xi_{\sigma} ,
\]

\[
\xi \in H^{2}(k_{\mathrm{pl}}, G)
\]

\[
n(\sigma) = \operatorname{card}(\sigma)\cdot\operatorname{ord}(\xi_{\sigma})
\]

\[
0 \longrightarrow G \xrightarrow{\ u\ } \underline{Z}^{*}
\xrightarrow{\ N_{Z/k}\ } \underline{G}_{m} \longrightarrow 0 ,
\]

\[
\operatorname{Ker}\bigl(u : H^{2}(k,G) \longrightarrow H^{2}(k, \underline{Z}^{*})\bigr)
\]

\[
0 \longrightarrow G' \longrightarrow G \xrightarrow{\ n\ } G \longrightarrow 0
\]

\[
F_{M} : M^{\sigma} \longrightarrow M
\]

\[
F_{M} : M^{\sigma} = M \otimes_{W} (W, \sigma) \longrightarrow M ,
\]

\[
F_{E} : E^{\sigma} = E \otimes_{K} (K, \sigma) \longrightarrow E .
\]

\[
X \longmapsto H^{*}_{\mathrm{cris}}(X)
\]

\[
H^{*}_{\mathrm{cris}} : \text{schémas propres lisses sur } k \longrightarrow
\add{\mathrm{Grad}}\,(\mathrm{FCriso}^{+}(k)) ,
\]

\[
H^{2d}_{\mathrm{cris}}(X) \simeq W(-d)
\]

\[
\prod(k)_{\underline{\mathbb{Q}}_{p}} \longleftarrow \underline{G}(k) ,
\qquad \add{\text{associé au } \otimes\text{-foncteur }
(\mathrm{Mot}(k) \longrightarrow \mathrm{FCriso}(k))}
\]

\[
M = V \otimes_{\underline{\mathbb{Z}}_{p}} W ,
\]

\[
\underline{G}(k) \longrightarrow \underline{\Pi}_{p}(k) ,
\]

\[
\underline{G}(k) \longrightarrow \underline{\Pi}_{p}(k) \times
[\underline{G}_{m}]_{\underline{\mathbb{Q}}_{p}}
\]

\[
(V_{i})_{c} \longmapsto \sum V_{i} \otimes \underline{\mathbb{Q}}_{p}(i)
\]

\[
k \longrightarrow k' ,
\]

\[
\mathrm{FCriso}(k) \longrightarrow \mathrm{FCriso}(k') ,
\]

\[
\underline{G}(k') \longrightarrow \underline{G}(k) .
\]

\[
0 \longrightarrow \underline{G}(k') \longrightarrow \underline{G}(k)
\longrightarrow [g]_{\underline{\mathbb{Q}}_{p}} \longrightarrow 0 .
\]

\[
\add{(6.2.1)} \qquad 0 \longrightarrow \underline{G}(\bar{k}) \longrightarrow
\underline{G}(k) \longrightarrow \underline{\Pi}_{p}(k) \longrightarrow 0 ;
\]

\[
\underline{G}(k) \simeq \underline{G}(\underline{\mathbb{F}}_{p})
\times_{\underline{\Pi}_{p}(\underline{\mathbb{F}}_{p})}
\underline{\Pi}_{p}(k) ,
\]

\[
\Gamma = \bar{\underline{\mathbb{Q}}}^{*}_{p} ,
\]

\[
0 \longrightarrow H \longrightarrow G \longrightarrow G' \longrightarrow 0 ,
\]

\[
\bar{\underline{\mathbb{Q}}}^{*}/\text{unités} \longrightarrow
\underline{\mathbb{Q}}
\]

\[
v_{p}(p) = 1 .
\]

\[
H = D(\underline{\mathbb{Q}}) .
\]

\[
\underline{\mathbb{Z}} \longrightarrow \bar{\underline{\mathbb{Q}}}^{*} ,
\qquad 1 \longmapsto p ,
\]

\[
\underline{\mathbb{Z}} \hookrightarrow \underline{\mathbb{Q}}
\]

\[
\add{(8.2.1)} \qquad i : \underline{G}(\underline{\mathbb{F}}_{p})
\longrightarrow \underline{\Pi}_{p}(\underline{\mathbb{F}}_{p})
\]

\[
i : \underline{\mathbb{Q}}_{p}\text{-faisceaux constructibles}
\hookrightarrow \mathrm{FCriso}(\underline{\mathbb{F}}_{p})
\]

\[
\add{(8.2.2)} \qquad \pi_{1}(\underline{\mathbb{F}}_{p}) = \hat{Z}
\longrightarrow G'(\underline{\mathbb{Q}}_{p}) , \qquad \text{i.e.}\quad
(\pi_{1}(\underline{\mathbb{F}}_{p}))_{\underline{\mathbb{Q}}_{p}}
\longrightarrow G'
\]

\[
\text{torseurs sous } (\hat{Z})_{\underline{\mathbb{Q}}_{p}}
\longrightarrow \text{torseurs sous } G'
\]

\[
\xi \in H^{2}(\underline{\mathbb{Q}}_{p}, \struck{\ill{}}\ H)
= \operatorname{Hom}(\underline{\mathbb{Q}}, \mathrm{Br}(\underline{\mathbb{Q}}_{p}))
= \operatorname{Hom}(\underline{\mathbb{Q}}, \underline{\mathbb{Q}}/\underline{\mathbb{Z}})
\]

\[
\xi = \partial(\eta)
\]

\[
\eta \in H^{1}(\underline{\mathbb{Q}}_{p}, G')
\]

\[
\begin{array}{ccccccccc}
0 &\longrightarrow& H &\longrightarrow& G &\longrightarrow& G' &\longrightarrow& 0 \\
 && \big\| && \downarrow && \downarrow && \\
0 &\longrightarrow& H &\longrightarrow& H' &\longrightarrow& (\hat{\underline{\mathbb{Z}}})_{\underline{\mathbb{Q}}_{p}} &\longrightarrow& 0 ,
\end{array}
\]

\[
\xi = \partial(\eta_{o}) ,
\]

\[
\eta_{o} \in H^{1}(\underline{\mathbb{Q}}_{p}, \hat{\underline{\mathbb{Z}}})
\]

\[
\mathrm{Br}(\underline{\mathbb{Q}}_{p}) \simeq
\underline{\mathbb{Q}}/\underline{\mathbb{Z}}
\]

\[
(8.4.1) \qquad \underline{G}(\bar{\underline{\mathbb{F}}}_{p})
\longrightarrow \underline{G}(\underline{\mathbb{F}}_{p}) .
\]

\[
(8.4.2) \qquad \underline{G}(\bar{\underline{\mathbb{F}}}_{p})
\longrightarrow \underline{H} .
\]

\[
\underline{G}(\gamma) \approx \underline{G}(\bar{\underline{\mathbb{F}}}_{p})
\add{\simeq \underline{H}} \quad ).
\]

\[
\mathrm{FCriso}(k) \longrightarrow \operatorname{Rep}(\underline{H})
\]

\[
\underline{H} \longrightarrow \underline{G}(k)
\]

\[
H \longrightarrow G(k) ,
\]

\[
E = \sum_{i \in \underline{\mathbb{Z}}} E_{i}(-i) ,
\]

\[
\mathrm{FCriso}(\underline{\mathbb{F}}_{p})
  \otimes_{\underline{\mathbb{Q}}_{p}} K_{a}
\longrightarrow
\mathrm{FCriso}(\underline{\mathbb{F}}_{q})
  \otimes_{\underline{\mathbb{Q}}_{p}} K_{a}
\]

\[
\underline{H} \longrightarrow
G_{a} = \underline{G}(\underline{\mathbb{F}}_{q}) ,
\]

\[
\operatorname{Rep}(G_{a}) \simeq
\mathrm{FCriso}(\underline{\mathbb{F}}_{q}) \longrightarrow
\operatorname{Rep}(\underline{H}) ,
\]

\[
\mathrm{FCriso}(\underline{\mathbb{F}}_{q})
  \otimes_{\underline{\mathbb{Q}}_{p}} K_{a}
\longrightarrow
\operatorname{Rep}(\underline{H})
  \otimes_{\underline{\mathbb{Q}}_{p}} K_{a} .
\]

\[
\underline{H} \xrightarrow{\;u_{a}\;} G_{a} \longrightarrow G ,
\]

\[
u_{a}^{*}(\lambda) = \frac{v_{p}(\lambda)}{a} .
\]

\[
G = \varprojlim G_{i}
\]

\[
\mathcal{M} \longrightarrow
\text{faisceaux localement libres sur } S .
\]

\[
M \rightsquigarrow \operatorname{gr} F(M) .
\]

\[
G = \underline{\operatorname{Aut}}_{S}(F)
\]

\[
\mathbb{G}_{m_{S}} \xrightarrow{\;i\;} G^{P} = G' . )
\]

\[
g' x_{p}^{(p)} \equiv x_{p}^{(p)} \qquad
\bigl( F'^{(p+1)}(M) \bigr)
\]

\[
\boxed{\;\mathfrak{h}' = \sum_{n \geq 1} \mathfrak{g}'_{n}\;}
\]

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

\[
\xi : G' \longrightarrow \mathbb{G}_{m}
\]

\[
\xi\, i_{1}(\lambda) = \lambda
\]

\[
F''_{\mathbb{C}} \;\overset{\text{Gaga}}{\simeq}\; F_{\mathbb{C}}
\;\overset{\substack{\text{th. de Hodge}\\ \text{des intégrales harmoniques}}}
{\simeq}\; F'_{\mathbb{C}}
\]

\[
F^{p,q}_{\mathbb{C}} = F^{(p)}_{\mathbb{C}} \cdot
\sigma\bigl( F^{(q)}_{\mathbb{C}} \bigr) \dots
\]

\[
U \subset P \subset G \qquad
P = \underline{\operatorname{Aut}}_{\otimes,\,\mathrm{filt}}\, T , \quad
U = \underline{\operatorname{Aut}}_{\otimes,\,\mathrm{filt},\,
\mathrm{id.\ sur\ Gr}}(T)
\]

\[
\operatorname{Fil}^{p}(V) = \sum_{i \geq p} \operatorname{Gr}^{i}(V)
\]

\[
[\mathfrak{g}^{\alpha}, \mathfrak{g}^{\beta}] \subset
\mathfrak{g}^{\alpha + \beta}
\qquad
\mathfrak{g}^{\alpha} \cdot V^{\beta} \subset V^{\alpha + \beta}
\]

\[
x V^{\beta} \subset \sum_{\beta' \geq \beta} V^{\beta'}
\qquad \text{pour tout } \beta \in \mathbb{Z} .
\]

\[
x \in \sum_{\alpha \geq 0} \mathfrak{g}^{\alpha} ,
\]

\[
x^{\mu} V^{\alpha} \subset V^{\alpha + \mu}
\]

\[
\mathfrak{p} = \sum_{\alpha \geq 0} \mathfrak{g}^{\alpha} , \qquad
\mathfrak{u} = \sum_{\alpha > 0} \mathfrak{g}^{\alpha} .
\]

\[
P = G \cap P' , \qquad U = G \cap U'
\]

\[
G/P \hookrightarrow G'/P'
\]

\[
V^{T \cdot U} \left[ \; = \bigcap_{u \in U'} V^{u T u^{-1}} \right]
\]

\[
V^{G} = V^{P} , \qquad P \text{ étant parabolique,}
\]

\[
j\,i(\lambda) = \lambda^{2} .
\]

\[
\underline{G}_{m}^{2} \longrightarrow G ,
\]

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

\[
i(\lambda) = i_{1}(\lambda)\, i_{2}(\lambda) \, ;
\]

\[
H'_{i_{1}} = \underline{\operatorname{Aut}}\ \underline{\operatorname{filt}}^{1}(F')
\subset \underline{\operatorname{Aut}}(F') = G'
\]

\[
Q = \underline{\operatorname{Isom}}\ \underline{\operatorname{filt}}^{1}(F,F')
\subset P = \underline{\operatorname{Isom}}(F,F')
\]

\[
i_{1} : \underline{G}_{m} \longrightarrow G' ,
\]

\[
F = F' \overset{H'_{i_{1}}}{\wedge} Q' ,
\]

\[
V_{\underline{C}} = \coprod_{p,q} V^{p,q} ,
\]

\[
i_{1}, i_{2} : \underline{G}_{m\,\underline{C}} \longrightarrow G_{\underline{C}} .
\]

\[
i_{\underline{C}} = i_{1} i_{2} : \lambda \mapsto i_{1}(\lambda) i_{2}(\lambda)
\]

\[
i : \underline{G}_{m} \longrightarrow G ,
\]

\[
i_{2} = \overline{i_{1}} \quad \text{i.e.} \quad
i_{2}(\lambda) = \overline{i_{1}(\bar{\lambda})} \quad \text{pour tout}\quad
\lambda \in \underline{C} .
\]

\[
\varphi : V \times V \longrightarrow \underline{\mathbb{Q}}(n) ,
\]

\[
\psi(x,y) = \varphi(x, \bar{y})\, (-i)^{p-q}
\quad \text{pour $x$ de bidegré } (p, n-p)
\]

\[
\underline{C}(G) \overset{\approx}{\longrightarrow} \operatorname{Ind} \underline{C}_{f}(G)
\]

\[
G \longrightarrow \prod_{T/S} \underline{\operatorname{Aut}}(\underline{F}) ,
\quad \text{or}
\]

\[
\prod_{T/S} \underline{\operatorname{Aut}}(\underline{F}) \simeq
\underline{\operatorname{Aut}}_{\mathcal{B}}\Bigl(\prod_{T/S} \underline{F}\Bigr)
= \underline{\operatorname{Aut}}_{\mathcal{B}}(\underline{E}) .
\]

\[
G(A) \longrightarrow \operatorname{Aut}_{\otimes}(\varphi_{G}^{f})
\]

\[
(X,Y) \mapsto X \otimes Y
\]

\[
\varphi(X,Y,Z) : (X \otimes Y) \otimes Z \overset{\sim}{\longrightarrow}
X \otimes (Y \otimes Z)
\]

\[
\psi(X,Y) : X \otimes Y \overset{\sim}{\longrightarrow} Y \otimes X
\]

\[
u(X) : \underline{1} \otimes X \simeq X \qquad \text{d'où} \quad
\psi(X,\underline{1})\,u(X) : X \otimes \underline{1} \simeq X ,
\]

\[
u(\underline{1}) = v(\underline{1}) : \underline{1} \otimes \underline{1}
\simeq \underline{1}
\]

\[
\underline{1} \xrightarrow{\ v'(\underline{1})\ } \underline{1} \otimes
\underline{1}' \xrightarrow{\ u(\underline{1}')\ } \underline{1}'
\]

\[
c(X,Y) : F(X \otimes Y) \overset{\sim}{\longrightarrow} F(X) \otimes F(Y)
\]

\[ \Delta(\pi_{i}) = \sum_{j+k=i} \pi_{j} \otimes \pi_{k} . \]

\[ h : \underline{V} \longrightarrow \operatorname{Grad}(\underline{M})
   \quad \text{(structure de Koko sur } \operatorname{Grad}(\underline{M})) , \]

\[ (0) \qquad \sigma \in (U \mathbin{\widehat{\otimes}} U)^{*} \]

\[ (1) \quad
\begin{cases}
\sigma'\sigma = 1 \ \text{(i.e. } \sigma \text{ et } \sigma' \text{ inverses
l'un de l'autre : on aura aussi } \sigma\sigma' = 1 , \\
\qquad \text{puisque } \sigma \text{ est supposé inversible)} \\[2pt]
\sigma(\Delta u)\sigma^{-1} = (\Delta u)' \quad \text{pour tout } u \in U
\end{cases} \]

\[ (2) \qquad (\operatorname{id}_{U} \otimes \varepsilon_{U})(\sigma)
 = (\varepsilon_{U} \otimes \operatorname{id}_{U})(\sigma) = 1 ,
 \quad \text{i.e.} \quad
 \varepsilon \in 1 + U^{+} \mathbin{\widehat{\otimes}} U^{+} . \]

\[ (3) \qquad \struck{\ill{}} \quad
 (\operatorname{id}_{U} \mathbin{\widehat{\otimes}} \Delta)(\sigma)
 \struck{\ill{}} = \sigma^{12}\sigma^{13} \]

\[ G : \underline{C} \to \struck{\operatorname{Modf}}
   \operatorname{Projf}(k') \]

\[ G(M) = L \otimes_{U} M . \]

\[ L \mathbin{\widehat{\otimes}}_{\uncertain{k'}} L \xleftarrow{\ \sim\ }
   L \mathbin{\widehat{\otimes}}_{k'} U'
   \quad \bigl( \simeq L \otimes_{(U,\Delta)}
   (U \mathbin{\widehat{\otimes}}_{k} U)
   \xrightarrow{\ \sim\ } L \otimes_{(U',\Delta')}
   (U' \mathbin{\widehat{\otimes}}_{k} U') \bigr) . \]

\[ (*) \qquad B \otimes_{k'} B \xleftarrow{\ \sim\ } B \otimes_{k'} A' . \]

\[ (**) \qquad B \otimes_{k'} B \longrightarrow B \]

\[ V = G(U) = L \mathbin{\widehat{\otimes}}_{U} (U,\operatorname{int}) , \]

\[ Z_{g} = \sum_{g'g''=g} X_{g'} \otimes Y_{g''} . \]

\[ (*) \qquad M_{gg'} \simeq M_{g} \otimes M_{g'} . \]

\[ F : \underline{C} \longrightarrow \operatorname{Modf}(k) \]

\[ K' \xrightarrow{\ \varphi\ } \operatorname{End}_{\mathbb{Q}}(E_{0})
   \xrightarrow{\ \ill{}\ }
   \operatorname{End}_{\mathbb{Q}}(\check{E}_{0}) . \]

\[ \operatorname{End}_{\mathbb{Q}}(\check{E}_{0})
   \simeq \operatorname{End}(E_{0}(\rho)) \simeq \operatorname{End}(E_{0}) , \]

\[ \boxed{\ (E_{\varphi})^{\vee} = (E_{\varphi'})(\rho) \ } \]

\[ (L_{\chi})^{\vee} = L_{\check{\chi}} , \quad \text{où} \quad
   \check{\chi}(g) = \chi(g^{-1}) . \]

\[ (1) \qquad A^{u} : A^{G'} \longrightarrow A^{G} \]

\[ (1') \qquad A^{\mathrm{id}_{G}} = 1_{A^{G}} , \qquad A^{uv} = A^{v} A^{u} . \]

\[ (2) \qquad A^{G} \times A^{G'} \longrightarrow A^{G \times G'} \]

\[ (2') \qquad M \boxtimes M' = \struck{\ill{}} \ A^{p_{1}}(M) \otimes A^{p_{2}}(M') . \]

\[ (3) \qquad M \otimes M' = A^{\Delta_{G}}(M \boxtimes M') . \]

\[ (4) \qquad I^{u} : A^{G} \longrightarrow A^{G'} \]

\[ (4') \qquad I^{\mathrm{id}_{G}} = 1_{A^{G}} , \qquad I^{uv} = I^{u} I^{v} . \]

\[ (5) \qquad A^{u}\bigl(I^{u}(M)\bigr) = M^{[G' : G]} \qquad
   (M \in A^{G} , \ G \subset G') \]

\[ (6) \qquad M \longrightarrow A^{u} I^{u}(M) \]

\[ (7) \qquad M' \longrightarrow I^{u}\bigl(A^{u}(M')\bigr) \]

\[ (8) \qquad \text{si $u$ est un isom., alors } I^{u} = A^{u^{-1}} , \]

\[ (10) \qquad u_{p_{1}, \ldots, p_{h}} : \mathfrak{S}_{p_{1}} \times \cdots
   \times \mathfrak{S}_{p_{h}} \longrightarrow \mathfrak{S}_{p} \]

\[ (11) \qquad M * M' = I^{u_{p,q}}(M \boxtimes M') . \]

\[ (12) \qquad M * M' \simeq M' * M , \]

\[ (13) \qquad (M * M') * M'' \simeq M * (M' * M'') \simeq
   I^{u_{p,q,r}}(M \boxtimes M' \boxtimes M'') . \]

\[ (15) \qquad T^{n} : A^{G} \longrightarrow (A^{G})^{\mathfrak{S}_{n}} \]

\[ (16) \qquad \boxed{\ T^{n}(M + M') \simeq \sum_{p+q=n} T^{p}(M) * T^{q}(M')\ } \]

\[ (17) \]

\[ (18) \qquad u_{p,q} : \mathfrak{S}_{p} \times \mathfrak{S}_{q}
   \longrightarrow \mathfrak{S}_{pq} \]

\[ (19) \qquad \begin{cases}
   \alpha_{p,q}(\sigma) \text{ est croissant dans les intervalles }
     \bigl] iq, (i+1)q \bigr] \\
   \text{et applique ceux-ci sur les intervalles }
     \bigl] \sigma(i) q, (\sigma(i)+1) q \bigr]
\end{cases} \]

\[ (20) \qquad T^{m}\bigl(T^{n}(M)\bigr) \simeq \struck{\ill{}}\
   A^{u_{m,n}}\bigl(T^{mn}(M)\bigr) \]

\[ (21) \qquad T^{n}(M \otimes M') \simeq T^{n}(M) \otimes T^{n}(M')
   \ \in \ A^{G \times \mathfrak{S}_{n}} . \]

\[ \mathbf{1} = K \in A^{\mathfrak{S}_{0}} \qquad
   \bigl( \text{par définition } \mathfrak{S}_{0} = \{e\} , \text{ et }
   u_{0,p} = u_{p,0} = \ill{} \bigr) . \]

\[ (22) \qquad A(G) = K(A^{G}) . \]

\[ (23) \qquad A(u) : A(G') \longrightarrow A(G) \]

\[ (24) \qquad \alpha \boxdot \beta = \struck{\ill{}}\
   A(p_{1})(\alpha) \cdot A(p_{2})(\beta) \ \in \ A(G \times G') . \]

\[ (25) \qquad I(u) : A(G) \longrightarrow A(G') \]

\[ (26) \qquad \begin{cases}
   A(u) I(u) = \text{multiplication par } [G' : G] \\
   I(u) A(u) = \text{multiplication par } [G' : G]
\end{cases} \]

\[ (27) \qquad A(\sigma_{g}) = I(\sigma_{g}) = \text{identité de } A(G)
   \qquad (g \in G) . \]

\[ (28) \qquad A^{n} = A(\mathfrak{S}_{n}) \quad (n \geq 1) , \quad
   A^{0} = \mathbb{Z} \ ; \qquad A^{\bullet} = \coprod_{n=0}^{\infty} A^{n} \]

\[ (29) \qquad A(u_{p,q})(\alpha * \beta) = \frac{(p+q)!}{p!\, q!}\
   \alpha \boxdot \beta \ \in \ A^{\mathfrak{S}_{p} \times \mathfrak{S}_{q}} , \]

\[ (30) \qquad A(u_{p,p})(\alpha * \beta) = \frac{(2p)!}{p!\, q!}\
   \alpha \beta \]

\[ (31) \qquad \begin{cases}
   T = T_{G} : A(G) \longrightarrow 1 + \widehat{A(G)^{\bullet}}^{\,+} \\
   T(\alpha) = \sum_{n} T^{n}(\alpha) \qquad (T^{0}(\alpha) = 1)
\end{cases} \]

\[ (32) \qquad T\bigl(\gamma(M)\bigr) = \sum_{n} T^{n}(M) \]

\[ (33) \qquad T^{n}(\alpha + \beta) = \sum_{p+q=n} T^{p}(\alpha) * T^{q}(\beta) \]

\[ (33) \qquad T^{n}(\alpha \beta) = T^{n}(\alpha)\, T^{n}(\beta) \]

\[ (34) \]

\[ \frac{(2p)!}{(p!)^{2}}\, \alpha \beta = A(u_{p})(\alpha * \beta) \]

\[ T : A^{\bullet} \longrightarrow \widetilde{A^{\bullet}}
         = \mathbb{Z} \times \bigl( 1 + \widehat{A^{\bullet +}} \bigr) \]

\[ T^{n}(\alpha + \beta) = \sum_{p+q=n} T^{p}(\alpha)\, T^{q}(\alpha) \]

\[ T^{n}(\alpha \cdot \beta) = T^{n}(\alpha)\, T^{n}(\beta) . \]

\[ (1) \qquad A(T) \simeq \mathbb{Z}(\hat{T}) \longrightarrow
   \mathbb{Z} \times \bigl( 1 + \mathbb{Z}[[\hat{T}]]^{+} \bigr) ; \]

\[ \tilde{C}(\rho_{\chi}) = \bigl( 1 ,\ 1 + [\chi] \bigr) . \]

\[ \varphi : \mathbb{Z} \times \bigl( 1 + \mathbb{Z}[[\hat{T}]]^{+} \bigr)
   \longrightarrow \mathbb{Q}[[\hat{T}]] \]

\[ \sigma^{1}\bigl((\alpha^{i})\bigr) = \cdots =
   \sigma^{n-1}\bigl((\alpha^{i})\bigr) = 0 , \qquad
   \sigma^{n}\bigl((\alpha^{i})\bigr) = C^{n} \ \text{polynôme donné} . \]

\[ (1) \qquad K(G) \xrightarrow{\ c_{G}\ }
   \mathbb{Z} \times \bigl( 1 + CK(G)^{+} \bigr) . \]

\[ K(G) \longrightarrow K(X) \]

\[ (2) \qquad CK(G) \longrightarrow CK(X) = A(X) \]

\[ (3) \]

\[ (4) \]

\[ K(T) \simeq \mathbb{Z}(\hat{T}) \qquad (\text{algèbre du groupe } \hat{T}) \]

\[ \bigl\{ \simeq \mathbb{Z}[x^{1}, \ldots, x^{r}]_{x^{1} \cdots x^{r}}
   \ \struck{\text{associé}}\ \add{\text{avec}}\ \varepsilon(x^{i}) = 1 ,
   \ \lambda x^{i} = 1 - x^{i} t \bigr\} \]

\[ c_{T}(\chi) = 1 + \chi \qquad \text{i.e.} \qquad
   c_{T}^{i}(\chi) = \begin{cases}
   1 & \text{si } i = 0 \\ \chi & \text{si } i = 1 \\ 0 & \text{si } i > 1
\end{cases} \]

\[ CK(G) \longrightarrow CK(T) \simeq \mathbb{Z}[\hat{T}] \]

\[ (5) \qquad CK(G) \longrightarrow \mathbb{Z}[\hat{T}]^{W} \]

\[ (5\ \mathrm{bis}) \qquad C\bigl( K(G) \otimes_{\mathbb{Z}} k \bigr)
   \longrightarrow k[\hat{T}]^{W} \]

\[ \varphi u_{i} = u_{j} \varphi . \]

\[ \boxed{\ \text{Les } \underline{\mathcal{U}}\text{-modules permis sont les }
   \ill{} \text{ sur } A \ \ill{} \text{ qui sont de dim.\ finie sur } k . \ } \]

\[ \check{F}(V) = V' \]

\[ F^{\vee} F^{\vee} = \text{identité} . \]

\[ \varepsilon(x) = \varepsilon(\check{x}) , \]

\[ \varphi(\check{x}) = \check{\overline{\varphi(x)}} , \]

\[ \overline{\varphi}(V)' = \overline{\varphi}(V') , \]

\[ T(V,W) = V \otimes_{k} W \]

\[ \underline{\mathcal{U}} \mathbin{\hat{\otimes}_{k}} \underline{\mathcal{U}} \]

\[ \underline{\mathcal{U}}_{V} = \underline{\mathcal{U}}/J_{V} . \]

\[ \underline{\mathcal{U}} \otimes_{k} \underline{\mathcal{U}}(V) \]

\[ W = \Sigma\, V_i \otimes V_j . \]

\[ \underline{\mathcal{U}}/J \otimes_{k} \underline{\mathcal{U}}/J
   \subset L_{k}(V) \otimes_{k} L_{k}(W) = L_{k}(W \otimes_{k} V) . \]

\[ (V \otimes_{k} W)' \simeq V' \otimes_{k} W' \]

\[ P(v \otimes 1)\Delta = 1 \quad\text{et}\quad P(1 \otimes v)\Delta = 1 \]

\[ P(v \otimes 1)\Delta = P(1 \otimes v)\Delta = 1 \]

\[ \varepsilon(s) = \varepsilon(s)^{2} , \]

\[ A = \underline{\mathcal{U}}' \]

\[ u \to \langle u \cdot a,\; b \rangle , \]

\[ A \otimes 1 + 1 \otimes B \]

\[ x' \otimes x' \in V' \otimes V' ; \]

\[ \underline{A} = k[x]/x^{p} \]

\[ f(xy) \in A_{0} \otimes A_{0} . \]

\[ A \to A \otimes A , \]

\[ \langle u \otimes v,\; f \rangle = \langle uv,\; f \rangle \]

\[ G^{(k)} = k(G) , \]

\[ G \to G'(A_{0}) \]

\[ V = \Sigma\, V_i . \]

\[ K\text{-catégorie} \iff K\text{-bigèbre } \underline{\mathcal{U}}
   \iff \text{hyperalgèbre commutative } A \]

\[ \check{e} = e . \]

\[ n(i) = \dim M_{i} , \]

\[ n(\check{\imath}) = n(i) . \]

\[ P(x+y) = P(x)\,P(y) . \]

\[ f(\lambda,\mu)(x) = \lambda x^{p^{h}} + \mu \qquad
   ( h \text{ fixé, } x \text{ l'indéterminée} ) : \]

\begin{align*}
\tau_{y} f(\lambda,\mu)(x)
 &= \lambda (x+y)^{p^{h}} + \mu \\
 &= \lambda x^{p^{h}} + \mu + \lambda y^{p^{h}}
  = f\bigl(\lambda,\ \mu + \lambda y^{p^{h}}\bigr) , \quad\text{i.e.}
\end{align*}

\[ E_{h}(y) = \begin{pmatrix} 1 & y^{p^{h}} \\ 0 & 1 \end{pmatrix} ; \]

\[ d(E_{n}) = n , \qquad E_{1} \text{ est l'unité} . \]

\[ F_{n} = k[X_{1}, \ldots, X_{n}]\,/\,(X_{1}^{2}, \ldots, X_{n}^{2}) , \]

\[ \dot{S}^{k} = k! \sum_{1 \leq i_{1} < \ldots < i_{k} \leq n}
      \dot{X}_{i_{1}} \ldots \dot{X}_{i_{k}} . \]

\[ \xi^{n} = L_{n}(E_{1}, \ldots, E_{n}) = \sum_{i=1}^{n} c_{ni} E_{i}
   \qquad \underline{c_{nn} = 1} \]

\begin{align*}
\xi^{0} &= E_{1}                    &  E_{1} &= 1 \\
\xi^{1} &= E_{2}                    &  E_{2} &= \xi \\
\xi^{2} &= E_{1} + E_{3}            &  E_{3} &= \xi^{2} - 1 \\
\xi^{3} &= 2 E_{2} + E_{4}          &  E_{4} &= \xi^{3} - 2\xi \\
\xi^{4} &= 2 E_{1} + 3 E_{3} + E_{5} & E_{5} &= \xi^{4} - 3\xi^{2} + 5 \\
        &\qquad \ldots               &        &\qquad \ldots
\end{align*}

\[ P(\lambda) = \sum_{-n}^{n} \lambda^{i} c_{i} \qquad ( c_{i} \in L ) \]

\[ \sum_{i} \lambda^{i} \lambda'^{i} c_{i}
 = \sum_{i,j} \lambda^{i} \lambda'^{j} c_{i} c_{j} \ , \]

\[ c_{i} = c_{i}^{2} , \qquad c_{i} c_{j} = 0 \ \text{ si } i \neq j , \]

\[ \sigma_{n}(\lambda)\, t = \lambda^{n} t . \]

\[ \sigma_{n} \sigma_{m} = \sigma_{n+m} , \qquad
   \check{\sigma}_{n} = \sigma_{-n} \]

\[ P(\lambda) = \sum_{0}^{n} c_{i} \lambda^{i} \qquad ( c_{i} \in L(V) ) \]

\[ c_{i} c_{j} = 0 \ \text{ si } i \neq j , \qquad c_{i}^{2} = c_{i} \]

\[ \sigma_{p}(\lambda) t = \lambda^{p} t , \]

\begin{equation*}
(1) \qquad [\,Y, X\,] = p X
\end{equation*}

\[ V = \sum V_{n} \]

\[ (j - i)\, X_{ji} = p\, X_{ji} \]

\[ V = V' + V'' , \]

\[ V' = \sum_{\substack{n \equiv \alpha \\ (p)}} V_{n} , \qquad
   V'' = \sum_{\substack{n \not\equiv \alpha \\ (p)}} V_{n} \]

\[ V_{n+pk} \qquad k = 0, 1, \ldots \ . \]

\[ u_{i} = X_{n+p(i+1),\, n+pi} , \qquad E_{i} = V_{n+pi} , \]

\[ E_{0} \xrightarrow{\ u_{0}\ } E_{1} \xrightarrow{\ u_{1}\ } E_{2}
   \ \ldots \ \xrightarrow{\ u_{k-1}\ } E_{k} . \]

\[ E'_{j} = E_{j} \ \text{ si } \ j \leq \ell' , \qquad
   E'_{j} = E_{j} \cap F \ \text{ si } \ \ell' \leq j \leq \ell \]

\[ E''_{j} = 0 \ \text{ si } \ j \leq \ell' \ \text{ ou } \ j > \ell ,
   \qquad E''_{j} = D \ \text{ si } \ \ell' \leq j \leq \ell \]

\[ ( n \in \mathbf{Z} , \ k \in \mathbf{Z} , \ k \geq 1 ) \]

\[ e_{n,k} = \sigma_{n} \otimes e_{0,k} , \]

\[ k[Z]/Z^{k} \]

\[ Y . Z^{\alpha} = \alpha p\, Z^{\alpha} \qquad (Z^{k}) \]

\[ \check{e}_{n,k} = e_{-n-p(k-1),\, k} . \]

\[ A(H) \xrightarrow{\ j^{*}\ } A(G) \xrightarrow{\ i^{*}\ } A(F) \]

\[ \sigma_{n} \sigma_{m} = \sigma_{n+m} , \qquad \sigma_{0} = 1 . \]

\[ j^{*}(\sigma_{n}) = e_{n,1} \]

\[ i^{*}(e_{n,k}) = e_{k} \]

\[ A(G) = \Lambda[\xi] \]

\begin{equation*}
(1) \qquad \xi^{n} = \sum_{i=1}^{n+1} c_{ni}\, e_{0i} \qquad
   ( c_{ni} \in \Lambda , \ c_{n,n+1} = 1 )
\end{equation*}

\[ \xi^{n} = \sum_{i} c_{ni}\, e_{0i} \]

\[ e_{2}^{\,n} = \sum_{i} i^{*}(c_{ni})\, e_{i} \]

\[ i^{*}(\lambda) = d(\lambda) \qquad \lambda \in \Lambda \]

\[ e_{2}^{\,n} = \sum_{i} d(c_{ni})\, e_{i} \]

\[ \varepsilon(c_{ni}) = 0 \ \text{ si } \ i > n+1 , \qquad
   \varepsilon(c_{n,n+1}) = 1 \]

\[ c_{ni} = 0 \ \text{ si } \ i > n+1 , \qquad
   c_{n,n+1} \ \text{est de la forme} \ \sigma_{\alpha} \]

\[ c_{p,p+1} = \sigma_{\alpha} \]

\[ \sigma_{\alpha}\, e_{0,p+1} = e_{\alpha,p+1} \qquad \alpha + pn , \]

\[ p\alpha \leq pn , \qquad \text{d'où} \quad \alpha \leq 0 . \]

\[ \check{\xi} = e_{-p,2} = \sigma_{-p} \cdot \xi . \]