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

\[
0 \to \mathbb{V}(\underline{t}_{G^*}) \to \tilde G \to G \to 0
\]

\[
\begin{array}{ccccccccc}
0 & \to & \underline{t}_{G^*}^{\vee} & \to & \underline{t}_{\tilde G} & \to & \underline{t}_G & \to & 0 \\
 & & \| & & {\scriptstyle V_0}\downarrow\;\uparrow{\scriptstyle F_0} & & & & \\
 & & \omega_{G^*} & & & & & & \\
0 & \to & \underline{t}_{G^*}^{\vee(p)} & \to & \underline{t}_{\tilde G}^{(p)} & \to & \underline{t}_G^{(p)} & \to & 0 \\
 & & \| & & & & & & \\
 & & \omega_{G^*}^{(p)} & & & & & &
\end{array}
\]

\[
\begin{aligned}
\operatorname{Ker} F_0 &= \operatorname{Im} V_0 = \underline{t}_G^{\vee(p)} = \omega_G^{(p)} \\
\operatorname{Ker} V_0 &= \operatorname{Im} F_0 = \text{ce que ça \ill{}}
\end{aligned}
\]

\[
\underline{t}_{G^*}^{(p)} \xrightarrow{F_0} H(G) \to \underline{t}_{G^*}
\]

\[
\langle x, y \rangle = \varphi_n(xy) \qquad \varphi_n : A_n \to W_n
\]

\[
\varphi_n\Bigl(a + \sum_{1}^{\to} \lambda_i F^i + \sum_{1}^{\to}
\struck{\ill{}}\, V^i \mu_i\Bigr) = \varphi_n(z) = a
\]

\[
\begin{aligned}
\varphi_n(\lambda z) &= \lambda \varphi_n(z) \qquad \text{OK} \\
\varphi_n(z\lambda) &= \varphi_n\Bigl(\sum \lambda_{ij} \lambda^{\pi^{i-j}}
F^i V^j\Bigr) = \varphi_n(z)\lambda \qquad \text{OK.}
\end{aligned}
\]

\[
A_n = A/(F^{n+1}, V^{n+1})
\]

\[
I = \varinjlim_n I_n \qquad I_n \xrightarrow{\ p\ } I_{n+1}
\]

\[
D' : M \mapsto \operatorname{Hom}_{W_n\text{-}\mathrm{mod.}}(M, W_n)
\]

\[
I_n = A_n \simeq \operatorname{Hom}_{W_n}(A_n, W_n)
\]

\[
\begin{aligned}
u \cdot F &= \pi^{-1}(u \circ F) \\
u \cdot V &= \pi(u \circ F) \\
u \cdot \lambda &= u \circ \lambda
\end{aligned}
\qquad\Bigg|\qquad
\begin{aligned}
u \cdot F \cdot V &= \pi\pi^{-1}\, u \circ F \circ V = p\,u \\
u\,VF &= p\,u
\end{aligned}
\]

\[
D'(A_n) = \operatorname{Hom}_{W_n}\bigl((A_n)_g, W_n\bigr)
\]

\[
\begin{aligned}
u \cdot F &= \pi^{-1}(u \circ F_g) \\
u \cdot V &= \pi(u \circ V_g) \\
u \cdot \lambda &= u \circ \lambda_g \\
Fu &= u \circ F_d \\
Vu &= u \circ F_g \\
\lambda u &= u \circ \lambda_d
\end{aligned}
\qquad
u\lambda = \lambda \cdot u = u \circ \lambda = \lambda \circ u
\]

\[
A_n \overset{?}{\xrightarrow{\ \sim\ }} \operatorname{Hom}_{W_n}\bigl((A_n)_g, W_n\bigr)
\qquad \text{\uncertain{isom.}\ de \uncertain{bimodules}}
\]

\[
\boxed{A_n \times A_n \longrightarrow W_n}
\qquad
(x, y) \mapsto \langle x, y \rangle
\qquad \text{\uncertain{bimodule}}
\]

\[
\begin{aligned}
&\langle \lambda x, y \rangle = \lambda \langle x, y \rangle & \lambda &\in W_n \\
&\langle x, y\lambda \rangle = \langle \lambda x, y \rangle & \lambda &\in W_n \\
&\langle x, yF \rangle = \pi^{-1} \langle Fx, y \rangle \\
&\langle x, yV \rangle = \pi \langle Vx, y \rangle \\
&\langle x, \lambda y \rangle = \langle x\lambda, y \rangle \\
&\langle x, Fy \rangle = \langle xF, y \rangle \\
&\langle x, Vy \rangle = \langle xV, y \rangle
\end{aligned}
\]

\[
A_n \underset{A_n}{\otimes} A_n \longrightarrow W_n \qquad
W_n\text{-bilinéaire}
\]

\[
\struck{\pi\langle x, yF\rangle =}\ \langle Fx, y \rangle = \pi \langle x, yF \rangle ,
\qquad \langle x, yV \rangle = \pi \langle Vx, y \rangle
\]

\[
\mathcal{G} \xrightarrow{\ i\ } \underline{\mathrm{Hom}}^{\bullet}(K^{\bullet}, L^{\bullet})^{-1}
= \prod_i \underline{\mathrm{Hom}}(K^{i}, L^{i-1})
\]

\[
\theta(g \cdot p) = \theta(p) + \underbrace{\bigl(d \circ i(g) + i(g) \circ d\bigr)}_{\delta(i(g))}
\]

\[
\varphi = \varphi_{K, L}(\mathcal{G}, P, i, \theta) : K^{\bullet} \longrightarrow L^{\bullet}
\]

\[
K_{BT}(K; K') \longrightarrow K_{BT}(S'; G) , \tag{1}
\]

\[
K_{BT}(S', G) \to K_{BT}(s', G) , \tag{2}
\]

\[
\mathrm{BT}(s') \to \mathrm{Modlib}(W(k')) ,
\]

\[
\mathrm{BT}(s', G) \to \mathrm{Modlib}(W(k'), G) ,
\]

\[
K_{BT}(s', G) \longrightarrow K(W(k'), G) . \tag{3}
\]

\[
K_{BT}(K; K') \longrightarrow K(W(k'), G) . \tag{4}
\]

\[
K(W(k'), G) = R_{W(k')}(G) ;
\]

\[
K(W(k'), G) \to K(W(k'), I) \simeq R_{W(k')}(I)^{G/I} ,
\]

\[
K_{BT}(K, K') \to R_{W(k')}(I)^{G/I} \longrightarrow
\operatorname{Cent}(I, W(k'))^{G} , \tag{5}
\]

\[
\mathcal{H}^{(p)} \overset{F}{\underset{V}{\rightleftarrows}} \mathcal{H} .
\]

\[
\begin{array}{ccccccc}
0 \to & H^{10} & \to & H & \to & H^{01} & \to 0 \\
 & & & {\scriptstyle V}\downarrow\;\uparrow{\scriptstyle F} & & & \\
0 \to & H^{10(p)} & \to & H^{(p)} & \to & H^{01(p)} & \to 0
\end{array}
\]

\[
\operatorname{Im} V = \operatorname{Ker} F = H^{10(p)} , \qquad
\operatorname{Ker} V = \operatorname{Im} F ,
\]

\[
H^{10(p)} = \operatorname{Ker} F ,
\]

\[
u : E_1 \xrightarrow{\ \sim\ } E_2 , \qquad \text{d'où} \qquad
u^{(p)} : E_1^{(p)} \to E_2^{(p)} ,
\]

\[
u^{(p)}(F_1^{(p)}) = \struck{\ill{}}\, F_2^{(p)} , \quad \text{i.e.} \quad
u(F_1)^{(p)} = F_2^{(p)} .
\]

\[
T' \xrightarrow{\ \pi\ } T , \qquad T' = T , \ \pi = \text{\uncertain{Frobenius} \uncertain{absolu}.}
\]

\[
H \simeq H^{10} + H^{01}
\]

\[
H^{10} = \underline{t}_A^{\vee} \quad \text{et} \quad H^{01} = \underline{t}_{A^*} ,
\]

\[
\mathrm{DR}^1(A/S) \simeq \underline{t}_A^{\vee} + \underline{t}_{A^*}
\]

\[
\underline{t}_A \simeq \mathcal{L}_A \otimes_{\mathbb{F}_p} \underline{\mathcal{O}}_S ,
\qquad
\underline{t}_{A^*} \simeq \mathcal{L}_{A^*} \otimes_{\mathbb{F}_p} \underline{\mathcal{O}}_S
\]

\[
\mathcal{L}_{A^*} = (\underline{t}_{A^*})^{F} = H^1(A, \underline{\mathcal{O}}_A)^{F}
= H^1(A, \mathbb{Z}/p\mathbb{Z})
= \operatorname{Hom}({}_{p}A, \mathbb{Z}/p\mathbb{Z}) = \struck{\ill{}}\ {}_{p}A(k)^{\vee}
\]

\[
\left\{
\begin{aligned}
\mathcal{L}_{A^*}(k) &= {}_{p}A(k)^{\vee} \\
\mathcal{L}_{A}(k) &= {}_{p}A^*(k)^{\vee}
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
\underline{t}_A^{\vee} &= \mathcal{L}_A^{\vee} \otimes_{\mathbb{F}_p} k = {}_{p}A^*(k) \otimes_{\mathbb{F}_p} k \\
\underline{t}_{A^*} &= {}_{p}A(k)^{\vee} \otimes_{\mathbb{F}_p} k
\end{aligned}
\right.
\]

\[
{}_{p}A \to S
\]

\[
{}_{p}A \to ({}_{p}A)_{\text{ét}} \to S
\]

\[
\left\{
\begin{aligned}
\mathcal{L}_A &= ({}_{p}A^*)_{\text{ét}}^{\vee} \\
\mathcal{L}_{A^*} &= ({}_{p}A)_{\text{ét}}^{\vee}
\end{aligned}
\right.
\]

\[
\bigwedge^2 \mathrm{DR}^1(A/S) \simeq \bigwedge^2 \underline{t}_A^{\vee}
+ \underline{t}_A^{\vee} \otimes \underline{t}_{A^*} + \bigwedge^2 \underline{t}_{A^*}
\]

\[
\underline{t}_A^{\vee} \otimes \underline{t}_{A^*} \longrightarrow \underline{\mathcal{O}}_S
\quad \text{i.e.} \quad \underline{t}_{A^*} \longrightarrow \underline{t}_A .
\]

\[
\underbrace{\operatorname{Hom}(\underline{t}_A, \underline{t}_{A^*})}
\]

\[
\mathcal{L}_A \to \mathcal{L}_{A^*} \quad \text{\uncertain{homom.}}\ \ldots
\]

\[
({}_{p}A)_{\text{ét}} \to ({}_{p}A^*)_{\text{ét}} \quad \text{\uncertain{induit}}
\ \text{par}\ {}_{p}\tilde\varphi : {}_{p}A \to {}_{p}A^* \ldots
\]

\[
\omega_A = \struck{A}\, \det(\underline{t}_A)^{-1} = \bigwedge^{g} \underline{t}_A^{\vee}
\]

\[
\underline{t}_A^{(p)} \xrightarrow{\ \pi\ } \underline{t}_A
\]

\[
(\det \underline{t}_A)^{\otimes p} \longrightarrow \det \underline{t}_A
\]

\[
\sigma_A \in \Gamma(\det \underline{t}_A)^{1-p} = \Gamma\, \omega_A^{(p-1)}
\]

\[
0 \to \underline{t}_A^{\vee} \to \mathrm{DR}^1(A/S) \to \underline{t}_{A^*} \to 0
\]

\[
\det \underline{t}_A^{\vee} \otimes \det \underline{t}_{A^*} \simeq
\begin{array}{c}
\det \mathrm{DR}^1(A/S) \\
\wr \\
\mathrm{DR}^{2g}(A/S) \simeq \underline{\mathcal{O}}_S
\end{array}
\]

\[
\omega_A \cdot \omega_{A^*}^{\vee} \simeq \underline{\mathcal{O}}_S ,
\]

\[
\omega_A \simeq \omega_{A^*} .
\]

\[
\det({}_{p}A)_{\text{ét}} \simeq \det({}_{p}A^*)_{\text{ét}} .
\]

\[
T_p(A)_{\text{ét}} = \varprojlim \struck{\ill{}}\, ({}_{p^{\nu}}A)_{\text{ét}}
\]

\[
0 \to T_p(A)_{\mathrm{rad}} \to T_p(A) \to T_p(A)_{\text{ét}} \to 0 .
\]

\[
0 \to T_p(A^*)_{\mathrm{rad}} \to T_p(A^*) \to T_p(A^*)_{\text{ét}} \to 0
\]

\[
\begin{aligned}
T_p(A)_{\mathrm{rad}} &\simeq D(T_p(A^*)_{\text{ét}}) \\
T_p(A^*)_{\mathrm{rad}} &\simeq D(T_p(A)_{\text{ét}})
\end{aligned}
\]

\[
\det T_p(A)_{\text{ét}} \xrightarrow{\ \sim\ } \det T_p(A^*)
\]

\[
\det T_\ell(A) \overset{\alpha_\ell}{\simeq} \det T_\ell(A^*) \simeq
\mathbb{Z}_\ell(g) \quad ]
\]

\[
\det T_\ell(\tilde\varphi) : \det T_\ell(A)
\xrightarrow{\ \beta_\ell(\varphi)\ } \det T_\ell(A^*) ,
\]

\[
\beta_p(\varphi) = \det\bigl(T_p(\tilde\varphi)_{\text{ét}}\bigr) :
\det T_p(A)_{\text{ét}} \to \det T_p(A^*)_{\text{ét}}
\]

\[
\alpha_p(\varphi) = \beta_p(\varphi)\big/\sqrt{\deg\tilde\varphi} :
\det T_p(A)_{\text{ét}} \otimes \mathbb{Q}_p \to
\det T_p(A^*)_{\text{ét}} \otimes \mathbb{Q}_\ell
\]

\[
\Bigl[\det T_p(u)_{\text{ét}} : \det T_p(A)_{\text{ét}} \to
\det T_p(A)_{\text{ét}}\Bigr] \in \mathbb{Z}_p .
\]

\[
\left\{
\begin{array}{l}
\deg u \text{ est un carré parfait} \\
\det\bigl(T_p(u)_{\text{ét}}\bigr) = \sqrt{\deg u}
\end{array}
\right.
\]

\[
\deg u = \bigl(\det(T_p(u)_{\text{ét}})\bigr)^2 \quad \text{et} \quad
\det T_p(u)_{\text{ét}} \text{ est un entier} \geq 0 .
\]

\[
\begin{array}{c}
u'\tilde\varphi' = -\tilde\varphi u \\
\| \\
-u'\tilde\varphi
\end{array}
\qquad \text{i.e.} \qquad u' = \tilde\varphi u \tilde\varphi^{-1} .
\]

\[
\begin{array}{l}
\Delta(\tilde\psi)/\delta(\tilde\psi) = \Delta(\tilde\varphi)/\delta(\tilde\varphi) \\
\| \\
\Delta(\tilde\varphi)\cdot\Delta(u)/\delta(\tilde\varphi)\,\delta(u)
\end{array}
\]

\[
\Delta(u)/\delta(u) = 1 .
\]

\[
\Delta(u)\,\Delta(u^*) = \delta(u)^2 = \deg u
\]

\[
\Delta(u)^2 = \delta(u)^2
\]

\[
\Delta(u) = +\delta(u)
\]

\[
(*) \qquad \deg u = \det T_p(u)_{\text{ét}}\, \det T_p(u^*)_{\text{ét}} ,
\]

\[
F^* \to (+1, -1) ,
\]

\[
\Delta(u) = \Delta(v^2) = \Delta(v)^2 \in \mathbb{R}^2 = \mathbb{R}^+ ,
\]

\[
\text{i.e.}\quad
v_p\Bigl(\operatorname{card}\bigl(\operatorname{Coker}
T_p(\tilde\varphi)_{\text{ét}}\bigr)\Bigr) = v_p\bigl(\sqrt{\deg\tilde\varphi}\bigr) .
\]

\[
\operatorname{Coker} T_p(\tilde\varphi)_{\text{ét}} \simeq
\bigl(\operatorname{Ker} \tilde\varphi\bigr)(p)_{\text{ét}}
\]

\[
\begin{array}{c}
\bigl(\operatorname{card} (\operatorname{Ker}\tilde\varphi)(p)_{\text{ét}}\bigr)^2
= \operatorname{card} \operatorname{Ker}(\tilde\varphi(p)) \\
\hspace{7em}\| \\
\hspace{7em}(\deg\tilde\varphi)(p)
\end{array}
\]

\[
(*) \qquad \boxed{\det T_p(A)_{\text{ét}} \xrightarrow[\alpha_A]{\ \sim\ }
\det T_p(A^*)_{\text{ét}}}
\]

\[
\begin{array}{c}
\det T_p(u)_{\text{ét}} = \det T_p(u^{*-1})_{\text{ét}} \\
\hspace{7em}\| \\
\hspace{7em}\bigl(\det T_p(u^*)\bigr)^{-1}
\end{array}
\]

\[
\boxed{\det T_p(u)_{\text{ét}}\,\det T_p(u^*)_{\text{ét}} = 1}
\qquad \text{\emph{$u$ un automorphisme de $A$}} \quad ]
\]

\[
\det T_p(u) \neq \det T_p(u^*)
\]

\[
\boxed{\det T_p(u)_{\text{ét}}\, \det T_p(u^*)_{\text{ét}} = \deg u}
\]

\[
\alpha_{A^*} = \alpha_A^{-1}
\]