Cote n° 129 · pages 79–114
· 81 displayed formulas · Notes séminaire Montréal (Delale, Labute, Hakim, Messing) : copies de tapuscrits annotés (s.d.), notes manuscrites (s.d.).
Inventory dating : [1970]
Édition de démonstration
\[R_n : \underline{W}_{n+1} \longrightarrow \underline{W}_n, \qquad R_n(x_1, \dots, x_{n+1}) = (x_1, \dots, x_n)\]
LaTeX source
\[
R_n : \underline{W}_{n+1} \longrightarrow \underline{W}_n, \qquad R_n(x_1, \dots, x_{n+1}) = (x_1, \dots, x_n)
\]\[\Phi = (\phi_1, \dots, \phi_n) : \underline{W}_n \longrightarrow \underline{O}^n \qquad (= \underline{E}^n \text{ comme schéma})\]
LaTeX source
\[
\Phi = (\phi_1, \dots, \phi_n) : \underline{W}_n \longrightarrow \underline{O}^n \qquad (= \underline{E}^n \text{ comme schéma})
\]\[\phi_i(x_1, \dots, x_n) = \phi_i(x_1, \dots, x_i) = x_1^{p^{i-1}} + p x_2^{p^{i-2}} + \dots + p^{i-1} x_i\]
LaTeX source
\[
\phi_i(x_1, \dots, x_n) = \phi_i(x_1, \dots, x_i) = x_1^{p^{i-1}} + p x_2^{p^{i-2}} + \dots + p^{i-1} x_i
\]\[V_n = V : \underline{W}_n \longrightarrow \underline{W}_n, \qquad V(x_1, \dots, x_n) = (0, x_1, \dots, x_{n-1})\]
LaTeX source
\[
V_n = V : \underline{W}_n \longrightarrow \underline{W}_n, \qquad V(x_1, \dots, x_n) = (0, x_1, \dots, x_{n-1})
\]\[V : \underline{W} \longrightarrow \underline{W},\]
LaTeX source
\[
V : \underline{W} \longrightarrow \underline{W},
\]\[R_n \text{ ou } R : \underline{W}_{n+1} \longrightarrow \underline{W}_n, \qquad R(x_1, \dots, x_{n+1}) = (x_1, \dots, x_n)\]
LaTeX source
\[
R_n \text{ ou } R : \underline{W}_{n+1} \longrightarrow \underline{W}_n, \qquad R(x_1, \dots, x_{n+1}) = (x_1, \dots, x_n)
\]\[T_n \text{ ou } T : \underline{W}_n \longrightarrow \underline{W}_{n+1}, \qquad T(x_1, \dots, x_n) = (0, x_1, \dots, x_n)\]
LaTeX source
\[
T_n \text{ ou } T : \underline{W}_n \longrightarrow \underline{W}_{n+1}, \qquad T(x_1, \dots, x_n) = (0, x_1, \dots, x_n)
\]\[R_n T_n = V_n, \qquad T_n R_n = V_{n+1} .\]
LaTeX source
\[
R_n T_n = V_n, \qquad T_n R_n = V_{n+1} .
\]\[F \text{ ou } F_n : \underline{W}_n \longrightarrow \underline{W}_n, \qquad F(x_1, \dots, x_n) = (x_1^p, \dots, x_n^p),\]
LaTeX source
\[
F \text{ ou } F_n : \underline{W}_n \longrightarrow \underline{W}_n, \qquad F(x_1, \dots, x_n) = (x_1^p, \dots, x_n^p),
\]\[F : \underline{W} \longrightarrow \underline{W}\]
LaTeX source
\[
F : \underline{W} \longrightarrow \underline{W}
\]\[F_{\underline{F}_p} : \underline{W}_{F_p} \longrightarrow \underline{W}_{F_p} \qquad (\text{et variante avec } \underline{W}_{n\,\underline{F}_p})\]
LaTeX source
\[
F_{\underline{F}_p} : \underline{W}_{F_p} \longrightarrow \underline{W}_{F_p} \qquad (\text{et variante avec } \underline{W}_{n\,\underline{F}_p})
\]\[FV = p.\mathrm{id}, \qquad VF = p.\mathrm{id} \qquad (\emph{en car.}\ p) .\]
LaTeX source
\[
FV = p.\mathrm{id}, \qquad VF = p.\mathrm{id} \qquad (\emph{en car.}\ p) .
\]\[\underline{W}_{\to} = \varinjlim \underline{W}_n, \quad \text{morphismes de transition du type } D .\]
LaTeX source
\[
\underline{W}_{\to} = \varinjlim \underline{W}_n, \quad \text{morphismes de transition du type } D .
\]\[\underline{W}_{\to} = \underline{E}^{(\underline{N})} .\]
LaTeX source
\[
\underline{W}_{\to} = \underline{E}^{(\underline{N})} .
\]\[u : \underline{W}_{\to\,\mathbb{F}_p} \longrightarrow \varinjlim\, (\underline{W}_{n\,\mathbb{F}_p}, p),\]
LaTeX source
\[
u : \underline{W}_{\to\,\mathbb{F}_p} \longrightarrow \varinjlim\, (\underline{W}_{n\,\mathbb{F}_p}, p),
\]\[W = \underline{W}(k) = \varprojlim W_n \qquad (W_n = \underline{W}_n(k))\]
LaTeX source
\[
W = \underline{W}(k) = \varprojlim W_n \qquad (W_n = \underline{W}_n(k))
\]\[K = W[1/p]\]
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\[ K = W[1/p] \]
\[u : \underline{W}_{\to}(k) \xrightarrow{\ \sim\ } K/W .\]
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\[
u : \underline{W}_{\to}(k) \xrightarrow{\ \sim\ } K/W .
\]\[G \simeq G_{\mathrm{et}} \times G_{\mathrm{mult}} \times G_{\mathrm{bi}}\]
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\[
G \simeq G_{\mathrm{et}} \times G_{\mathrm{mult}} \times G_{\mathrm{bi}}
\]\[\left\lbrace
\begin{array}{ll}
F_G \text{ nilpotent} \Longleftrightarrow G_{\mathrm{et}} = 0 \text{ i.e.\ } G^{*} \text{ unipotent} & \quad F_G \text{ isom} \Longleftrightarrow G \text{ ét.} \\
V_G \text{ nilpotent} \Longleftrightarrow G_{\mathrm{mult}} = 0 \text{ i.e.\ } G \text{ unipotent} & \quad V_G \text{ isom} \Longleftrightarrow G \text{ de m.}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{ll}
F_G \text{ nilpotent} \Longleftrightarrow G_{\mathrm{et}} = 0 \text{ i.e.\ } G^{*} \text{ unipotent} & \quad F_G \text{ isom} \Longleftrightarrow G \text{ ét.} \\
V_G \text{ nilpotent} \Longleftrightarrow G_{\mathrm{mult}} = 0 \text{ i.e.\ } G \text{ unipotent} & \quad V_G \text{ isom} \Longleftrightarrow G \text{ de m.}
\end{array}
\right.
\]\[p\text{-}\mathrm{Grf}(k) \xrightarrow{\;\approx\;} \left\lbrace
\begin{array}{l}
\text{catégorie des modules de longueur} \\
\text{finie } M \text{ sur } W, \text{ munis de} \\
F_M, V_M : M \to M \text{ satisfaisant} \\
\quad F\lambda = \lambda^{\sigma} F,\ \lambda V = V \lambda^{\sigma} \\
\quad FV = p\cdot\mathrm{id},\ FV = p\cdot\mathrm{id}
\end{array}
\right.
\qquad G \longmapsto D^{*}(G)\]
LaTeX source
\[
p\text{-}\mathrm{Grf}(k) \xrightarrow{\;\approx\;} \left\lbrace
\begin{array}{l}
\text{catégorie des modules de longueur} \\
\text{finie } M \text{ sur } W, \text{ munis de} \\
F_M, V_M : M \to M \text{ satisfaisant} \\
\quad F\lambda = \lambda^{\sigma} F,\ \lambda V = V \lambda^{\sigma} \\
\quad FV = p\cdot\mathrm{id},\ FV = p\cdot\mathrm{id}
\end{array}
\right.
\qquad G \longmapsto D^{*}(G)
\]\[F_{D^{*}(G)} = D^{*}(F_G), \qquad V_{D^{*}G} = D^{*}(V_G)\]
LaTeX source
\[
F_{D^{*}(G)} = D^{*}(F_G), \qquad V_{D^{*}G} = D^{*}(V_G)
\]\[\left\lbrace
\begin{array}{lcl}
G \text{ unipotent} & \Longleftrightarrow & V_M \text{ nilpotent} \\
G^{*} \text{ unipotent} & \Longleftrightarrow & F_M \text{ nilpotent} \\
G \text{ étale} & \Longleftrightarrow & F_M \text{ isom} \\
G^{*} \text{ étale} & \Longleftrightarrow & V_M \text{ isom}
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{lcl}
G \text{ unipotent} & \Longleftrightarrow & V_M \text{ nilpotent} \\
G^{*} \text{ unipotent} & \Longleftrightarrow & F_M \text{ nilpotent} \\
G \text{ étale} & \Longleftrightarrow & F_M \text{ isom} \\
G^{*} \text{ étale} & \Longleftrightarrow & V_M \text{ isom}
\end{array}
\right.
\]\[G \longmapsto \underline{\mathrm{Hom}}_{\mathrm{gr}}(\mu, G) \overset{\mathrm{dfn}}{=} \bigl(\underline{\mathrm{Hom}}_{\mathrm{gr}}(\mu(n), G(n))\bigr)_{n \geq 1} \quad \text{(abus de notation)} .\]
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\[
G \longmapsto \underline{\mathrm{Hom}}_{\mathrm{gr}}(\mu, G) \overset{\mathrm{dfn}}{=} \bigl(\underline{\mathrm{Hom}}_{\mathrm{gr}}(\mu(n), G(n))\bigr)_{n \geq 1} \quad \text{(abus de notation)} .
\]\[0 \to G' \to G \to G'' \to 0\]
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\[ 0 \to G' \to G \to G'' \to 0 \]
\[0 \to G'(1) \to G(1) \to G''(1) \to G'/pG' \to G/pG \to G''/pG'' \to 0\]
LaTeX source
\[ 0 \to G'(1) \to G(1) \to G''(1) \to G'/pG' \to G/pG \to G''/pG'' \to 0 \]
\[0 \to G'(1) \to G(1) \to G''(1) \to 0 ,\]
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\[ 0 \to G'(1) \to G(1) \to G''(1) \to 0 , \]
\[\overline{G} \overset{\mathrm{dfn}}{=} \mathrm{Inf}^{\infty}(G) = \varinjlim G(n)[n]\]
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\[
\overline{G} \overset{\mathrm{dfn}}{=} \mathrm{Inf}^{\infty}(G) = \varinjlim G(n)[n]
\]\[\mathrm{Inf}^k G = \mathrm{Inf}^k \overline{G} = \mathrm{Inf}^k G(n) \subset G(n)[n] \subset G(n) .\]
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\[
\mathrm{Inf}^k G = \mathrm{Inf}^k \overline{G} = \mathrm{Inf}^k G(n) \subset G(n)[n] \subset G(n) .
\]\[(*) \qquad E(G_0, S) \longrightarrow E(G_0(n), S)\]
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\[ (*) \qquad E(G_0, S) \longrightarrow E(G_0(n), S) \]
\[\begin{array}{ccc}
X & \longleftarrow & X' \\
{\scriptstyle f}\downarrow & & \downarrow \\
S & \xleftarrow{\;g\;} & S'
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
X & \longleftarrow & X' \\
{\scriptstyle f}\downarrow & & \downarrow \\
S & \xleftarrow{\;g\;} & S'
\end{array}
\]\[\begin{array}{ccc}
X & \xleftarrow{\;f_0\;} & Y_0 \\
\big| & \overset{f\,?}{\nwarrow} & \big\uparrow \\
S & \longleftarrow & Y
\end{array}
\qquad Y_0 = V(\mathcal{J}),\ \mathcal{J}^2 = 0\]
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\[
\begin{array}{ccc}
X & \xleftarrow{\;f_0\;} & Y_0 \\
\big| & \overset{f\,?}{\nwarrow} & \big\uparrow \\
S & \longleftarrow & Y
\end{array}
\qquad Y_0 = V(\mathcal{J}),\ \mathcal{J}^2 = 0
\]\[\left\lbrace
\begin{array}{l}
\mathrm{Ext}_{\mathcal{O}_S}(\mathcal{O}_X, \mathcal{J}) \simeq \mathrm{Ext}^1_{\mathcal{O}_X}(L^{X/S}_{\bullet}, \mathcal{J}) \\[2pt]
\text{autom.\ d'une extension} \simeq \mathrm{Ext}^0_{\mathcal{O}_X}(L^{X/S}_{\bullet}, \mathcal{J}) = \mathrm{Hom}(\Omega^1_{X/S}, \mathcal{J})
\end{array}
\right.\]
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\[
\left\lbrace
\begin{array}{l}
\mathrm{Ext}_{\mathcal{O}_S}(\mathcal{O}_X, \mathcal{J}) \simeq \mathrm{Ext}^1_{\mathcal{O}_X}(L^{X/S}_{\bullet}, \mathcal{J}) \\[2pt]
\text{autom.\ d'une extension} \simeq \mathrm{Ext}^0_{\mathcal{O}_X}(L^{X/S}_{\bullet}, \mathcal{J}) = \mathrm{Hom}(\Omega^1_{X/S}, \mathcal{J})
\end{array}
\right.
\]\[\begin{array}{ccc}
X_0 & \longleftarrow & X\,? \\
\downarrow & & \downarrow \\
S_0 & \longleftarrow & S
\end{array}
\qquad S_0 = V(\mathcal{J}),\ \mathcal{J}^2 = 0\]
LaTeX source
\[
\begin{array}{ccc}
X_0 & \longleftarrow & X\,? \\
\downarrow & & \downarrow \\
S_0 & \longleftarrow & S
\end{array}
\qquad S_0 = V(\mathcal{J}),\ \mathcal{J}^2 = 0
\]\[\begin{array}{ll}
\ell^{G/S}_{\bullet} = \mathbb{L}e^{*}(L^{G/S}_{\bullet}) & \quad \text{complexe de co-Lie} \\
\check{\ell}^{G/S}_{\bullet} = \mathbb{R}\underline{\mathrm{Hom}}(\ell^{G/S}_{\bullet}, \mathcal{O}_S) & \quad \text{— id. — Lie}
\end{array}\]
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\[
\begin{array}{ll}
\ell^{G/S}_{\bullet} = \mathbb{L}e^{*}(L^{G/S}_{\bullet}) & \quad \text{complexe de co-Lie} \\
\check{\ell}^{G/S}_{\bullet} = \mathbb{R}\underline{\mathrm{Hom}}(\ell^{G/S}_{\bullet}, \mathcal{O}_S) & \quad \text{— id. — Lie}
\end{array}
\]\[\left\lbrace
\begin{array}{ll}
\ell^{G/S}_{\bullet} \text{ parfait d'ampl.\ parfaite} & \subset [-1,0] \\
\check{\ell}^{G/S}_{\bullet} \quad \text{— id. —} & \subset [0,1]
\end{array}
\right.\]
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\[
\left\lbrace
\begin{array}{ll}
\ell^{G/S}_{\bullet} \text{ parfait d'ampl.\ parfaite} & \subset [-1,0] \\
\check{\ell}^{G/S}_{\bullet} \quad \text{— id. —} & \subset [0,1]
\end{array}
\right.
\]\[\left\lbrace
\begin{array}{l}
\underline{\omega}_G = \underline{H}_0(\ell^{G/S}_{\bullet}) \\
\underline{n}_G = \underline{H}_1(\ell^{G/S}_{\bullet}) \\
\underline{t}_G = \underline{H}^0(\check{\ell}^{G/S}_{\bullet}) \\
\underline{\nu}_G = \underline{H}^1(\check{\ell}^{G/S}_{\bullet})
\end{array}
\right.
\qquad t_G = \check{\omega}_G, \quad \underline{n}_G = \check{\nu}_G\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\underline{\omega}_G = \underline{H}_0(\ell^{G/S}_{\bullet}) \\
\underline{n}_G = \underline{H}_1(\ell^{G/S}_{\bullet}) \\
\underline{t}_G = \underline{H}^0(\check{\ell}^{G/S}_{\bullet}) \\
\underline{\nu}_G = \underline{H}^1(\check{\ell}^{G/S}_{\bullet})
\end{array}
\right.
\qquad t_G = \check{\omega}_G, \quad \underline{n}_G = \check{\nu}_G
\]\[\mathrm{rg}\, \ell^{G}_{\bullet} = \dim G \qquad (= \mathrm{rg}\, \underline{\omega}_G - \mathrm{rg}\, n_G \text{ quand } \underline{\omega}_G, \underline{n}_G \text{ sont loc.\ libres}).\]
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\[
\mathrm{rg}\, \ell^{G}_{\bullet} = \dim G \qquad (= \mathrm{rg}\, \underline{\omega}_G - \mathrm{rg}\, n_G \text{ quand } \underline{\omega}_G, \underline{n}_G \text{ sont loc.\ libres}).
\]\[0 \to G' \to G \to G'' \to 0\]
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\[ 0 \to G' \to G \to G'' \to 0 \]
\[\begin{array}{ccc}
& \ell^{G'}_{\bullet} & \\
{\scriptstyle 1}\swarrow & & \nwarrow \\
\ell^{G''}_{\bullet} & \longrightarrow & \ell^{G}_{\bullet}
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
& \ell^{G'}_{\bullet} & \\
{\scriptstyle 1}\swarrow & & \nwarrow \\
\ell^{G''}_{\bullet} & \longrightarrow & \ell^{G}_{\bullet}
\end{array}
\]\[0 \to n_{G''} \to n_{G} \to n_{G'} \to \omega_{G''} \to \omega_{G} \to \omega_{G'} \to 0\]
LaTeX source
\[
0 \to n_{G''} \to n_{G} \to n_{G'} \to \omega_{G''} \to \omega_{G} \to \omega_{G'} \to 0
\]\[\boxed{\mathbb{R}\underline{\mathrm{Hom}}(\ell^{G}_{\bullet}, \mathcal{J}) \simeq \tau_{\leq 1}\, \mathbb{R}\underline{\mathrm{Hom}}(G^{*}, \mathcal{J})}\]
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\[
\boxed{\mathbb{R}\underline{\mathrm{Hom}}(\ell^{G}_{\bullet}, \mathcal{J}) \simeq \tau_{\leq 1}\, \mathbb{R}\underline{\mathrm{Hom}}(G^{*}, \mathcal{J})}
\]\[\begin{array}{cccccccccccccc}
0 \to & ({}_F M)^{(p)} & \to & ({}_p M)^{(p)} & \xrightarrow{F} & {}_V M & \to & M_F & \xrightarrow{V} & (M_p)^{(p)} & \to & (M_V)^{(p)} & \to 0 \\
& \wr\| & & \wr\| & & \wr\| & & \wr\| & & \wr\| & & \wr\| & \\
0 \to & \underline{n}_G & \to & D^{*}(G_p)^{\vee} & \to & t_{G^{*}} & \to & \omega_G & \to & D^{*}(G_p) & \to & \nu_{G^{*}} & \to 0 ,
\end{array}\]
LaTeX source
\[
\begin{array}{cccccccccccccc}
0 \to & ({}_F M)^{(p)} & \to & ({}_p M)^{(p)} & \xrightarrow{F} & {}_V M & \to & M_F & \xrightarrow{V} & (M_p)^{(p)} & \to & (M_V)^{(p)} & \to 0 \\
& \wr\| & & \wr\| & & \wr\| & & \wr\| & & \wr\| & & \wr\| & \\
0 \to & \underline{n}_G & \to & D^{*}(G_p)^{\vee} & \to & t_{G^{*}} & \to & \omega_G & \to & D^{*}(G_p) & \to & \nu_{G^{*}} & \to 0 ,
\end{array}
\]\[\ell^{G}_{\bullet} \simeq \mathcal{D}/F\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M, \qquad \mathcal{D}/F\mathcal{D} \simeq k_{\sigma}[V]\]
LaTeX source
\[
\ell^{G}_{\bullet} \simeq \mathcal{D}/F\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M, \qquad \mathcal{D}/F\mathcal{D} \simeq k_{\sigma}[V]
\]\[\begin{array}{ccc}
& \check{\ell}^{G^{*}}_{\bullet} & \\
{\scriptstyle ?}\swarrow & & \nwarrow \\
\ell^{G}_{\bullet}[-1] & \longrightarrow & \Delta^{*}(G)
\end{array}
\qquad \text{et} \qquad
\begin{array}{ccc}
& \mathcal{D}/V\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M & \\
{\scriptstyle 1}\swarrow & & \nwarrow \\
\mathcal{D}/F\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M & \longrightarrow & \mathcal{D}/p\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M
\end{array}\]
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\[
\begin{array}{ccc}
& \check{\ell}^{G^{*}}_{\bullet} & \\
{\scriptstyle ?}\swarrow & & \nwarrow \\
\ell^{G}_{\bullet}[-1] & \longrightarrow & \Delta^{*}(G)
\end{array}
\qquad \text{et} \qquad
\begin{array}{ccc}
& \mathcal{D}/V\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M & \\
{\scriptstyle 1}\swarrow & & \nwarrow \\
\mathcal{D}/F\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M & \longrightarrow & \mathcal{D}/p\mathcal{D} \overset{\mathbb{L}}{\otimes}_{\mathcal{D}} M
\end{array}
\]\[\begin{array}{ccc}
G_0 & \xrightarrow{\;u_0\;} & H_0 \\
& \searrow \swarrow & \\
& S_0 &
\end{array}
\qquad\qquad
\begin{array}{ccc}
G & \overset{u\,?}{\dashrightarrow} & H \\
& \searrow \swarrow & \\
& S &
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
G_0 & \xrightarrow{\;u_0\;} & H_0 \\
& \searrow \swarrow & \\
& S_0 &
\end{array}
\qquad\qquad
\begin{array}{ccc}
G & \overset{u\,?}{\dashrightarrow} & H \\
& \searrow \swarrow & \\
& S &
\end{array}
\]\[\begin{array}{ccc}
G_0 & \longleftarrow & G\,? \\
\big| & & \big| \\
S_0 & \longleftarrow & S
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
G_0 & \longleftarrow & G\,? \\
\big| & & \big| \\
S_0 & \longleftarrow & S
\end{array}
\]\[\underline{\omega}_{G(n)} = \underline{H}_0(L^{G(n)}_{\bullet}) \quad\text{et}\quad \underline{n}_{G(n)} = \underline{H}_1(L^{G(n)}_{\bullet})\]
LaTeX source
\[
\underline{\omega}_{G(n)} = \underline{H}_0(L^{G(n)}_{\bullet}) \quad\text{et}\quad \underline{n}_{G(n)} = \underline{H}_1(L^{G(n)}_{\bullet})
\]\[\underline{\omega}_{G(n)} \xrightarrow{\;\sim\;} \underline{\omega}_{G(n')}\]
LaTeX source
\[
\underline{\omega}_{G(n)} \xrightarrow{\;\sim\;} \underline{\omega}_{G(n')}
\]\[\underline{n}_{G(n)} \xrightarrow{\;0\;} \underline{n}_{G(n')}\]
LaTeX source
\[
\underline{n}_{G(n)} \xrightarrow{\;0\;} \underline{n}_{G(n')}
\]\[\underline{n}_{G(n')} \xrightarrow{\;\sim\;} \underline{n}_{G(n)}
\qquad\text{et}\qquad
\underline{\omega}_{G(n')} \xrightarrow{\;0\;} \underline{\omega}_{G(n)}\]
LaTeX source
\[
\underline{n}_{G(n')} \xrightarrow{\;\sim\;} \underline{n}_{G(n)}
\qquad\text{et}\qquad
\underline{\omega}_{G(n')} \xrightarrow{\;0\;} \underline{\omega}_{G(n)}
\]\[\left\lbrace
\begin{array}{l}
\underline{\mathrm{Ext}}^1_{\Lambda_n}(G, M) = 0 \\
\underline{\mathrm{Ext}}^2_{\Lambda_n}(G, M) \xrightarrow{\;\sim\;} \underline{\mathrm{Ext}}^2_{\mathbb{Z}}(G, M)
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\underline{\mathrm{Ext}}^1_{\Lambda_n}(G, M) = 0 \\
\underline{\mathrm{Ext}}^2_{\Lambda_n}(G, M) \xrightarrow{\;\sim\;} \underline{\mathrm{Ext}}^2_{\mathbb{Z}}(G, M)
\end{array}
\right.
\]\[\mathrm{gr}^i_f(A) = f^i A / f^{i+1} A\]
LaTeX source
\[
\mathrm{gr}^i_f(A) = f^i A / f^{i+1} A
\]\[\theta : \mathrm{gr}^0_f(A)[t] \underset{\text{surj}}{\longrightarrow} \mathrm{gr}^{\bullet}_f(A)
\qquad
\theta^i : \mathrm{gr}^0_f(A) \underset{\text{surj}}{\xrightarrow{\;f^i\;}} \mathrm{gr}^i_f(A)\]
LaTeX source
\[
\theta : \mathrm{gr}^0_f(A)[t] \underset{\text{surj}}{\longrightarrow} \mathrm{gr}^{\bullet}_f(A)
\qquad
\theta^i : \mathrm{gr}^0_f(A) \underset{\text{surj}}{\xrightarrow{\;f^i\;}} \mathrm{gr}^i_f(A)
\]\[\begin{array}{ccc}
\text{(ii)} & \Longrightarrow & \text{(ii bis)} \\
\Downarrow & & \Downarrow \\
\text{(i)} \Longleftrightarrow \text{(i bis)} & \Longrightarrow & \text{(ii bis)}
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
\text{(ii)} & \Longrightarrow & \text{(ii bis)} \\
\Downarrow & & \Downarrow \\
\text{(i)} \Longleftrightarrow \text{(i bis)} & \Longrightarrow & \text{(ii bis)}
\end{array}
\]\[\mathrm{Ker}\, f^j \subset \mathrm{Ker}\, f^{n-1} \subset \mathrm{Im}\, f \quad \text{donc} \quad \mathrm{Ker}\, f^j = f(\mathrm{Ker}\, f^{j+1})\]
LaTeX source
\[
\mathrm{Ker}\, f^j \subset \mathrm{Ker}\, f^{n-1} \subset \mathrm{Im}\, f \quad \text{donc} \quad \mathrm{Ker}\, f^j = f(\mathrm{Ker}\, f^{j+1})
\]\[\mathrm{Ker}\, f^j = f\, \mathrm{Ker}\, f^{j+1} = f^2\, \mathrm{Ker}\, f^{j+2} = \cdots = f^{n-j-1}\, \mathrm{Ker}\, f^{n-1} = f^{n-j-1+1} A\]
LaTeX source
\[
\mathrm{Ker}\, f^j = f\, \mathrm{Ker}\, f^{j+1} = f^2\, \mathrm{Ker}\, f^{j+2} = \cdots = f^{n-j-1}\, \mathrm{Ker}\, f^{n-1} = f^{n-j-1+1} A
\]\[\mathrm{Ker}\, f^{j-1} \subset \mathrm{Ker}\, f^{j} \subset \mathrm{Im}\, f^{n-j} \subset \mathrm{Im}\, f \quad \text{d'où}\]
LaTeX source
\[
\mathrm{Ker}\, f^{j-1} \subset \mathrm{Ker}\, f^{j} \subset \mathrm{Im}\, f^{n-j} \subset \mathrm{Im}\, f \quad \text{d'où}
\]\[\mathrm{Ker}\, f^{j-1} = f\bigl(\mathrm{Ker}\, f^{j}\bigr) \subset f\bigl(\mathrm{Im}\, f^{n-j}\bigr) = \mathrm{Im}\, f^{n-j+1}\]
LaTeX source
\[
\mathrm{Ker}\, f^{j-1} = f\bigl(\mathrm{Ker}\, f^{j}\bigr) \subset f\bigl(\mathrm{Im}\, f^{n-j}\bigr) = \mathrm{Im}\, f^{n-j+1}
\]\[x \longmapsto x = f^{n-j}(y), \qquad f^{j-1}\bigl(f^{n-j} y\bigr) = 0, \qquad f^{n-1-j}\, y = f^{n-j} z\]
LaTeX source
\[
x \longmapsto x = f^{n-j}(y), \qquad f^{j-1}\bigl(f^{n-j} y\bigr) = 0, \qquad f^{n-1-j}\, y = f^{n-j} z
\]\[S_j \Longrightarrow S_1 \quad \text{i.e.\ } \mathrm{Ker}\, f \subset \mathrm{Im}\, f^{n-1}\]
LaTeX source
\[
S_j \Longrightarrow S_1 \quad \text{i.e.\ } \mathrm{Ker}\, f \subset \mathrm{Im}\, f^{n-1}
\]\[A/f^{n-1}A \xrightarrow{\;f\;} fA .\]
LaTeX source
\[
A/f^{n-1}A \xrightarrow{\;f\;} fA .
\]\[G_2/G_1 \xrightarrow{\ \sim\ } G_1 .\]
LaTeX source
\[
G_2/G_1 \xrightarrow{\ \sim\ } G_1 .
\]\[G'_2/G'_1 \xrightarrow{\ \sim\ } G'_1 .\]
LaTeX source
\[
G'_2/G'_1 \xrightarrow{\ \sim\ } G'_1 .
\]\[\mathrm{Ker}\, \alpha'' \simeq \underbrace{(G'_2 \cap G_1)}_{\mathrm{Ker}\, p_{G'_2}}/G'_1
\simeq \mathrm{Ker}\,(G'_2/G'_1 \to G'_1)\]
LaTeX source
\[
\mathrm{Ker}\, \alpha'' \simeq \underbrace{(G'_2 \cap G_1)}_{\mathrm{Ker}\, p_{G'_2}}/G'_1
\simeq \mathrm{Ker}\,(G'_2/G'_1 \to G'_1)
\]\[G'_2 = \overline{G'_{2\eta}}, \qquad
G'_1 = \overline{G'_{1\eta}} = \overline{G'_{2\eta} \cap G_{1\eta}}
\subset \overline{G'_{2\eta}} \cap \overline{G_{1\eta}}
= G'_2 \cap G_1 = \mathrm{Ker}(p_{G'_2})\]
LaTeX source
\[
G'_2 = \overline{G'_{2\eta}}, \qquad
G'_1 = \overline{G'_{1\eta}} = \overline{G'_{2\eta} \cap G_{1\eta}}
\subset \overline{G'_{2\eta}} \cap \overline{G_{1\eta}}
= G'_2 \cap G_1 = \mathrm{Ker}(p_{G'_2})
\]\[\mathrm{Ker}\, f^j = \mathrm{Im} \qquad
\mathrm{Ker}\, f = \mathrm{Im}\, f^{n-1}\]
LaTeX source
\[
\mathrm{Ker}\, f^j = \mathrm{Im} \qquad
\mathrm{Ker}\, f = \mathrm{Im}\, f^{n-1}
\]\[\mathrm{Ker}\, f^j = \mathrm{Im}\, f^{n-j}
\quad\overset{?}{\Longleftarrow}\quad
\mathrm{Ker}\, f^{j+1} = \mathrm{Im}\, f^{n-j-1}\]
LaTeX source
\[
\mathrm{Ker}\, f^j = \mathrm{Im}\, f^{n-j}
\quad\overset{?}{\Longleftarrow}\quad
\mathrm{Ker}\, f^{j+1} = \mathrm{Im}\, f^{n-j-1}
\]\[(f^j)^{-1}(f^{n-j-1}A) = f^{n-j-1}A\]
LaTeX source
\[
(f^j)^{-1}(f^{n-j-1}A) = f^{n-j-1}A
\]\[(\mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z})_\eta \simeq G_{2\eta}\]
LaTeX source
\[
(\mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z})_\eta \simeq G_{2\eta}
\]\[V = uF, \qquad F = u^{-1}V, \qquad V^2 = up, \qquad F^2 = u^{-1}p\]
LaTeX source
\[
V = uF, \qquad F = u^{-1}V, \qquad V^2 = up, \qquad F^2 = u^{-1}p
\]\[0 \to N \to G \xrightarrow{u} H \to 0
\quad \not\Longrightarrow \quad
\mathrm{Ker}\ \text{\uncertain{est}}\ \text{\uncertain{lisse}}\]
LaTeX source
\[
0 \to N \to G \xrightarrow{u} H \to 0
\quad \not\Longrightarrow \quad
\mathrm{Ker}\ \text{\uncertain{est}}\ \text{\uncertain{lisse}}
\]\[\Phi(G) \to \Phi(H) \quad \text{est surjectif}\]
LaTeX source
\[
\Phi(G) \to \Phi(H) \quad \text{est surjectif}
\]\[G(n) = {}_{p^n}A, \qquad G = {}_{p^\infty}A .\]
LaTeX source
\[
G(n) = {}_{p^n}A, \qquad G = {}_{p^\infty}A .
\]\[S_0 \hookrightarrow S \qquad A \longmapsto (A_0, {}_{p^\infty}A, \varphi)\]
LaTeX source
\[
S_0 \hookrightarrow S \qquad A \longmapsto (A_0, {}_{p^\infty}A, \varphi)
\]\[\left\lbrace
\begin{array}{l}
(x+y)^{(n)} = \sum_{i+j=n} x^{(i)} y^{(j)} \\
(\lambda x)^{(n)} = \lambda^n x^{(n)} \\
\dots
\end{array}\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
(x+y)^{(n)} = \sum_{i+j=n} x^{(i)} y^{(j)} \\
(\lambda x)^{(n)} = \lambda^n x^{(n)} \\
\dots
\end{array}\right.
\]\[\mathrm{BT}(S)^{\circ} \longrightarrow \mathrm{Cris\,Mod\,loc\,lib}(S),
\qquad G \longmapsto \mathbb{D}(G)\]
LaTeX source
\[
\mathrm{BT}(S)^{\circ} \longrightarrow \mathrm{Cris\,Mod\,loc\,lib}(S),
\qquad G \longmapsto \mathbb{D}(G)
\]\[F_M V_M = p\,\mathrm{id}, \qquad V_M F_M = p\,\mathrm{id}\]
LaTeX source
\[
F_M V_M = p\,\mathrm{id}, \qquad V_M F_M = p\,\mathrm{id}
\]\[\mathrm{BT}(S)^{\circ} \longrightarrow \mathrm{Cris\,Dieud}(S)\]
LaTeX source
\[
\mathrm{BT}(S)^{\circ} \longrightarrow \mathrm{Cris\,Dieud}(S)
\]\[0 \to \omega_{G^{\ill{}}} \to \mathbb{D}(G)_S \to \check{\omega}_{G^*} \to 0\]
LaTeX source
\[
0 \to \omega_{G^{\ill{}}} \to \mathbb{D}(G)_S \to \check{\omega}_{G^*} \to 0
\]\[G \longmapsto (G_0,\ \text{filtration sur}\ \mathbb{D}(G_0)_A)\]
LaTeX source
\[
G \longmapsto (G_0,\ \text{filtration sur}\ \mathbb{D}(G_0)_A)
\]