Cote n° 31 · pages 2–100
· 166 displayed formulas · SGA A [SGA 4] (Résidus de rédaction) : notes manuscrites (s.d.).
Inventory dating : [1963-1973]
Édition de démonstration
\[G' \subset G'' \subset G\]
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\[ G' \subset G'' \subset G \]
\[0 \longrightarrow G \longrightarrow T \longrightarrow T' \longrightarrow 0\]
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\[ 0 \longrightarrow G \longrightarrow T \longrightarrow T' \longrightarrow 0 \]
\[0 \longrightarrow {}_{p}T \longrightarrow T \xrightarrow{\;p\;} T \longrightarrow 0\]
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\[
0 \longrightarrow {}_{p}T \longrightarrow T \xrightarrow{\;p\;} T \longrightarrow 0
\]\[H^i({}_pT) \longrightarrow {}_pH^i(T) \longrightarrow 0\]
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\[
H^i({}_pT) \longrightarrow {}_pH^i(T) \longrightarrow 0
\]\[0 \longrightarrow H^i(T)_p \longrightarrow H^{i+1}({}_pT)\]
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\[
0 \longrightarrow H^i(T)_p \longrightarrow H^{i+1}({}_pT)
\]\[H^i(k, \textstyle\prod_{k'/k} G') \simeq H^i(k, G),\]
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\[
H^i(k, \textstyle\prod_{k'/k} G') \simeq H^i(k, G),
\]\[\textstyle\prod_{k'/k} G_{k'} \longrightarrow G\]
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\[
\textstyle\prod_{k'/k} G_{k'} \longrightarrow G
\]\[H^{n+1}(\textstyle\prod_{k'/k} G_{k'}) \longrightarrow H^{n+1}(G) \longrightarrow 0
\quad\text{surjectif,}\]
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\[
H^{n+1}(\textstyle\prod_{k'/k} G_{k'}) \longrightarrow H^{n+1}(G) \longrightarrow 0
\quad\text{surjectif,}
\]\[\simeq H^{n+1}(G_{k'})\]
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\[
\simeq H^{n+1}(G_{k'})
\]\[\cdot \longrightarrow \mu_p \longrightarrow G_m \longrightarrow G_m \longrightarrow 0\]
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\[ \cdot \longrightarrow \mu_p \longrightarrow G_m \longrightarrow G_m \longrightarrow 0 \]
\[cd_p^{\text{add}}(k) =
\begin{cases}
0 & \text{si } k \text{ parfait} \\
1 & \text{si } k \text{ imparfait}
\end{cases}\]
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\[
cd_p^{\text{add}}(k) =
\begin{cases}
0 & \text{si } k \text{ parfait} \\
1 & \text{si } k \text{ imparfait}
\end{cases}
\]\[0 \longrightarrow {}_{f^{n-1}}E_n / {}_fE_n \longrightarrow E_n / {}_fE_n
\longrightarrow E_n / {}_{f^{n-1}}E_n \longrightarrow 0\]
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\[
0 \longrightarrow {}_{f^{n-1}}E_n / {}_fE_n \longrightarrow E_n / {}_fE_n
\longrightarrow E_n / {}_{f^{n-1}}E_n \longrightarrow 0
\]\[E_n / {}_{f^{n-1}}E_n \simeq A/fA = B\]
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\[
E_n / {}_{f^{n-1}}E_n \simeq A/fA = B
\]\[E_n = {}_{f^n}E_{n+1},\]
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\[
E_n = {}_{f^n}E_{n+1},
\]\[f(x,h) = \lambda(\xi_n, f)\]
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\[ f(x,h) = \lambda(\xi_n, f) \]
\[\begin{align*}
E_{n+1}/fE_{n+1} &\simeq (E_n \times A) \big/ \big( f(E_n \times A) + A(\xi_n, f) \big) \\
&\simeq (E_n/fE_n \times A/fA) \big/ A(\bar{\xi}_n, f) \\
&\simeq \big( [B \times (B/gB)^{n-1}] \times B \big) \big/ B(g, 0, 0) \\
&\simeq B/gB \times (B/gB)^{n-1} \times B \simeq B \times (B/gB)^{n}.
\end{align*}\]
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\begin{align*}
E_{n+1}/fE_{n+1} &\simeq (E_n \times A) \big/ \big( f(E_n \times A) + A(\xi_n, f) \big) \\
&\simeq (E_n/fE_n \times A/fA) \big/ A(\bar{\xi}_n, f) \\
&\simeq \big( [B \times (B/gB)^{n-1}] \times B \big) \big/ B(g, 0, 0) \\
&\simeq B/gB \times (B/gB)^{n-1} \times B \simeq B \times (B/gB)^{n}.
\end{align*}\[E_n \xrightarrow{\;u_n\;} E_{n+1}\]
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\[
E_n \xrightarrow{\;u_n\;} E_{n+1}
\]\[u_n \otimes_A B : E_n/fE_n \longrightarrow E_{n+1}/fE_{n+1}.\]
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\[
u_n \otimes_A B : E_n/fE_n \longrightarrow E_{n+1}/fE_{n+1}.
\]\[\varinjlim Q_n = \varinjlim\, Q_n \otimes_B B/gB \simeq \varinjlim\, (B/gB)^{n-1}
\simeq (B/gB)^{(\mathbb{N})}.\]
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\[
\varinjlim Q_n = \varinjlim\, Q_n \otimes_B B/gB \simeq \varinjlim\, (B/gB)^{n-1}
\simeq (B/gB)^{(\mathbb{N})}.
\]\[P = \underline{\mathrm{Isom}}_{S\text{-gr}}(H,G) \subset
\underline{\mathrm{Hom}}_{S\text{-gr}}(H,G) = Q\]
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\[
P = \underline{\mathrm{Isom}}_{S\text{-gr}}(H,G) \subset
\underline{\mathrm{Hom}}_{S\text{-gr}}(H,G) = Q
\]\[Q \simeq \underline{\mathrm{Hom}}_{S\text{-gr}}(G, \underline{G}_m^{\,r})
\times \underline{\mathrm{Hom}}_{S\text{-gr}}(G,N) \simeq R^r \times E,\]
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\[
Q \simeq \underline{\mathrm{Hom}}_{S\text{-gr}}(G, \underline{G}_m^{\,r})
\times \underline{\mathrm{Hom}}_{S\text{-gr}}(G,N) \simeq R^r \times E,
\]\[cd(X) < 2\dim X + 1, \qquad cd(X_{\mathbb{Q}}) \leqslant 2\dim X_{\mathbb{Q}} + 2,
\qquad cd(X_{\mathbb{R}}) \leqslant 2\dim X\]
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\[
cd(X) < 2\dim X + 1, \qquad cd(X_{\mathbb{Q}}) \leqslant 2\dim X_{\mathbb{Q}} + 2,
\qquad cd(X_{\mathbb{R}}) \leqslant 2\dim X
\]\[H^i(X,F) \simeq H^i(X(\mathbb{C})/\mathbb{Z}/2\mathbb{Z},\, F)\]
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\[
H^i(X,F) \simeq H^i(X(\mathbb{C})/\mathbb{Z}/2\mathbb{Z},\, F)
\]\[cd_2(X) \leqslant 2\dim X\]
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\[ cd_2(X) \leqslant 2\dim X \]
\[cd_r(k') = cd_r(k) + \deg.\ \mathrm{tr.}\ k'/k\]
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\[
cd_r(k') = cd_r(k) + \deg.\ \mathrm{tr.}\ k'/k
\]\[cd_r(K) \geqslant cd_r(k) + \dim A\]
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\[ cd_r(K) \geqslant cd_r(k) + \dim A \]
\[\check{H}^i(\mathcal{U},F) = 0 \quad \text{pour } 0 < i \leqslant n\]
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\[
\check{H}^i(\mathcal{U},F) = 0 \quad \text{pour } 0 < i \leqslant n
\]\[\varinjlim_{\mathcal{U}} \check{H}^i(\mathcal{U},F) = 0 \quad \text{pour }
0 < i \leqslant n\]
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\[
\varinjlim_{\mathcal{U}} \check{H}^i(\mathcal{U},F) = 0 \quad \text{pour }
0 < i \leqslant n
\]\[H^{*}(S,F) \Longleftarrow \check{H}^p(\mathcal{U}, \mathcal{H}^q(F)) =
E_2^{pq}\]
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\[
H^{*}(S,F) \Longleftarrow \check{H}^p(\mathcal{U}, \mathcal{H}^q(F)) =
E_2^{pq}
\]\[H^i(S,F) \overset{\simeq}{\Longleftarrow} \check{H}^i(\mathcal{U},F)
\quad \text{pour } i \leqslant n\]
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\[
H^i(S,F) \overset{\simeq}{\Longleftarrow} \check{H}^i(\mathcal{U},F)
\quad \text{pour } i \leqslant n
\]\[H^{*}(S,F) \Longleftarrow \varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U},
\mathcal{H}^q(F)) = E_2^{pq}\]
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\[
H^{*}(S,F) \Longleftarrow \varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U},
\mathcal{H}^q(F)) = E_2^{pq}
\]\[\varinjlim_{\mathcal{U}} \check{H}^i(\mathcal{U},F) \simeq H^i(S,F), \qquad
i \leqslant n\]
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\[
\varinjlim_{\mathcal{U}} \check{H}^i(\mathcal{U},F) \simeq H^i(S,F), \qquad
i \leqslant n
\]\[\varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U}; \mathcal{H}^q(F)) = 0
\quad \text{pour } \uncertain{p < q \leqslant n}\]
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\[
\varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U}; \mathcal{H}^q(F)) = 0
\quad \text{pour } \uncertain{p < q \leqslant n}
\]\[\varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U}, \overline{H}^q(F)) = 0
\quad \text{pour } \uncertain{p < q \leqslant n} \quad \struck{\text{et }
q \neq 0}\]
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\[
\varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U}, \overline{H}^q(F)) = 0
\quad \text{pour } \uncertain{p < q \leqslant n} \quad \struck{\text{et }
q \neq 0}
\]\[\overline{H}^q(F)(T) = \varinjlim_{\mathcal{V}} \check{H}^q(\mathcal{V},
\uncertain{F})\]
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\[
\overline{H}^q(F)(T) = \varinjlim_{\mathcal{V}} \check{H}^q(\mathcal{V},
\uncertain{F})
\]\[\begin{array}{ccc}
\text{(ii)} & \Longleftrightarrow & \text{(iii bis)} \\
& & \Downarrow \\
\text{(iii ter)} & \Longrightarrow & \text{(iii)} \\
& & \Downarrow \\
& & \text{(i)}
\end{array}\]
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\[
\begin{array}{ccc}
\text{(ii)} & \Longleftrightarrow & \text{(iii bis)} \\
& & \Downarrow \\
\text{(iii ter)} & \Longrightarrow & \text{(iii)} \\
& & \Downarrow \\
& & \text{(i)}
\end{array}
\]\[H^{*}(S,F) \Longleftarrow \varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U},
\mathcal{H}^q(F)) = E_2^{pq}.\]
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\[
H^{*}(S,F) \Longleftarrow \varinjlim_{\mathcal{U}} \check{H}^p(\mathcal{U},
\mathcal{H}^q(F)) = E_2^{pq}.
\]\[E_2^{pq} = 0 \quad \text{si } p = 0,\ q > 0\]
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\[
E_2^{pq} = 0 \quad \text{si } p = 0,\ q > 0
\]\[\mathcal{H}^i(F) \overset{\simeq}{\Longleftarrow} \overline{H}^i(F) \quad
\text{pour } i < n \quad \struck{\ill{}}\]
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\[
\mathcal{H}^i(F) \overset{\simeq}{\Longleftarrow} \overline{H}^i(F) \quad
\text{pour } i < n \quad \struck{\ill{}}
\]\[E_2^{pq} = 0 \quad \text{si } \struck{p + q \leqslant n-1 \text{ et } q
\neq 0}\ 0 < q \leqslant n-1\]
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\[
E_2^{pq} = 0 \quad \text{si } \struck{p + q \leqslant n-1 \text{ et } q
\neq 0}\ 0 < q \leqslant n-1
\]\[\varinjlim_{\mathcal{U}} \check{H}^n(\mathcal{U},F) \simeq H^n(S,F)\]
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\[
\varinjlim_{\mathcal{U}} \check{H}^n(\mathcal{U},F) \simeq H^n(S,F)
\]\[\begin{array}{ccccc}
\uncertain{\text{(ii bis)}} & \struck{\Longleftrightarrow} & & &
\text{(iii bis)} \\
\Downarrow & & & & \Downarrow \\
\text{(ii)} & \Longrightarrow & \text{(i)} & \Longleftarrow & \text{(iii)}
\end{array}\]
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\[
\begin{array}{ccccc}
\uncertain{\text{(ii bis)}} & \struck{\Longleftrightarrow} & & &
\text{(iii bis)} \\
\Downarrow & & & & \Downarrow \\
\text{(ii)} & \Longrightarrow & \text{(i)} & \Longleftarrow & \text{(iii)}
\end{array}
\]\[\begin{gather}
\underline{S} = \operatorname{gr}_{\mathcal{J}} \underline{O}_{X'} =
\coprod_{k \in \underline{N}} \mathcal{J}^k / \mathcal{J}^{k+1} \tag{2.5} \\
\operatorname{gr}_{\mathcal{J}}(\underline{F}) = \coprod_{k \in
\underline{N}} \mathcal{J}^k \underline{F} / \mathcal{J}^{k+1}
\underline{F} \tag{2.6} \\
\underline{K}^i = R^i f_{*}(\operatorname{gr}_{\mathcal{J}}(\underline{F}))
= \coprod_{k \in \underline{N}} R^i f_{*}(\mathcal{J}^k \underline{F} /
\mathcal{J}^{k+1} \underline{F}), \tag{2.7}
\end{gather}\]
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\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}\[\begin{equation}
F_0 = \underline{F} / \mathcal{J} \underline{F}, \tag{2.8}
\end{equation}\]
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\begin{equation}
F_0 = \underline{F} / \mathcal{J} \underline{F}, \tag{2.8}
\end{equation}\[\pi_1(B) / \operatorname{im} p_{*}(\pi_1(E)) = G/S,\]
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\[
\pi_1(B) / \operatorname{im} p_{*}(\pi_1(E)) = G/S,
\]\[p\gamma = (p\gamma_1, p g_1 \gamma_2) = (p\gamma_1, p\gamma_2) \struck{\ill{}}
\quad \text{d'où } \chi(g_1 g_2) = \chi(g_1)\chi(g_2)\]
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\[
p\gamma = (p\gamma_1, p g_1 \gamma_2) = (p\gamma_1, p\gamma_2) \struck{\ill{}}
\quad \text{d'où } \chi(g_1 g_2) = \chi(g_1)\chi(g_2)
\]\[\theta = \sum_{g \in G,\, i \in I} g\, a_i \tau_i\]
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\[
\theta = \sum_{g \in G,\, i \in I} g\, a_i \tau_i
\]\[\varphi_{M}(\operatorname{Tr}_{G} b) = \operatorname{Tr}_{G}\varphi_{M}(b),\]
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\[
\varphi_{M}(\operatorname{Tr}_{G} b) = \operatorname{Tr}_{G}\varphi_{M}(b),
\]\[0 \to M \to M \otimes_{\mathbb{Z}} \mathbb{Z}(G) \to N \to 0,
\qquad m \mapsto \sum_{g} m \otimes g,\]
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\[
0 \to M \to M \otimes_{\mathbb{Z}} \mathbb{Z}(G) \to N \to 0,
\qquad m \mapsto \sum_{g} m \otimes g,
\]\[A'/\operatorname{Tr}_{G'} B \longrightarrow A/\operatorname{Tr}_{G} B\]
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\[
A'/\operatorname{Tr}_{G'} B \longrightarrow A/\operatorname{Tr}_{G} B
\]\[\operatorname{Tr}_{G/G'} : \hat{H}(G',M) \to \hat{H}(G,M)
\ \text{est surjective},\]
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\[
\operatorname{Tr}_{G/G'} : \hat{H}(G',M) \to \hat{H}(G,M)
\ \text{est surjective},
\]\[\operatorname{Tr}_{G'/G} : \hat{H}(G,M) \to \hat{H}(G',M)
\ \text{est \uncertain{injective}},\]
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\[
\operatorname{Tr}_{G'/G} : \hat{H}(G,M) \to \hat{H}(G',M)
\ \text{est \uncertain{injective}},
\]\[\hat{H}(G,M) = \hat{H}(G_{i},M)^{G/G_{i}} .\]
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\[
\hat{H}(G,M) = \hat{H}(G_{i},M)^{G/G_{i}} .
\]\[\overline{\hat{H}(G,M)} = \hat{H}(G,\bar{M})
= \hat{H}\Bigl(G, \bigoplus_{s \in G/G_{d}} s\,\bar{M}_{\mathfrak{m}_{0}}\Bigr)
= \hat{H}(G_{d}, \bar{M}_{\mathfrak{m}_{0}})
= \overline{\hat{H}(G_{d}, M_{\mathfrak{m}_{0}})}\,.\]
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\[
\overline{\hat{H}(G,M)} = \hat{H}(G,\bar{M})
= \hat{H}\Bigl(G, \bigoplus_{s \in G/G_{d}} s\,\bar{M}_{\mathfrak{m}_{0}}\Bigr)
= \hat{H}(G_{d}, \bar{M}_{\mathfrak{m}_{0}})
= \overline{\hat{H}(G_{d}, M_{\mathfrak{m}_{0}})}\,.
\]\[0 \to G_{k}^{0} \to G_{k} \to \mathfrak{P}_{k} \to 0\]
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\[
0 \to G_{k}^{0} \to G_{k} \to \mathfrak{P}_{k} \to 0
\]\[0 \to \mathbb{Z}_{2}^{r_{1}} \to G_{k}^{0} \to
\mathcal{U}_{K}^{0}/\operatorname{Im} E \to 0\]
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\[
0 \to \mathbb{Z}_{2}^{r_{1}} \to G_{k}^{0} \to
\mathcal{U}_{K}^{0}/\operatorname{Im} E \to 0
\]\[0 \to G_{k}^{0,S} \to G_{k}^{S} \to \mathfrak{P}_{k} \to 0\]
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\[
0 \to G_{k}^{0,S} \to G_{k}^{S} \to \mathfrak{P}_{k} \to 0
\]\[0 \to \mathbb{Z}_{2}^{S_{\infty}} \Big/ \bigcap_{k}
\operatorname{Im}\bigl(E^{(S,k)}\bigr)
\to G_{k}^{0,S} \to \mathcal{U}_{K}^{S,0}/\operatorname{Im} E \to 0\]
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\[
0 \to \mathbb{Z}_{2}^{S_{\infty}} \Big/ \bigcap_{k}
\operatorname{Im}\bigl(E^{(S,k)}\bigr)
\to G_{k}^{0,S} \to \mathcal{U}_{K}^{S,0}/\operatorname{Im} E \to 0
\]\[\Gamma \supset \ell\Gamma \supset \ell^{2}\Gamma \supset \cdots
\supset \ell^{k}\Gamma = 0 ,\]
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\[
\Gamma \supset \ell\Gamma \supset \ell^{2}\Gamma \supset \cdots
\supset \ell^{k}\Gamma = 0 ,
\]\[0 \to G_{u} \to G \to G/G_{u} \to 0 .\]
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\[
0 \to G_{u} \to G \to G/G_{u} \to 0 .
\]\[H^{i}(\mathfrak{g}, G) \simeq H^{i}(\mathfrak{g}, \mathcal{U})
\quad \text{pour } i \geq 1\]
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\[
H^{i}(\mathfrak{g}, G) \simeq H^{i}(\mathfrak{g}, \mathcal{U})
\quad \text{pour } i \geq 1
\]\[H^{i}(\mathfrak{g}, \mathcal{U}) = \varinjlim_{n}
H^{i}(\mathfrak{g}, {}_{n}\mathcal{U})\]
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\[
H^{i}(\mathfrak{g}, \mathcal{U}) = \varinjlim_{n}
H^{i}(\mathfrak{g}, {}_{n}\mathcal{U})
\]\[G \supset pG \supset \cdots \supset p^{\ell}G = 0 .\]
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\[
G \supset pG \supset \cdots \supset p^{\ell}G = 0 .
\]\[0 \to \mathbb{Z}/p \to \tilde{K} \xrightarrow{\ \wp\ } \tilde{K} \to 0 .\]
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\[
0 \to \mathbb{Z}/p \to \tilde{K} \xrightarrow{\ \wp\ } \tilde{K} \to 0 .
\]\[e \to G' \to G \to G'' \to e ,\]
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\[ e \to G' \to G \to G'' \to e , \]
\[H^{i}\bigl(G'', H^{j}(G',\Gamma)\bigr) \Longrightarrow H^{*}(G,\Gamma).\]
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\[
H^{i}\bigl(G'', H^{j}(G',\Gamma)\bigr) \Longrightarrow H^{*}(G,\Gamma).
\]\[H^{i}(Y, F|Y) = 0 \quad \text{pour } i > 0\]
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\[
H^{i}(Y, F|Y) = 0 \quad \text{pour } i > 0
\]\[\bigl[H^{i}(V, F|V) = 0 \ \text{pour tout \ill{} voisinage } V
\text{ de } Y,\ i > 0\bigr].\]
LaTeX source
\[
\bigl[H^{i}(V, F|V) = 0 \ \text{pour tout \ill{} voisinage } V
\text{ de } Y,\ i > 0\bigr].
\]\[\varinjlim_{U \supset Y} H^{0}(U,F) \longrightarrow H^{0}(Y,F)\]
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\[
\varinjlim_{U \supset Y} H^{0}(U,F) \longrightarrow H^{0}(Y,F)
\]\[\struck{F_{Y} \to \textstyle\coprod_{i} F_{Y_{i}} \to G \to \ill{}}\]
LaTeX source
\[
\struck{F_{Y} \to \textstyle\coprod_{i} F_{Y_{i}} \to G \to \ill{}}
\]\[\struck{H^{0}(Y_{2}, F_{Y_{i}}) \to H^{1}(Y,G) \to H^{1}(Y, F_{Y}) = 0}\]
LaTeX source
\[
\struck{H^{0}(Y_{2}, F_{Y_{i}}) \to H^{1}(Y,G) \to H^{1}(Y, F_{Y}) = 0}
\]\[H^{0}(Y_{1}, F_{1}) \times H^{0}(Y_{2}, F_{Y_{2}}) \to
H^{0}(Y_{12}, F_{12}) \to H^{1}(Y, F_{Y}) \to 0\]
LaTeX source
\[
H^{0}(Y_{1}, F_{1}) \times H^{0}(Y_{2}, F_{Y_{2}}) \to
H^{0}(Y_{12}, F_{12}) \to H^{1}(Y, F_{Y}) \to 0
\]\[[\,F_{i} = F_{Y_{i}},\quad F_{12} = F_{Y_{12}}\,]\]
LaTeX source
\[
[\,F_{i} = F_{Y_{i}},\quad F_{12} = F_{Y_{12}}\,]
\]\[H^{0}(Y_{12}, F_{12}) = F_{a} \times F_{b}\]
LaTeX source
\[
H^{0}(Y_{12}, F_{12}) = F_{a} \times F_{b}
\]\[\varphi_{1x} = \varphi'_{1x} + 1 \quad \text{pour } x \in U \cap U'
\cap Y_{2},\]
LaTeX source
\[
\varphi_{1x} = \varphi'_{1x} + 1 \quad \text{pour } x \in U \cap U'
\cap Y_{2},
\]\[\varphi_{1x} = \varphi'_{1x} + 1, \qquad \varphi_{1x} = \varphi'_{2x},\]
LaTeX source
\[
\varphi_{1x} = \varphi'_{1x} + 1, \qquad \varphi_{1x} = \varphi'_{2x},
\]\[Y_{1} = X_{1} \cap Y .\]
LaTeX source
\[
Y_{1} = X_{1} \cap Y .
\]\[H^{1}\bigl(C(U,V); F\bigr) = H^{0}(U \cap V, F) \big/
\operatorname{Im}\bigl(H^{0}(U,F) \times H^{0}(V,F)\bigr)\]
LaTeX source
\[
H^{1}\bigl(C(U,V); F\bigr) = H^{0}(U \cap V, F) \big/
\operatorname{Im}\bigl(H^{0}(U,F) \times H^{0}(V,F)\bigr)
\]\[\delta(L/K,\ k/\ell) = 0 \quad \text{i.e.}\]
LaTeX source
\[
\delta(L/K,\ k/\ell) = 0 \quad \text{i.e.}
\]\[k \subset \ell(K^{p}) \qquad \text{i.e.} \quad k(K^{p}) \subset
\ell(K^{p})\]
LaTeX source
\[
k \subset \ell(K^{p}) \qquad \text{i.e.} \quad k(K^{p}) \subset
\ell(K^{p})
\]\[K \to L = K(x), \qquad x^{p} \in K,\quad x^{p} \notin \ell(K^{p})\]
LaTeX source
\[
K \to L = K(x), \qquad x^{p} \in K,\quad x^{p} \notin \ell(K^{p})
\]\[N_{L/K/k} = 0, \qquad N_{L/K/\ell} \simeq L, \qquad N_{L/K}\]
LaTeX source
\[
N_{L/K/k} = 0, \qquad N_{L/K/\ell} \simeq L, \qquad N_{L/K}
\]\[\Omega^{1}_{K/k} \otimes_{K} L \longrightarrow \Omega^{1}_{L/k}\]
LaTeX source
\[
\Omega^{1}_{K/k} \otimes_{K} L \longrightarrow \Omega^{1}_{L/k}
\]\[K_{0} \ \big| \ K \qquad k(K_{0}^{p}) \subset K_{0}^{p}\]
LaTeX source
\[
K_{0} \ \big| \ K \qquad k(K_{0}^{p}) \subset K_{0}^{p}
\]\[\boxed{\ K''_{\infty} \subset K_{\infty}(K_{0}'^{p})\ }\]
LaTeX source
\[
\boxed{\ K''_{\infty} \subset K_{\infty}(K_{0}'^{p})\ }
\]\[\cdots \longrightarrow F \longrightarrow C^{0} \longrightarrow C^{1}
\longrightarrow \cdots\]
LaTeX source
\[
\cdots \longrightarrow F \longrightarrow C^{0} \longrightarrow C^{1}
\longrightarrow \cdots
\]\[\varphi(Y) = \operatorname{Sup}_{x \in Y} \varphi(x)
= \operatorname{Max}_{x \text{ maximal dans } Y} \varphi(x)\]
LaTeX source
\[
\varphi(Y) = \operatorname{Sup}_{x \in Y} \varphi(x)
= \operatorname{Max}_{x \text{ maximal dans } Y} \varphi(x)
\]\[H^{n+1}\bigl(Y, i_{*}(G)\bigr) = 0 \qquad
\text{si } n = \varphi(X) = \varphi(Y).\]
LaTeX source
\[
H^{n+1}\bigl(Y, i_{*}(G)\bigr) = 0 \qquad
\text{si } n = \varphi(X) = \varphi(Y).
\]\[\varphi(y) < \varphi(x) - \operatorname{cd}_{\ell}
\bigl(\underline{O}_{X,\bar{y}} \otimes k(x)\bigr)\]
LaTeX source
\[
\varphi(y) < \varphi(x) - \operatorname{cd}_{\ell}
\bigl(\underline{O}_{X,\bar{y}} \otimes k(x)\bigr)
\]\[H^{i}(X,F) = 0 \qquad \text{si } i > \operatorname{Sup}_{x \in
\operatorname{Supp} F} \varphi(x) = \varphi(F).\]
LaTeX source
\[
H^{i}(X,F) = 0 \qquad \text{si } i > \operatorname{Sup}_{x \in
\operatorname{Supp} F} \varphi(x) = \varphi(F).
\]\[H^{*}\bigl(\overline{\{x\}}, G\bigr) \Longleftarrow
H^{p}\bigl(X, R^{q} g_{*}(G)\bigr)\]
LaTeX source
\[
H^{*}\bigl(\overline{\{x\}}, G\bigr) \Longleftarrow
H^{p}\bigl(X, R^{q} g_{*}(G)\bigr)
\]\[K_{\bar{y}} = \underline{O}_{X,\bar{y}} \otimes_{X} k(x) = \ill{}\]
LaTeX source
\[
K_{\bar{y}} = \underline{O}_{X,\bar{y}} \otimes_{X} k(x) = \ill{}
\]\[\varphi(y) < \varphi(x) - \operatorname{cd}_{\ell}(K_{\bar y})
\leqslant \varphi(x) - q\]
LaTeX source
\[
\varphi(y) < \varphi(x) - \operatorname{cd}_{\ell}(K_{\bar y})
\leqslant \varphi(x) - q
\]\[\boxed{\ \varphi\bigl(R^{q} g_{*}(G)\bigr) \leqslant \varphi(x) - q\ }\]
LaTeX source
\[
\boxed{\ \varphi\bigl(R^{q} g_{*}(G)\bigr) \leqslant \varphi(x) - q\ }
\]\[H^{p}\bigl(X, R^{q} g_{*}(G)\bigr) = 0\]
LaTeX source
\[
H^{p}\bigl(X, R^{q} g_{*}(G)\bigr) = 0
\]\[\varphi(x) \leqslant \varphi(y) + 2 \deg\!\operatorname{tr}
\bigl(k(x)/k(y)\bigr)\]
LaTeX source
\[
\varphi(x) \leqslant \varphi(y) + 2 \deg\!\operatorname{tr}
\bigl(k(x)/k(y)\bigr)
\]\[\operatorname{cd}_{\ell} K_{x,\bar y} \leqslant \dim \underline{O}_{X,\bar y}
\quad \bigl[= \dim \underline{O}_{Y,y}\bigr].\]
LaTeX source
\[
\operatorname{cd}_{\ell} K_{x,\bar y} \leqslant \dim \underline{O}_{X,\bar y}
\quad \bigl[= \dim \underline{O}_{Y,y}\bigr].
\]\[\boxed{\ \varphi(x) = \operatorname{Sup}_{y \in \bar{x}}
\Bigl(\operatorname{cd}_{\ell}\bigl(k(y)\bigr) + 2 \dim
\underline{O}_{\bar{x},y}\Bigr)\ }\]
LaTeX source
\[
\boxed{\ \varphi(x) = \operatorname{Sup}_{y \in \bar{x}}
\Bigl(\operatorname{cd}_{\ell}\bigl(k(y)\bigr) + 2 \dim
\underline{O}_{\bar{x},y}\Bigr)\ }
\]\[\varphi(Y) = \operatorname{Sup}_{y \in Y}\Bigl(
\operatorname{cd}_{\ell}\bigl(k(y)\bigr) + 2 \dim
\underline{O}_{Y,y}\Bigr), \qquad
\operatorname{cd}_{\ell}(X) \leqslant \varphi(X).\]
LaTeX source
\[
\varphi(Y) = \operatorname{Sup}_{y \in Y}\Bigl(
\operatorname{cd}_{\ell}\bigl(k(y)\bigr) + 2 \dim
\underline{O}_{Y,y}\Bigr), \qquad
\operatorname{cd}_{\ell}(X) \leqslant \varphi(X).
\]\[H^{*}(G) \Longleftarrow E_{2}^{pq} = H^{p}\bigl(X, R^{q} i_{*}(G)\bigr)\]
LaTeX source
\[
H^{*}(G) \Longleftarrow E_{2}^{pq} = H^{p}\bigl(X, R^{q} i_{*}(G)\bigr)
\]\[\underline{O}_{X,\bar y} \otimes_{X} k(x), \qquad
q \leqslant \operatorname{cd}\bigl(\underline{O}_{X,\bar y} \otimes \ill{}
\bigr), \qquad
q \leqslant \operatorname{cd}_{p}\operatorname{Frac}
\underline{O}_{X,\bar y}\]
LaTeX source
\[
\underline{O}_{X,\bar y} \otimes_{X} k(x), \qquad
q \leqslant \operatorname{cd}\bigl(\underline{O}_{X,\bar y} \otimes \ill{}
\bigr), \qquad
q \leqslant \operatorname{cd}_{p}\operatorname{Frac}
\underline{O}_{X,\bar y}
\]\[\operatorname{cd}(\bar{y}) + q \ \operatorname{cd}_{p}\operatorname{Frac}
\underline{O}_{X,\bar y} < n\]
LaTeX source
\[
\operatorname{cd}(\bar{y}) + q \ \operatorname{cd}_{p}\operatorname{Frac}
\underline{O}_{X,\bar y} < n
\]\[H^{*}(G) \longrightarrow H^{*}f' \longrightarrow H^{k+1}(X_{T})
\longrightarrow H^{k+1}(G)\]
LaTeX source
\[
H^{*}(G) \longrightarrow H^{*}f' \longrightarrow H^{k+1}(X_{T})
\longrightarrow H^{k+1}(G)
\]\[\boxed{
\begin{array}{l}
\operatorname{cd}_{\ell}\bigl(k(x)\bigr) \leqslant n \\[2pt]
\forall\, Y \ \uncertain{\text{réunion}} \text{ dans } X, \quad
\operatorname{cd}_{\ell}(Y) < n \\[2pt]
\forall\, y \in X,\ y \ne x, \quad
\operatorname{cd}_{\ell}(\bar{y}) + \operatorname{cd}_{\ell}
\operatorname{Frac} \underline{O}_{X,\bar y} < n
\end{array}}\]
LaTeX source
\[
\boxed{
\begin{array}{l}
\operatorname{cd}_{\ell}\bigl(k(x)\bigr) \leqslant n \\[2pt]
\forall\, Y \ \uncertain{\text{réunion}} \text{ dans } X, \quad
\operatorname{cd}_{\ell}(Y) < n \\[2pt]
\forall\, y \in X,\ y \ne x, \quad
\operatorname{cd}_{\ell}(\bar{y}) + \operatorname{cd}_{\ell}
\operatorname{Frac} \underline{O}_{X,\bar y} < n
\end{array}}
\]\[2 \dim \bar{y} + \dim \underline{O}_{X,\bar y} \quad
\bigl[\oplus \dim X - \dim \bar{y}\bigr] \qquad
\dim X + \dim \bar{y} \leqslant 2 \dim X\]
LaTeX source
\[
2 \dim \bar{y} + \dim \underline{O}_{X,\bar y} \quad
\bigl[\oplus \dim X - \dim \bar{y}\bigr] \qquad
\dim X + \dim \bar{y} \leqslant 2 \dim X
\]\[X \times_{S} G \longrightarrow X \times_{S} X\]
LaTeX source
\[
X \times_{S} G \longrightarrow X \times_{S} X
\]\[G \longrightarrow X\]
LaTeX source
\[ G \longrightarrow X \]
\[x \in \operatorname{Supp} F \ \Longrightarrow\ \dim \underline{O}_{X,x}
\geqslant m\]
LaTeX source
\[
x \in \operatorname{Supp} F \ \Longrightarrow\ \dim \underline{O}_{X,x}
\geqslant m
\]\[\operatorname{cd}_{\ell}(X_{f}) \leqslant n\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X_{f}) \leqslant n
\]\[\cdots \longrightarrow H^{i}(\bar{S},F) \longrightarrow H^{i}(S,F)
\longrightarrow H^{i+1}_{T}(\bar{S},F)\]
LaTeX source
\[
\cdots \longrightarrow H^{i}(\bar{S},F) \longrightarrow H^{i}(S,F)
\longrightarrow H^{i+1}_{T}(\bar{S},F)
\]\[H^{i}(\bar{S},F) = H^{i}(\bar{S}_{0},F) = H^{i}(\mathbf{P}^{1}_{k}, F_{0}),
\quad (\text{nul si } i \geqslant 3)\]
LaTeX source
\[
H^{i}(\bar{S},F) = H^{i}(\bar{S}_{0},F) = H^{i}(\mathbf{P}^{1}_{k}, F_{0}),
\quad (\text{nul si } i \geqslant 3)
\]\[H^{i+1}_{T}(\bar{S},F) = H^{i}\bigl(\bar{S}_{x} - T_{x}, F_{(x)}\bigr)
\quad \text{si } i \geqslant 2\]
LaTeX source
\[
H^{i+1}_{T}(\bar{S},F) = H^{i}\bigl(\bar{S}_{x} - T_{x}, F_{(x)}\bigr)
\quad \text{si } i \geqslant 2
\]\[\dim \bar{S}_{x} + \operatorname{codim}\bigl(\bar{S}_{x}, \bar{X}_{x}\bigr)
\leqslant \dim \bar{X}_{x} = n+1\]
LaTeX source
\[
\dim \bar{S}_{x} + \operatorname{codim}\bigl(\bar{S}_{x}, \bar{X}_{x}\bigr)
\leqslant \dim \bar{X}_{x} = n+1
\]\[H^{i+1}_{T}(\bar{S},F) = 0 \qquad \text{si } i > n+1-m\]
LaTeX source
\[
H^{i+1}_{T}(\bar{S},F) = 0 \qquad \text{si } i > n+1-m
\]\[H^{i}(\bar{S},F) = H^{i}(\bar{S}_{0},F_{0}) = 0 \quad \text{si }
i > n+1-m \geqslant 2\]
LaTeX source
\[
H^{i}(\bar{S},F) = H^{i}(\bar{S}_{0},F_{0}) = 0 \quad \text{si }
i > n+1-m \geqslant 2
\]\[\struck{1'_{n} \Rightarrow 2'_{n+2}} \qquad
1'_{n} + 2'_{n+1} \Longrightarrow 2'_{n+2}\]
LaTeX source
\[
\struck{1'_{n} \Rightarrow 2'_{n+2}} \qquad
1'_{n} + 2'_{n+1} \Longrightarrow 2'_{n+2}
\]\[k\langle x_{1}, \ldots, x_{n+2}\rangle, \quad f = x_{1}\ldots \ill{}\]
LaTeX source
\[
k\langle x_{1}, \ldots, x_{n+2}\rangle, \quad f = x_{1}\ldots \ill{}
\]\[H^{p}\bigl(U, R^{q} f_{*}(F)\bigr) \ne 0 \ \Longrightarrow\ \struck{\ill{}}
\ p \leqslant n+1-(q-1), \ \text{i.e.\ } p+q \leqslant n+2,\]
LaTeX source
\[
H^{p}\bigl(U, R^{q} f_{*}(F)\bigr) \ne 0 \ \Longrightarrow\ \struck{\ill{}}
\ p \leqslant n+1-(q-1), \ \text{i.e.\ } p+q \leqslant n+2,
\]\[V' = \struck{V} \otimes_{B} B' \quad \add{\text{Affine}} \ill{}\]
LaTeX source
\[
V' = \struck{V} \otimes_{B} B' \quad \add{\text{Affine}} \ill{}
\]\[H^{q}(V',F') = 0 \quad \text{si } q > i+1, \ \text{i.e.\ }
\underline{i < q-1},\]
LaTeX source
\[
H^{q}(V',F') = 0 \quad \text{si } q > i+1, \ \text{i.e.\ }
\underline{i < q-1},
\]\[\operatorname{cd}_{\ell}(X) \leqslant \dim X\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant \dim X
\]\[\operatorname{cd}_{d}\bigl(E^{d}_{k}\bigr) \leqslant d\]
LaTeX source
\[
\operatorname{cd}_{d}\bigl(E^{d}_{k}\bigr) \leqslant d
\]\[R^{q} f_{*}(F)_{\bar y} \ne 0 \ \Longrightarrow\
\dim \underline{O}_{Y,y} + 1 = \struck{i+1} \ \text{i.e.\ }
R^{q} f_{*}(F)_{\bar y} \ \ill{}\]
LaTeX source
\[
R^{q} f_{*}(F)_{\bar y} \ne 0 \ \Longrightarrow\
\dim \underline{O}_{Y,y} + 1 = \struck{i+1} \ \text{i.e.\ }
R^{q} f_{*}(F)_{\bar y} \ \ill{}
\]\[H^{p}\bigl(Y, R^{q} f_{*}(F)\bigr) \ne 0 \ \Longrightarrow\
p \leqslant n - (q-1), \quad \text{i.e.\ } p+q \leqslant n+1.\]
LaTeX source
\[
H^{p}\bigl(Y, R^{q} f_{*}(F)\bigr) \ne 0 \ \Longrightarrow\
p \leqslant n - (q-1), \quad \text{i.e.\ } p+q \leqslant n+1.
\]\[V = X_{f}, \qquad U = X_{f\pi} = V_{\pi}, \qquad
Z = V - U = V \cap V(\pi) = V(\pi)_{f}\]
LaTeX source
\[
V = X_{f}, \qquad U = X_{f\pi} = V_{\pi}, \qquad
Z = V - U = V \cap V(\pi) = V(\pi)_{f}
\]\[H^{i}_{Z}(V,F) \longrightarrow H^{i}(V,F) \longrightarrow H^{i}(U,F)\]
LaTeX source
\[
H^{i}_{Z}(V,F) \longrightarrow H^{i}(V,F) \longrightarrow H^{i}(U,F)
\]\[H^{*}_{Z}(V,F) \Longleftarrow H^{p}\bigl(Z, \underline{H}^{q}_{Z}(F)\bigr)\]
LaTeX source
\[
H^{*}_{Z}(V,F) \Longleftarrow H^{p}\bigl(Z, \underline{H}^{q}_{Z}(F)\bigr)
\]\[\underline{H}^{q}_{Z}(F)_{\bar x} \ne 0 \ \Longrightarrow\
\dim \underline{O}_{Z,x} \geqslant q-1\]
LaTeX source
\[
\underline{H}^{q}_{Z}(F)_{\bar x} \ne 0 \ \Longrightarrow\
\dim \underline{O}_{Z,x} \geqslant q-1
\]\[H^{p}\bigl(Z, \underline{H}^{q}_{Z}(F)\bigr) \ne 0 \ \Longrightarrow\
p \leqslant (n-1)-(q-1) = n-q, \quad \text{i.e.\ } p+q \leqslant n,\]
LaTeX source
\[
H^{p}\bigl(Z, \underline{H}^{q}_{Z}(F)\bigr) \ne 0 \ \Longrightarrow\
p \leqslant (n-1)-(q-1) = n-q, \quad \text{i.e.\ } p+q \leqslant n,
\]\[q-1 > \dim \underline{O}_{Z,x} \ \Longrightarrow\
\underline{H}^{q}_{Z}(F)_{\bar x} = 0.\]
LaTeX source
\[
q-1 > \dim \underline{O}_{Z,x} \ \Longrightarrow\
\underline{H}^{q}_{Z}(F)_{\bar x} = 0.
\]\[\underline{H}^{q}_{Z}(F)_{\bar x} = H^{q-1}\bigl(X_{\bar x} -
V(\pi)_{\bar x}, F\bigr)\]
LaTeX source
\[
\underline{H}^{q}_{Z}(F)_{\bar x} = H^{q-1}\bigl(X_{\bar x} -
V(\pi)_{\bar x}, F\bigr)
\]\[d = \operatorname{Sup}\bigl[\dim X_{0},\ \dim X' + 1\bigr].\]
LaTeX source
\[
d = \operatorname{Sup}\bigl[\dim X_{0},\ \dim X' + 1\bigr].
\]\[\operatorname{cd}_{\ell}(X) \leqslant \operatorname{Sup}
\bigl[2 \dim X_{0},\ 2 \dim X' + 1\bigr]\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant \operatorname{Sup}
\bigl[2 \dim X_{0},\ 2 \dim X' + 1\bigr]
\]\[\operatorname{cd}_{\ell}(X) \leqslant 2 \dim Y - 1 + 2d \qquad
\bigl(\text{si } \underline{\dim Y \ne 0}\bigr)\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant 2 \dim Y - 1 + 2d \qquad
\bigl(\text{si } \underline{\dim Y \ne 0}\bigr)
\]\[\operatorname{codim}(X,E) \geqslant n+r-\nu \quad \text{i.e.\ }
\nu \geqslant \boxed{n+r-\operatorname{codim}(X,E)}\]
LaTeX source
\[
\operatorname{codim}(X,E) \geqslant n+r-\nu \quad \text{i.e.\ }
\nu \geqslant \boxed{n+r-\operatorname{codim}(X,E)}
\]\[\operatorname{codim}(X',E') = \dim E' - \dim X' = \bigl((n-1)+r\bigr) -
\dim \ill{}\]
LaTeX source
\[
\operatorname{codim}(X',E') = \dim E' - \dim X' = \bigl((n-1)+r\bigr) -
\dim \ill{}
\]\[\dim X' \geqslant (n-1+r) - (\nu-1) = n+r-\nu \qquad \text{ok}\]
LaTeX source
\[
\dim X' \geqslant (n-1+r) - (\nu-1) = n+r-\nu \qquad \text{ok}
\]\[\operatorname{codim}(X,E) = \operatorname{codim}(X_{0},E_{0}) +
\operatorname{codim}(E_{0},E) = (r-\nu) + n \qquad \text{ok.}\]
LaTeX source
\[
\operatorname{codim}(X,E) = \operatorname{codim}(X_{0},E_{0}) +
\operatorname{codim}(E_{0},E) = (r-\nu) + n \qquad \text{ok.}
\]\[\operatorname{cd}_{\ell}(X) \leqslant (n+r) - \operatorname{codim}(X,E)\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant (n+r) - \operatorname{codim}(X,E)
\]\[E^{r}_{Y} \longrightarrow E^{r-1}_{Y} \qquad \cdots\]
LaTeX source
\[
E^{r}_{Y} \longrightarrow E^{r-1}_{Y} \qquad \cdots
\]\[n = \operatorname*{Sup}_{x \in X}\bigl[\operatorname{cd}_{\ell}(k(x)) +
\dim \mathcal{O}_{X,x}\bigr].\]
LaTeX source
\[
n = \operatorname*{Sup}_{x \in X}\bigl[\operatorname{cd}_{\ell}(k(x)) +
\dim \mathcal{O}_{X,x}\bigr].
\]\[\operatorname{cd}_{\ell}(X) \leqslant n .\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant n .
\]\[H^{*}(X,F) \Longleftarrow H^{p}\bigl(S_{\ell}, R^{q} f_{*}(F)\bigr)\]
LaTeX source
\[
H^{*}(X,F) \Longleftarrow H^{p}\bigl(S_{\ell}, R^{q} f_{*}(F)\bigr)
\]\[R^{q} f_{*}(F)_{\mathfrak{g}} \ne 0 \Longrightarrow q \leqslant
\struck{1 + } d, \qquad
R^{q} f_{*}(F)_{s} \ne 0 \Longrightarrow q \leqslant 1 + d\]
LaTeX source
\[
R^{q} f_{*}(F)_{\mathfrak{g}} \ne 0 \Longrightarrow q \leqslant
\struck{1 + } d, \qquad
R^{q} f_{*}(F)_{s} \ne 0 \Longrightarrow q \leqslant 1 + d
\]\[\operatorname{cd}_{\ell}(X) \leqslant d + 2 = \dim X + 1 =
\uncertain{\operatorname{cd}'_{\ell} \mathcal{R}(X)}\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant d + 2 = \dim X + 1 =
\uncertain{\operatorname{cd}'_{\ell} \mathcal{R}(X)}
\]\[\boxed{\operatorname{cd}_{\ell}(X) \leqslant \dim X + 1}\]
LaTeX source
\[
\boxed{\operatorname{cd}_{\ell}(X) \leqslant \dim X + 1}
\]\[H^{i}(G,F) = 0 \qquad \text{pour } i > 0,\]
LaTeX source
\[
H^{i}(G,F) = 0 \qquad \text{pour } i > 0,
\]\[H^{i}(G',F') = H^{i}(G,F) \otimes_{A} A',\]
LaTeX source
\[
H^{i}(G',F') = H^{i}(G,F) \otimes_{A} A',
\]\[\operatorname{cd}_{\ell}(X) \leqslant \operatorname{cd}_{\ell}(k(x))\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant \operatorname{cd}_{\ell}(k(x))
\]\[\operatorname{cd}_{\ell}(X) \leqslant
\operatorname{Sup}\bigl(\operatorname{cd}_{\ell}(k(x)),\;
\nu(V(\ell)) + 3\bigr)\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant
\operatorname{Sup}\bigl(\operatorname{cd}_{\ell}(k(x)),\;
\nu(V(\ell)) + 3\bigr)
\]\[\nu(Z) = \operatorname*{Sup}_{z \text{ maximaux dans } Z} \nu(k(z))\]
LaTeX source
\[
\nu(Z) = \operatorname*{Sup}_{z \text{ maximaux dans } Z} \nu(k(z))
\]\[\operatorname{cd}_{\ell}(X) \leqslant \operatorname{cd}_{\ell}(k(x))\]
LaTeX source
\[
\operatorname{cd}_{\ell}(X) \leqslant \operatorname{cd}_{\ell}(k(x))
\]\[\operatorname{cd}_{\ell}(U) \leqslant \dim A + \nu(k) +
\operatorname{cd}_{\ell}(k)\]
LaTeX source
\[
\operatorname{cd}_{\ell}(U) \leqslant \dim A + \nu(k) +
\operatorname{cd}_{\ell}(k)
\]\[H^{i}(U,F) = 0 \quad \text{si } i > n .\]
LaTeX source
\[
H^{i}(U,F) = 0 \quad \text{si } i > n .
\]\[H^{i-2}(Y, F_{Y} \otimes \check{T}_{\ell}) \longrightarrow H^{i}(X,F)\]
LaTeX source
\[
H^{i-2}(Y, F_{Y} \otimes \check{T}_{\ell}) \longrightarrow H^{i}(X,F)
\]\[H^{2n}(X, \mu_{N}^{\otimes n}) \simeq \mathbb{Z}/N\mathbb{Z}\]
LaTeX source
\[
H^{2n}(X, \mu_{N}^{\otimes n}) \simeq \mathbb{Z}/N\mathbb{Z}
\]\[H^{2n}(X, \mu_{N}^{\otimes n}) \simeq H^{2n-2}(Y, \mu_{N}^{\otimes n} \otimes
\check{T})\]
LaTeX source
\[
H^{2n}(X, \mu_{N}^{\otimes n}) \simeq H^{2n-2}(Y, \mu_{N}^{\otimes n} \otimes
\check{T})
\]\[H^{j}(X,F) \longrightarrow H^{j}(Y,F)\]
LaTeX source
\[
H^{j}(X,F) \longrightarrow H^{j}(Y,F)
\]\[X' \longrightarrow X\]
LaTeX source
\[ X' \longrightarrow X \]
\[B \simeq A[T]/(T^{n} - x), \qquad B \otimes_{A} A' = A'[T]/(T^{n} - x'),\]
LaTeX source
\[
B \simeq A[T]/(T^{n} - x), \qquad B \otimes_{A} A' = A'[T]/(T^{n} - x'),
\]\[H^{i}(X,F) \;\overset{\sim}{\longrightarrow}\; H^{i}(X',F')\]
LaTeX source
\[
H^{i}(X,F) \;\overset{\sim}{\longrightarrow}\; H^{i}(X',F')
\]\[h : V \longrightarrow S, \qquad i \ill{}\]
LaTeX source
\[
h : V \longrightarrow S, \qquad i \ill{}
\]\[F' = \operatorname{Hom}(F, I_{X/S}) \struck{\ill{}}\]
LaTeX source
\[
F' = \operatorname{Hom}(F, I_{X/S}) \struck{\ill{}}
\]\[R^{i} h_{*}(G_{1}) \Longleftarrow \ill{}\]
LaTeX source
\[
R^{i} h_{*}(G_{1}) \Longleftarrow \ill{}
\]\[\ill{} \; R^{i} h_{*}(G).\]
LaTeX source
\[
\ill{} \; R^{i} h_{*}(G).
\]\[c) \Longleftarrow R^{i} g_{*}\bigl(R^{i} i_{*}(G')\bigr), \qquad
R^{i} g_{*}\bigl(R^{i} i_{*}(G)\bigr)\]
LaTeX source
\[
c) \Longleftarrow R^{i} g_{*}\bigl(R^{i} i_{*}(G')\bigr), \qquad
R^{i} g_{*}\bigl(R^{i} i_{*}(G)\bigr)
\]