Cote n° 128 · pages 1–29
· 67 displayed formulas · Descente non plate. Modules formels. Cf Notes Murre-Levelt : notes manuscrites (s.d.).
Inventory dating : s.d.
Édition de démonstration
\[\begin{aligned}
\lambda (b_0 \otimes \cdots \otimes b_n) &= (\lambda b_0) \otimes b_1 \cdots \otimes b_n \\
&= (\lambda_0 \otimes 1) \otimes b_2 \cdots \otimes b_n
&& \text{pour } \lambda_0 \in I \text{ conv.} \\
&= 1_B \otimes \bigl[\lambda_1 (b_2 \otimes \cdots \otimes b_n)\bigr] = \\
&= 1_B \otimes \mu (1_B \otimes \cdots \otimes 1_B)
&& \mu \in I \text{ conv.} \\
&= \mu (1_B \otimes \cdots \otimes 1_B)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\lambda (b_0 \otimes \cdots \otimes b_n) &= (\lambda b_0) \otimes b_1 \cdots \otimes b_n \\
&= (\lambda_0 \otimes 1) \otimes b_2 \cdots \otimes b_n
&& \text{pour } \lambda_0 \in I \text{ conv.} \\
&= 1_B \otimes \bigl[\lambda_1 (b_2 \otimes \cdots \otimes b_n)\bigr] = \\
&= 1_B \otimes \mu (1_B \otimes \cdots \otimes 1_B)
&& \mu \in I \text{ conv.} \\
&= \mu (1_B \otimes \cdots \otimes 1_B)
\end{aligned}
\]\[I \to \bigotimes_A B \to B\]
LaTeX source
\[ I \to \bigotimes_A B \to B \]
\[0 \to I \to \bigotimes^E_A B \to \bigotimes^E_{A_0} B_0 \to 0 .\]
LaTeX source
\[
0 \to I \to \bigotimes^E_A B \to \bigotimes^E_{A_0} B_0 \to 0 .
\]\[0 \to C_I \to C(B/A) \to C(B_0/A_0) \to 0\]
LaTeX source
\[ 0 \to C_I \to C(B/A) \to C(B_0/A_0) \to 0 \]
\[H^i(B/A) \xrightarrow{\ \sim\ } H^i(B_0/A_0) \qquad \text{pour } i \neq 0\]
LaTeX source
\[
H^i(B/A) \xrightarrow{\ \sim\ } H^i(B_0/A_0) \qquad \text{pour } i \neq 0
\]\[0 \to I \to H^0(B/A) \to H^0(B_0/A_0) \to 0\]
LaTeX source
\[ 0 \to I \to H^0(B/A) \to H^0(B_0/A_0) \to 0 \]
\[0 \to C_{M \otimes_A I} \to K(B/A, M) \to K(B_0/A_0, M_0) \to 0\]
LaTeX source
\[
0 \to C_{M \otimes_A I} \to K(B/A, M) \to K(B_0/A_0, M_0) \to 0
\]\[H^i(B/A, M) \simeq H^i(B_0/A_0, M_0) \qquad i \neq 0\]
LaTeX source
\[ H^i(B/A, M) \simeq H^i(B_0/A_0, M_0) \qquad i \neq 0 \]
\[0 \to M \otimes_A I \to H^0(B/A, M) \to H^0(B_0/A_0, M_0) \to 0 .\]
LaTeX source
\[ 0 \to M \otimes_A I \to H^0(B/A, M) \to H^0(B_0/A_0, M_0) \to 0 . \]
\[\begin{cases}
H^i(B/A, M) = 0 & i \neq 0 \\
H^0(B/A, M) \xleftarrow{\ \sim\ } M
\end{cases}\]
LaTeX source
\[
\begin{cases}
H^i(B/A, M) = 0 & i \neq 0 \\
H^0(B/A, M) \xleftarrow{\ \sim\ } M
\end{cases}
\]\[\begin{cases}
H^i(B/A) = 0 & i \neq 0 \\
H^0(B, A) \xleftarrow{\ \sim\ } A
\end{cases}\]
LaTeX source
\[
\begin{cases}
H^i(B/A) = 0 & i \neq 0 \\
H^0(B, A) \xleftarrow{\ \sim\ } A
\end{cases}
\]\[0 \to I C^{*}_D(A'/A, E') \to C^{*}_D(A'/A, E') \to C^{*}_D(A'_0/A_0, E'_0) \to 0\]
LaTeX source
\[
0 \to I C^{*}_D(A'/A, E') \to C^{*}_D(A'/A, E') \to C^{*}_D(A'_0/A_0, E'_0) \to 0
\]\[(*) \qquad I C^n_D(A'/A, E') \simeq E^{(n)}_0 \otimes_{A^{(n)}_0}
\underset{\substack{\| \\ I}}{I A^{(n)}} \sim\]
LaTeX source
\[
(*) \qquad I C^n_D(A'/A, E') \simeq E^{(n)}_0 \otimes_{A^{(n)}_0}
\underset{\substack{\| \\ I}}{I A^{(n)}} \sim
\]\[\bigl(E_0 \otimes_{A_0} A^{(n)}_0\bigr) \otimes_{A^{(n)}_0} I A^{(n)}
\simeq E_0 \otimes_{A_0} \bigl(\underset{\substack{\| \\ I}}{I A^{(n)}}\bigr)
\simeq E_0 \otimes_{A_0} I\]
LaTeX source
\[
\bigl(E_0 \otimes_{A_0} A^{(n)}_0\bigr) \otimes_{A^{(n)}_0} I A^{(n)}
\simeq E_0 \otimes_{A_0} \bigl(\underset{\substack{\| \\ I}}{I A^{(n)}}\bigr)
\simeq E_0 \otimes_{A_0} I
\]\[0 \to C^{*}_{E_0 \otimes_{A_0} I} \to C^{*}_D(A'/A, E') \to C^{*}_D(A'_0/A_0, E'_0) \to 0\]
LaTeX source
\[
0 \to C^{*}_{E_0 \otimes_{A_0} I} \to C^{*}_D(A'/A, E') \to C^{*}_D(A'_0/A_0, E'_0) \to 0
\]\[0 \to E_0 \otimes_{A_0} I \to
\underset{\substack{\| \\ E}}{H^0_D(A'/A, E')} \to
\underset{\substack{\| \\ E_0}}{H^0_D(A'_0/A_0, E'_0)} \to 0\]
LaTeX source
\[
0 \to E_0 \otimes_{A_0} I \to
\underset{\substack{\| \\ E}}{H^0_D(A'/A, E')} \to
\underset{\substack{\| \\ E_0}}{H^0_D(A'_0/A_0, E'_0)} \to 0
\]\[H^i\bigl(C_{E_0 \otimes_{A_0} I}\bigr) =
\begin{cases}
E_0 \otimes_{A_0} I & \text{pour } i = 0 \\
0 & \text{pour } i = 1 \\
\end{cases}
\quad (\text{et } \uncertain{\text{même}}\ i \neq 0)\]
LaTeX source
\[
H^i\bigl(C_{E_0 \otimes_{A_0} I}\bigr) =
\begin{cases}
E_0 \otimes_{A_0} I & \text{pour } i = 0 \\
0 & \text{pour } i = 1 \\
\end{cases}
\quad (\text{et } \uncertain{\text{même}}\ i \neq 0)
\]\[E \otimes_A A' \to E'\]
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\[ E \otimes_A A' \to E' \]
\[I^n = 0\]
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\[ I^n = 0 \]
\[\Delta^{(n)}_{B/A} \xrightarrow{\ \sim\ } \Delta^{(n)}_{B_0/A_0}\]
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\[
\Delta^{(n)}_{B/A} \xrightarrow{\ \sim\ } \Delta^{(n)}_{B_0/A_0}
\]\[\left\lbrace
\begin{aligned}
J &\mapsto p_n(J) \\
p_n^{-1}(J^0) \cap \Delta^{(n)}_{B/A} &\leftarrow J^0
\end{aligned}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{aligned}
J &\mapsto p_n(J) \\
p_n^{-1}(J^0) \cap \Delta^{(n)}_{B/A} &\leftarrow J^0
\end{aligned}
\right.
\]\[0 \to A \to A' \rightrightarrows A' \otimes_A A'\]
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\[ 0 \to A \to A' \rightrightarrows A' \otimes_A A' \]
\[A \to A' \overset{p_1}{\underset{p_2}{\rightrightarrows}} A''\]
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\[
A \to A' \overset{p_1}{\underset{p_2}{\rightrightarrows}} A''
\]\[\delta(x') = p_2(x') - p_1(x')\]
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\[ \delta(x') = p_2(x') - p_1(x') \]
\[\mathfrak{m}' \subset A\]
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\[
\mathfrak{m}' \subset A
\]\[\mathfrak{m}' = \mathfrak{m} = \mathfrak{m}A'\]
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\[
\mathfrak{m}' = \mathfrak{m} = \mathfrak{m}A'
\]\[\begin{aligned}
\Delta_0 = \mathrm{Ker}(A''_0 \to A'_0)
&\simeq \mathrm{Ker}\bigl(A''/\mathfrak{m}A'' \to A'/\mathfrak{m}A'\bigr) , \\
&\simeq \mathrm{Ker}\bigl(A''/\mathfrak{m}A'' \to A''/\Delta + \mathfrak{m}A''\bigr) \\
&\simeq (\Delta + \mathfrak{m}A'')/\mathfrak{m}A''
\simeq \Delta/\Delta \cap \mathfrak{m}A'' ,
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
\Delta_0 = \mathrm{Ker}(A''_0 \to A'_0)
&\simeq \mathrm{Ker}\bigl(A''/\mathfrak{m}A'' \to A'/\mathfrak{m}A'\bigr) , \\
&\simeq \mathrm{Ker}\bigl(A''/\mathfrak{m}A'' \to A''/\Delta + \mathfrak{m}A''\bigr) \\
&\simeq (\Delta + \mathfrak{m}A'')/\mathfrak{m}A''
\simeq \Delta/\Delta \cap \mathfrak{m}A'' ,
\end{aligned}
\]\[A'_0 \simeq \underbrace{k \times \cdots \times k}_{n} , \quad
\text{avec } n \geqslant 2 ,\]
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\[
A'_0 \simeq \underbrace{k \times \cdots \times k}_{n} , \quad
\text{avec } n \geqslant 2 ,
\]\[\Omega = \Delta/\Delta^2\]
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\[ \Omega = \Delta/\Delta^2 \]
\[d(x') = \delta(x') \bmod \Delta^2 \in \Omega
= \varphi\bigl(d_{A'/A}(x')\bigr)\]
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\[
d(x') = \delta(x') \bmod \Delta^2 \in \Omega
= \varphi\bigl(d_{A'/A}(x')\bigr)
\]\[d(x'y') = x'\,dy' + y'\,dx'\]
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\[ d(x'y') = x'\,dy' + y'\,dx' \]
\[d(\mathfrak{m}'^2) \subset \mathfrak{m}'\Omega\]
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\[
d(\mathfrak{m}'^2) \subset \mathfrak{m}'\Omega
\]\[(*) \qquad \mathfrak{m}'^2 \subset A \quad \text{d'où }
\mathfrak{m}'^2 \subset \mathfrak{m}\]
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\[
(*) \qquad \mathfrak{m}'^2 \subset A \quad \text{d'où }
\mathfrak{m}'^2 \subset \mathfrak{m}
\]\[A' = A + \mathfrak{m}'\]
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\[
A' = A + \mathfrak{m}'
\]\[\mathfrak{m}A' = \mathfrak{m}A + \mathfrak{m}\mathfrak{m}'
\subset \mathfrak{m} + \mathfrak{m}'^2\]
LaTeX source
\[
\mathfrak{m}A' = \mathfrak{m}A + \mathfrak{m}\mathfrak{m}'
\subset \mathfrak{m} + \mathfrak{m}'^2
\]\[\mathfrak{m}A' = \mathfrak{m} ,\]
LaTeX source
\[
\mathfrak{m}A' = \mathfrak{m} ,
\]\[\underset{\substack{\| \\ k}}{A_0} \to A'_0 \rightrightarrows A''_0\]
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\[
\underset{\substack{\| \\ k}}{A_0} \to A'_0 \rightrightarrows A''_0
\]\[A' \simeq D_k(V) \simeq k + V\]
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\[ A' \simeq D_k(V) \simeq k + V \]
\[A' \otimes_k A' \simeq k\, 1 \otimes 1 + V \otimes 1 + 1 \otimes V
+ V \otimes V\]
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\[ A' \otimes_k A' \simeq k\, 1 \otimes 1 + V \otimes 1 + 1 \otimes V + V \otimes V \]
\[\delta(v) = v \otimes 1 - 1 \otimes v\]
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\[ \delta(v) = v \otimes 1 - 1 \otimes v \]
\[\begin{aligned}
B &= H^0(A'/A) = \mathrm{Ker}(A' \rightrightarrows A'') \\
C_0 &= H^0(A'_0/A_0) = \mathrm{Ker}(A'_0 \rightrightarrows A''_0)
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
B &= H^0(A'/A) = \mathrm{Ker}(A' \rightrightarrows A'') \\
C_0 &= H^0(A'_0/A_0) = \mathrm{Ker}(A'_0 \rightrightarrows A''_0)
\end{aligned}
\]\[C_0 = C_0^{*} + \mathrm{Im}(B \to C_0)\]
LaTeX source
\[
C_0 = C_0^{*} + \mathrm{Im}(B \to C_0)
\]\[(*) \quad
\mathrm{Ker}\bigl(H^1(A'/A, G_m) \to H^1(A'_0/A_0, G_m)\bigr)
\simeq
\mathrm{Ker}\bigl(H^1(A'/A, G_a) \to H^1(A'_0/A_0, G_a)\bigr)\]
LaTeX source
\[
(*) \quad
\mathrm{Ker}\bigl(H^1(A'/A, G_m) \to H^1(A'_0/A_0, G_m)\bigr)
\simeq
\mathrm{Ker}\bigl(H^1(A'/A, G_a) \to H^1(A'_0/A_0, G_a)\bigr)
\]\[0 \to I C^{\cdot}(A'/A) \to C^{\cdot}(A'/A) \to C^{\cdot}(A'_0/A_0) \to 0\]
LaTeX source
\[
0 \to I C^{\cdot}(A'/A) \to C^{\cdot}(A'/A) \to C^{\cdot}(A'_0/A_0) \to 0
\]\[0 \to I C^{\cdot}(A'/A) \to C^{\cdot}(A'/A ; G_m) \to C^{\cdot}(A'_0/A_0 ; G_m) \to 0\]
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\[
0 \to I C^{\cdot}(A'/A) \to C^{\cdot}(A'/A ; G_m) \to C^{\cdot}(A'_0/A_0 ; G_m) \to 0
\]\[\mathrm{Coker}\bigl(H^0(A'_0/A_0, G_m) \to H^1(I C^{\cdot}(A'/A))\bigr)\]
LaTeX source
\[
\mathrm{Coker}\bigl(H^0(A'_0/A_0, G_m) \to H^1(I C^{\cdot}(A'/A))\bigr)
\]\[\mathrm{Coker}\bigl(H^0(A'_0/A_0, G_a) \to H^1(I C^{\cdot}(A'/A))\bigr)\]
LaTeX source
\[
\mathrm{Coker}\bigl(H^0(A'_0/A_0, G_a) \to H^1(I C^{\cdot}(A'/A))\bigr)
\]\[C_0 = C_0^{*} + \mathrm{Im}(B \to C_0) .\]
LaTeX source
\[
C_0 = C_0^{*} + \mathrm{Im}(B \to C_0) .
\]\[H^1(A'_0/A_0, G_a) = 0 \qquad H^1(A'_0/A_0, G_m) = 0\]
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\[ H^1(A'_0/A_0, G_a) = 0 \qquad H^1(A'_0/A_0, G_m) = 0 \]
\[H^1(A'/A, G_m) \simeq H^1(A'/A, G_a)\]
LaTeX source
\[ H^1(A'/A, G_m) \simeq H^1(A'/A, G_a) \]
\[\mathrm{Ker}\bigl(H^1(A'/A, \mathrm{Gl}_n) \to H^1(A'_0/A_0, \mathrm{Gl}_n)\bigr)
\to
\mathrm{Ker}\bigl(H^1(A'/A, \mathbb{M}_n) \to H^1(A'_0/A_0, \mathbb{M}_n)\bigr) .\]
LaTeX source
\[
\mathrm{Ker}\bigl(H^1(A'/A, \mathrm{Gl}_n) \to H^1(A'_0/A_0, \mathrm{Gl}_n)\bigr)
\to
\mathrm{Ker}\bigl(H^1(A'/A, \mathbb{M}_n) \to H^1(A'_0/A_0, \mathbb{M}_n)\bigr) .
\]\[H^1(A'/A, \mathrm{Gl}_n) \xrightarrow{\ \sim\ } H^1(A'/A, \mathbb{M}_n)
= \mathbb{M}_n\bigl(H^1(A'/A)\bigr)\]
LaTeX source
\[
H^1(A'/A, \mathrm{Gl}_n) \xrightarrow{\ \sim\ } H^1(A'/A, \mathbb{M}_n)
= \mathbb{M}_n\bigl(H^1(A'/A)\bigr)
\]\[H^1\bigl(Z C^{\cdot}(A'/A, \mathbb{M}_n)\bigr) / \mathrm{Gl}(n, C_0)\]
LaTeX source
\[
H^1\bigl(Z C^{\cdot}(A'/A, \mathbb{M}_n)\bigr) / \mathrm{Gl}(n, C_0)
\]\[H^1\bigl(Z C^{\cdot}(A'/A, \mathbb{M}_n)\bigr) / \mathbb{M}_n(C_0)\]
LaTeX source
\[
H^1\bigl(Z C^{\cdot}(A'/A, \mathbb{M}_n)\bigr) / \mathbb{M}_n(C_0)
\]\[\mathrm{End}_{C_0}(E_{C_0}) = \mathrm{Aut}_{C_0}(E_{C_0})
+ \mathrm{Im}\bigl(\mathrm{End}_{B_0}(E_{B_0}) \to \mathrm{End}_{C_0}(E_{C_0})\bigr) .\]
LaTeX source
\[
\mathrm{End}_{C_0}(E_{C_0}) = \mathrm{Aut}_{C_0}(E_{C_0})
+ \mathrm{Im}\bigl(\mathrm{End}_{B_0}(E_{B_0}) \to \mathrm{End}_{C_0}(E_{C_0})\bigr) .
\]\[\mathrm{Ker}\bigl(H^1(A'/A, \underline{\mathrm{Aut}}(E)) \to H^1(A'_0/A_0, \underline{\mathrm{Aut}}(E))\bigr)
\to\]
LaTeX source
\[
\mathrm{Ker}\bigl(H^1(A'/A, \underline{\mathrm{Aut}}(E)) \to H^1(A'_0/A_0, \underline{\mathrm{Aut}}(E))\bigr)
\to
\]\[\mathrm{Ker}\bigl(H^1(A'/A, \underline{\mathrm{End}}(E)) \to H^1(A'_0/A_0, \underline{\mathrm{End}}(E))\bigr)\]
LaTeX source
\[
\mathrm{Ker}\bigl(H^1(A'/A, \underline{\mathrm{End}}(E)) \to H^1(A'_0/A_0, \underline{\mathrm{End}}(E))\bigr)
\]\[\simeq \Bigl[\bigl(\mathrm{Ker}(H^1(A'/A) \to H^1(A'_0/A_0))\bigr)^{(N)}\Bigr]^{N}\]
LaTeX source
\[
\simeq \Bigl[\bigl(\mathrm{Ker}(H^1(A'/A) \to H^1(A'_0/A_0))\bigr)^{(N)}\Bigr]^{N}
\]\[H^1(A'/A, \underline{\mathrm{Aut}}(E)) \simeq H^1(A'/A, \underline{\mathrm{End}}(E))
\simeq \Bigl[H^1(A'/A)^{(N)}\Bigr]^{N} .\]
LaTeX source
\[
H^1(A'/A, \underline{\mathrm{Aut}}(E)) \simeq H^1(A'/A, \underline{\mathrm{End}}(E))
\simeq \Bigl[H^1(A'/A)^{(N)}\Bigr]^{N} .
\]\[\mathbb{M}_m(B^{(n)}) = \mathbb{M}_m(B^{(n)})^{*} + \mathrm{Im}\,\mathbb{M}_m(B^{(n)})\]
LaTeX source
\[
\mathbb{M}_m(B^{(n)}) = \mathbb{M}_m(B^{(n)})^{*} + \mathrm{Im}\,\mathbb{M}_m(B^{(n)})
\]\[C^{\cdot}(A'/A) = \varprojlim_n C^{\cdot}(A'_n/A_n) ,\]
LaTeX source
\[
C^{\cdot}(A'/A) = \varprojlim_n C^{\cdot}(A'_n/A_n) ,
\]\[H^1(A'/A) = H^1\bigl(C^{\cdot}(A'/A)\bigr)
= \varprojlim H^1\bigl(C^{\cdot}(A'_n/A_n)\bigr)
= \varprojlim_n H^1(A'_n/A_n) = 0 .\]
LaTeX source
\[
H^1(A'/A) = H^1\bigl(C^{\cdot}(A'/A)\bigr)
= \varprojlim H^1\bigl(C^{\cdot}(A'_n/A_n)\bigr)
= \varprojlim_n H^1(A'_n/A_n) = 0 .
\]\[E = H^0_D(A'/A, E') .\]
LaTeX source
\[ E = H^0_D(A'/A, E') . \]
\[C^{\cdot}_D(A'/A, E') \simeq \varprojlim_n C^{\cdot}_D(A'_n/A_n, E'_n)\]
LaTeX source
\[
C^{\cdot}_D(A'/A, E') \simeq \varprojlim_n C^{\cdot}_D(A'_n/A_n, E'_n)
\]\[E = H^0_D(A'/A, E') = \varprojlim_n H^0(A'_n/A_n, E'_n) .\]
LaTeX source
\[ E = H^0_D(A'/A, E') = \varprojlim_n H^0(A'_n/A_n, E'_n) . \]
\[E^{(n)} = H^0(A'_n/A_n, E'_n)\]
LaTeX source
\[
E^{(n)} = H^0(A'_n/A_n, E'_n)
\]\[A \longrightarrow A' \rightrightarrows A''\]
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\[ A \longrightarrow A' \rightrightarrows A'' \]