\[
  X' \simeq \mathbf{V}(\mathcal{O}_X(1)) - \sigma(X)
  = \operatorname{Spec} \struck{\ill{}} \Bigl(\coprod_{n \in \mathbf{Z}}
  \mathcal{O}(-n)\Bigr).
\]

\[
\begin{aligned}
  \operatorname{Ext}^{\bullet}_{Z' \cap U' = T'}(X'; F'; G')
  &= H^{\bullet}\, \struck{\ill{}}\; \underline{\operatorname{Hom}}_{Z' \cap U'\,
  T'}(F'; C(G'))\\
  &= H^{\bullet}\, \struck{\ill{}}\; \underline{\operatorname{Hom}}_{Z \cap U\,
  T}(F, p_{U*} C(G'))\\
  &= H^{\bullet}\, \underline{\operatorname{Hom}}_{Z \cap U\, T}(F,
  C(p_{U*} G'))\\
  &= \operatorname{Ext}^{\bullet}_{Z \cap U\, T}\Bigl(F, \coprod_{n \in
  \mathbf{Z}} G(n)\Bigr)
\end{aligned}
\]

\[
  T = U \cap Z \ \text{(loc. fermé dans $X$)}, \qquad T' = U' \cap Z',
\]

\[
  p : X' \longrightarrow X, \qquad p_U : U' \longrightarrow U.
\]

\[
  \operatorname{Ext}^i_{\struck{\ill{}}\,T',\, \mathcal{O}_{X'}}(X'; F', G')
  \simeq \coprod_{n \in \mathbf{Z}} \operatorname{Ext}^i_{T\,
  \struck{\ill{}}}(F, G(n)).
\]

\[
  \underline{\operatorname{Ext}}^{\bullet}_{\overline{T'}\,
  \struck{\ill{}},\, \mathcal{O}_{\overline{X'}}}(\overline{X'}/Y;
  \overline{F'}, \overline{G'}) \simeq \coprod_{n \in \mathbf{Z}}
  \underline{\operatorname{Ext}}^{i}_{\struck{\ill{}},\, \mathcal{O}_U}(X/Y;
  F, G(n))
\]

\[
  \operatorname{Ext}^{\bullet}_{\overline{T'}}(\overline{X'}; \overline{F'},
  \overline{G'})
  \ \Big\uparrow\
  \operatorname{Ext}^{\bullet}_{\widehat{T'}}(\widehat{X'}; \widehat{F'},
  \widehat{G'})
\]

\[
  H^p\bigl(\overline{Z'} - \overline{R'},\ \underline{\operatorname{Ext}}^q_{\overline{Z'}}
  (\overline{F'}, \overline{G'})\bigr)
\]

\[
  H^p(\overline{Z'} - \overline{R'}, \overline{E})
  \overset{(2)}{\longleftrightarrow}
  H^p(\widehat{Z'} - \widehat{R'}, \widehat{E})
\]

\[
\begin{aligned}
  \operatorname{Ext}^{\bullet}_{T, \mathcal{O}_X}(X, F, G(n))
  &\Longleftarrow H^p\bigl(U, \underline{\operatorname{Ext}}^q_{T,
  \mathcal{O}_U}(F, G(n))\bigr)\\
  \operatorname{Ext}_{T', \mathcal{O}_{X'}}(\overline{X'}; \overline{F'},
  \overline{G'})
  &\Longleftarrow H^p\bigl(U', \underline{\operatorname{Ext}}_{T',
  \mathcal{O}_{U'}}(\overline{F'}, \overline{G'})\bigr)
\end{aligned}
\]

\[
\begin{aligned}
  \operatorname{Ext}^{\bullet}_{T, \mathcal{O}_X}(X; F, G(n))
  &\Longleftarrow H^p_T\bigl(\struck{X}\, X,
  \underline{\operatorname{Ext}}^q(F, G)(n)\bigr)\\
  \operatorname{Ext}_{T', \mathcal{O}_{X'}}(\overline{X'}, \overline{F'},
  \overline{G'})
  &\Longleftarrow H^p_{T'}\bigl(\overline{X'},
  \underline{\operatorname{Ext}}^q(F', G')(n)\bigr)
\end{aligned}
\]

\[
  \operatorname{Ext}^i(\mathfrak{X}; \mathcal{M}, \mathcal{N}) \simeq
  \operatorname{Ext}^i_A(M, N).
\]

\[
  \underline{\operatorname{Ext}}^i(\mathcal{M}, \mathcal{N}) \simeq
  \operatorname{Ext}^i_A(M, N)^{\Delta}\,\struck{\ill{}}.
\]

\[
  \operatorname{Ext}^i_{\mathfrak{X}}(\mathcal{M}, \mathcal{N}) \simeq
  H^0\bigl(\mathfrak{X}, \underline{\operatorname{Ext}}^i(\mathcal{M},
  \mathcal{N})\bigr)
\]

\[
  H^0\bigl(\mathfrak{X}, \underline{\operatorname{Ext}}^i(\mathcal{M},
  \mathcal{N})\bigr) \simeq \operatorname{Ext}^i_A(M, N) \quad \text{pour
  tout } i,
\]

\[
  \underline{\operatorname{Ext}}^i_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{M},
  \mathcal{N}) \simeq
  \underline{\operatorname{Ext}}^i_{\mathcal{O}_X}(\widetilde{M},
  \widetilde{N})^{\wedge} \quad \text{\ill{}}\ \
  \underline{\operatorname{Ext}}^i(\widetilde{M}, \widetilde{N})
  = \operatorname{Ext}^i_A(M, N)^{\sim} \ \ (\text{\uncertain{cq}}),
\]

\[
  \underline{\operatorname{Ext}}^i_{\mathcal{O}_{\mathfrak{X}}}(\mathcal{M},
  \mathcal{N}) \simeq \operatorname{Ext}^i_A(M, N)^{\Delta}
\]

\[
  \operatorname{Ext}^i_Y(\mathfrak{X}; \mathcal{F}, \mathcal{G})
  \Longleftarrow H^p_Y\bigl(\mathfrak{X},
  \underline{\operatorname{Ext}}^q(\mathcal{F}, \mathcal{G})\bigr),
\]

\[
  \varprojlim_n \operatorname{Ext}^i_Y(\mathfrak{X}, \mathcal{F},
  \mathcal{G}_n).
\]

\[
\left\{
\begin{aligned}
  \underline{E}_{U/k}(G, {}_{\infty}\mu) &\simeq \underline{E}_k(N^i_{U/k}\,G,
  \mathbf{Q}/\mathbf{Z})\\
  \underline{E}_{S/k}(G, {}_{\infty}\mu) &\simeq
  \underline{E}_k(N^i_{Y/k}(G), \mathbf{Q}/\mathbf{Z})\\
  \underline{E}_{Y/k}(G, {}_{\infty}\mu) &\simeq E_k(N^i_{S/k}(G),
  \mathbf{Q}/\mathbf{Z}).
\end{aligned}
\right.
\]

\[
\begin{array}{ccc}
  0 & & \\
  \downarrow & & \\
  H^0_Y(M) & \text{---} & \operatorname{Ext}^1_Y(S; M, A)\\
  \downarrow & & \uparrow\\
  H^0(S, M) & \text{---} & \operatorname{Ext}^i_Y(S; M, A)\\
  \downarrow & & \\
  H^0(U, M) & &
\end{array}
\]

\[
\begin{array}{ccccc}
  0 & & & & \\
  \downarrow & & & & \\
  H^0(M) & \times & \operatorname{Ext}^{\ill{}}(M, A) & \longrightarrow &
  I\\
  \downarrow & & \uparrow & & \\
  M & \times & \operatorname{Ext}^{\ill{}}(M, A) & \longrightarrow & I\\
  \downarrow & & \uparrow & & \\
  M \otimes K & \times & \operatorname{Hom}(M_K, K) & \longrightarrow & I\\
  \downarrow & & \uparrow & & \\
  H^1(M) & \times & \operatorname{Ext}^0(M, A) & \longrightarrow & I\\
  \downarrow & & \uparrow & & \\
  0 & & 0 & &
\end{array}
\]

\[
  \operatorname{Ext}_Y(M, A) \simeq \operatorname{Hom}(H^i_Y(X, M), I)
\]

\[
  H^i_x(F) \times \operatorname{Ext}^{n-i}(X; F, \mathcal{O}_X)
  \longrightarrow H^n_x(\mathcal{O}_X) = I \qquad (I \text{ module
  \uncertain{dualisant}})
\]

\[
  \boxed{\operatorname{Ext}^{n-i}(X; F, \mathcal{O}_X) \longrightarrow
  \operatorname{Hom}(H^i_x(F), I)}
\]

\[
  \operatorname{Ext}^{n-i}(U; \mathcal{O}_X, \mathcal{O}_X)
  \xrightarrow{\ \sim\ } \operatorname{Hom}(H^i_x(\mathcal{O}_{U,X}), I),
\]

\[
\left\{
\begin{aligned}
  &H^i_x(\mathcal{O}_{U,X}) = 0 \quad \text{si } i \neq n,\\
  &D\bigl(H^n_x(\mathcal{O}_{U,X})\bigr) \simeq \struck{\operatorname{Hom}}\,
  \Gamma(U, \mathcal{O}_X) = A_f.
\end{aligned}
\right.
\]

\[
  \operatorname{Hom}(K, K/A) \simeq K
\]

\[
  u : M_0 \longrightarrow M_1
\]

\[
  H^0_x(F) = \operatorname{Ker} u, \quad H^1_x(F) = \operatorname{Coker} u
  \qquad [H^0(X, F) \simeq M_0,\ H^i(X, F) = 0 \ i \neq 0]
\]

\[
  \mathcal{O}_X \longrightarrow \underbrace{\underline{K}_X \longrightarrow
  \underline{K}_X/\mathcal{O}_X \longrightarrow 0 \longrightarrow 0
  \longrightarrow \cdots}_{K^{\bullet}}
\]

\[
  \operatorname{Ext}^{*}(X; F, \mathcal{O}_X) \simeq H^{*}\bigl(0
  \longrightarrow \underset{0}{\operatorname{Hom}(M_1, K)} \longrightarrow
  \underset{1}{\operatorname{Hom}(M_0, K/A)} \longrightarrow 0 \cdots\bigr)
\]

\[
\begin{aligned}
  \operatorname{Ext}^0(X; F, \mathcal{O}_X) &\simeq
  \operatorname{Ker}\bigl(\operatorname{Hom}(M_1, K) \to
  \operatorname{Hom}(M_0, K/A)\bigr)\\
  \operatorname{Ext}^1(X; F, \mathcal{O}_X) &\simeq
  \operatorname{Coker}(\quad \text{id} \quad)
\end{aligned}
\]

\[
\begin{aligned}
  E^0 &\overset{\varphi^0}{\longrightarrow}
  \operatorname{Hom}(\operatorname{Coker} u, K/A)\\
  E^1 &\overset{\varphi^1}{\longrightarrow}
  \operatorname{Hom}(\operatorname{Ker} u, K/A)
\end{aligned}
\]

\[
  0 \longrightarrow \underset{\substack{\wr\\ K}}{\operatorname{Hom}_A(K,
  K)} \longrightarrow \operatorname{Hom}(K, K/A) \longrightarrow 0,
\]

\[
  \operatorname{Ext}^i(K, A) = 0 \qquad \operatorname{Ext}^i(K, A)
\]

\[
  0 \longrightarrow K \longrightarrow \coprod_{p} K/A_p \longrightarrow I
  \longrightarrow 0
\]

\[
  0 \longrightarrow K \longrightarrow \prod_{p}{}^{\wedge}\, \widehat{K}_p
  \longrightarrow \operatorname{Hom}(K, I) \longrightarrow 0
\]

\[
  \operatorname{Ext}^i_A(K, A) \qquad \operatorname{Ext}^i(A,\ \ )
  \qquad \operatorname{Hom}(K, A) = 0
\]

\[
  \operatorname{Ext}^1(K, A) \qquad 0 \longrightarrow A \longrightarrow E
  \longrightarrow K \longrightarrow 0
\]

\[
  0 \longrightarrow A \longrightarrow K \longrightarrow \coprod
  \qquad \operatorname{Hom}(K, A/\mathfrak{m}^n)
\]

\[
  \operatorname{Ext}^1(K,\ \ ) \qquad \operatorname{Ext}^i \qquad
  \operatorname{Ext}^1(\underline{K}, A^{(I)})
\]

\[
\begin{array}{ccc}
  K & \longrightarrow & \prod \widehat{K}_p\\
  \uparrow & & \uparrow\\
  A & \longrightarrow & \prod \widehat{A}_p
\end{array}
\qquad
\left|
\begin{array}{ll}
  0 & \text{Classes d'idèles}\\[4pt]
  1 & \text{Classes d'\uncertain{idéaux}}
\end{array}
\right.
\]

\[
  H^0_x(F),\ H^1_x(F) = \operatorname{Ker} \text{ et } \operatorname{Coker}
  \text{ de } \underset{\substack{\|\\ 0}}{H^0(X, F)} \longrightarrow
  \underset{\substack{\|\\ 0}}{H^0(X', F)} \quad \text{\uncertain{nuls}}.
\]

\[
  H^2_x(F) \simeq H^1(X', F).
\]

\[
  0 \longrightarrow F \longrightarrow \mathcal{O}_{X'} \longrightarrow G
  \longrightarrow 0,
\]

\[
  0 \to \underset{\substack{\|\\ A}}{H^0(X', \mathcal{O}_{X'})}
  \xrightarrow{\alpha^0}
  \underset{\substack{\|\\ \prod \mathcal{O}_{x_i}\\ \|\\ \varinjlim_g A
  g^{-1}}}{H^0(Y'; G)}
  \xrightarrow{\partial}
  \underset{\substack{\|\\ H^2_x(F)}}{H^1(X'; F)}
  \xrightarrow{\alpha^1}
  \underset{\substack{\|\\ I}}{H^1(X'; \mathcal{O}_{X'})} \to 0
\]

\[
  d_g : h \longmapsto [h]\bigl[\tfrac{1}{g}\bigr].
\]

\[
  H^n(\Omega) \simeq H^n(A) \otimes \Omega \xrightarrow{\ \eta\ } I
\]

\[
  H^n(M) \xleftarrow{\ \sim\ } H^n(A) \otimes \underline{M}\,]
\]

\[
  H^n(\Omega) \simeq \struck{\ill{}}\, H^n(A) \otimes \Omega
  \xrightarrow{\ \varphi\ } I
\]

\[
  \ill{}\quad H^n(A) \xrightarrow{\varphi_1} D(\Omega).
  \quad \uncertain{Un} \ill{} \ill{}
\]

\[
  H^p(M) \times \operatorname{Ext}^{n-p}(M, \Omega) \longrightarrow
  H^n(\Omega) \rightsquigarrow I
\]

\[
  H^p(M) \longrightarrow D(\operatorname{Ext}^{n-p}(M, \Omega))
\]

\[
  H^n(M) \simeq \struck{\ill{}}\, H^n(A) \otimes M \longrightarrow
  D(\operatorname{Hom}(M, \Omega)) \simeq D(\Omega) \otimes M
\]

\[
  \struck{H^n(\Omega) \ill{}(A)}
\]

\[
  H^n(\Omega) \times \widehat{\operatorname{Hom}(\Omega, \Omega)}
  \longrightarrow H^n(\Omega) \longrightarrow I
\]

\[
  \operatorname{Hom}(\Omega, \Omega) \xrightarrow{\ \sim\ } D(H^n(\Omega))
\]

\[
  H^i(A) = 0 \quad \text{pour } k \leqslant i < n.
\]

\[
\left\{
\begin{aligned}
  &\operatorname{Ext}^i(k, \Omega) = 0 \quad \text{si } i \neq n\\
  &\operatorname{Ext}^n(k, \Omega) \simeq k
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
  &H^i(\Omega_{\ill{}}) = 0 \quad \text{si } i \neq n\\
  &H^n(\Omega_{\ill{}}) \text{ \uncertain{est} \uncertain{dualisant}}
\end{aligned}
\right.
\]

\[
\left\{
\begin{aligned}
  &\operatorname{Ext}^i(\Omega, \Omega) = 0 \quad \text{si } i \neq 0\\
  &\struck{\operatorname{Ext}}\ \operatorname{Hom}(\Omega, \Omega) \simeq A
\end{aligned}
\right.
\]

\[
  H^i(\Omega) = 0 \quad \text{pour } i < n ,
\]

\[
  H^i(\Omega) \simeq D\bigl(\mathrm{Ext}^{n-i}(\Omega, \Omega)\bigr)
\]

\[
  \mathrm{Ext}^{n-i}(\Omega, \Omega) = 0 \quad \text{si } \uncertain{i \neq n},
  \ \text{i.e. } n - i \neq 0 .
\]

\[
  D\bigl(\mathrm{Hom}(\Omega, \Omega)\bigr) \simeq H^n(\Omega)
\]

\[
  \mathrm{Ext}^i(M, \Omega) = 0 \quad \text{si } i > n
\]

\[
  \boxed{\mathfrak{D}_1 : H^n(A) \xrightarrow{\ \sim\ } D(\Omega)}
\]

\[
  \boxed{H^n(A) \otimes \Omega = H^n(\Omega) \xrightarrow{\ \mathfrak{D}\ } I ,
  \qquad \mathfrak{D} \in D\bigl(H^n(\Omega)\bigr)}
\]

\[
  \mathfrak{D}_2 : \hat{\Omega} \xrightarrow{\ \sim\ } D\bigl(H^n(A)\bigr)
\]

\[
  \boxed{H^n(M) \xleftarrow[\ \sim\ ]{\mathrm{can}} H^n(A) \otimes M}
  \xrightarrow[\ \sim\ ]{\mathfrak{D}_1 \otimes \mathrm{id}_M}
  D(\Omega) \otimes M \xrightarrow[\ \sim\ ]{\mathrm{can}}
  D\bigl(\mathrm{Hom}(M, \Omega)\bigr)
\]

\[
  \boxed{\begin{array}{l}
    \mathfrak{D}_{1,M} : H^n(M) \xrightarrow{\ \sim\ }
      D\bigl(\mathrm{Hom}(M, \Omega)\bigr) \\
    \mathfrak{D}_{2,M} : \mathrm{Hom}(M, \Omega)^{\wedge}
      \xrightarrow{\ \sim\ } D\bigl(H^n(M)\bigr)
  \end{array}}
\]

\[
  \begin{array}{l}
    \mathfrak{D}_{1,\Omega} : H^n(\Omega) \xrightarrow{\ \sim\ }
      D\bigl(\mathrm{Hom}(\Omega, \Omega)\bigr) \\
    \mathfrak{D}_{2,\Omega} : \mathrm{Hom}(\Omega, \Omega)^{\wedge}
      \xrightarrow{\ \sim\ } D\bigl(H^n(\Omega)\bigr)
  \end{array}
\]

\[
  {}^t\mathfrak{D} : \hat{A} \longrightarrow \mathrm{Hom}(\Omega, \Omega)^{\wedge}
\]

\[
  0 \longrightarrow \mathfrak{N} \longrightarrow A \longrightarrow A'
  \longrightarrow 0
\]

\[
  H^n(A) \simeq H^n(A')
\]

\[
  \mathrm{Hom}_A(\Omega, \Omega) = \bigcap_{\substack{\mathfrak{p}\ \text{premier}\\
  \text{de corang 1}}} A_{\mathfrak{p}} =
  \struck{\text{normalisé}}\ \text{clôture intégrale de } A .
\]

\[
  0 \longrightarrow A \xrightarrow{\ x\ } A \longrightarrow A/xA
  \longrightarrow 0
\]

\[
  H^{n-1}(A) \xrightarrow{\ x\ } H^{n-1}(A) \longrightarrow H^{n-1}(A/xA)
  \longrightarrow H^n(A) \xrightarrow{\ x\ } H^n(A)
\]

\[
  H^{n-1}(A/xA) \simeq \struck{\mathrm{Ker}\ \text{de}\ x\ \text{dans}}\
  \text{noyau de } x \text{ dans } H^n(A) .
\]

\[
  H^{n-1}(A/xA) \simeq D(\Omega/x\Omega) , \quad \text{cqfd.}
\]

\[
  A \simeq A'/\mathfrak{N}
\]

\[
  (1) \qquad H^i(M) \simeq
  D\bigl(\mathrm{Ext}^{\ill{}-i}_{A'}(M, \Omega_{A'})\bigr)
  \qquad \text{i.e.}
\]

\[
  (2) \qquad H^n(A) \simeq
  D\bigl(\mathrm{Ext}^{n'-n}_{A'}(M, \Omega_{A'})\bigr)
\]

\[
  (2\ \text{bis}) \qquad \Omega_A \simeq
  \mathrm{Ext}^{n'-n}_{A'}(M, \Omega_{A'}) .
\]

\[
  (\Omega_A)_{\mathfrak{p}} \simeq
  \mathrm{Ext}^{n'-n}_{A'_{\mathfrak{p}'}}\bigl(M_{\mathfrak{p}^*},
  (\Omega_{A'})_{\mathfrak{p}'}\bigr) .
\]

\[
  \mathrm{Ext}^{n'-i}_{A'}(M, \Omega_{A'}) = 0 \quad \text{pour}
  \ 0 \leq i < k .
\]

\[
  (13) \qquad \Omega_A^{(i)} =
  \mathrm{Ext}^{n'-n+i}_{A'}(M, \Omega_{A'}) :
\]

\[
  (13\ \text{bis}) \qquad D(\Omega_A^{(i)}) \simeq H^{\uncertain{n-i}}(A)
\]

\[
  \Omega_A^{(0)} = \Omega_A
\]

\[
  \Omega_A^{(i)} = 0 \ \text{pour} \ i \geq 1 \Longleftrightarrow A \
  \text{est Coh.\ Mac.}
\]

\[
  \mathrm{Ext}^{\bullet}_{A'}(N, M) \Longleftarrow
  \mathrm{Ext}^p_A\bigl(N, \mathrm{Ext}^q_{A'}(A, M)\bigr)
\]

\[
  \mathrm{Hom}_{A'}(N, M) = \mathrm{Hom}_A\bigl(N, \mathrm{Hom}_{A'}(B, M)\bigr)
\]

\[
  \mathcal{C}^{A'} \xrightarrow{\ h^{A'}_A\ } \mathcal{C}^{A}
  \xrightarrow{\ h^A_N\ } \mathcal{G}^{\uncertain{A}} ,
  \qquad h^A_N \circ h^{A'}_A = h^{A'}_N
\]

\[
  \boxed{\bigl(E^{i}(M)\bigr) = \bigl(H^{\ill{}i}(M)\bigr) \Longleftarrow
  E_2^{pq}(M) = D\bigl(\mathrm{Ext}^p_A(M, \Omega_A^{(q)})\bigr)}
\]

\[
  \mathrm{Ext}^{n-1}(k, M) = 0 .
\]

\[
  \struck{0 < \mathrm{lg}_{\mathfrak{p}} M \leq \mathrm{lg}_{\mathfrak{p}} A}
  \quad \text{et} \ \ill{} \qquad
  \boxed{0 < \mathrm{lg}_{\mathfrak{p}} M \leq \mathrm{lg}_{\mathfrak{p}} A}
\]

\[
  \boxed{\mathrm{Ext}^{n-1}_A(k, M) = 0}
\]

\[
  H^n(M) \simeq D(A)^m \ \text{pour un entier} \ m
\]

\[
  \mathrm{lg}_{\mathfrak{p}}(M) = m\, \mathrm{lg}_{\mathfrak{p}}(A)
  \qquad
  \begin{array}{l}
    \mathrm{Ext}^n(k, M) \simeq k^m \\
    \mathrm{Ext}^i(k, M) = 0 \ \text{si} \ i > n
  \end{array}
\]

\[
  M \simeq \Omega^m , \ \text{où} \ \Omega \ \text{est un module
  fondamental}
\]

\[
  \struck{0 \to A \xrightarrow{\ x\ } A \to A/xA \to 0}
\]

\[
  \struck{\ill{} \to H^{n-1}(\ill{}) \to H^n(\ill{})}
  \qquad
  \struck{\mathrm{Ext}^{\bullet}(N, A/x) \Longleftarrow
  \mathrm{Ext}^{\bullet}_{A/x}\bigl(N, \mathrm{Ext}^{\bullet}_A(A/x, A)\bigr)}
\]

\[
  \mathrm{Ext}^{n'-n}_{A'}(A, A') \simeq A
\]

\[
  \boxed{A \ \text{est Coh.\ Mac.\ et} \ M \simeq \Omega_A}
\]

\[
  \begin{array}{l}
    \mathrm{Ext}^n(A/\mathfrak{N}, M) = \mathrm{Hom}(A/\mathfrak{N}, M) \\
    \mathrm{Ext}^n(k, M) = \mathrm{Hom}(k, M)
  \end{array}
\]

\[
  \mathrm{Ext}^{\bullet}_A(N, M) \Longleftarrow
  \mathrm{Ext}^{\bullet}_{A/x}\bigl(N, \mathrm{Ext}^{\bullet}_A(A/x, M)\bigr)
\]

\[
  \mathrm{Ext}^i_A(N, M) \simeq \mathrm{Ext}^{i-1}_{A/x}(N, M/xM)
\]

\[
  \begin{array}{c}
    D(M/xM) = \text{annulateur de } x \text{ dans } D(M) \\
    \wr| \\
    H^n(A/xA) = \text{annulateur de } x \text{ dans } H^n(A) .
  \end{array}
\]

\[
  H^n(A) \longrightarrow D(M)
\]

\[
  M \longrightarrow \Omega_A
\]

\[
      \left\{\begin{array}{l}
        \struck{\ill{}} \\
        \mathrm{Ext}^{n+1}(\underline{k}, M) = 0 \\
        H^n(M) \ \text{dualisant} \\
        H^n(M) \ \text{injectif}
      \end{array}\right.
    \]

\[
  \boxed{A \ \text{est C.-M., et} \ M \ \text{est} \ \simeq \Omega_A}
\]

\[
  N \mapsto \operatorname{Ext}^i(N, M) \qquad (N \text{ de longueur finie})
\]

\[
  \operatorname{Ext}^i(N, M) \simeq \operatorname{Hom}(N, H^i(M))
\]

\[
  \operatorname{Ext}^{\uncertain{i}}(k, M) = 0 \Longleftrightarrow H^i(M) = 0 .
\]

\[
  \operatorname{Ext}^i(N, M) \simeq \operatorname{Hom}(N, H^i(M))
\]

\[
  0 \to M \to J \to M' \to 0
\]

\[
  \overline{\Omega}_{A/xA} \simeq \Omega_A / x\,\Omega_A
\]

\[
  \overline{\Omega}_{A/xA} = \struck{H^{\ill{}}(A/xA) = \ill{}}
  \operatorname{Ext}^{\uncertain{n-1}}_{A'}(A/x, A')
\]

\[
  \struck{H^{\ill{}}(A/xA) = \ill{}}
\]

\[
  \operatorname{Ext}^{*}_{A'}(A/x, A') \Longleftarrow
  \operatorname{Ext}^i_A\bigl(A/x, \underbrace{\operatorname{Ext}^{\uncertain{j}}_{A'}(A, A')}_{\Omega^{(j)}_A}\bigr)
  \qquad \ill{}
\]

\[
  \Omega^{(*)}(A/x) \Longleftarrow \operatorname{Ext}^i_A(A/x, \Omega^{(j)}_A)
\]

\[
  0 \to \operatorname{Ext}^1_A(A/x, \Omega_A) \to \Omega_{A/x} \to
  \operatorname{Hom}_A(A/x, \Omega^{(1)}_A) \to 0 .
\]