\[
  \mathbf{P}^{n} \supset X^{r} \supset Y^{m}, \qquad r > m > 0
\]

\[
  H^{p}_{Y}(X, F) = 0 \quad \text{pour } p > \ill{}
\]

\[
  H^{p}(X^{r}, F) \longrightarrow H^{p}(X^{r} - Y, F) \longrightarrow
  H^{p+1}_{Y}(X^{r}, F)
\]

\[
  H^{p}(X^{r} - Y, F) = 0 \quad \text{pour } p \geqslant m
\]

\[
  H^{p}(X - Y, \mathcal{O}(n)) = 0 \quad \text{pour } n \text{ grand},\ p \geqslant m .
\]

\[
  H^{p}_{Y}(X^{r}, F) \longrightarrow H^{p}(X^{r}, F) \quad
  \begin{cases}
    \text{bijectif pour } p > m \\
    \text{surjectif pour } p = m
  \end{cases}
\]

\[
  \mathrm{Ext}^{q}_{\mathcal{O}_{\hat{X}^{r}}}
    \!\left(\hat{X}^{r}, \hat{F}, \hat{\Omega}^{r}_{X^{r}/k}\right)
  \;\xleftarrow{\ \sim\ }\;
  \mathrm{Ext}^{q}_{\mathcal{O}_{X^{r}}}
    \!\left(X^{r}, F, \Omega^{r}_{X^{r}/k}\right)
\]

\[
  \begin{cases}
    \text{bijectif pour } q < r - m \\
    \text{injectif pour } q = r - m
  \end{cases}
\]

\[
  H^{q}\!\left(X^{r}, \Omega(n)\right) \longrightarrow
  H^{q}\!\left(\hat{X}^{r}, \hat{\mathcal{O}}(n)\right) \quad
  \begin{cases}
    \text{bijectif pour } q < r - m \\
    \text{injectif pour } q = r - m
  \end{cases}
  \quad n \text{ grand}
\]

\[
  H^{q}(X^{r}, F) \longrightarrow H^{q}(\hat{X}^{r}, \hat{F}) \quad
  \begin{cases}
    \text{bij. pour } q < r - m \\
    \text{inj. pour } q = r - m
  \end{cases}
  \quad \text{pour } F \text{ loc. libre}
\]

\[
  u : A^{n} \longrightarrow B .
\]

\[
  H^{p}_{Y}(X, F) \simeq H^{p}_{Y}(P, F) = \varinjlim\;
  \mathrm{Ext}^{p}\!\left(P; \mathcal{O}_{Y_{n}}, F\right)
\]

\[
  \left[\,H^{p}(X - Y, F) = \varinjlim\;
  \mathrm{Ext}^{p}\!\left(P; J^{n}, F\right)\,\right]
\]

\[
  \varprojlim\; \mathrm{Ext}^{r-p}\!\left(P; F,
  \mathcal{O}_{Y_{n}} \otimes \Omega^{r}_{P/k}\right)
  \;\xrightarrow{\ \sim\ }\;
  \mathrm{Ext}^{r-p}\!\left(\hat{Y}, \hat{F}, \hat{\Omega}^{r}_{P^{r}}\right)
\]

\[
  H^{p}_{Y}(X, F) \to H^{p}(X, F) \to H^{p}(X - Y, F) \to H^{p+1}_{Y}(X, F)
\]

\[
  \mathrm{Ext}^{r-p}_{\mathcal{O}_{P}}\!\left(\hat{Y}, \hat{F}, \hat{\Omega}^{r}\right)
  \leftarrow \mathrm{Ext}^{r-p}\!\left(P, F, \Omega^{r}\right)
  \leftarrow \varprojlim\; \mathrm{Ext}^{r-p}\!\left(P, F, J^{n}\Omega\right)
  \leftarrow
\]

\[
  \mathrm{Ext}^{r}
\]

\[
  \mathrm{Ext}^{i}_{Y}(X; F, G) = \varinjlim\; \mathrm{Ext}^{i}(X, F_{n}, G)
\]

\[
  \varprojlim\; \mathrm{Ext}^{r-i}_{a}(X, G, F_{n}) \simeq
  \mathrm{Ext}^{r-i}(\hat{X}, \hat{G}, \hat{F})
\]

\[
  \mathrm{Ext}^{i}_{Y'}(X'; F', G') = \varinjlim\; \mathrm{Ext}^{i}(X'; F'_{n}, G')
\]

\[
  \varprojlim\; \mathrm{Ext}^{r-1-i}(X', G', F'_{n})
\]

\[
  \mathrm{Ext}^{r-1-i}(\hat{X}', \hat{G}', \hat{F}') \;?
\]

\[
  \mathrm{Ext}^{r-i}(\hat{X}, \hat{G}, \hat{F}) \simeq \mathrm{Ext}^{r-i}(X; G, F)
\]

\[
  \mathrm{Ext}^{i}_{Y}(X; F, G) \longrightarrow \mathrm{Ext}^{i}_{Z}(X; F, G)
  \longrightarrow \mathrm{Ext}^{i}_{Z-Y}(X; F, G)
\]

\[
  \mathrm{Ext}^{r-1-i}_{a}\!\left(X_{/Y}; G_{/Y}, F_{/Y}\right) \longleftarrow
  \mathrm{Ext}^{r-1-i}_{a}\!\left(X_{/Z}; G_{/Z}, F_{/Z}\right) \longleftarrow
\]

\[
  X_{/Y} \longrightarrow X_{/Z}
\]

\[
  (11) \qquad C^{p}(\mathfrak{U}, F) = \varinjlim_{n}\; C^{p}_{n}(M) .
\]

\[
  (12) \qquad C^{p}(\mathfrak{U}, F) \;\xrightarrow{\ \sim\ }\;
  C^{p+1}\!\left((f), M\right) .
\]

\[
  (13) \qquad H^{p}(\mathfrak{U}, F) \;\xrightarrow{\ \sim\ }\;
  H^{p+1}\!\left((f), M\right) .
\]

\[
  (14) \qquad 0 \to H^{0}\!\left((f), M\right) \to M \to
  H^{0}(\mathfrak{U}, F) \to H^{1}\!\left((f), M\right) \to 0 .
\]

\[
  H^{r-i}\!\left(\hat{P}, \Omega^{r}_{P}(-n)\right)
\]

\[
  H^{i}\!\left(\hat{P}, \Omega^{r}_{P}(-n)\right) = 0 \quad \text{si } 0 < i < r-m
\]

\[
  H^{r}\!\left(P, \Omega^{r}(-n)\right) \;\xrightarrow{\ \sim\ }\;
  H^{r}\!\left(\hat{P}, \hat{\Omega}^{r}(-n)\right)
\]