\[
  X^{[n]} = \underbrace{X \times_S \cdots \times_S X}_{n+1},
\]

\[
  \mathcal{P}^{0} = \mathcal{O}_X .
\]

\[
  (\mathcal{SP})^{i} = \mathcal{P}^{i+1}
\]

\[
  (\mathcal{SP})(I) = \mathcal{P}(I \sqcup e)
\]

\[
  \alpha^{*} : \mathcal{P}^{*} \to (\mathcal{SP})^{*}
  \qquad \text{i.e.}\quad \alpha^{i} : \mathcal{P}^{i} \to \mathcal{P}^{i+1} .
\]

\[
  \Gamma \mathcal{P}^{i} = \bigotimes_{k}^{i+1} A
  = \underbrace{A \otimes_k A \otimes \cdots \otimes_k A}_{i+1},
\]

\[
  \underbrace{A \otimes \cdots \otimes A}_{i+1}
  \longrightarrow
  \underbrace{A \otimes \cdots \otimes A}_{i+2},
  \qquad
  \alpha^{i}(x_0 \otimes \cdots \otimes x_i) = x_0 \otimes \cdots \otimes x_i \otimes 1 .
\]

\[
  \mathcal{P}^{i}[M] = \mathcal{P}^{i} \otimes_{\mathcal{O}_X} M ,
\]

\[
  \mathcal{Q}^{i}[M] = \mathcal{Q}^{i} \otimes_{\mathcal{O}_X} M
\]

\[
  X^{\widehat{[2]}} \rightrightarrows X^{\widehat{[1]}}
  \xrightarrow{\widehat{\mathrm{pr}}_2} X^{\widehat{[0]}} = X
\]

\[
  \mathcal{Q}^{*}[M] = \mathcal{Q}^{*} \otimes_{\mathcal{O}_X} M .
\]

\[
  H^{*}(X, M) \simeq \mathbb{H}^{*}(X, l\mathcal{Q}^{*}[M])
  \Longleftarrow
  E_2^{pq} = H^{p}\bigl(i \mapsto H^{q}(X, l\mathcal{Q}^{i}[M])\bigr)
\]

\[
  H^{q}(X, l\mathcal{Q}^{i}[M]) = \varprojlim_{n}
  H^{q}(X, \mathcal{Q}^{i}_{\uncertain{n}}[M]_{(n)})
  \;\bigl(\simeq H^{q}(X^{\widehat{[i+1]}}_{n}, M)\bigr)
\]

\[
  \mathcal{Q}^{i}[M] = \varprojlim_{n} \mathcal{Q}^{i}[M]_{(n)}, \quad
  \mathcal{Q}^{i}[M]_{(n)} = \mathcal{Q}^{i}_{(n)} \otimes_{\mathcal{O}_X} M, \quad
  \mathcal{Q}^{i}_{(n)} = \mathcal{P}^{i+1}_{(n)} = \cdots\;]
\]

\[
  \mathbb{H}^{*}(X, M^{\cdot}) \simeq \mathbb{H}^{*}(X, \mathcal{Q}^{*}[M^{\cdot}])
  \simeq \mathbb{H}^{*}(X_{\mathrm{str}}, \mathcal{Q}^{0}[M^{\cdot}]) .
\]

\[
  \mathbb{H}^{*}(X, M^{\cdot}) \simeq \mathbb{H}^{*}(X, \mathcal{Q}^{*}[M^{\cdot}])
  \Longleftarrow E_2^{pq} = \mathbb{H}^{p}(X, C^{\cdot}(V^{q}))
\]

\[
  \mathbb{H}^{p}_{\mathrm{str}}(X/S, \underline{V})
  = \mathbb{H}^{p}(X, C^{\cdot}_{\mathrm{str}}(\underline{V})) ,
\]

\[
  \boxed{\;
  \mathbb{H}^{*}(X, M^{\cdot}) \Longleftarrow
  E_2^{pq} = \mathbb{H}^{p}_{\mathrm{str}}(X, V^{q}(M^{\cdot})),
  \quad\text{où}\quad
  V^{q}(M^{\cdot}) = \mathcal{H}^{q}(\mathcal{Q}^{0}[M^{\cdot}])
  \;}
\]

\[
  V^{q}(M^{\cdot}) = 0 \quad \text{pour } q \neq q_0 ,
\]

\[
  \mathbb{H}^{n}(X, M^{\cdot}) \simeq
  \mathbb{H}^{n-q_0}_{\mathrm{str}}(X, V^{q_0}(M^{\cdot})) .
\]

\[
  \Omega^{\cdot}_{X/S} \otimes \underline{V} = \Omega^{\cdot}(\underline{V})
\]

\[
  H^{*}_{\mathrm{DR}}(X/S, \underline{V}) \simeq H^{*}_{\mathrm{str}}(X/S, \underline{V})
\]

\[
  \underline{V} \to \mathcal{H}^{0}(l\mathcal{Q}^{0}[\Omega^{\cdot}(\underline{V})])
\]

\[
  \underline{V} \longrightarrow l\mathcal{Q}^{0}[\Omega^{\cdot}(\underline{V})]
\]

\[
  \boxed{\;
  H^{*}_{\mathrm{str}}(X, \underline{V}) \longrightarrow
  H^{*}_{\mathrm{DR}}(X/S, \underline{V})
  \;}
\]

\[
  H^{*}(X_0/S_{\mathrm{cris}}, \mathcal{O}_{X_0\,\mathrm{cris}})
\]

\[
  H^{*}((X_0/S)_{\mathrm{cris}}, \mathcal{O}) = \varprojlim_{n}
  H^{*}((X_n/S_n)_{\mathrm{cris}}, \mathcal{O})
\]

\[
  H^{i}((X_0/S)_{\mathrm{cris}}, \mathcal{O}) = 0 \quad \text{si } i > \dim X_0\,!
\]

\[
  W \longrightarrow W[1/p] = K\,! .
\]

\[
  H^{*}(X, \mathbb{C}) \overset{\text{12}}{=}
  \mathbb{H}^{*}(X, \Omega^{\cdot}_{X})
  \Longleftarrow E_1^{pq} = H^{q}(X, \Omega^{p}_{X})
\]

\[
  \mathbb{H}^{*}(X, \Omega^{\cdot}_{X}) \qquad \text{(hypercohomologie)}
\]

\[
  \mathbb{H}^{*}(X, \Omega^{\cdot}_{X}) \longrightarrow
  \mathbb{H}^{*}(X^{\mathrm{an}}, \Omega^{\cdot}_{X^{\mathrm{an}}})
  \simeq H^{*}(X^{\mathrm{an}}, \mathbb{C})
\]

\[
  H^{0}_{\mathrm{DR}}(X) \simeq H^{0}(X, \mathcal{O}_X)^{p} ;
\]

\[
  \dim H^{1}_{\mathrm{DR}}(X) \geq 2 \dim \mathrm{Pic}_X ,
\]

\[
  f : X \to Y .
\]

\[
  H^{*}(X^{\mathrm{an}}, \mathbb{C}) \Longleftarrow
  E_2^{pq} = H^{p}\bigl(Y^{\mathrm{an}},
  R^{q} f^{\mathrm{an}}_{*}(\mathbb{C}_{X^{\mathrm{an}}})\bigr) .
\]

\[
  H^{i}(X, \mathbb{Z}_\ell) = \varprojlim_{\nu}
  H^{i}(X_{\mathrm{ét}}, \mathbb{Z}/\ell^{\nu}\mathbb{Z}) ,
\]

\[
  \boxed{\;H^{i}(X, \mathbb{Z}_\ell) \simeq
  H^{i}(X^{\mathrm{an}}, \mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{Z}_\ell\;}
\]

\[
  \ell \neq \mathrm{car}\, k ,
\]

\[
  X \longmapsto H^{\cdot}(X)
\]

\[
  \boxed{\;H^{\cdot}_{\mathrm{DR}}(X/S) =
  \mathbb{H}^{*}(X, \Omega^{\cdot}_{X/S})\;}
\]

\[
  H_{\mathrm{DR}}(X) = \mathbb{H}^{*}(X ; \Omega^{\cdot}_X)
\]

\[
  \left\lbrace
  \begin{array}{l}
    H_{\mathrm{DR}}(X/S) = \mathbb{H}^{*}(X, \Omega^{\cdot}_{X/S}) \\
    \text{et les } Rf_{*}(\Omega^{\cdot}_{X/S})
  \end{array}
  \right.
\]

\[
  H(X_0) = \mathbb{H}^{*}(\mathfrak{X}, \Omega^{\cdot}_{\mathfrak{X}/S})
  = H^{*}\bigl(\Gamma(\mathfrak{X}, \Omega^{\cdot}_{\mathfrak{X}/S})\bigr)
\]

\[
  \mathfrak{X} = \operatorname{Spf} W\{t\} ,
  \qquad W\{t\} = \varprojlim W_n[t]
\]

\[
  \mathbb{H}^{1}(\mathfrak{X}, \Omega^{\cdot}_{\mathfrak{X}/S}) = 0
\]

\[
  \begin{aligned}
  \mathbb{H}^{1}(\mathfrak{X}, \Omega^{\cdot}_{\mathfrak{X}/S})
  &= \Omega^{1}(\mathfrak{X}/S) / d\bigl(\mathcal{O}(\mathfrak{X}/S)\bigr) \\
  &\simeq W\{t\} / \mathrm{Im}\,(f \mapsto f')
  \end{aligned}
\]

\[
  \sum \frac{1}{n+1}\, a_n t^{n+1}
  \quad \Bigl(\text{ou encore } \sum \frac{1}{n}\, a_{n-1} t^{n}\Bigr)
\]

\[
  v_p(a_n) - v_p(n+1) \longrightarrow +\infty
\]

\[
  v_p(a_n) \longrightarrow +\infty\,!
\]

\[
  \frac{|a_n|_p}{|n+1|_p} \longrightarrow 0
\]

\[
  |a_n|_p = \varepsilon_n\, |n+1|_p , \qquad \varepsilon_n \longrightarrow 0
\]

\[
  |a_n| = O(1)\, \rho^{n} \qquad \text{pour un } \rho > 1
\]

\[
  \varphi : A \otimes_W k \simeq A_0 .
\]

\[
  \Omega^{*}(A/W) = \textstyle\bigwedge^{*}(A/W) ,
\]

\[
  u^{*} : H^{\cdot}\bigl(\Omega^{\cdot}(A'/W)\bigr) \longrightarrow
  H^{\cdot}\bigl(\Omega^{\cdot}(A/W)\bigr)
\]

\[
  (*) \qquad \mathcal{K}^{\cdot} = Rf_{*}(\Omega^{\cdot}_{Y/M})
\]

\[
  f_g^{(\Sigma)} : X_g^{(\Sigma)} \longrightarrow S
\]

\[
  Rf_{g*}(\Omega^{\cdot}_{X_g/S}) \xleftarrow{\ \simeq\ } g^{*}(\mathcal{K}^{\cdot})
\]

\[
  R^{i}f_{g*}(\Omega^{\cdot}_{X_g/S}) \xleftarrow{\ \simeq\ } g^{*}(\mathcal{H}^{i})
\]

\[
  \mathcal{K}^{\cdot} = Rf_{*}(\Omega^{\cdot}_{Y/M})
\]

\[
  \underline{E} \simeq h^{*}(\underline{E}_0)
\]

\[
  d\, i_X + i_X\, d = \theta_X
\]

\[
  \mathcal{H}^{i}\bigl(f_{*}(\Omega^{\cdot}_{X/S})\bigr) =
  R^{i}f_{*}(\Omega^{\cdot}_{X/S})
\]

\[
  Rf_{*}(\Omega^{\cdot}_{X/S})
\]

\[
  H^{*}_{\substack{\text{cris}/S \\ \text{ou strat}}}(X/S, \mathcal{O}_X)
  \simeq H^{*}_{\mathrm{DR}}(X/S)
\]

\[
  H^{*}_{\substack{\text{strat} \\ \text{ou crist.}}}(X/\mathbb{C},
  \mathcal{O}_X) \simeq H^{*}(X^{\mathrm{an}}, \mathbb{C})
\]

\[
  C^{n}(\mathcal{F}) = \mathcal{F}^{(n)} =
  \varprojlim_{i} \mathcal{F}_{\Delta^{n}_{X/S}(i)}
\]

\[
  \boxed{\;H^{*}(X_{\mathrm{str}}, \mathcal{F}) \simeq
  \mathbb{H}^{*}\bigl(X_{\mathrm{zar}}, C^{*}(\mathcal{F})\bigr)\;}
\]

\[
  \boxed{\;(*)\quad \mathcal{Q}^{i}[M] = \mathcal{P}^{i}
  \otimes_{\mathcal{O}_X} M\;}
\]

\[
  H^{*}(X_{\mathrm{strat}}, F) \Longleftarrow
  E_2^{pq} = H^{p}\bigl(\nu \mapsto H^{q}(\widetilde{X}^{\nu+1}, F)\bigr)
\]

\[
  \widetilde{X}^{\nu+1} = \varinjlim_{i} \Delta^{(\nu)}_{X}(i)
\]

\[
  H^{*}\bigl((U, X'), F\bigr) \simeq H^{*}(X', F_{(U, X')})
\]

\[
  H^{q}\bigl(\widetilde{\Delta^{(\nu)}_X(i)}, F\bigr) = 0
  \quad \text{si } q > 0
\]

\[
  H^{q}(\widetilde{X}^{(\nu+1)}, F) =
  H^{q}\Bigl(\varinjlim_{i} \widetilde{\Delta^{(\nu)}_X(i)}, F\Bigr) =
  \varprojlim H^{q}\bigl(\widetilde{\Delta^{(\nu)}_X(i)}, F\bigr)
\]

\[
  \left\lbrace
  \begin{array}{l}
    H^{q}(\widetilde{X}^{(\nu+1)}, F) = 0 \quad \text{si } q > 0 \\
    H^{0}(\widetilde{X}^{(\nu+1)}, F) = \varprojlim_{i}
    F\bigl(\Delta^{(\nu)}_X(i)\bigr)
  \end{array}
  \right.
\]

\[
  \boxed{\;H^{*}(X_{\mathrm{strat}}, F) \simeq
  H^{*}\bigl(\nu \mapsto F(X^{(\nu+1)}/X)\bigr)\;}
\]

\[
  \mathcal{F}^{(\nu)} : U \longmapsto F(U^{(\nu+1)}/U)
  = \varprojlim_{i} F\bigl(\Delta^{(\nu)}_{U}(i)\bigr)
\]

\[
  H^{*}(X_{\mathrm{strat}}, F) \simeq
  \mathbb{H}^{*}(X_{\mathrm{zar}}, \mathcal{F}^{*}) ,
  \qquad
  \mathcal{F}^{*} = (\nu \mapsto \mathcal{F}^{\nu})
\]

\[
  \boxed{\;H^{*}(X_{\mathrm{strat}}, F) \Longleftarrow
  E_{2}^{pq} = H^{p}\bigl(\nu \mapsto H^{q}(X_{\mathrm{zar}},
  \mathcal{F}^{(\nu)})\bigr)\;}
\]

\[
  \widetilde{Y} = \varinjlim_{i} \widetilde{Y(i)} .
\]

\[
  H^{*}(X_{\mathrm{cris}}, F) \Longleftarrow
  E_{2}^{pq} = H^{p}\bigl(\nu \mapsto H^{q}(\widetilde{Y}^{\nu+1}, F)\bigr)
\]

\[
  H^{*}(\widetilde{X'}, F) = H^{*}(X'_{\mathrm{zar}}, F_{X'})
\]

\[
  H^{*}(X_{\mathrm{cris}}, F) \simeq
  H^{*}\bigl(\nu \mapsto F(Y^{\nu+1}/X)\bigr)
\]

\[
  \mathcal{F}^{(\nu)} : U \longmapsto \varprojlim F\bigl(U, U(i)^{\nu+1}\bigr)
\]

\[
  \mathcal{F}^{(\nu)}(U) = F\bigl(V^{\nu+1}/U\bigr)
\]

\[
  H^{*}(X_{\mathrm{cris}}, F) \simeq
  \mathbb{H}^{*}(X_{\mathrm{zar}}, \mathcal{F}^{*})
\]

\[
  H^{*}(X_{\mathrm{cris}}, F) \Longleftarrow
  H^{p}\bigl(\nu \mapsto H^{q}(X, \mathcal{F}^{(\nu)})\bigr)
\]

\[
  \mathcal{F}^{(\nu)} \simeq \mathcal{F}_0^{(\nu)}
\]

\[
  H^{*}(X_{\mathrm{cris}}, F) \xrightarrow{\ \sim\ }
  H^{*}(X_{0\,\mathrm{cris}}, F_0) .
\]

\[
  H^{*}(X_{\mathrm{cris}}, \mathcal{O}_{X_{\mathrm{cris}}})
  = H^{*}(X_{0\,\mathrm{cris}}, \mathcal{O}_{X_{0\,\mathrm{cris}}})
\]

\[
  H^{*}(X_{\mathrm{strat}}, \mathcal{O}_{X_{\mathrm{strat}}})
  \Longleftarrow E_{2}^{pq} = H^{p}\bigl(\nu \mapsto
  H^{q}(\mathcal{O}_{X^{\nu+1}/X})\bigr)
\]

\[
  H^{q}\bigl(X^{\nu+1}/X, \mathcal{O}_{X^{\nu+1}/X}\bigr)
  \simeq \varprojlim_{i} H^{q}\bigl(\Delta^{(\nu)}(i),
  \mathcal{O}_{\Delta^{(\nu)}(i)}\bigr)
\]

\[
  E_{2}^{pq} = \varprojlim_{i} H^{p}\Bigl(\nu \mapsto
  H^{q}\bigl(\Delta^{(\nu)}(i), \mathcal{O}_{\Delta^{(\nu)}(i)}\bigr)\Bigr) .
\]

\[
  H^{*}(X_{\mathrm{strat}}, \mathcal{O}_{X_{\mathrm{strat}}})
  \simeq H^{*}(X^{\mathrm{an}}_{\mathrm{strat}},
  \mathcal{O}_{X^{\mathrm{an}}_{\mathrm{strat}}})
  \overset{\mathrm{déf}}{=}
  \mathbb{H}^{*}(X^{\mathrm{an}}, \mathcal{F}^{\mathrm{an}\,*}_{X^{\mathrm{an}}})
\]

\[
  \mathcal{F}^{\mathrm{an}\,*}_{X^{\mathrm{an}}}
  = \mathcal{F}^{*}_{X^{\mathrm{an}}}
  = (\nu \mapsto \mathcal{F}^{\nu}_{X^{\mathrm{an}}})
\]

\[
  \mathcal{F}^{\nu}_{X^{\mathrm{an}}} : U \longmapsto
  \Gamma\bigl(U^{\nu+1}/U, \mathcal{O}_{U^{\nu+1}/U}\bigr)
\]

\[
  H^{*}(X_{\mathrm{strat}}, \mathcal{O}_{X_{\mathrm{strat}}})
  \simeq H^{*}(X^{\mathrm{an}}, \mathbb{C})
\]

\[
  H^{*}(X_{\mathrm{cris}}, \mathcal{O}_{X_{\mathrm{cris}}}) \longrightarrow
  H^{*}(X_{\mathrm{strat}}, \mathcal{O}_{X_{\mathrm{strat}}})
\]

\[
  H^{*}(X^{\mathrm{an}}, \mathcal{F}^{*}_{X^{\mathrm{an}}})
  \simeq H^{*}(X^{\mathrm{an}}, \mathbb{C}) .
\]

\[
  H^{*}_{\mathrm{cris}}(X, \mathcal{O}_{X_{\mathrm{cris}}})
  = \varprojlim_{i} H^{*}_{\mathrm{cris}}\bigl(Y(i),
  \mathcal{O}_{Y(i)_{\mathrm{cris}}}\bigr)
\]

\[
  H^{*}(X_{\mathrm{cris}}, F_{X_{\mathrm{cris}}}) \xrightarrow{\ \sim\ }
  H^{*}(X_{\mathrm{strat}}, F\ (\text{dite stratifiée}))\ .
\]

\[
  H^{*}(X_{\mathrm{strat}}, F) \xrightarrow{\ \sim\ }
  H^{*}(X_{0\,\mathrm{strat}}, F)
\]