\[
  G \simeq \mathrm{Im}\bigl(F \to i_*(G_U)\bigr) .
\]

\[
  Y_i \to Y' = \mathrm{Spec}(\widehat{\mathcal{O}}_{Y,y}),
\]

\[
  \mathrm{Spec}\bigl(\widehat{\mathcal{O}}_{Y,y} / \mathfrak{m}_y^{i+1}\bigr),
  \qquad i \in \mathbb{N} .
\]

\[
  \left\lbrace
  \begin{array}{l}
    \mathcal{O}_{Y,y} \text{ réduit} \\
    \mathcal{O}_{X,x} \text{ réduit} \\
    f \text{ universellement ouvert (EGA IV 14, 15)} \\
    X_y \text{ réduit en les gén.\ maximales } x_i \text{ de } x \text{ dans } X_y \\
    (X_y)_{\mathrm{red}} \text{ est géom.\ normal}/k(y)
  \end{array}
  \right.
\]

\[
  \Downarrow
\]

\[
  \mathcal{O}_\Lambda^n \xrightarrow{\ u\ } \mathcal{O}_\Lambda
\]

\[
  \Lambda' \mapsto \mathrm{Ker}\bigl(u_{\Lambda'} \colon
  \Lambda'^{\,n} \to \Lambda'\bigr)
\]

\[
  H^*(\mathcal{C}_{/\delta}, F) \simeq H^*(\widehat{Z}, F|\widehat{Z})
\]

\[
  A = \varprojlim \bigl(A_\alpha = A/\mathfrak{J}^{\alpha+1}\bigr)
\]

\[
  \left.
  \begin{array}{l}
    M \to N \\
    M_\alpha \xrightarrow{\ \sim\ } N_\alpha
  \end{array}
  \right\rbrace
  \Longrightarrow
  \forall\, \mathfrak{p} \in \mathrm{Spec}\, A,\
  \mathfrak{J} \not\subset \mathfrak{p},\
  M_{\mathfrak{p}} \xrightarrow{\ \sim\ } N_{\mathfrak{p}}
\]

\[
  \exists \text{ solution} \Longleftrightarrow \exists \text{ solution
  après tt chgt de base } S' \to S,\ S' \text{ artinien ou trait}
\]

\[
  H^q(\mathcal{C}_{/\delta_i}, F) \simeq
  H^q_{\mathcal{M},\mathcal{C}}(Z_n, F) \quad \ldots
\]

\[
  0 \to {\varprojlim_n}^{(1)} H^{q-1}_{(\mathcal{M},\mathcal{C})}(X^{(i)}_n, F)
  \to H^q(X^{(i)}, F) \to \varprojlim_n H^q_{(\mathcal{M},\mathcal{C})}(X^{(i)}_n, F)
  \to 0
\]

\[
  \begin{array}{ccc}
    Z_U & \hookrightarrow & Z\,? \\
    \cap & & \cap \\
    U & \subset & X \\
    & \searrow & \downarrow \\
    & & Y
  \end{array}
\]

\[
  \Longleftrightarrow\ \forall y \in Y,\quad Z_y \cap U \supset
  \mathrm{Ass}\, Z_y \qquad \text{EGA IV 11.12\,?}
\]

\[
  \left\lbrace
  \begin{array}{l}
    Z'_i \cap U'_y \neq \emptyset \\
    \text{i.e.\ } Z'_i \cap X'_y \neq \{x'\}
  \end{array}
  \right.
\]

\[
  \begin{array}{llll}
    G_{U'} & U' \hookrightarrow X' = X^h & & X'_y \supset U'_y \\
    & \qquad\ \downarrow & & \ \downarrow \\
    G_U & U \subset X & & X_y \supset U_y
  \end{array}
\]

\[
  \exists\, J = \{i_1, \ldots, i_n\} \subset I, \qquad
  \mathfrak{A}_n = \mathrm{Ker}\Bigl(\underbrace{\widehat{\mathcal{O}}_{Y,y}}_{A}
  \to \prod_{\alpha \in J} \mathcal{O}_{Y_\alpha, y_\alpha}\Bigr)
  \subset \widehat{\mathfrak{m}}_y^{\,n}
\]

\[
  A \to A/\mathfrak{A}_n \hookrightarrow \prod \mathcal{O}_{Y_\alpha, y_\alpha} .
\]

\[
  \Gamma\bigl(S_\infty, C^{\cdot}(F)_{S_\infty}\bigr) =
  \varprojlim \Gamma\bigl(S_i, C^{\cdot}(F)_{S_i}\bigr)
\]

\[
  H^n(S_\infty, F) \to \varprojlim H^n(S_i, F)
\]

\[
  H^*(S_\infty, F) \Longleftarrow E_2^{pq} = {\varprojlim_i}^{(p)} H^q(S_i, F)
\]

\[
  0 \to {\varprojlim_i}^{(1)} H^{n-1}(S_i, F) \to H^n(S_\infty, F)
  \to \varprojlim_i H^n(S_i, F) \to 0
\]

\[
  F(Y) \to F(X) \rightrightarrows F(X \times_Y X) .
\]

\[
  F(Y) \to F(X) \rightrightarrows F(X \times_Y X) .
\]

\[
  X \xrightarrow{\ u'\ } X'_1 \xrightarrow{\ u''\ } Y ,
\]

\[
  0 \to F(S) \to F(S') \rightrightarrows F(S'')
\]