\[
  F_B \colon \mathrm{Mod}(B) \to (\mathrm{Ens})
\]

\[
  F \text{ représentable (resp.\ par un $A$-module de type fini, de prés.\ finie)}
\]

\[
  \Updownarrow
\]

\[
  F \text{ exact à gauche, et } F_B \text{ représentable (resp.\ par un $B$-module de type fini, de prés.\ finie).}
\]

\[
      F(M) \xrightarrow{\ \sim\ } \varprojlim F(M/f^{n+1}M)
    \]

\[
      F(\widehat{M}) \xrightarrow{\ \sim\ } \varprojlim F(M/\mathfrak{m}^{n+1}M),
    \]

\[
      F(M) \xrightarrow{\ \sim\ } \varprojlim F(M/\mathfrak{m}_{\mathfrak{p}}^{n+1}M) ;
    \]

\[
  \begin{cases}
    \underline{N}'_x = N_x \subset M_x \\
    \underline{N}'_s = N_s \subset M_s & \text{si } s \in U \cap V.
  \end{cases}
\]

\[
  \mathrm{Tor}_i^{\mathcal{O}_{S,s}}\bigl(F_s,\ \mathcal{O}_{S,s}/(f_1, \ldots, f_n)\bigr) = 0
  \quad \text{pour } i > 0 .
\]

\[
  \underline{\mathrm{Hom}}(P, \Omega)_s \otimes \mathcal{O}_{S,s}/(f_1, \ldots, f_n)
  \longrightarrow
  \underline{\mathrm{Hom}}_{\mathcal{O}_{S,s}}\bigl(P_s,\ \Omega_s/(f_1, \ldots, f_n)\Omega_s\bigr)
\]

\[
  L_1 \to L_0 \to P
\]

\[
  K^{\cdot} \colon\quad 0 \to \check{L}_0 \otimes Q \to \check{L}_1 \otimes Q \to 0
\]

\[
  H^0(K^{\cdot}) \otimes M \to H^0(K^{\cdot} \otimes M)
\]

\[
  \mathrm{Tor}_1(M, H^1(K^{\cdot})) = \mathrm{Tor}_1(M, Z^1(K^{\cdot})) = 0 .
\]

\[
      \mathrm{Hom}\bigl(R',\ \Omega_x/(f_1, \ldots, f_n)\Omega_x\bigr) \to
      \mathrm{Hom}\bigl(R,\ \Omega_x/(f_1, \ldots, f_n)\Omega_x\bigr)
    \]

\[
      F(\Omega) \otimes A_{\mathfrak{p}}/(f_1, \ldots, f_n) \to
      F\bigl(\Omega \otimes A_{\mathfrak{p}}/(f_1, \ldots, f_n)\bigr)
    \]

\[
  \mathrm{Hom}(P_{i_0}, \Omega/\underline{f}\Omega) \to
  \mathrm{Hom}(P_i, \Omega_{\mathfrak{p}}/\underline{f}\Omega_{\mathfrak{p}})
\]

\[
      F(M) \xrightarrow{\ \sim\ } \varprojlim F(M/\mathfrak{m}_{\mathfrak{p}}^{n+1}M) ;
    \]

\[
      F(M) \xrightarrow{\ \sim\ } \varprojlim F(M/f^{n+1}M) ;
    \]

\[
  F(\Omega) \otimes B_{\mathfrak{p}}/\underline{f} B_{\mathfrak{p}}
  \longrightarrow
  F(\Omega \otimes B_{\mathfrak{p}}/\underline{f} B_{\mathfrak{p}})
\]

\[
  \underline{\mathrm{Hom}}(P, M) \simeq
  f_{*}\,\underline{\mathrm{Hom}}(F,\ G \otimes_{\mathcal{O}_S} M)
\]

\[
  \mathrm{Hom}(P, M) \simeq \mathrm{Hom}(F,\ G \otimes_{\mathcal{O}_S} M),
\]

\[
  M \longmapsto R^{q} f_{*}\bigl(\underline{\mathrm{Ext}}^{i}(F,\ G \otimes_{\mathcal{O}_S} M)\bigr)
\]

\[
  M \longmapsto H^{q}\bigl(X,\ \underline{\mathrm{Ext}}^{i}(F,\ G \otimes_{\mathcal{O}_S} M)\bigr)
\]

\[
  \underline{\mathrm{Ext}}^{i}(F,\ G \otimes_{\mathcal{O}_S} \Omega) \otimes A_{\mathfrak{p}}/\underline{f} A_{\mathfrak{p}}
  \xrightarrow{\ \sim\ }
  \underline{\mathrm{Ext}}^{i}(F,\ G \otimes_{\mathcal{O}_S} \Omega \otimes A_{\mathfrak{p}}/\underline{f} A_{\mathfrak{p}})
\]

\[
  \underline{\mathrm{Ext}}^{i}(F,\ \underbrace{G \otimes_A \Omega}_{G'})
  = H^{i}\bigl(\underline{\mathrm{Hom}}^{\cdot}(L_{\cdot},\ G \otimes_{\mathcal{O}_S} \Omega)\bigr)
  = H^{i}\bigl(\underbrace{(\check{L}_{\cdot} \otimes_B G) \otimes_{\mathcal{O}_S} \Omega}_{K^{\cdot}}\bigr)
\]

\[
  \underline{\mathrm{Ext}}^{i}(F,\ (G \otimes_{\mathcal{O}_S} \Omega) \otimes_A M) = H^{i}(K^{\cdot} \otimes M),
\]

\[
  \underline{\mathrm{Ext}}^{i}_{\mathcal{O}_X}(F, G') \otimes_{\mathcal{O}_S} M
  \xrightarrow{\ \sim\ }
  \underline{\mathrm{Ext}}^{i}_{\mathcal{O}_X}(F,\ G' \otimes_A M)
\]

\[
  H^{i}(K^{\cdot}) \otimes_A M \xrightarrow{\ \sim\ } H^{i}(K \otimes_A M)
  \quad \text{si } i \leq N .
\]

\[
  \begin{cases}
    \mathrm{Tor}_i^{A}(K^{j}, M) = 0 & \text{pour } i > 0, \text{ tout } j \\
    \mathrm{Tor}_i^{A}(H^{j}(K^{\cdot}), M) = 0 & \text{pour } i > 0, \text{ tout } j.
  \end{cases}
\]

\[
  \mathrm{Tor}_i^{A}(F,\ A_{\mathfrak{p}}/\underline{f} A_{\mathfrak{p}}) = 0
  \quad \text{pour } i > 0 .
\]

\[
  R^{j} f_{*}(E)_s \otimes \mathcal{O}_{S,s}/\underline{f}\,\mathcal{O}_{S,s}
  \longrightarrow
  R^{j} f^{(s)}_{*}\bigl(E^{(s)} \otimes \mathcal{O}_{S,s}/\underline{f}\,\mathcal{O}_{S,s}\bigr)
\]

\[
  E^{(s)} \otimes \mathcal{O}_{S,s}/\underline{f}\,\mathcal{O}_{S,s}
  \simeq
  E^{(s)} \overset{\mathbb{L}}{\otimes} \mathcal{O}_{S,s}/\underline{f}\,\mathcal{O}_{S,s} .
\]

\[
  K^{\cdot} \otimes \mathcal{O}_{S,s}/\underline{f}\,\mathcal{O}_{S,s}
  = K^{\cdot} \overset{\mathbb{L}}{\otimes} \mathcal{O}_{S,s}/\underline{f}\,\mathcal{O}_{S,s} .
\]

\[
  Rf_{*}\bigl(\underbrace{E \overset{\mathbb{L}}{\otimes}_{\mathcal{O}_S} M}_{E \otimes_{\mathcal{O}_S} M}\bigr)
  \longrightarrow
  \underbrace{Rf_{*}(E) \overset{\mathbb{L}}{\otimes}_{\mathcal{O}_S} M}_{K^{\cdot} \overset{\mathbb{L}}{\otimes} M \,=\, K^{\cdot} \otimes M},
\]

\[
  H^{j}(K^{\cdot}) \otimes M \to H^{j}(K^{\cdot} \otimes M) .
\]

\[
  \mathrm{Tor}_i^{\mathcal{O}_S}(H^{j}(K^{\cdot}), M) = 0 \quad \text{pour } i > 0,\ \text{tout } j,
\]

\[
  f_{*}(Z) = \prod_{X/S} Z/X
\]

\[
  \underline{\mathrm{Hom}}(F, f^{*}(M)) \simeq \underline{\mathrm{Hom}}(P, M),
\]

\[
  Z \hookrightarrow Z' = V(\underline{F}),
\]

\[
  f_{*}(Z) \to f_{*}(Z')
\]

\[
  f_{*}(Y) = \prod_{X/S} Y/X .
\]

\[
  f_{*}(V(F)) \simeq V(P) :
\]