\[
  L_A\bigl(T^n_A(M)\bigr) \simeq T^n_A\bigl(L_A(M)\bigr) \supset TS^n_A\bigl(L_A(M)\bigr)
\]

\[
  N^F_{T/S} : \Gamma(G_T/T) \longrightarrow \Gamma(G/S)
\]

\[
  N^{\alpha}_{F/T/S}(\varphi)
\]

\[
  N^{\alpha}_{F/T/S} : \Gamma(G/T) \to \Gamma(F\ill{}
\]

\[
  \boxed{\;N^{\gamma}_{(G \otimes_T F)/U/S}(\varphi)
  = N^{\alpha}_{F/T/S}\bigl(N^{\beta}_{G/U/T}(\varphi)\bigr)\;}
\]

\[
  N^{\varphi}_{G/X/Z}(\alpha) : Z \longrightarrow G'
\]

\[
  N^{\psi}_{G/X/Y}(\alpha') : Y \longrightarrow G''
\]

\[
  N^{\chi}_{F/Y/Z}\bigl(N^{\psi}_{G/X/Y}(\alpha')\bigr) : Z \longrightarrow G'
\]

\[
  \boxed{\;N^{\varphi}_{G/X/Z}(c\alpha') = N^{\chi}_{F/Y/Z}\bigl(N^{\psi}_{G/X/Y}(\alpha')\bigr)\;}
\]

\[
  \mathrm{Can}_u = \mathrm{Can}_{X/Y} : Y \to \Sigma^r X
\]

\[
    \Sigma^r(X/Y) = \struck{\ill{}} \coprod_{\substack{i+j=r\\ i,j \geqslant 0}}
    \Sigma^i(X_1/Y) \times \Sigma^j(X_2/Y)
  \]

\[
  Y \to \Sigma^r(X/Y),
\]

\[
  \Phi_A(X) = \struck{\ill{}} \operatorname{Hom}_G(X, E)
\]

\[
  A = \Phi(G_s)
\]

\[
  \text{(A)}\qquad
  \boxed{\;H^*(G, A) \Longleftarrow H^p\bigl(T/e, \underline{\mathcal{H}}^q(A)\bigr)\;}
  \qquad (T \neq \emptyset)
\]

\[
  \underline{\mathcal{H}}^q(A)(S) = R^q_A\bigl(\struck{A}\,
  \struck{\operatorname{Hom}_{(G)}}(S, A)\bigr)
\]

\[
  \underline{\mathcal{H}}^q(A)(F\backslash G) = H^q(F, A)\;]
\]

\[
  \underline{\mathcal{H}}^q(\Phi)(S) = H^q(-/S, \Phi) =
  \left\{\begin{array}{l}
    \text{valeurs en } \Phi \text{ des foncteurs dérivés } q\text{-ièmes}\\
    \text{de } \Phi \to \Phi(S)
  \end{array}\right.
\]

\[
  \text{(B)}\qquad
  \boxed{\;H^*(-/S, \Phi) \Longleftarrow H^p\bigl(T/S, \underline{\mathcal{H}}^q(\Phi)\bigr)\;}
\]

\[
  H^*(F, A) \Longleftarrow H^p\bigl(F \bmod F' ; \underline{\mathcal{H}}^q(A)\bigr)
\]

\[
  \cdot \to H^2(T/S, \mathbf{G}_m) \to H^2(-/S, \mathbf{G}_m) \to
  H^0(T/S, \mathcal{H}^2(\mathbf{G}_m)) \xrightarrow{\ d_3\ }
  H^3(T/S, \mathbf{G}_m) \to H^3(-/S, \mathbf{G}_m)'
\]

\[
  \to H^1(T/S, \mathcal{H}^2(\mathbf{G}_m)) \xrightarrow{\ d_3\ }
  H^4(T/S, \mathbf{G}_m) \to E_4^{4,0} \to 0
\]

\[
  H^{*}(X) \Leftarrow H^p(Y, H^q(F))
\]

\[
  H^{*}(Y) \Leftarrow H^p(B_F, \{H^q(X)\})
\]

\[
  H^{*}(B_F) \Leftarrow H^p(B_{\struck{X}}, \{H^q(Y)\})
\]

\[
  H^{*}(B_G) \Leftarrow H^p(B_Y, H^q(B_F))
\]

\[
  0 \to F''_i \to F_i \to F'_i \to 0
\]

\[
  \begin{cases}
    A = \struck{\mathfrak{F}}\ \mathcal{E}(F), & F \text{ \uncertain{défini} sur } X \times_S S' \\
    B = \mathcal{E}(G), & G \text{ --- } X \times_S S''
  \end{cases}
\]

\[
  (F_{s'} \otimes_{k(s')} K) \otimes_{\struck{X}\, X_{s'} \otimes_{k(s)} K}
  (G_{s''} \otimes_{k(s'')} K) \simeq H_{s'''} .
\]

\[
  \struck{\ill{}\ \mathcal{E}(F) \otimes}\ \mathcal{E}_{X/S}(F) \otimes
  \mathcal{E}_{X/S}(G) \subset \mathcal{E}_{X/S}(H) .
\]

\[
  0 \to Z_0 \to L_0 \to \bar{F} \to 0
\]

\[
  H^q(K^{(i)} \otimes_{T_i} T') \longleftarrow H^q(K^{(i)}) \otimes_{T_i} T'
\]

\[
  K^{i-1} \xrightarrow{\ d^{i-1}\ } K^i \to K^{i+2}
\]

\[
  \mathcal{T}or_{*}^{\mathcal{O}_Y}(F_1, \ldots, F_n) \otimes_Y Y'
  \quad\text{et}\quad
  \mathcal{T}or_{*}^{\mathcal{O}_{Y'}}(F'_1, \ldots, F'_n)
\]

\[
  \mathrm{Tor}_p^{\mathcal{O}_S}\bigl(\mathrm{Tor}_q^{\mathcal{O}_Y}(F_1, \ldots, F_n)
  \otimes_{\ill{}} \ldots, \mathcal{O}_{S'}\bigr)
\]

\[
  \mathrm{Tor}_p^{\mathcal{O}_{S'}}(F'_1, \ldots, F'_n) \otimes_{S'} T
  \longrightarrow
  \mathrm{Tor}_p^{\mathcal{O}_T}(F'_{1T}, \ldots, F'_{nT})
\]

\[
  \mathrm{Tor}_i^{\mathcal{O}_S}(F_1, \ldots, F_n) \times_S T \longrightarrow
  \mathrm{Tor}_i^{\mathcal{O}_T}(F_{1(T)}, \ldots, F_{n(T)})
\]

\[
  K = \bigotimes_{\substack{B \\ 1 \le i \le n}} L^{(i)} .
\]

\[
  K' = K \otimes_A A' = \bigotimes_{\substack{B' \\ 1 \le i \le n}}
  L^{(i)} \otimes_B B' = \bigotimes_{\substack{B' \\ 1 \le i \le n}} L^{(i)\prime}
\]

\[
  H_{*}(K \otimes_A A') \Longleftarrow \mathrm{Tor}_p^A(H_q(K), A')
\]

\[
  \mathrm{Tor}_{*}^{B'}(M'_1, \ldots, M'_n) \Longleftarrow
  \mathrm{Tor}_p^A\bigl(\mathrm{Tor}_q^B(M_1, \ldots, M_n), A'\bigr) .
\]

\[
  \mathrm{Tor}_p^B(M_1, \ldots, M_n) \otimes_A A' \longrightarrow
  \mathrm{Tor}_p^{B'}(M'_1, \ldots, M'_n)
\]

\[
  \mathrm{Tor}_p^{\mathcal{O}_{S_\alpha}}(F_1^{(\alpha)}, \ldots, F_n^{(\alpha)})
  \otimes_{S_\alpha} T \longrightarrow
  \mathrm{Tor}_p^{\mathcal{O}_T}(F^{(\alpha)}_{1(T)}, \ldots, F^{(\alpha)}_{n(T)})
\]

\[
  \mathcal{E}xt^p_{\mathcal{O}_{S'}}(F', G') \otimes_{S'} T \longrightarrow
  \mathcal{E}xt^p_{\mathcal{O}_T}(F'_{(T)}, G'_{(T)})
\]

\[
  \mathcal{E}xt^i_{\mathcal{O}_X}(F, G) \times_S T \longrightarrow
  \mathcal{E}xt^i_{\mathcal{O}_{X_T}}(F_{(T)}, G_{(T)})
\]

\[
  M \otimes_A A' = M \otimes_B B' = M',
\]

\[
  \operatorname{Ext}^p_{B'}(M', N') = H^p\bigl(\operatorname{Hom}(L'_*, N')\bigr)
  = H^p\bigl(\operatorname{Hom}_{\uncertain{A}}(L_*, N) \otimes_A A'\bigr)
\]

\[
  \operatorname{Ext}^p_{B'}(M', N') = H^p(K^* \otimes_A A'), \qquad
  \operatorname{Ext}^p_{B}(M, N) = H^p(K^*) \qquad (p \le N).
\]

\[
  H^p(K^*) \otimes_A A' \longrightarrow H^p(K^* \otimes_A A')
\]

\[
  \operatorname{Ext}^p_{B'}(M', N') \simeq \operatorname{Ext}^p_B(M, N) \otimes_A A'
\]