\[ \int_C d\omega = \int_{\partial C} \omega \]

\[
C^\bullet_{\mathrm{DRS}\,\mathbb{R}\text{-alg}} \subset C^\bullet_{\mathrm{DRS}\,C^\infty} \subset C^\bullet_{\mathrm{DRS}},
\qquad C^\bullet_{\mathrm{DRS}\,C^\infty} \supset C^\bullet_{\mathrm{DR}}
\]

\[ H^q(C^\bullet_{\mathrm{DRS}\,\mathbb{Q}}(X,\mathbb{Q})) \simeq H^q(X,\mathbb{Q}) \]

\[ \mathrm{DRS}^\bullet_{[n]} = C^\bullet_{\mathbb{Q}\text{-DR}}(E^{[n]}) \]

\[ \mathrm{DRS}^{\bullet}_{*} = (\mathrm{DRS}^\bullet_{[n]})_{n \geq 0} \]

\[ C^\bullet_{\mathrm{DRS}}(X,\mathbb{Q}) \simeq \mathrm{Hom}(S_*(X), \mathrm{DRS}^\bullet_*) \]

\[ \mathrm{DRS}^\bullet_{[n]} \simeq \text{\struck{$\mathbb{Q}[X_i, dX_i]$}}\ C^\bullet_{\mathrm{DR}/\mathbb{Q}}\bigl(\mathbb{Q}[(X_i)_{0 \leq i \leq n}]/\textstyle\sum X_i - 1\bigr) \]

\[ = \mathbb{Q}[X_i, dX_i]_{0 \leq i \leq n} \big/ \bigl(\textstyle\sum X_i - 1,\ \sum dX_i\bigr) \simeq \mathbb{Q}[X_0, \ldots, X_{n-1}][dX_0, \ldots, dX_{n-1}] \]

\[ \text{\struck{$\mathbb{Q}$}}\ \mathbb{Z}\bigl(\{X_i\}[dX_i]\bigr)_{0 \leq i \leq n} \big/ \bigl(\textstyle\sum X_i - 1,\ \sum dX_i\bigr)_{\mathrm{pd}} \]

\[ \simeq \mathbb{Z}\{X_0, \ldots, X_{n-1}\}[dX_0, \ldots, dX_{n-1}] \]

\[ X_n = 1 - \sum_0^{n-1} X_i \]

\[ \sum X_i = t \quad \text{dans } S^{n+1} \]

\[ \bigl(C^\bullet_{\mathrm{DRpd}[n]}(S,J,t) = \Bigl(S\{X_i\}_{0 \leq i \leq n}[dX_i]_{0 \leq i \leq n}\Bigr) \Big/ \Bigl(\textstyle\sum X_i - t,\ \sum dX_i\Bigr)_{\mathrm{pd}} \]

\[ \Bigl(S\{X_i\}_{0 \leq i \leq n} \big/ (\textstyle\sum X_i - t)_{\mathrm{pd}}\Bigr) \otimes_S \overset{*}{\Lambda}\bigl(S^{[n]}/\mathrm{diag}\,S^{[n]}\bigr) \]

\[ C^\bullet_{\mathrm{DRpd}[n]}(S,J,t) \to S/J \]

\[ C^\bullet_{\mathrm{DRpd}*}(S,J,t) = \bigl(C^\bullet_{\mathrm{DRpd}[n]}(S,J,t)\bigr)_{n \geq 0} \]

\[ C^\bullet_{\mathrm{DRpd}}(\mathcal{X}_*; S,J,t) = \mathrm{Hom}\bigl(\mathcal{X}_*, C^\bullet_{\mathrm{DRpd}*}(S,J,t)\bigr) \]

\[ C^\bullet_{\mathrm{DRpd}}(X; \ ) = C^\bullet_{\mathrm{DRpd}}(S_*(X); \ldots) \doteq \mathrm{Hom}\bigl(S_*(X), C^\bullet_{\mathrm{DRpd}*}(\ )\bigr) \]

\[ S = k\{T\}, \quad J = k\{T\}^+ = \mathrm{Ker}\bigl(k\{T\} \to k\bigr), \quad t = T \]

\[ \begin{cases} C^\bullet_{\mathrm{DRpd}[n]}(S,J,t) \simeq \underbrace{k\{T, X_0, \ldots, X_n\}/(\sum X_i - T)_{\mathrm{pd}}}_{\simeq\, k\{X_0, \ldots, X_n\}} \otimes_k \overset{\bullet}{\Lambda}\, k^{[n]}/k \\ S/J \simeq k \end{cases} \]

\[ \Phi_{k*} = \bigl[[n] \mapsto k^{[n]}\bigr] \supset k_* = \bigl[[n] \to k\bigr] \]

\[ \Psi_{k*} = \Phi_{k*}/k_* = \bigl([n] \to k^{[n]}/\underbrace{k}_{\mathrm{diag}}\bigr) \]

\[ C^\bullet_{\mathrm{DRpd}*}\bigl(k\{T\}, k\{T\}^+, k\bigr) \simeq \Gamma^\bullet_k \Phi_{k*} \otimes_k \overset{\bullet}{\Lambda}\, \Psi_{k*} \]

\[ C^{\bullet\bullet}_{\mathrm{DRpd}*}(k) \simeq \Gamma^\bullet_k \Phi_{k*} \otimes_k \overset{\bullet}{\Lambda}\, \Psi_{k*} \]

\[ C^{\bullet\bullet}_{\mathrm{DRpd}}(\mathcal{X}_*, k) \overset{\mathrm{déf}}{=} C^\bullet_{\mathrm{DRpd}}\bigl(\mathcal{X}_*; k\{T\}, k\{T\}^+, T\bigr) = \mathrm{Hom}\bigl(\mathcal{X}_*, C^{\bullet\bullet}_{\mathrm{DRpd}*}(k)\bigr) \]

\[ 0 \to C^{n,0}_* \to C^{n-1,1}_* \to C^{n-2,2}_* \to \cdots \to C^{1,n-1}_* \to C^{0,n}_* \to 0, \qquad k_* \to C^{n,0}_* \]

\[ H^{p,q}_{\mathrm{DRpd}}(\mathcal{X}_*, k) \simeq \begin{cases} H^q(\mathcal{X}_*, k) & \text{si } q \leq p+q \text{ i.e.\ } p \geq 0 \\ 0 & \text{si } p < 0 \end{cases} \]

\[ H^{\bullet q}_{\mathrm{DRpd}}(\mathcal{X}_*, k) \simeq \tau_q\bigl(H^0(\mathcal{X}_*, k) \otimes_k k\{T\}\bigr)[q] \]

\[ H'^{\,n,q}_{\mathrm{DRpd}}(\mathcal{X}_*, k) = H^{\overset{p}{\overbrace{n-q}},\,q}_{\mathrm{DRpd}}(\mathcal{X}_*, k) \]

\[ H'^{\,\bullet,q}_{\mathrm{DRpd}}(\mathcal{X}_*, k) \simeq \tau_q\bigl(H^q(\mathcal{X}_*, k) \otimes_k k\{T\}\bigr) \]

\[ \mathbb{R}\Gamma(\mathcal{X}_*, k) \overset{L}{\otimes} \mathbb{R}\Gamma(\mathcal{X}_*, k) \to \mathbb{R}\Gamma(\mathcal{X}_*, k) \]

\[ C^\bullet_{\mathrm{DRS}}(\mathcal{X}_*, k) \simeq C^{\bullet\bullet}_{\mathrm{DRpd}}(\mathcal{X}_*, k)/(T-1) \]

\[ C^\bullet_{\mathrm{DRS}}(\mathcal{X}_*, \underbrace{k \otimes_{\mathbb{Z}} \mathbb{Q}}_{k_{\mathbb{Q}}}) \simeq C^{\bullet\bullet}_{\mathrm{DRpd}}(\mathcal{X}_*, k)/(T-1) \]

\[ C^\bullet_{\mathrm{DRpd}}(\mathcal{X}_*; S,J,t)/(t-1) \simeq C^\bullet_{\mathrm{DRS}}\bigl(\mathcal{X}_*, S_{\mathbb{Q}}/(t-1)\bigr) \]

\[ C^{\bullet\bullet}_{\mathrm{DRpd}}(\mathcal{X}_*, k) \to H^0(\mathcal{X}_*, k) = \mathrm{Hom}(\mathcal{X}_*, k_*) \]

\[ (\mathrm{Hot}) \to (\mathrm{DRpd}) \]

\[ C^\bullet(\mathcal{X}_*, k) \simeq \mathrm{Hom}^\bullet_{\mathbb{Z}}\bigl(C_\bullet(\mathcal{X}_*, \mathbb{Z}), k\bigr) \]

\[ C^{\bullet\bullet}_{\mathrm{DRpd}}(\mathcal{X}_*, k') \simeq \mathrm{Hom}_k\bigl(C^{\mathrm{DRpd}}_{\bullet\bullet}(\mathcal{X}_*, k), k'\bigr) \]

\[ C^{\mathrm{DRpd}}_{\bullet\bullet}(\mathcal{X}_*, k') \simeq C^{\mathrm{DRpd}}_{\bullet\bullet}(\mathcal{X}_*, k) \otimes_k k' \]

\[ C^{\bullet\bullet}_{\mathrm{DRpd}*}(k) \simeq \Gamma^\bullet \Phi_{*k} \otimes_k \overset{\bullet}{\Lambda}\, \Psi_{*k} \]

\[ \mathcal{C}^{p,q}_{*k} \simeq \mathrm{Ker}\bigl(\mathcal{D}^{p,q+1}_* \to \mathcal{D}^{p,q+2}_*\bigr) \]

\[ C^{p,q}_{\mathrm{DRpd}}(\mathcal{X}_*, k) \simeq \mathrm{Ker}\bigl(\mathcal{D}^{p,q+1}(\mathcal{X}_*, k) \to \mathcal{D}^{p,q+2}(\mathcal{X}_*, k)\bigr) \]

\[ i = \sum_{0 \leq \varepsilon \leq e} u_\varepsilon p^\varepsilon, \qquad j = \sum_{0 \leq \varepsilon \leq e} v_\varepsilon p^\varepsilon, \qquad 0 \leq u_\varepsilon, v_\varepsilon \leq p-1, \]

\[ c_{ij} = \frac{(i+j)!}{i!\,j!} \in \mathbb{Z} \qquad \text{(coeff.\ binomial)} \]

\[ c_{ij} \not\equiv 0 \ (p) \quad \text{ssi} \quad u_\varepsilon + v_\varepsilon \leq p-1 \quad \forall \varepsilon \ (0 \leq \varepsilon \leq \nu) \]

\[ c_{ij} \equiv \prod_{0 \leq \varepsilon \leq \nu} c_{u_\varepsilon, v_\varepsilon} \quad (p) \]

\[ (*) \quad \begin{cases} c_{ij} \not\equiv 0 \ (p) \quad \text{ssi} \quad u + v \leq p-1 \text{ et } c_{r,s} \not\equiv 0 \ (p) \\ c_{ij} \equiv c_{u,v}\, c_{r,s} \ (p) \end{cases} \]

\[ i + j = p\underbrace{(r+s)}_{t} + \underbrace{u+v}_{w} \]

\[ (X+Y)^{i+j} = \bigl((X+Y)^p\bigr)^t (X+Y)^w = (X^p + Y^p)^t (X+Y)^w \]

\[ = \sum_{\substack{0 \leq \alpha \leq w \\ 0 \leq \beta \leq t}} \binom{w}{\alpha} \binom{t}{\beta} X^{\alpha + p\beta}\, Y^{(w-\alpha) + p(t-\beta)}, \]

\[ c_{i,j} \equiv \binom{w}{u} \binom{t}{r} = c_{u,v}\, c_{r,s} \quad (p) \]

\[ (X+Y)^{i+j} = (X^p + Y^p)^{t'} (X+Y)^{w'} = \sum_{\substack{0 \leq \alpha \leq w' \\ 0 \leq \beta \leq t'}} \binom{w'}{\alpha} \binom{t'}{\beta} X^{\alpha + p\beta}\, Y^{(w'-\alpha) + p(t'-\beta)} . \]

\[ c_{N+1,\, p^\varepsilon - 1} \equiv c_{N'+1,\, p^{\varepsilon-1} - 1} \quad \begin{cases} \equiv 1 \ (p) & \text{si } N'+1 \equiv 0 \ (p^{\varepsilon-1}) \\ \equiv 0 \ (p) & \text{si } N'+1 \not\equiv 0 \ (p^{\varepsilon-1}) \end{cases} \]

\[ c_{N,\, p^\varepsilon} \equiv c_{N',\, p^{\varepsilon-1}} \quad \Bigl\{ \equiv v_{\varepsilon-1} + 1 \ (p) \quad \Bigl(\text{si } N' = \sum v_\alpha p^\alpha\Bigr) \]

\[ p^\varepsilon - 1 = (p-1) + (p-1)p + \cdots + (p-1)p^{\varepsilon-1} \]

\[ p^{\varepsilon-1} - 1 = (p-1) + \cdots + (p-1)p^{\varepsilon-2} \quad \text{si } \varepsilon \geq 2 \qquad (= 0 \text{ si } \varepsilon = 1) \]

\[ \left.\begin{aligned} N'+1 &= u_0 + u_1 p + \cdots + u_\nu p^\nu \\ N' &= v_0 + v_1 p + \cdots + v_\nu p^\nu \end{aligned}\right\} \quad (\nu \geq \varepsilon - 1) \]

\[ \begin{aligned} N+1 &= 0 + u_0 p + u_1 p^2 + \cdots + u_\nu p^{\nu+1} \\ N &= (p-1) + v_0 p + v_1 p^2 + \cdots + v_\nu p^{\nu+1} \end{aligned} \]

\[ \begin{cases} c_{N+1,\, p^\varepsilon - 1} \equiv c_{u_0, p-1}\, c_{u_1, p-1} \cdots c_{u_{\varepsilon-2}, p-1} \ (p) & (\equiv 1 \ (p) \text{ si } \varepsilon = 1) \\ c_{N'+1,\, p^{\varepsilon-1} - 1} \equiv c_{u_0, p-1}\, c_{u_1, p-1} \cdots c_{u_{\varepsilon-2}, p-1} \ (p) & (\equiv 1 \ (p) \text{ si } \varepsilon = 1) \end{cases} \]

\[ \begin{cases} c_{N,\, p^\varepsilon} \equiv c_{v_{\varepsilon-1}, 1} \equiv v_{\varepsilon-1} + 1 \\ c_{N',\, p^{\varepsilon-1}} \equiv c_{v_{\varepsilon-1}, 1} \equiv v_{\varepsilon-1} + 1 \end{cases} \]

\[ \binom{j}{i}(x_j - x_i) = 0 \qquad i = N, N+1,\ i < j \in I \quad \Bigl(\text{donc } j = N + p^\varepsilon,\ 0 \leq \varepsilon \leq e\Bigr) \]

\[
  y_\varepsilon = x_{N+p^\varepsilon} - x_N \qquad 0 \leq \varepsilon \leq e ,
\]

\[
  (S) = S\bigl(N, e, (y_\varepsilon)_{0 \leq \varepsilon \leq e}\bigr)
  \quad
  \begin{cases}
    c_{N, p^\varepsilon}\, y_\varepsilon = 0 & 0 \leq \varepsilon \leq e \\
    c_{N+1, p^\varepsilon - 1}\, (y_\varepsilon - y_0) = 0 & 0 \leq \varepsilon \leq e
  \end{cases}
\]

\[
  (\bar S) = \bar S\bigl(N, e, (y_\varepsilon)_{0 \leq \varepsilon \leq e}\bigr)
  \quad
  \begin{cases}
    y_\varepsilon = 0 \ \text{si}\ c_{N, p^\varepsilon} \not\equiv 0 \ (p) & 0 \leq \varepsilon \leq e \\
    y_\varepsilon - y_0 = 0 \ \text{si}\ c_{N+1, p^\varepsilon - 1} \not\equiv 0 \ (p) & 1 \leq \varepsilon \leq e .
  \end{cases}
\]

\[
  \text{\struck{$\bar S'$}}
  \quad
  \begin{cases}
    y'_\varepsilon = 0 \ \text{si}\ c_{N', p^\varepsilon} \not\equiv 0 \ (p) & 0 \leq \varepsilon \leq e' \\
    y'_\varepsilon - y_0 = 0 \ \text{si}\ c_{N'+1, p^\varepsilon - 1} \not\equiv 0 \ (p) & 0 \leq \varepsilon \leq e'
  \end{cases}
\]

\[
  \binom{j}{i} (x_j - x_i) = 0 \qquad \text{pour } j > i \geq N .
\]

\[
  \omega_N(M) = M \otimes_k S^{+(N)} = \sum_{i \geq N} M\, T^{(i)} .
\]

\[
  M \xrightarrow{\ \varphi\ } \prod_{i \geq N} \omega_N(M)_i = \prod_{i \geq N} M\, T^{(i)}
\]

\[
  (*) \qquad T^{(j)} x_i = c_{ij}\, x_{i+j} \qquad \forall i \geq N,\ j \geq 1 .
\]

\[
  \xi_i\, T^{(i)} T^{(j)} = c_{ij}\, \xi_{i+j}\, T^{(i+j)} ,
  \qquad \xi_i\, T^{(i)} T^{(j)} = c_{ij}\, \xi_i\, T^{(i+j)} ,
\]

\[
  u_i : M \to M' \qquad
  \bigl( M \simeq M\, T^{(i)} = \omega_N(M)_i ,\ \ M' \simeq M'\, T^{(i)} = \omega_N(M')_i \bigr)
\]

\[
  \operatorname{Hom}_k(M, M') \xrightarrow{\ \sim\ }
  \operatorname{Hom}_{S\text{-mod.\ gr.}}\bigl(\omega_N(M), \omega_N(M')\bigr) .
\]

\[
  \operatorname{Hom}_{S\text{-mod.\ gr.}}\bigl(\omega_N(M), \mathcal{M}\bigr)
  \simeq \operatorname{Hom}_k\bigl(M, \pi_N(\mathcal{M})\bigr)
\]

\[
  \pi_N(\mathcal{M}) \subset \prod_{i \geq N} \mathcal{M}_i ,\qquad
  \pi_N(\mathcal{M}) = \bigl\{ (x_i)_{i \geq N} \bigm| x_i\, T^{(j)} = c_{ij}\, x_{i+j}
  \ \ \forall i \geq N,\ j \geq 1 \bigr\} .
\]

\[
  \omega_N(M) \simeq \sum_{i \geq N} \omega_{N'}(M)_i
  \overset{\text{déf}}{=} \tau_{\geq N}\bigl(\omega_{N'}(M)\bigr)
  \qquad \text{(« troncation »)} .
\]

\[
  \tau_N(\mathcal{M}) \hookrightarrow \mathcal{M}
\]

\[
  \operatorname{Hom}_{\mathrm{Omb}}\bigl(\mathcal{N}, \tau_N(\mathcal{M})\bigr) \simeq
  \operatorname{Hom}_{N\text{-}\mathrm{Omb}\ \text{ou}\ \mathrm{Omb}}\bigl(\tau_N(\mathcal{N}), \tau_N(\mathcal{M})\bigr) .
\]

\[
  M \otimes_k S \to \varphi_N \tau_N (M \otimes_k S) .
\]

\[
  (*) \qquad M \otimes_k S \xrightarrow{\ \sim\ } \varphi^{\circ}_N \tau_N (M \otimes_k S)
  \qquad (\forall N \geq 0) .
\]

\[
  (M \otimes_k S)_i \xrightarrow{\ \sim\ }
  \operatorname{Hom}^i\bigl(\tau_{N-i}(S), M \otimes_k S\bigr) ,
  \qquad (M \otimes S)_i \simeq M ,\quad \xi\, T^{(i)} \leftarrow\!\shortmid\ \xi
\]

\[
  (1) \qquad c_{l,k}\, \xi_k = c_{l, k-i}\, \xi_{k+l} \qquad \text{pour } k \geq N,\ l \geq 0 .
\]

\[
  (2) \qquad \xi_k = c_{i, k-i}\, \xi \quad \Bigl(= \binom{k}{i} \xi\Bigr)
  \qquad \text{pour tout } k \geq N .
\]

\[
  \sum_{\alpha=1}^{r} m_\alpha\, c_{k_\alpha, k_\alpha - i} = 1 .
\]

\[
  \xi = \sum_{1}^{r} m_\alpha\, \xi_{k_\alpha} ,
\]

\[
  \xi_k = c_{k, k-i} \sum_{1}^{r} m_\alpha\, \xi_{k_\alpha} .
\]

\[
  (1') \quad \xi_{k_0} = 0 , \qquad (3) \quad c_{i, k_0 - i} \not\equiv 0 \ (p) ,
\]

\[
  (4) \qquad c_{i,j} \not\equiv 0 \ (p) \quad \text{et} \quad c_{i,j'} \not\equiv 0 \ (p) .
\]

\[
  (5) \qquad c_{j'-j, j} \not\equiv 0 \ (p) \iff c_{j'-j, j+i} \not\equiv 0 \ (p) ,
\]

\[
  (6) \qquad \xi_k = 0 \iff \xi_{k'} = 0 .
\]

\[
  (a) \quad \frac{c_{j'-j, j+i}}{c_{j'-j, j}}
  = \frac{(j'+i)!}{(j'-j)!\,(j+i)!} : \frac{j'!}{(j'-j)!\, j!} ,
  \qquad
  (b) \quad \frac{c_{i, j'}}{c_{i, j}} = \frac{(i+j')!}{i!\, j'!} : \frac{(i+j)!}{i!\, j!} .
\]

\[
  c_{j'-j, j+i}\, \xi_k = c_{j'-j, j}\, \xi_{k'} ,
\]

\[
  i = i_0 + i_1 p + \cdots + i_r p^r \qquad 0 \leq i_\alpha \leq p - 1 \quad (0 \leq \alpha \leq r) ,
\]

\[
  j = j_0 + j_1 p + \cdots + j_r p^r \qquad 0 \leq j_\alpha \leq p - 1 \quad (0 \leq \alpha \leq r) .
\]

\[
  \delta_\alpha = (p-1) - (i_\alpha + j_\alpha) , \quad 0 \leq \delta_\alpha \leq p-1 ;
  \qquad
  j'_\alpha \overset{\text{déf}}{=} j_\alpha + \delta_\alpha = (p-1) - i_\alpha
\]

\[
  c_{i, j'} \not\equiv 0 \ (p) \quad \text{car } i_\alpha + j'_\alpha \ (= p - 1) \leq p - 1 \ \ \forall \alpha ,
\]

\[
  c_{j'-j, j} = c_{\delta, j} \not\equiv 0 \ (p) \quad \text{car } \delta_\alpha + j_\alpha = j'_\alpha \leq p - 1 ,
\]

\[
  k' = i + j' = \sum_0^r (p-1) p^\alpha = p^{r+1} - 1 ;
\]

\[
  \xi_k = 0 \iff \xi_{p^{r+1} - 1} = 0 .
\]

\[
  c_{l,k}\, \xi_k = c_{l, k-i}\, \xi_{k+l} \qquad \forall l \geq 0 ,
\]

\[
  i = i_0 + i_1 p + \cdots + i_r p^r , \qquad j = j_0 + j_1 p + \cdots + j_r p^r ;
\]

\[
  \sigma = \sigma_0 + \sigma_1 p + \cdots + \sigma_{r+1} p^{r+1}
\]

\[
  (a) \quad \rho_{-1} = 0 ,
\]

\[
  (b) \quad i_\alpha + j_\alpha + \rho_{\alpha - 1} = \sigma_\alpha + p\, \rho_\alpha ,
  \qquad 0 \leq \sigma_\alpha, \rho_\alpha \leq p - 1
\]

\[
  (c) \quad \sigma_{r+1} = \rho_r .
\]

\[
  l_\alpha = (p-1) - \sigma_\alpha \quad (0 \leq \alpha \leq r+1) ,
  \qquad l = \sum_0^{r+1} l_\alpha p^\alpha ,
\]

\[
  \sigma + l = \sum_0^{r+1} (p-1) p^\alpha = p^{r+2} - 1 ,
  \quad \text{i.e.} \quad l = p^{r+2} - 1 - \sigma .
\]

\[
  (a') \quad \rho_{-1} = 0 ,
\]

\[
  (b') \quad (i_\alpha + j_\alpha + l_\alpha) + \rho_{\alpha-1} = (p-1) + p\, \rho_\alpha
  \quad \text{si } 0 \leq \alpha \leq r
\]

\[
  (b')^{\text{bis}} \quad i_{r+1} + j_{r+1} + l_{r+1} + \rho_r = (p-1) + p \cdot 0
  \qquad (i_{r+1} = 0,\ j_{r+1} = 0) ;
\]

\[
  i + j + l = p^r - 1 = \sum_{\alpha=0}^{r-1} (p-1) p^\alpha
  \qquad [\,(i+j) + l = i + (j+l)\,] .
\]

\[
  \boxed{\ v_p(n!) = \frac{n - \mathrm{chif}_p\, n}{p-1}\ }
  \qquad n = \sum_{\alpha \geq 0} n_\alpha p^\alpha , \quad
  \mathrm{chif}_p\, n = \sum n_\alpha
\]

\[
  n - \mathrm{chif}_p\, n = \sum n_\alpha (p^\alpha - 1) , \qquad
  \frac{n - \mathrm{chif}_p\, n}{p-1} = \sum_\alpha n_\alpha (1 + \cdots + p^{\alpha-1})
\]

\[
  \text{\struck{$v_p(n!)$}}\ \
  v_p\Bigl(\frac{p^n}{n!}\Bigr) = n - \frac{n - \mathrm{chif}_p\, n}{p-1}
  = \frac{\mathrm{chif}_p(n) + n(p-2)}{p-1}
\]

\[
  v_p\bigl((n+1)!\bigr) =
\]

\[
  v_p\Bigl(\frac{p^{r+1}!}{(p^r!)^p\, p!}\Bigr)
  = \underbrace{v_p(p^{r+1}!)}_{\frac{p^{r+1}-1}{p-1}}
  - p\, \underbrace{v_p(p^r!)}_{\frac{p^r - 1}{p-1}}
  - \underbrace{v_p(p!)}_{1}
  = 0
\]

\[
  \begin{aligned}
    \operatorname{Hom}^d(M^*, \varphi_N N^*)
    &= \operatorname{Hom}(M^*[-d], \varphi_N N^*) \\
    &\simeq \operatorname{Hom}\bigl(\tau_N(M^*[-d]), N^*\bigr) \\
    &\simeq \operatorname{Hom}\bigl((\tau_{N-d} M^*)[-d], N^*\bigr) \\
    &\simeq \operatorname{Hom}^d(\tau_{N-d} M^*, N^*)
  \end{aligned}
\]

\[
  N^* \in \operatorname{Ob} \mathcal{M}_N \qquad
  \boxed{\ \operatorname{Hom}^d(M^*, \varphi_N N^*) \simeq \operatorname{Hom}^d(\tau_{N-d} M^*, N^*)\ }
\]

\[
  M^* \in \mathcal{M}^*\ [\text{i.e. } M^j = 0 \text{ si } j < -d] ,
\]

\[
  N^* = \tau_N N \otimes_k S_k \simeq \tau_N N_{S_k} \quad (N \text{ un } k\text{-module}) ,
  \qquad N \geq 0 .
\]

\[
  \operatorname{Hom}^d(M^*, \varphi_N \tau_N N_{S_k})
  = \operatorname{Hom}^d(M^*, \underbrace{\varphi_N \tau_N N_{S_k}}_{= \tau_N \varphi_N \tau_N N_{S_k}})
  = \operatorname{Hom}^d(M^*, \tau_N \ldots
\]

\[
  \boxed{\ \operatorname{Hom}^d(M^*, N \otimes_k S_k)
  \simeq \operatorname{Hom}^d\bigl(\tau_{N-d} M, \tau_N (N \otimes_k S_k)\bigr)\ }
  \qquad \text{si }
  \begin{cases} M^* \in \mathcal{M}_{-d} \\ N \geq 0 \end{cases}
\]

\[
  \boxed{
  \begin{aligned}
    \operatorname{Hom}^d(M \otimes_k S_k, N \otimes_k S_k)
    &\xrightarrow{\ \sim\ } \operatorname{Hom}^d(\tau_{N-d} M_{S_k}, \tau_N N_{S_k}) \\
    &\simeq \operatorname{Hom}_k(M, N)
    \qquad \text{si } d \geq 0,\ N \geq 0
  \end{aligned}}
\]

\[
  \tau_\alpha M_{S_k} \to \tau_\beta N_{S_k} \qquad (\beta \geq \alpha) .
\]

\[
  \begin{aligned}
    \operatorname{Hom}^d(\tau_\alpha M_{S_k}, \tau_\beta N_{S_k})
    &\xleftarrow{\ \sim\ } \operatorname{Hom}^d\bigl(\tau_\alpha M_{S_k},
      \underbrace{\tau_{\alpha+d} \tau_\beta N_{S_k}}_{= \tau_{\alpha+d} N_{S_k}}\bigr) \\
    &\xleftarrow{\ \sim\ } \operatorname{Hom}^d(M_{S_k}, N_{S_k}) \simeq \operatorname{Hom}_k(M, N)
  \end{aligned}
\]

\[
  m\, T^{(j)} \longmapsto u(m)\, T^{(d)} T^{(j)} = c_{d,j}\, u(m)\, T^{(d+j)} ,
  \qquad j \geq \alpha \ ) .
\]

\[
  \text{\struck{$\operatorname{Hom}^d(\tau_\alpha M_{S_k}, \tau_\beta N_{S_k})
  \to \operatorname{Hom}^d(\tau_{\beta-d} M_{S_k}, \tau_\beta N_{S_k})
  \simeq \operatorname{Hom}_k(M, N)$}}
\]

\[
  \operatorname{Hom}^d\bigl(\tau_\alpha M_{S_k}, \underbrace{\tau_\beta N_{S_k}}_{\cap\ \tau_{\alpha+d} N_{S_k}}\bigr)
  \hookrightarrow \operatorname{Hom}^d(\tau_\alpha M_{S_k}, \tau_{\alpha+d} N_{S_k})
  \simeq \operatorname{Hom}_k(M, N)
\]

\[
  \bigl( m\, T^{(j)} \longmapsto c_{d,j}\, u(m)\, T^{(d+j)} \bigr)
\]

\[
  \begin{cases}
    c_{d, \alpha}\, u = 0 \\
    c_{d, \alpha+1}\, u = 0 \\
    \cdots \\
    c_{d, \beta-d}\, u = 0
  \end{cases}
\]

\[
  N \otimes S_k = \varphi_{0 \beta} \tau_\beta N_{S_k} = \tau_0 \varphi_{\beta} \tau_\beta N_{S_k}
  \subset \varphi_\beta N_{S_k} ,
\]

\[
  \operatorname{Hom}^d(M^*, N_{S_k}) \subset \operatorname{Hom}^d(M^*, \varphi_\beta \tau_\beta N_{S_k})
  = \operatorname{Hom}(M^*[-d], \varphi_\beta N_{S_k})
  = \operatorname{Hom}(\tau_\beta M^*[-d], N_{S_k}
\]

\[
  T_r^{\,p} = p!\,\bigl((T^{(p^r)})^{(p)}\bigr) = p!\,\gamma_r\, T^{(p^{r+1})} = p\,\nu'_r\, T_{r+1},
  \qquad \gamma_r = \frac{p^{r+1}!}{(p^r!)^p\,p!} \in \mathbb{Z}_p^{*},
\]

\[
  \nu'_r = (p-1)!\,\frac{p^{r+1}!}{(p^r!)^p\,p!} \in \mathbb{Z}_p^{*} .
\]

\[
  \boxed{T_r^{\,p} = p\,\nu'_r\, T_{r+1}}, \quad \nu'_r \in \mathbb{Z}_p^{*}
  \qquad\qquad \boxed{S_k = k\{T\}}
\]

\[
  \boxed{T_s^{\,p-1}\, t_s = p\,\nu'_s\, t_{s+1} \quad \text{si } s \geq r .}
\]

\[
  u : \tau_{p^r} S_k \longrightarrow M \otimes_k S_k
\]

\[
  \boxed{u(t_s) = \eta_s \in S^{(p^s - d)} \otimes_k M}
\]

\[
  (R_s) \quad \boxed{T_s^{\,p-1}\,\eta_s = p\,\nu'_s\,\eta_{s+1}}
\]

\[
  d = d_\nu p^\nu + \cdots + d_{r-1}p^{r-1} = p^\nu\bigl(d_\nu + d_{\nu+1}p + \cdots + d_{r-1}p^{r-1-\nu}\bigr),
  \qquad d_\nu \neq 0 \text{ i.e. } d_\nu \geq 1 .
\]

\begin{align*}
  p^s - d &= p^\nu\Bigl(p^{s-\nu} - \frac{d}{p^\nu}\Bigr)
           = p^\nu\Bigl(p^{s-\nu} - 1 - \underbrace{\Bigl(\frac{d}{p^\nu} - 1\Bigr)}_{(d_\nu - 1) + d_{\nu+1}p + \cdots + d_{r-1}p^{r-1-\nu}}\Bigr) \\
          &= p^\nu\bigl(\delta_\nu + \delta_{\nu+1}p + \cdots + \delta_{r-1}p^{r-\nu-1} \\
          &\qquad + \underbrace{(p-1)p^{r-\nu} + (p-1)p^{r-\nu+1} + \cdots + (p-1)p^{s-\nu-1}}_{\text{si } s > r}\bigr),
\end{align*}

\[
  \delta_\nu = (p-1) - (d_\nu - 1), \quad \delta_{\nu+1} = (p-1) - d_{\nu+1}, \quad \ldots, \quad \delta_{r-1} = (p-1) - d_{r-1} .
\]

\[
  T_\nu^{\delta_\nu}\, T_{\nu+1}^{\delta_{\nu+1}} \cdots T_{r-1}^{\delta_{r-1}}\,\bigl(T_r \cdots T_{s-1}\bigr)^{p-1}
\]

\[
  \eta_s = \xi_s\, T_\nu^{\delta_\nu}\, T_{\nu+1}^{\delta_{\nu+1}} \cdots T_{r-1}^{\delta_{r-1}}\,\bigl(T_r \cdots T_{s-1}\bigr)^{p-1}
  \qquad \text{avec } \xi_s \in M .
\]

\[
  \eta_s T_s^{\,p-1} = \xi_s\, T_\nu^{\delta_\nu} \cdots T_{r-1}^{\delta_{r-1}}\,\bigl(T_r \cdots T_s\bigr)^{p-1} .
\]

\[
  p\,\nu'_s\,\eta_{s+1} = p\,\nu'_s\,\xi_{s+1}\, T_\nu^{\delta_\nu} \cdots T_{r-1}^{\delta_{r-1}}\,\bigl(T_r \cdots T_s\bigr)^{p-1},
\]

\[
  \boxed{\xi_s = p\,\nu'_s\,\xi_{s+1}}
\]

\[
  \xi_s = p^h\, \nu'_s \cdots \nu'_{s+h-1}\, \xi_{s+h} \qquad \forall\, h \geq 1,
\]

\[
  \xi_s \in \bigcap_h \bigl(p^h M\bigr),
\]

\[
  \mathrm{Hom}^{-d}(\tau_\alpha M_{S_k}, N_{0\,S_k}) \xrightarrow{\ \sim\ } \mathrm{Hom}^{-d}(\tau_\alpha M_{S_k}, N_{0\,S_k}) .
\]

\[
  C^{*} : \widehat{\Delta}^{\circ} \longrightarrow \ldots
\]

\[
  C^{n}(X_\bullet) = \mathrm{Hom}(X_\bullet, C^{n}_\bullet) .
\]

\[
  M_\bullet = \underline{k} \otimes M \ldots
\]

\[
  H^{0}(X_\bullet, C^{0}_\bullet) \to H^{0}(X_\bullet, Z^{1}_\bullet) \to H^{1}(X_\bullet, M)
  \to H^{1}(X_\bullet, C^{0}_\bullet) \to H^{1}(X_\bullet, Z^{1}_\bullet)
\]

\[
  M \mapsto C^{*}(K_\bullet, M) \overset{\text{déf}}{=} \mathrm{Hom}_{\widehat{\Delta}}(K_\bullet, M)
  = \mathrm{Hom}(\mathbb{Z}^{(K_\bullet)}, M) \simeq \mathrm{Hom}_{\widehat{\Delta}}(K_\bullet, \ldots)
\]

\[
  C^{*}(K_\bullet, M) \simeq \mathrm{Hom}(K_\bullet, C^{*}_{\bullet M})
\]

\[
  C^{n}_{\bullet M} = \ldots \simeq k^{\mathrm{Hom}(\Delta_n, \Delta_m)} \otimes_k M
\]

\[
  C^{n}_{\bullet M} = \bigcap \mathrm{Ker}\bigl(C^{n}_{\bullet M} \to \ldots\bigr) = \ldots
\]

\[
  C^{*!}(X_\bullet) = \mathrm{Hom}(X_\bullet, C^{*!}_{\bullet M})
\]

\begin{align*}
  (C^{n!}_{\bullet M})_m &\simeq M^{\mathrm{Hom\,mono}(\Delta_n, \Delta_m)} \simeq k^{\mathrm{Hom}(\Delta_n, \Delta_m)} \otimes_k M \\
  &\simeq \Bigl(\bigwedge^{n+1} k^{\Delta_m}\Bigr) \otimes_k M
\end{align*}

\[
  C^{n!}_{\bullet M} \simeq \bigwedge^{n+1} \ldots \otimes_k M
\]

\[
  \partial : A_n^{*} \to A_{n-1}^{*}, \qquad
  \partial(\omega) = \sum_{0 \leq i \leq n} \partial_{n,i}^{*}(\omega)(-1)^{i}
\]

\[
  {}^{*}A_n^{*} \xrightarrow{\ K_n\ } A_0
\]

\[
  \varpi^{N} = d\omega^{N-1} \quad (1)
\]

\[
  K_N(d\omega^{N-1}) = K_{N-1}(\partial\omega) .
\]

\[
  d\omega^{N-1} = 0 \;\Longrightarrow\; K_{N-1}(\partial\omega^{N-1}) = 0 .
\]

\[
  K_{N-1}(\partial\omega^{N-1}) = K_{N-1}(\partial d\alpha^{N-2})\ldots
\]

\[
  K_{N-1}(\partial\omega^{N-1}) = K_{N-1}(d\partial\alpha^{N-2})
  \overset{\text{Stokes}}{=} K_{N-2}\bigl(\underbrace{\partial\partial\alpha^{N-2}}_{0}\bigr) = 0
\]

\[
  K_{N-1}(\partial\omega^{0}) = K_{N-1}\bigl(\underbrace{\partial\varepsilon_1(\alpha)}_{=0}\bigr) = 0
\]

\[
  \begin{cases}
    H^{N}(A_N^{*}) = 0 & \text{si } N \geq 1 \\[2pt]
    H^{N-1}(A_N^{*}) = \begin{cases} 0 & \text{si } N \geq 2 \\ \varepsilon_1(A_0) & \text{si } N = 1 \end{cases}
  \end{cases}
\]

\[
  \begin{cases}
    A_N^{i} = 0 & \text{si } i \notin [0, N] \\[2pt]
    H^{N}(A_N^{*}) = 0 & \text{si } N \geq 1 \\[2pt]
    H^{N-1}(A_N^{*}) \begin{cases} = 0 & \text{si } N \geq 2 \\ = \mathrm{Ker}\,\delta & \text{si } N = 1 \end{cases}
  \end{cases}
\]

\[
  (SS) = \widehat{\Delta} = \underline{\mathrm{Hom}}(\Delta^{\circ}, (\mathrm{Ens})) .
\]

\[
  (SS)/X \simeq \widehat{\Delta/X} \simeq \underline{\mathrm{Hom}}\bigl((\Delta/X)^{\circ}, \mathrm{Ens}\bigr) .
\]

\[
  A \times X \longrightarrow X
\]

\[
  (\Delta/X)^{\circ} \longrightarrow (\Delta)^{\circ} \xrightarrow{\ A\ } (\mathrm{Ens}) .
\]

\[
  \mathbb{R}\Gamma(X, A_\bullet^{*}) \overset{\text{déf}}{=} \mathbb{R}^{*}\Gamma_X(\underline{A_X})
\]

\[
  C^{n}(X, A) \overset{\text{déf}}{=} \prod_{\sigma \in \mathrm{Ob}\,\Delta_n/X} A(\sigma),
  \qquad \mathrm{Ob}\,\Delta_n/X = X_n .
\]

\[
  \text{\struck{$(b \in B) \longmapsto \sum_{x \in C_b} F(x) \longrightarrow F(i_*(x'))$}}
\]

\[
  \text{\struck{$(b' \in B) \longmapsto \sum_{x' \in C_{b'}} F(x') \longrightarrow F(x')$}}
\]

\[
  \text{\struck{$(b \in B) \longmapsto \prod_{x \in C_b} F(x) \longrightarrow F(i^*(x'))$}}
\]

\[
  \text{\struck{$(b' \in B) \longmapsto \prod_{x \in C_{b'}} F(x) \longrightarrow F(x')$}}
\]

\[
  C^*(X_*, F) \simeq \Gamma\bigl(X_*, \underbrace{(C^*_*|X_*)}_{C^*_{X_*}} \otimes_{k_*} F\bigr)
\]

\[
  H^*\bigl(C^*(X_*, F)\bigr) \simeq H^*(X_*, F),
  \qquad
  C^*(X_*, F) \overset{\text{déf}}{=} \Gamma(X_*, C^*_{X_*} \otimes_k F),
\]

\[
  R\Gamma(X_*, F^*) \simeq \underbrace{C^*(X_*, F^*)}_{\text{par diagonale associé aux deux degrés}}
\]

\[
  0 \to A^0_* \to A^1_* \to A^2_* \to \cdots
\]

\[
  0 \to C^0_{*M} \to C^1_{*M} \to C^2_{*M} \to \cdots
\]

\[
  A^n_* \xrightarrow{u_n} C^n_{*M},
\]

\[
  A^n_n \xrightarrow{k_n} M .
\]

\[
  d_C \circ u_n : A^n_* \longrightarrow C^n_{*M} \longrightarrow C^{n+1}_{*M}
\]

\[
  A^n_{n+1} \xrightarrow{\;k_n \partial_{n+1}\;} M,
  \qquad
  A^n_{n+1} \xrightarrow{\partial_{n+1}} A^n_n \xrightarrow{k_n} M,
\]

\[
  u_{n+1} \circ d_A : A^n_* \longrightarrow A^{n+1}_* \longrightarrow C^{n+1}_{*M}
\]

\[
  A^n_{n+1} \longrightarrow M,
  \qquad
  A^n_{n+1} \xrightarrow{d^n_A} A^{n+1}_{n+1} \xrightarrow{k_{n+1}} M,
\]

\[
  k_n \partial_{n+1} = k_{n+1} d :
\]

\[
  \Gamma(X_*, A^*_{X_*}) \Longrightarrow C^*(X_*, M_*) \simeq \Gamma(X_*, C^*_{*M}|X_*),
\]

\[
  \Gamma(X_*, A^*_{X_*}) \longrightarrow R\Gamma_{X_*}(M)
\]

\[
  \underset{\textstyle \mathrm{Hom}(X_*, A^*)}{\Gamma(X_*, A^*_{X_*})}
  \longrightarrow C(X_*, M)
\]

\[
  \mathrm{Hom}(X_*, A^n_*) = \Gamma(X_*, A^n_{X_*}) \xrightarrow{u^n} C^n(X_*, M)
\]

\[
  f_n : X_n \to A^n_n
\]

\[
  u^n(f) = k_n \circ f_n \in \mathrm{Hom}(X_n, M).
\]

\[
  k_0 : A^0_0 = A_0 \longrightarrow M
\]

\[
  k_0\bigl(\partial_1(\mathrm{Ker}(A^0_1 \xrightarrow{d} A^1_1))\bigr) = 0
\]

\[
  A_0 / \partial_1\bigl(\mathrm{Ker}(A^0_1 \xrightarrow{d} A^1_1)\bigr) ;
\]

\[
  A^*_* \longrightarrow C^*_{*M}
\]

\[
  k_0 : A^0_0 \longrightarrow C^0_{0M} = M
\]

\[
  A^*_* \longrightarrow C^{*!}_{*M} \simeq \Lambda^{*+1} \Phi_{*k} \otimes_k M,
\]

\[
  C^{n!}_{*M} = \bigcap_{\substack{\sigma : \Delta_m \to \Delta_n \\ m < n}}
  \mathrm{Ker}\bigl(C^n_{*M} \xrightarrow{\sigma^*} C^m_{*M}\bigr).
\]

\[
  \mathrm{Hom}(A^n_*, C^{n!}_{*M}) = \bigl\{ A^n_* \xrightarrow{f} C^n_{*M} \bigm|
  \sigma^* f = 0 \ \ \forall \sigma : \Delta_m \to \Delta_n,\ m < n \bigr\}.
\]

\[
  A^n_n \xrightarrow{k_n} M,
\]

\[
  A^n_m \longrightarrow M
\]

\[
  A^n_m \xrightarrow{\sigma^A} A^n_n \xrightarrow{k_n} M,
\]

\[
  f^n(A^n_*) \subset C^{n!}_* \iff k_n \circ \sigma^* = 0 \quad
  \forall \sigma : \Delta_n \to \Delta_m,\ m < n
\]

\[
  A^*_{*S} = \bigl(\Gamma^*(\Phi_{*S}) \otimes_S \Lambda^* \Psi_{*S}\bigr) / (T - t)_{pd},
\]

\[
  A^*_* \xrightarrow{\;\sim\;} \bigl(\Gamma^*(\Phi_{*S}) \otimes_S \Lambda \Psi_{*S} / (T - t)_{pd}\bigr)
  \longrightarrow \Lambda^{*+1} \Phi_{*S}
\]

\[
    \sigma^* : \varinjlim_{U \supset \operatorname{Supp}\sigma} \Omega^*(U) \longrightarrow A^*_n
  \]

\[
  \sigma \longmapsto \varinjlim_{U \supset \operatorname{Supp}\sigma} \Omega^*(U)
\]

\[
  \left\{\begin{array}{l}
    T \in \Gamma(\mathcal{S}s, \Phi_{*k}) \ \ [= \Gamma(\mathcal{S}s, C^{1,0}_*)] \\
    k_* \hookrightarrow \Phi_{*k}
  \end{array}\right.
\]

\[
  C^{**}_{*k} = \mathcal{C}^{**}_{*k} = \Gamma^*_k \Phi_* \otimes_k \Lambda^* \Phi_*
\]

\[
  C^{00}_{*k} = k_*,
\]

\[
  0 \to \Gamma^n \Phi_* \to \Gamma^{n-1} \Phi_* \otimes \Lambda^1 \Phi_* \to
  \Gamma^{n-2} \Phi_* \otimes \Lambda^2 \Phi_* \to
\]

\[
  \cdots \to \Phi_* \otimes \Lambda^{n-1} \Phi_* \to \Lambda^n \Phi_* \to 0 \to 0 \cdots
\]

\[
  dT \in \Gamma(\mathcal{S}s, C^{01}_*) \simeq \Gamma(\mathcal{S}s, \overline{\Phi}_*),
  \qquad dT = \overline{T},
\]

\[
  \overline{T}_\wedge : C^{**}_* \longrightarrow C^{**}_k ,
\]

\[
  d \quad (d(\overline{T} \cdot x) = \underbrace{d\overline{T}}_{=0} \cdot x - \overline{T}\, dx = -\overline{T}\, dx),
\]

\begin{align*}
  DR^{pq}_k = DR^{pq}_* &= \mathrm{Ker}\bigl(C^{p,q+1}_* \xrightarrow{\overline{T}_\wedge} C^{p,q+2}_*\bigr) \\
  &= \mathrm{Ker}\bigl(\Gamma^p \Phi_* \otimes \Lambda^{q+1} \Phi_* \to \Gamma^p \Phi_* \otimes \Lambda^{q+2} \Phi_*\bigr) \\
  &\simeq \text{\struck{$\mathrm{Ker}$}}\ \Gamma^p \Phi_* \otimes
  \underbrace{\mathrm{Ker}\bigl(\Lambda^{q+1} \Phi_* \to \Lambda^{q+2} \Phi_*\bigr)}_{\Lambda^q \Psi_*} ,
  \qquad \Psi_* = \Phi_* / k_* \\
  &\simeq \Gamma^p \Phi_* \otimes \Lambda^q \Psi_*
\end{align*}

\[
  DR^{**}_* = (DR^{pq}_*)_{(p,q) \in \mathbb{Z} \times \mathbb{Z}},
\]

\[
  u = \text{\struck{$\overline{T}_\wedge$}} : C^{**}_* \xrightarrow{\;u\;} DR^{**}_*
  \overset{i}{\hookrightarrow} C^{**}_*
\]

\[
  \left|\begin{array}{l}
    i(x\, u(y)) = i(x)\, y \\
    i(u(y)\, x) = y\, i(x)
  \end{array}\right.
  \qquad i(x)\, i(y) = 0
\]

\[
  0 \to DR^{p,q}_* \overset{i}{\longrightarrow} C^{p,q+1}_*
  \xrightarrow{\;(\overline{T}_\wedge) = u\;} DR^{p,q+1}_* \to 0,
  \qquad DR^{p,q+1}_* \subset C^{p,q+2}_*
\]

\[
  C^{**} = C^{**}_*(X_*) = C^{**}(X_*, k)
\]

\[
  \text{(A)} \quad
  \left\{\begin{array}{l}
    C^{00} \simeq k^{\pi_0(X_*)} \quad (= k \text{ si } X_* \text{ connexe}),
    \qquad = H^{00} \\
    H^{pq}(C^{**}) = 0 \ \text{ si } pq \neq 0,
  \end{array}\right.
\]

\[
  \text{(B)} \quad
  \left\{\begin{array}{l}
    0 \to C^{p,0} \xrightarrow{\overline{T}_\wedge} C^{p,1} \xrightarrow{\overline{T}_\wedge} C^{p,2} \to \cdots
    \to C^{p,q} \xrightarrow{\overline{T}_\wedge} C^{p,q+1} \to \\
    \text{est une suite exacte si } p > 0
  \end{array}\right.
\]

\[
  \underset{\textstyle k}{\overset{0}{\Lambda} \Phi_*}
  \xrightarrow{\overline{T}_\wedge} \overset{1}{\Lambda} \Phi_*
  \xrightarrow{\overline{T}_\wedge} \overset{2}{\Lambda} \Phi_* \to
\]

\[
  H^n\bigl(0 \to C^{0,0} \xrightarrow{\overline{T}_\wedge} C^{0,1} \to \cdots \to
  C^{0,q} \xrightarrow{\overline{T}_\wedge} C^{0,q+1} \to \cdots\bigr)
  \simeq \left\{\begin{array}{ll}
    0 & \text{si } n \leq 1 \\
    H^{n-1}(X_*, k) & \text{si } n \geq 2
  \end{array}\right.
\]

\[
  DR^{pq} = DR^{pq}(X_*) = \mathrm{Ker}\bigl(C^{p,q+1} \xrightarrow{\overline{T}_\wedge} C^{p,q+2}\bigr),
  \qquad
  DR^{**} = (DR^{pq})_{p,q \in \mathbb{Z}},
\]

\[
  C^{p,q} \xrightarrow{\;\overline{T}_\wedge =: u\;} DR^{pq}, \ \subset C^{p,q+1},
  \quad \text{épi si } p \neq 0,
  \qquad \text{d'où} \quad
  C^{**} \xrightarrow{\;u\;} DR^{**} \overset{i}{\hookrightarrow} C^{**}
\]

\[
  \text{(C)} \quad i(x) \cdot i(y) = 0
\]

\[
  u : C^{**} \longrightarrow DR^{**}
\]

\[
  u(x \cdot y) = u(x)\, u(y), \qquad i(x\, u(y)) = i(x)\, y, \qquad i(u(y)\, x) = y\, i(x)
\]

\[
  \overline{T} \cdot (xy) = (\overline{T} x)\, y
\]

\[
  \mathrm{Im}\, u \times DR \to DR \quad \text{et} \quad DR \times \mathrm{Im}\, u \to DR
\]

\[
  u(x)^{(i)} = u(x^{(i)})
\]

\[
  x'^{(i)} = x^{(i)} + x^{(i-1)} y + x^{(i-2)} y^{(2)} + \cdots + y^{(i)},
\]

\[
  \text{(E)} \quad
  \left\{\begin{array}{lll}
    \text{multiplication} & DR^{0,p+} \times DR^{0,q+} \longrightarrow DR^{0,p+q+} & (p, q \geq 1) \\
    \text{puiss. division} & DR^{0,p} \xrightarrow{\;\gamma^i\;} DR^{0,pi} & (i \geq 2)
  \end{array}\right.
\]

\[
  \begin{array}{ccc}
    0 & & \\
    \downarrow & & \\
    \Gamma^p \Phi_* \otimes \Lambda^q \Psi_* & & \cdots \to \Lambda^{q+1} \Phi_* \\
    \uparrow & \text{?} & \\
    \Gamma^p \Phi_* \otimes \Lambda^q \overline{\Phi}_* & & \\
    \uparrow & & \\
    \Gamma^p \Phi_* \otimes \Lambda^{q-1} \Psi_* & & \cdots \to \Lambda^q \Phi_* \\
    \uparrow & & \\
    0 & &
  \end{array}
\]

\[
  0 \to \underset{\textstyle \Phi_*}{\Phi_* \otimes k} \longrightarrow
  \Phi_* \otimes \overline{\Phi}_* \longrightarrow \Phi_* \otimes \Psi_* \to 0
\]

\[
  \mathrm{Hom}_{k\text{-Mod}}\bigl(\Phi_* \otimes \Psi_*, \underset{\textstyle i_{0*}(k)}{\Phi_*}\bigr)
  \simeq \mathrm{Hom}_{k\text{-Mod}}\bigl((\Phi_* \otimes \Psi_*)_0, k\bigr) = 0,
  \qquad \text{car } (\Phi_* \otimes \Psi_*)_0 = \Phi_0 \otimes \Psi_0 = 0,
\]

\[
  \Phi_* \otimes \overline{\Phi}_* \simeq (\Phi_* \otimes \Psi_*) \oplus \Phi_*
\]

\[
  s : \Phi_* \otimes \Psi_* \longrightarrow \Phi_* \otimes \overline{\Phi}_*
\]

\[
  \mathbb{Z} \xrightarrow{\;u\;} \Phi
\]

\[
  \mathrm{Coker}\, i = \Psi
\]

\[
    \mathbb{Z}\{T\} \longrightarrow \mathcal{H}^{*0}(K^{**}_u)
  \]

\[
  H^{**}(X, u, k) \overset{\text{déf}}{=} H^{**}\bigl(K^{\cdot\cdot}(X, u, k)\bigr)
  \longrightarrow H^{**}(X, k\{T\}) \simeq H^*(X, k) \otimes_{\mathbb{Z}} \mathbb{Z}\{T\}
\]

\[
  f_* K^{\cdot\cdot}_{u,k} \longrightarrow Rf_*(k\{T\}) \simeq Rf_*(k) \otimes_{\mathbb{Z}} \mathbb{Z}\{T\}
\]

\[
  u'\colon L\longrightarrow\Phi'
\]

\[
  H^{*i}(K^{\cdot\cdot})=
  \begin{cases}
    \Gamma^{*}L & \text{si } i=0\\
    0 & \text{si } i>0
  \end{cases}
\]

\[
  k\xrightarrow{\ \lambda_{0}\ }k'\qquad\text{hom.\ d'Anneaux}
\]

\[
  \Phi\xrightarrow{\ \lambda_{\Phi}\ }\Phi'
\]

\[
  \Phi\overset{\lambda_{\Phi},\,\mu_{\Phi}}{\rightrightarrows}\Phi'
\]

\[
  \begin{cases}
    \text{(a)}\ \Psi=\operatorname{Coker}u\ \ k\text{-plat}\\
    \text{(b)}\ \forall\ k\text{-modules } N, M,\ \text{et } i>0,\\
    \qquad\Gamma^{i}(N\otimes_{k}\Phi)\otimes M\ \text{est flasque}
  \end{cases}
\]

\[
  \Gamma^{i}_{k}\bigl(\check N\,\text{\struck{$\otimes_{k}\Phi$}}\bigr)\otimes_{k}M
  \quad\text{flasque}
\]

\[
  \mathrm{DR}^{**}_{pd}(k,u)=\Gamma\bigl(X,\mathcal{DR}^{**}_{pd}(k,u)\bigr).
\]

\[
  \begin{aligned}
    &\mathrm{DR}(\Phi)\ \cdots\\
    &\mathrm{DR}(\Phi'')\xrightarrow{\ \simeq\ }\mathrm{DR}(N'')\\
    &\mathrm{DR}(\Phi'')\nearrow\simeq
  \end{aligned}
\]

\[
  \overline{A^{**}}=\mathcal{DR}^{**}_{pd}(A^{**})
  \overset{\mathrm{déf}}{=}\mathcal{DR}^{**}_{pd}(k,u)\otimes_{k\{T\}}A^{**},
\]

\[
  A^{**}\longrightarrow\overline{A^{**}}
\]

\[
  A^{ij}\neq0\ \text{\struck{$\Leftarrow$}}\Rightarrow 0\leq j\leq i
\]

\[
  d(xy)=(dx)y+(-1)^{\deg\mathrm{ext}\,x}x\,dy\ ]
\]

\[
  \sum_{i+j>0}A^{ij}=A^{+}
\]

\[
  \begin{cases}
    k\{T\}\xrightarrow{\ \sim\ }H^{*0}(A^{**})\\
    H^{*i}(A^{**})=0\quad(i>0)\\
    \text{les } A^{ij}\text{ sont $k$-plats}
  \end{cases}
\]

\[
  \mathrm{DR}(\Phi,T)=\underbrace{\Gamma^{*}(\Phi)}_{\text{deg compl}}
  \otimes_{k}\underbrace{\textstyle\bigwedge^{*}\Psi}_{\text{deg ext}}
\]

\[
  \text{flasque}+\text{plat}\Longrightarrow\text{homotope à $0$, donc}
\]

\[
  \Downarrow
\]

\[
  i_{n}\colon P\longrightarrow\widehat{\Delta}(\mathrm{S}_{s})
\]

\[
  u_{n}^{*}(K)=K_{n}=K(\Delta_{n}).
\]

\[
  i_{0*}(M)\simeq\Phi_{*}\otimes_{\mathbb{Z}}M\simeq\Phi_{*k}\otimes_{k}M
\]

\[
  i_{0*}(M)(I)\simeq M(i_{0}^{*}(I))\simeq M^{I}
\]

\[
  \mathrm{Hom}(I,i_{0*}(M))=\mathrm{Hom}\bigl(\underbrace{i_{0}^{*}(I)}_{\mathrm{Hom}(\Delta_{0},I)\simeq I},M\bigr)
\]

\[
  \mathrm{Hom}(A_{*},\Phi_{*}\otimes_{k}M)\simeq\mathrm{Hom}_{k}(A_{0},M)
\]

\[
  \text{\struck{$\mathrm{Hom}$}}\ A_{*}\longrightarrow\Phi_{*}\otimes_{\mathbb{Z}}A_{0}
  \simeq\Phi_{*k}\otimes_{k}A_{0}\quad(\simeq i_{0*}i_{0}^{*}A_{*})
\]

\[
  k_{*}\overset{u}{\hookrightarrow}A_{*}=A^{1,0}_{*}
\]

\[
u : \widehat{A}^{**} \to \widehat{A}'^{**},
\qquad
\widehat{A}^{**} = \prod A^{ij},
\quad
\widehat{A}'^{**} = \prod A'^{ij}
\]

\[
\begin{array}{ccc}
\mathrm{DR}(A^{1,0}_{*}, T) & \longrightarrow & A^{**} \\
\big\downarrow & & \\
\mathrm{DR}(\mathfrak{G}_{*} \otimes_{\mathbb{Z}} A_0, T) & & \\
\big\uparrow & &
\end{array}
\]

\[
\Gamma^{*}(A_{**}) \mathbin{\widehat{\otimes}_k} \widehat{\Lambda}^{*}(B_{**}),
\qquad
B_{**} = A_{*}/k_{*}
\]

\[
A_{*} \xrightarrow{\;u\;} A'^{**+},
\qquad
T \longmapsto T
\]

\[
v(p,q) : A_{*} \to A'^{pq}
\qquad (p, q \geqslant 0,\ p + q > 0)
\]

\[
v(1,0)(T) = T,
\qquad
v(p,q)(T) = 0 \ \text{si}\ (p,q) \neq (1,0)
\]

\[
\bar{v}(p,q) : \mathfrak{B} \to A'^{pq}
\qquad (\text{si}\ (p,q) \neq (1,0))
\]

\[
\xi_0 = \beta_1(\ldots),
\qquad
\xi_0 \in \widehat{A}^{**+}_{1}
\]

\[
\boxed{\alpha^{*}(\xi_0) = 1, \quad \beta^{*}(\xi_0) = 1}
\qquad (\alpha, \beta : \Delta_0 \to \Delta_1)
\]

\[
\boxed{\xi_0 \, d\xi_0 = d\xi_0 \cdot \xi_0}
\qquad
\boxed{d\xi_0 \cdot \xi_0 \ \ldots}
\]

\[
q_0^{*}(\xi_0)\, q_1^{*}(\xi_0) = q_1^{*}(\xi_0)\, q_0^{*}(\xi_0)
\]

\[
q_0^{*}(\xi_0)\, q_1^{*}(d\xi_0) \ldots = \ldots\, q_0^{*}(d\xi_0)\, q_1^{*}(\xi_0)
\]

\[
q_1^{*}(\xi_0)\, q_0^{*}(d\xi_0) \ldots = \ldots\, q_0^{*}(\xi_0)\, q_1^{*}(d\xi_0)
\]

\[
q_0^{*}(d\xi_0)\, q_1^{*}(d\xi_0) = - q_1^{*}(d\xi_0)\, q_0^{*}(d\xi_0)
\]

\[
C^{**} : (\mathrm{Ss})^{\circ} \longrightarrow \mathfrak{D}
\]

\[
\Delta^{\circ} \longrightarrow \mathfrak{D}
\]

\[
C^{**} = C^{**}_{*} = (C^{**}_{n})_{n \geqslant 0} \ \ldots
\]

\[
\mathcal{H}^{**}_{C}(\mathcal{X}) = H^{**}(C^{**}(\mathcal{X}))
\]

\[
\tau_{\omega}(H^{**}(\mathcal{X}))
\]

\[
\mathcal{H}^{**}_{C}(\mathcal{X}) = H^{*}_{c}(X) \otimes_k S_k
\]

\[
S_k \to C^{**}_{n} .
\]

\[
\begin{array}{ccc}
\mathbb{R}^{*}\Gamma(\mathcal{X}, S_k) & \longrightarrow & \mathbb{R}^{*}\Gamma(\mathcal{X}, C^{**}) \\
 & & \big\uparrow \\
(1) & & C^{**}(\mathcal{X}) = \Gamma(\mathcal{X}, C^{**})
\end{array}
\]

\[
\begin{array}{ccc}
\tau_{\omega}(\mathbb{R}^{*}\Gamma(\mathcal{X}, S_k)) & \longrightarrow & \tau_{\omega}\mathbb{R}^{*}\Gamma(\mathcal{X}, C^{**}) \\
 & & \big\uparrow \\
(2) & & \tau_{\omega} C^{**}(\mathcal{X})
\end{array}
\]

\[
\begin{array}{ccc}
H^{**}(\mathcal{X}, S_k) & \longrightarrow & H^{**}(\mathcal{X}, C^{**}) \\
 & & \big\uparrow \\
(1') & & H^{**}_{C}(X)
\end{array}
\]

\[
(2') \quad
\begin{array}{ccc}
\tau_{\omega}(H^{*}(\mathcal{X}, k) \otimes_k S_k) & \xrightarrow{\ \sim\ } & \tau_{\omega} H^{**}(\mathcal{X}, C^{**}) \\
 & & \big\uparrow \wr \\
 & & \tau_{\omega}(H^{**}_{c}(X))
\end{array}
\]

\[
\begin{array}{ccc}
\mathbb{R}^{*}\Gamma(\Delta_n, S_k) \simeq S_k & \longrightarrow & \mathbb{R}^{*}\Gamma(\Delta_n, C^{**}) = \mathcal{C}^{**}_{n} \\
 & & \big\uparrow \ \mathrm{id}_{C^{**}_{n}} \\
 & & C^{**}(\Delta_n) = C^{**}_{n}
\end{array}
\]

\[
(3) \qquad
H^{**}_{C}(\Delta_n) = H^{**}(C^{**}_{n})
\quad
\text{\struck{est isom.}}
\]

\[
\uparrow \varepsilon \ \text{augmentation}
\qquad
S_k
\]

\[
\text{3a)} \qquad
\text{\struck{$\ldots$}}\ S^{i}_{k} \to H^{i0}(C^{**}_{n})
= \mathrm{Ker}(C^{i,0}_{n} \to C^{i1}_{n})
\]

\[
\text{3b)} \qquad
\forall p > 0, \quad H^{ip}(C^{**}_{n}) = 0
\ \text{pour}\ i\ \text{grand (dépendant de}\ p\text{)}
\]

\[
H^{ip}(C^{**}_{n}) = 0 \quad \text{si} \quad i \geqslant \lambda(p)
\]

\[
C''^{**}_{*} \to C^{**}_{*},
\qquad
C''^{**}_{*} \to C'^{**}_{*} .
\]

\[
C'^{**}_{*} \rightrightarrows C^{**}_{*}
\]

\[
C''^{**}_{*} \to C'^{**}_{*}
\]

\[
  \gamma_i : V_k(s^+)\longrightarrow V_k(s^+) \qquad i\geqslant 1
\]

\[
  \gamma_i^{*} : \operatorname{Sym}_k(s^+)\longrightarrow \operatorname{Sym}^i_k(s^+)
\]

\[
  \gamma_i^{*} : s^+\longrightarrow \operatorname{Sym}^i_k(s^+)
\]

\[
  \boxed{\gamma_i^{*} : s^+\longrightarrow \operatorname{Sym}^i_k(s^+)\quad (i\geqslant 1)}
\]

\[
  (k\text{-alg. comm.})\longrightarrow
\]

\[
  \mathcal{A}^{*}_{*}(s,s^+,\gamma_*,t)(k')
  =\mathcal{A}^{*}_{*}\bigl(s^{k'},(s^{k'})^+,\gamma_*^{k'},t^{k'}\bigr)
\]

\[
  \mathcal{A}_*(s,s^+,\gamma_*,t)(k')(I)\simeq
  \widehat{\Gamma}_{s^{k'}_*}\{\{k'^{I}\}\}\otimes_{k'}
  \Lambda^{*}(k'^{I}/k')\big/\text{idéal dif. engendré par }\textstyle\sum X_i-t
\]

\[
  \mathcal{A}_n(s,s^+,\gamma_*,t)(k')\simeq
  s^{k'}\{\{X_0,\dots,X_n\}\}[dX_0,\dots,dX_n]\big/
  \bigl(\textstyle\sum X_i-t,\ \sum dX_i\bigr)_{\mathrm{pd}}
\]

\begin{align*}
  &\simeq s^{k'}\{\{X_0,\dots,X_{n-1}\}\}[dX_0,\dots,dX_{n-1}]\\
  &\simeq \prod\bigl(s^{k'}\bigr)^{\mathbb{N}^n\times\mathfrak{P}([0,n-1])}
  \qquad\text{comme foncteur en } k'\\
  &\simeq \operatorname{Hom}_k(s,k')^{\mathbb{N}^n\times\mathfrak{P}([0,\dots])}
  \qquad\text{comme foncteur en } k'\\
  &\simeq \operatorname{Hom}_k\bigl(s^{(\mathbb{N}^n\times\mathfrak{P}([0,\dots]))},k'\bigr)\\
  &= V\bigl(s^{(\mathbb{N}^n\times\mathfrak{P}([0,\dots]))}\bigr)(k')\\
  &= V\bigl(s\otimes_k k[Y_0,\dots,Y_{n-1}]\otimes_k
     {\textstyle\Lambda^{*}}(y_0,\dots,y_{n-1})\bigr)(k')
\end{align*}

\[
  \mathcal{A}_n(s,s^+,\gamma_*,t)(k')\big/\mathcal{A}_n(\ )(k')^+
  \simeq s^{k'}\big/(s^{k'})^+\hookrightarrow \bar s^{k'}
\]

\[
  \mathcal{A}_n(\ )(k')\longrightarrow \bar s^{k'}=V(\bar s)(k')
\]

\[
  V\bigl(s\otimes_k k[Y_0,\dots,Y_{n-1}]\otimes_k{\textstyle\Lambda^{*}}(y_0,\dots,y_{n-1})\bigr)(k')
  \longrightarrow V(\bar s)(k')
\]

\[
  s\otimes_k k[Y_0,\dots,Y_{n-1}]\otimes_k{\textstyle\Lambda^{*}}(y_0,\dots,y_{n-1})
  \longleftarrow \bar s
\]

\[
  \mathcal{A}^{n}_{*}(s,s^+,\gamma_*,t)
  =s\otimes_k k[Y_0,\dots,Y_{n-1}]\otimes_k{\textstyle\Lambda^{*}}(y_0,\dots,y_{n-1})
\]

\[
  \mathcal{A}_*(s,s^+,\gamma_*,t)=V_k\bigl(\mathcal{A}^*(s,s^+,\gamma_*,t)\bigr)
\]

\[
  k'\longmapsto\operatorname{Hom}\bigl(\mathcal{X},\mathcal{A}_*(-)(k')\bigr)
  \overset{\mathrm{df}}{=}
  C^{*}_{DR}\bigl(\mathcal{X};s^{k'},(s^{k'})^+,\gamma_*,t\bigr)
\]

\begin{align*}
  &\simeq \varprojlim_{\sigma\text{ dans }\Delta/\mathcal{X}}
     \operatorname{Hom}\bigl(\sigma,\mathcal{A}_*(\ )(k')\bigr)\\
  &\simeq \varprojlim_{\sigma\text{ dans }\Delta/\mathcal{X}}
     \mathcal{A}_{\dim\sigma}(\ )(k')
     \qquad\bigl(=\operatorname{Hom}_k(\mathcal{A}^{\dim\sigma}_*(-),k')\bigr)\\
  &\simeq \operatorname{Hom}_k\Bigl(\varinjlim_{\sigma\text{ dans }\Delta/\mathcal{X}}
     \mathcal{A}^{\dim\sigma}(-),k'\Bigr)\\
  &\simeq \operatorname{Hom}_k\bigl(C^{DR}_*(\mathcal{X};s,s^+,\gamma_*,t),k'\bigr)
\end{align*}

\[
  \boxed{C^{DR}_*(\mathcal{X};s,s^+,\gamma_*,t)\overset{\mathrm{df}}{=}
  \varinjlim_{\sigma\text{ dans }\Delta/\mathcal{X}}
  \mathcal{A}^{\dim\sigma}(s,s^+,\gamma_*,t)}
\]

\[
  C^*_{DR}\bigl(\mathcal{X};s^{k'},s^{+k'},\gamma_*,t^{k'}\bigr)
  =\operatorname{Hom}\bigl(C^{DR}_*(\mathcal{X},-),k'\bigr)
\]

\[
  \mathcal{A}_n(\ )(k')=s^{k'}\{\{X_0,\dots,X_n\}\}[dX_0,\dots,dX_n]
  \big/\bigl(\textstyle\sum X_i-t,\ \sum dX_i\bigr)_{\mathrm{pd}}
\]

\[
  \left\{
  \begin{aligned}
    &\deg\mathrm{tot}\,X_r=\deg\mathrm{tot}\,dX_r=\delta
      \quad\text{\struck{degré}}\\
    &\deg\mathrm{tot}\,X_r^{(n)}=\delta,\ \ \deg\mathrm{tot}_{\mathcal{A}_n}\lambda
      =\deg\mathrm{tot}_{s^{k'}}\lambda \qquad \lambda\in s^{k'}
  \end{aligned}\right.
\]

\begin{align*}
  s&=s_k=k[u] &&\text{donc } s^{k'}=k'\{\{T\}\}\\
  \bar s&=\bar s_k=k &&\text{donc } \bar s^{k'}=k'
\end{align*}

\[
  \bar s\longrightarrow s \ \text{l'inclusion}\ k\to k[u]
  \qquad\text{donc } s^{k'}\to\bar s^{k'}\ \text{l'augm. ordinaire}\
  k'\{\{T\}\}\to k'
\]

\[
  s^+=k[u]/k\simeq k[u]^+
\]

\[
  t(u^n)=\begin{cases}1 & \text{si } n=1\\ 0 & \text{si } n\neq1\end{cases}
\]

\begin{align*}
  \mathcal{A}_*(s,s^+,\gamma^*,t)(k')
  &\simeq\mathcal{A}_*\bigl(s^{k'},s^{+k'},\gamma^{*k'},t^{k'}\bigr)
   \simeq\mathcal{A}_*\bigl(k'\{\{T\}\},k'\{\{T\}\}^+,\gamma^{k'},T\bigr)\\
  &\simeq\mathcal{A}_*(k')^{\wedge}
\end{align*}

\[
  \text{\struck{$\mathcal{A}_k^{*}$}}\ \mathcal{A}^{*}_{k*}
  =\mathcal{A}^{*}_{*}(s_k,\bar s_k,\gamma_k^{*},t_k)
\]

\[
  \mathcal{A}_*(k')^{\wedge}\simeq\operatorname{Hom}_k(\mathcal{A}^{*}_{k},k')
\]

\[
  \text{\struck{$C^{DR}_*(\mathcal{X};k)$}}\quad
  C^{DR,\mathrm{pd}}_*(\mathcal{X},k)\overset{\mathrm{df}}{\simeq}
  C^{DR}_*\bigl(\mathcal{X},k[u],k[u]^+,\gamma^{*}_k,t_k\bigr)
\]

\[
  C^{DR,\mathrm{pd}}_*(\mathcal{X})\overset{\mathrm{df}}{=}
  C^{DR,\mathrm{pd}}_*(\mathcal{X};\mathbb{Z})
\]

\[
  C^{DR,\mathrm{pd}}_*(\mathcal{X},k)\simeq
  C^{DR,\mathrm{pd}}_*(\mathcal{X})\otimes_{\mathbb{Z}}k ,
\]

\[
  C^{*}_{DR,\mathrm{pd}}(\mathcal{X},k)\simeq
  \operatorname{Hom}\bigl(C^{DR,\mathrm{pd}}_*(\mathcal{X}),k\bigr)
\]

\begin{align*}
  C^{DR,\mathrm{pd}}_*(\mathcal{X};M)&\overset{\mathrm{df}}{=}
  C^{DR,\mathrm{pd}}_*(\mathcal{X})\otimes_{\mathbb{Z}}M\\
  C^{*}_{DR,\mathrm{pd}}(\mathcal{X};M)&\overset{\mathrm{df}}{=}
  \operatorname{Hom}\bigl(C^{DR,\mathrm{pd}}_*(\mathcal{X}),M\bigr)
\end{align*}

\[
  \text{\struck{$\tau^{\alpha}(F)=|\alpha|^{*}(\sigma^{\alpha}(F))\ \dots$}}
\]

\[
  \text{\struck{$\mathcal{X}'=\varprojlim\mathcal{X}_{\mathcal{U}}\to X,\quad
  \mathcal{X}_{\mathcal{U}}\to\mathcal{X},\quad
  \varinjlim F/X_i,\quad \Delta/\mathcal{X}_i,\ \Delta/\mathcal{X}_j,\quad
  \mathcal{X}_i\hookrightarrow\mathcal{X}_j,\quad
  \mathcal{F}_i\leftarrow\mathcal{F}_j$}}
\]

\[
  \pi_n : \mathcal{A}^n_n(s,s^+,\gamma^*,t)(k')
  =\mathcal{A}^{\wedge}_n\bigl(s^{k'},(s^{k'})^+,\gamma^{k'},t^{k'}\bigr)^n
  \longrightarrow s^{k'}
\]

\[
  (\mathcal{A}^{*}_{*})_n\qquad
  s\xrightarrow{\ \kappa_n\ } s[Y_0,\dots,Y_n][y_0,\dots,y_n]^{(n)}
\]

\[
  K^n_{\ill{}}(d\omega)=K^{n-1}_{\ill{}}(\partial\omega)
  \quad\Longleftrightarrow\quad
  d_{\mathcal{A}^*}\,\kappa_n=\partial_{ss}\,\kappa_{n-1}
\]

\[
  C^{\mathrm{sing}}_*(\mathcal{X},s)\longrightarrow
  C^{DR}_*(\mathcal{X};s,s^+,\gamma_*,t)
\]

\[
  C^{\mathrm{sing}}_*(\mathcal{X},k[u])\longrightarrow
  C^{DR,\mathrm{pd}}_*(\mathcal{X},k)
\]

\[
  \mathcal{A}^{0}_n\simeq k\{\{T\}\}\{\{X_0,\dots,X_n\}\}\big/
  \bigl(\textstyle\sum X_i-T\bigr)_{\mathrm{pd}}
  \xleftarrow{\ \text{épi}\ } k\{\{T\}\}\{\{X_0,\dots,X_n\}\}
\]

\[
  \simeq k\{\{X_0,\dots,X_n\}\}
  \longleftarrow k\{\{T,X_0,\dots,X_n\}\}
  \qquad X_i\mapsto X_i,\ \ \textstyle\sum_{0\leqslant i\leqslant n}X_i\leftarrow T
\]

\[
  (\mathcal{A}^n)_0\simeq k[U+Y_0,U+Y_1,\dots,U+Y_n]
  \xrightarrow{\ \text{mono}\ } k[U][Y_0,\dots,Y_n]\simeq k[U,Y_0,\dots,Y_n]
\]

\[
  k[Z_0,Z_1,\dots,Z_n]\longrightarrow k[U,Y_0,\dots,Y_n],\qquad
  Z_i\mapsto U+Y_i
\]

\[
  \bigl\langle X_0^{(i_0)}\cdots X_n^{(i_n)},\,Z_0^{j_0}\cdots Z_n^{j_n}\bigr\rangle
  =\prod_{0\leqslant\alpha\leqslant n}\delta_{i_\alpha,j_\alpha}
  =\begin{cases}0 & \text{si } (i_\alpha)\neq(j_\alpha)\\
  1 & \text{si } (i_\alpha)=(j_\alpha)\end{cases}
\]

\[
  \bigl\langle T^{(\lambda)}X_0^{(i_0)}\cdots X_n^{(i_n)},\,
  U^{\mu}Y_0^{j_0}\cdots Y_n^{j_n}\bigr\rangle
  =\delta_{\lambda,\mu}\prod_\alpha\delta_{i_\alpha,j_\alpha}
  =\begin{cases}0 & \text{si } (\lambda,(i_\alpha))\neq(\mu,(j_\alpha))\\
  1 & \text{si } \dots=\dots\end{cases}
\]

\[
  \mathcal{A}_n^{*=*}\simeq
  {\textstyle\Lambda}^{*}_k[dX_0,\dots,dX_n]\big/\bigl(\textstyle\sum dX_i\bigr)
  \xleftarrow{\ \text{épi}\ }{\textstyle\Lambda}^{*}_k[dX_0,\dots,dX_n]
\]

\[
  (\mathcal{A}^n)_{*=*}\simeq
  \bigl({\textstyle\Lambda}^{*}_k[y_0,\dots,y_n]\bigr)^{!}
  \simeq{\textstyle\Lambda}^{*}_k\bigl([y_0,\dots,y_n]^{!}\bigr)
  \longrightarrow{\textstyle\Lambda}^{*}_k[y_0,\dots,y_n]
\]

\[
  {\textstyle\Lambda}^{n}_k\bigl([y_0,\dots,y_n]'\bigr)\simeq
  {\textstyle\Lambda}^{n+1}_k\bigl([y_0,\dots,y_n]\bigr)\simeq k
\]

\[
  0\longrightarrow[y_0,\dots,y_n]'\longrightarrow[y_0,\dots,y_n]
  \longrightarrow k\longrightarrow0
\]

\[
  (\mathcal{A}^n)_n\overset{\text{iso can}}{\simeq}(\mathcal{A}^n)_0
  \simeq k[Z_0,\dots,Z_n]
\]

\[
  \kappa_n : k[U]\longrightarrow k[Z_0,\dots,Z_n]
\]

\[
  s,\ s^+,\ \gamma^*,\ t
\]

\[
  (\mathcal{A}^{*}_{n})^{n}\simeq
  S\{\{X_0,\dots,X_n\}\}\big/\bigl(\textstyle\sum X_i-t\bigr)_{\mathrm{pd}}
  \otimes_S S[dX_0,\dots,dX_n]\big/\bigl(\textstyle\sum dX_i\bigr)
\]

\[
  \simeq S\{X_0,\dots,X_{n-1}\}[dX_0,\dots,dX_{n-1}]
  \xrightarrow{\ \kappa^n\ (S\text{-linéaire})\ } S
\]

\[
  \kappa^n\bigl(X_0^{(i_0)}\cdots X_{n-1}^{(i_{n-1})}\,dX_0\wedge\dots\wedge dX_{n-1}\bigr)
  =\text{\struck{\ill{}}}\ t^{(n+\sum i_\alpha)}\,\gamma_{i_0,\dots,i_{n-1}}
\]

\[
  \gamma_{i_0\dots i_{n-1}}=\frac{(n+\sum i_\alpha)!}{\prod_\alpha(i_\alpha!)}
  \int\!\!\cdots\!\!\int_{\sigma_n}X_0^{i_0}\cdots X_{n-1}^{i_{n-1}}\,
  dX_0\wedge\dots\wedge dX_{n-1}
\]

\[
  \sigma_n=\{X_0,\dots,X_{n-1}\geqslant0,\ X_0+\dots+X_{n-1}\leqslant1\}
\]

\[
  \kappa^{n}\bigl(\underbrace{X_0^{(i_0)}\cdots X_{n-1}^{(i_{n-1})}\,
  dX_0\wedge\dots\wedge dX_{n-1}}_{d\left(X_{n-1}^{(i_{n-1}+1)}\,
  dX_0^{(i_0+1)}\wedge\dots\wedge dX_{n-2}^{(i_{n-2}+1)}\right)}\bigr)
  =\kappa^{n-1}\bigl(\partial_{ss}\,X_{n-1}^{(i_{n-1}+1)}\,
  dX_0^{(i_0+1)}\cdots\bigr)
\]

\[
  \kappa_{n-1}\Bigl(\bigl(t-(X_0+\dots+X_{n-2})\bigr)^{(i_{n-1}+1)}
  X_0^{(i_0)}\cdots X_{n-2}^{(i_{n-2})}\,dX_0\wedge\dots\wedge dX_{n-2}\Bigr)
\]

\[
  \bigl(t-(X_0+\dots+X_{n-2})\bigr)^{(i_{n-1}+1)}
  =\sum_{\lambda+l_0+\dots+l_{n-2}=i_{n-1}+1}
  (-1)^{\sum l_\alpha}X_0^{(l_0)}\cdots X_{n-2}^{(l_{n-2})}\,t^{(\lambda)}
\]

\begin{align*}
  &=\kappa_{n-1}\Bigl(\sum_{\lambda+\sum_0^{n-2}l_\alpha=i_{n-1}+1}
    (-1)^{\sum l_\alpha}c_{i_0,l_0}c_{i_1,l_1}\cdots c_{i_{n-2},l_{n-2}}\\
  &\qquad\qquad t^{(\lambda)}X_0^{(i_0+l_0)}\cdots X_{n-2}^{(i_{n-2}+l_{n-2})}\,
    dX_0\wedge\dots\wedge dX_{n-2}\Bigr)\\
  &=t^{(n+\sum i_\alpha)}\Bigl[(-1)^{i_{n-1}+1}
    \sum_{\lambda+\sum l_\alpha=i_{n-1}+1}(-1)^{\sum l_\alpha}
    c_{i_0,l_0}\cdots c_{i_{n-2},l_{n-2}}\\
  &\qquad\qquad c_{\lambda,\,n-1+\sum_0^{n-2}(l_\alpha+i_\alpha)}\,
    \gamma_{i_0+l_0,\dots,i_{n-2}+l_{n-2}}\Bigr]
\end{align*}

\[
  \begin{aligned}
  \gamma_{i_0,\dots,i_{n-1}}&=\text{\struck{$(-1)^{i_{n-1}+1}$}}
  \sum_{\lambda+\sum_0^{n-2}l_\alpha=i_{n-1}+1}
  (-1)^{\sum_0^{n-2}l_\alpha}c_{i_0,l_0}\cdots c_{i_{n-2},l_{n-2}}\\
  &\qquad c_{\lambda,\,n-1+\sum_0^{n-2}(l_\alpha+i_\alpha)}\,
  \gamma_{i_0+l_0,\,i_1+l_1,\dots,\,i_{n-2}+l_{n-2}}
  \end{aligned}
\]

\[
  C_k \longrightarrow \mathrm{Ens}, \qquad A \longmapsto A
\]

\[
  V(M)(A) = \operatorname{Hom}_{k\text{-mod}}(M, A) = \operatorname{Hom}_{A\text{-mod}}(M_A, A).
\]

\[
  \operatorname{Hom}_k(M, N) \xrightarrow{\ \sim\ } \operatorname{Hom}_{\mathcal{O}_k}(V(N), V(M)).
\]

\[
  V(N) \longrightarrow \text{\struck{$V(M), \operatorname{Sym}(B)$}}\ F \qquad (\text{où } F \in \operatorname{ob} \widehat{C}_k)
\]

\[
  \operatorname{Hom}_{\widehat{C}_k}(V(N), F) \simeq F(\operatorname{Sym}^*(N)),
\]

\[
  \operatorname{Hom}_{\widehat{C}_k}(V(N), \operatorname{Spec} B) \simeq \operatorname{Hom}_{k\text{-alg}}(B, \operatorname{Sym}^*(N)) ;
\]

\[
  \operatorname{Hom}_{\widehat{C}_k}(V(N), V(M)) \simeq \operatorname{Hom}_{k\text{-mod}}(M, \operatorname{Sym}^* N).
\]

\[
  u^* : V(N) \longrightarrow V(M)
\]

\[
  u : M \longrightarrow \operatorname{Sym}^*(N).
\]

\[
  u(x) = \sum u_i(x), \qquad u_i(x) \in \operatorname{Sym}^i(N)
\]

\[
  u_i : M \longrightarrow \operatorname{Sym}^i(N)
\]

\[
  u^*(x^*) = \widetilde{x^*} \circ u \in \operatorname{Hom}_{k\text{-mod}}(M, A), \qquad M \xrightarrow{\ u\ } \operatorname{Sym}^* N \xrightarrow{\ \widetilde{x^*}\ } A,
\]

\[
  u^*(x^*)(x) = \sum_{i \geqslant 0} \widetilde{x^*}^{\,i}(u_i(x)), \qquad x \in M
\]

\[
  u^*(\lambda x^*)(x) = \sum_{i \geqslant 0} \lambda^i\, \widetilde{x^*}^{\,i}(u_i(x)) = \sum_{i \geqslant 0} \widetilde{x^*}^{\,i}(\lambda^i u_i(x)).
\]

\[
  u = \sum u_i \longmapsto u^\lambda = \sum \lambda^i u_i .
\]

\[
  u^* \circ \lambda_{V(N)} = \lambda^n_{V(M)} \circ u^* \text{)}
\]

\[
  M \longrightarrow \operatorname{Sym}^n N.
\]

\[
  M \longrightarrow N
\]

\[
  \prod_{i \in I} V(N_i) \longrightarrow F \qquad \text{\struck{$V(\textstyle\coprod_i N_i)$}}
\]

\[
  \prod_{\alpha \in I} V(N_\alpha) \simeq V\Bigl(\bigoplus N_\alpha\Bigr)
\]

\[
  M \xrightarrow{\ u\ } \operatorname{Sym}^*\Bigl(\bigoplus N_\alpha\Bigr) = \bigotimes_\alpha \operatorname{Sym}^*(N_\alpha).
\]

\[
  u(x) = \sum_{(i_\alpha)} \underbrace{u_{(i_\alpha)}(x)}_{\in\, \bigotimes_\alpha \operatorname{Sym}^{i_\alpha}(N_\alpha)}
\]

\[
  u^*\bigl((\lambda_\alpha x^*_\alpha)_{\alpha \in I}\bigr) = (u^\lambda)^*(x^*_\alpha)
\]

\[
  u^* \prod_\alpha \lambda_{\alpha\, V(N_\alpha)} = (u^\lambda)^* ,
\]

\[
  u^\lambda = \sum_{(i_\alpha)} \Bigl(\prod_\alpha \lambda_\alpha^{i_\alpha}\Bigr) u_{(i_\alpha)} .
\]

\[
  u : M \longrightarrow \bigotimes_\alpha \operatorname{Sym}^{n_\alpha}(N_\alpha).
\]

\[
  u : M \longrightarrow \bigotimes_\alpha N_\alpha
\]

\[
  \prod V(N_\alpha) \longrightarrow \text{\struck{\ill{}}}\ V(M)
\]

\[
  M \longrightarrow \bigotimes_\alpha N_\alpha ,
\]

\[
  V(N) \times V(N') \longrightarrow V(M)
\]

\[
  M \longrightarrow N \otimes N' .
\]

\[
  \Delta : M \longrightarrow M \otimes M
\]

\[
  M \xrightarrow{\ \varepsilon\ } k
\]

\[
  \Delta e = e \otimes e .
\]

\[
  V\bigl(\varinjlim_i M_i\bigr) = \varprojlim_i V(M_i)
\]

\[
  M \simeq \bigoplus_{\alpha \in Z} M_\alpha \qquad ((M_\alpha) \text{ sous-modules de } M)
\]

\[
  V(M) \simeq \prod_{\alpha \in Z} V(M_\alpha)
\]

\[
  D(Z) = G : A \longmapsto \operatorname{Hom}_{\mathrm{Ab}}(Z, A^*) \simeq \operatorname{Hom}_{k\text{-alg}}(k(Z), A),
\]

\[
  G = \operatorname{Spec} k(Z)
\]

\[
  \Delta : M \longrightarrow M \otimes M
\]

\[
  \underset{V(M)^i}{V(M_i)} \times \underset{V(M)^j}{V(M_j)} \longrightarrow \underset{V(M)^{i+j}}{V(M_{i+j})} .
\]

\[
  \text{\struck{\ill{}}}\ Z \longrightarrow \mathbb{Z}/2\mathbb{Z}
\]

\[
  V(M) = V(M)^{\mathrm{pair}} \times V(M)^{\mathrm{impair}}
\]

\[
  xy = (-1)^{\deg x \deg y}\, yx ,
\]

\[
  \begin{cases}
    xy = yx & \text{si } x \text{ ou } y \text{ de degré pair}  \\
    x^2 = 0 & \text{si } x \text{ de degré impair.}
  \end{cases}
\]

\[
  x^2 = y^2 = (x+y)^2 = 0 ,
\]

\[
  \underset{V(M^{-i})}{V(M_i)} = V(M)^i = V(M)_{-i} .
\]

\[
  W(C) = (k' \longmapsto C \otimes_k k') \quad \text{sous} \quad V(s) = (k' \longmapsto s^{k'} = \operatorname{Hom}_{k\text{-mod}}(s, k')) \text{.]}
\]

\[
  C \xrightarrow{\ \Delta_C\ } C \otimes s \xrightarrow{\ \mathrm{id}_C \otimes u\ } \prod_{u \in s^k} C \otimes k
\]

\[
  C \otimes s \longrightarrow \prod_{u \in s^k} C \otimes k
\]

\[
  C \otimes_k k^{(I)} \longrightarrow (C \otimes_k k)^I = C^I
\]

\[
  \operatorname{Hom}(k^{(I)}, C)
\]

\[
  s = k[U] , \qquad s^k = k\{\{T\}\} \quad \text{structure de \uncertain{k-alg.\ ordinaire}}
\]

\[
  C = k[U][Y_0, \ldots, Y_n] = k[U, Y_0, \ldots, Y_n] , \qquad C^k = k\{\{T\}\}\{\{X_0, \ldots, X_n\}\}
\]

\[
  T^{(i)} X_0^{(i_0)} \cdots X_n^{(i_n)} \quad \text{opérant par l'op.\ diff.} \quad D_U^{(i)} D_{Y_0}^{(i_0)} \cdots D_{Y_n}^{(i_n)} .
\]

\[
  C \longrightarrow C \otimes_k s \hookrightarrow \operatorname{Hom}_k(s^k, C)
\]

\[
  \text{\struck{$C \otimes_k \operatorname{Hom}_k((s^k)^k, k)$}} \longrightarrow \operatorname{Hom}_k(s^k, C \otimes k)
\]

\[
  i_x : u \longmapsto u_C\, x , \qquad s^k \longrightarrow C
\]

\[
  u_C\, x = \Bigl(\underbrace{\sum x_\alpha \otimes e_\alpha}_{\substack{\text{somme finie,} \\ x_\alpha \in C}}\Bigr)(u) = \sum u(e_\alpha)\, x_\alpha
\]

\[
  u(e_\alpha) = u_\alpha \quad \text{dans} \quad u \longmapsto (u_\alpha)_{\alpha \in A} ,\ s^k \simeq k^A ,
\]

\[
  u . x = \sum u_\alpha x_\alpha .
\]

\[
  s^k \longrightarrow \operatorname{Hom}_k(C, C)
\]

\[
  s^k = \prod_{n \in \mathbb{Z}} {s_n}^k = \prod_{n \in \mathbb{Z}} (s^k)_n .
\]

\[
  \text{\struck{\ill{}}}\ (s^k)^n = (s_{-n})^k
\]

\[
  u : \underset{V(C_{-m})}{V(C_m)^m} \longrightarrow \underset{V(C_{-m-n})}{V(C)^{m+n}}
\]

\[
\begin{cases}
\varphi_{0}\colon S\to S' \ \text{un hom.\ d'anneaux}\\
A'_{1}\in S'\{X_{0}\}^{++}=\operatorname{Ker}\bigl(\alpha_{S'}\colon S'\{X_{0}\}\to S'\bigr)\\
B'_{i}\in S'\{X_{0}\}
\end{cases}
\]

\[
\begin{cases}
\varphi_{0} = \text{composante en degré } 0 \text{ de } \varphi_{*},\quad
\varphi_{0}\colon (\mathcal{A}_{S})_{0}\to(\mathcal{A}_{S'})_{0},\\
A'_{1} = k_{0}\,\varphi_{1}(X_{0}),\\
B'_{i} = k_{1}\,\varphi_{1}(X_{0}^{(i)}),
\end{cases}
\]

\[
\begin{cases}
\varphi_{1}(\lambda)=\varphi_{0}(\lambda) & \text{si } \lambda\in S\\
\varphi_{1}(X_{0}^{(i)})=A_{1}'^{(i)}+B'_{i}\,dX_{0}\\
\varphi_{1}(dX_{0})=dA'_{1}
\end{cases}
\]

\[
U_{0}=X_{0},\ U_{1}=X_{0}+X_{1},\ \ldots,\ U_{n-1}=X_{0}+X_{1}+\cdots+X_{n-1}.
\]

\[
\begin{cases}
\varphi_{n}(\lambda)=\varphi_{0}(\lambda) & \text{si } \lambda\in S\\
\varphi_{n}(U_{r}^{(i)})=A_{1}'^{(i)}\{U_{r}\}+B'_{i}\{U_{r}\}\,dU_{r} & (0\leq r\leq n-1,\ i\geq 1)\\
\varphi_{n}(dU_{r})=dA'_{1}\{U_{r}\}=A_{1}'^{\,\mathrm{dér}}\{U_{r}\}\,dU_{r} & (0\leq r\leq n-1)
\end{cases}
\]

\[
B'_{i}=A_{1}'^{(i-1)}B'_{1}\qquad\forall i\geq 1
\]

\[
A'_{1}=\underbrace{a'_{1}}_{\lambda}X_{0},\qquad
B'_{i}=b'_{i0}+b'_{i1}X_{0}+\cdots+b'_{i,i-1}X_{0}^{(i-1)}
\]

\[
(2')\quad a'_{1}t'=\varphi_{0}(t)
\]

\[
(3')\quad (2b'_{ij})J'=0\qquad (i\geq 1,\ 0\leq j\leq i-1)\ ].
\]

\[
A'_{1}=a'_{1}X_{0},\qquad B'_{i}=0\quad i\geq 0,
\]

\[
S=k\{T\},\ S^{+}=(T)_{\mathrm{pd}},\ t=T;\qquad
S'=k'\{T\},\ S'^{+}=(T)_{\mathrm{pd}},\ t'=T.
\]

\[
A'_{1}=\sum_{i\geq 1,\ j\geq 0}\alpha_{ij}X_{0}^{(i)}X_{1}^{(j)}.
\]

\[
\varphi_{0}(T)=\sum_{i\geq 1}\alpha_{i,0}T^{(i)}.
\]

\[
\mathcal{A}_{k*}=\mathcal{A}(k\{T\},k\{T\}^{+},T)\to\mathcal{A}_{k'*}=\mathcal{A}(k'\{T\},k'\{T\}^{+},T)
\]

\[
\begin{cases}
\varphi_{00}\colon k\to k'\{T\}\ \text{hom.\ d'anneaux}\\
A'_{1}\in k'\{T\}\{X_{0}\}^{++}\simeq k'\{X_{0},X_{1}\}\\
B'_{i}\in k'\{T\}\{X_{0}\}\simeq k'\{X_{0},X_{1}\}\quad (i\geq 1)
\end{cases}
\]

\[
A'_{1}=\sum_{j\geq 1}a'_{j}X_{0}^{(j)}\quad (a'_{j}\in k'\{T\}=S_{k'}),\qquad
B'_{i}=\sum_{j\geq 0}b'_{ij}X_{0}^{(j)}\quad (b'_{ij}\in k'\{T\}=S_{k'}).
\]

\[
\mathrm{Fil}^{\mathrm{gross}}_{d}\mathcal{A}_{kn}
=\Bigl\{\textstyle\sum F_{i_{0}\ldots i_{r}}\,dX_{i_{0}}\wedge\cdots\wedge dX_{i_{r}}\Bigr\}
\]

\[
F_{i_{0}\ldots i_{r}}\in k\{X_{0},\ldots,X_{n}\} \text{ de « degré » }\leq d-r,
\]

\[
(\mathcal{A}_{k})_{n}\simeq k\{X_{0},\ldots,X_{n}\}[dX_{0},\ldots,dX_{n}]\big/\textstyle\sum dX_{i}
\]

\[
\begin{cases}
\varphi_{0}(k)\subset k'\\
A'_{1}=\alpha'_{0}X_{0},\quad \alpha'_{0}\in k'\\
B'_{i}=\sum\limits_{\substack{j,k\geq 0\\ j+k\leq i-1}}\alpha'_{ijk}X_{0}^{(j)}X_{1}^{(k)}\quad (\forall i\geq 1)
\end{cases}
\]

\[
\varphi_{0}(T)=\alpha'_{0}T.
\]

\[
\begin{cases}
\varphi_{0}k\subset k'\\
A'_{1}=\alpha'_{0}X_{0}\quad (\alpha'_{0}\in k')\\
B'_{i}=\sum\limits_{\substack{j,k\geq 0\\ j+k=i-1}}\alpha'_{i,j,k}X_{0}^{(j)}X_{1}^{(k)},\quad i\geq 1
\end{cases}
\]

\[
\begin{cases}
\varphi_{0}k\subset k'\\
A'_{1}=\alpha'_{0}X_{0}\quad (\alpha'_{0}\in k')\\
B'_{i}=0\quad (i\geq 1)
\end{cases}
\]

\[
\mathcal{A}_{*}(S_{k},S_{k}^{+},T)\to\mathcal{A}_{*}(S_{k'},S_{k'}^{+},T)
\]

\[
(u_{\varphi_{00},\alpha'_{0}})\colon (S_{k},S_{k}^{+},T)\to(S_{k'},S_{k'}^{+},T)
\]

\[
u_{\varphi_{00},\alpha'_{0}}\colon k\{T\}\to k'\{T\}
\]

\[
B'_{i}=\sum_{\substack{j,k\geq 0\\ j+k\geq i-1}}\alpha'_{i,j,k}X_{0}^{(j)}X_{1}^{(k)}
\]

\[
\widehat{\mathcal{A}}_{k*}=\bigl(\widehat{\mathcal{A}}_{k,n}\bigr),
\]

\[
\widehat{\mathcal{A}}_{k,n}\simeq k\{\{X_{0},\ldots,X_{n}\}\}[dX_{0},\ldots,dX_{n}]\big/\textstyle\sum dX_{i}
\]

\[
\simeq k\{\{T\}\}\{\{X_{0},\ldots,X_{n}\}\}[dX_{0},\ldots,dX_{n}]\big/\bigl(\textstyle\sum X_{i}-T,\ \sum dX_{i}\bigr)_{\mathrm{pd}}
\]

\[
\simeq k\{\{T\}\}\{\{X_{0},\ldots,X_{n-1}\}\}[dX_{0},\ldots,dX_{n-1}].
\]

\[
\mathcal{A}_{n}(S,S^{+},t)^{\wedge}\simeq S\{\{X_{0},\ldots,X_{n}\}\}[dX_{0},\ldots,dX_{n}]
\]

\[
\simeq S\{\{X_{0},\ldots,X_{n-1}\}\}[dX_{0},\ldots,dX_{n-1}]
\]

\[
A,\ \struck{B}\,B_{i}\in\widehat{S}_{k}\{\{X_{0}\}\}\simeq k\{\{T\}\}\{\{X_{0}\}\}\simeq k\{\{X_{0},X_{1}\}\}\qquad (i\geq 1).
\]

\[
\begin{cases}
A=\sum\limits_{j\geq 1}a_{j}X_{0}^{(j)} & (a_{j}\in\widehat{S}_{k}\simeq k\{\{T\}\},\ j\geq 0)\\
\text{avec } a_{j}=\sum\limits_{k\geq 0}a_{j,k}T^{(k)}, & a_{jk}\in k \text{ pour } j\geq 1,\ k\geq 0.
\end{cases}
\]

\[
A=\sum_{i\geq 1,\ j\geq 0}\alpha_{ij}X_{0}^{(i)}X_{1}^{(j)},\qquad \alpha_{ij}\in k.
\]

\[
\varphi_{0}(T)=\sum_{j\geq 1}\lambda_{j}T^{(j)},\qquad \lambda_{j}\in k.
\]

\[
\boxed{a_{1,0}=\alpha_{1,0}=\lambda_{1}}\in k.
\]

\[
\Omega(k)=\operatorname{Aut}_{k\text{-alg.\ diff.\ }\frac{1}{2}\text{simpl.}}\bigl(\widehat{\mathcal{A}}_{k*}\bigr)
\]

\[
A=\sum_{j\geq 1}a_{j}X_{0}^{(j)},\qquad
B_{i}=\sum_{j\geq 0}b_{ij}X_{0}^{(j)}\qquad\Bigm|\qquad a_{j}, b_{ij}\in\widehat{S}_{k}=k\{\{T\}\}
\]

\[
a_{10}\in k^{*},\qquad 2b_{ijk}=0.
\]

\[
\mathbf{G}_{m,\mathbf{Z}}\times{}_{2}\mathbf{G}_{a}^{\mathbf{N}^{*}\times\mathbf{N}}\times{}_{2}\mathbf{G}_{a}^{\mathbf{N}^{*}\times\mathbf{N}\times\mathbf{N}}
\]

\[
\Omega\simeq\operatorname{Spec}\Bigl(\mathbf{Z}\bigl[(A_{jk})_{j\geq 1,k\geq 0},\ (B_{ijk})_{i\geq 1,j\geq 0,k\geq 0}\bigr]\Bigr)
\]

\[
\Delta\colon\Omega\to\mathbf{G}_{m}
\]

\[
\delta\colon\mathbf{G}_{m}\to\Omega
\]

\[
A=\lambda X_{0}\quad\text{i.e.}\quad \varphi_{1}(X_{0})=\lambda X_{0},\ \text{d'où}\ \varphi_{1}(X_{1})=\lambda X_{1},
\]

\[
\varphi_{0}(T)=\lambda T,\qquad \varphi_{n}(X_{i})=\lambda X_{i}\quad\forall\, 0\leq i\leq n).
\]

\[
  \tau_\omega : (\text{Mod gr de } S_h \Longrightarrow)\ M \longrightarrow M_\omega ,
\]

\[
  A_1 = k\{X_0, X_1, dX_0, dX_1\}/(dX_0 + dX_1)_{\mathrm{pd}}
      = k\{T\}\{X_0, dX_0\} \simeq k\{T\}\{X_1, dX_1\},
\]

\[
  X_0 + X_1 = T .
\]

\[
  \varphi_1(X_0^{(i)}) = \xi_i ,
\]

\[
  (1)\qquad \boxed{\xi_i\,\xi_j = c_{ij}\,\xi_{i+j}} \qquad (i, j \geqslant 1)
\]

\[
  \varphi_1(dX_0) = d\xi_1
\]

\[
  \varphi_1(dX_0^{(i)}) \overset{?}{=} d\varphi_1(X_0^{(i)}) ,
\]

\[
  (2)\qquad \boxed{d\xi_i = \xi_{i-1}\,d\xi_1} \qquad (i \geqslant 2).
\]

\[
  \alpha(X_0^{(i)}) = 0 \ \ \forall i \geqslant 1, \quad \alpha(dX_0) = 0 ;
  \qquad
  \beta(X_0^{(i)}) = T^{(i)} \ \ \forall i \geqslant 1, \quad \beta(dX_0) = 0 .
\]

\[
  \begin{cases}
    \alpha \circ \varphi_1 = \alpha \\
    \beta \circ \varphi_1 = \beta
  \end{cases}
\]

\[
  (3\mathrm{a})\quad \boxed{\alpha(\xi_i) = 0}
  \qquad\qquad
  (3\mathrm{b})\quad \boxed{\beta(\xi_i) = T^{(i)}}
\]

\[
  \xi_i = \overbrace{\sum_{j \geqslant 0} a_{ij} X_0^{(j)}}^{A_i}
        + \Bigl(\overbrace{\sum_{j \geqslant 0} b_{ij} X_0^{(j)}}^{B_i}\Bigr) dX_0 ,
  \qquad a_{ij}, b_{ij} \in k\{T\} .
\]

\[
  (3'\mathrm{a})\qquad \xi_i = \sum_{j \geqslant 1} a_{ij} X_0^{(j)} + \cdots
  \qquad \forall i \geqslant 1 ,
\]

\[
  (3'\mathrm{b})\qquad \sum_{j \geqslant 1} a_{ij}\,T^{(j)} = T^{(i)} \qquad \forall i \geqslant 1 .
\]

\[
  d\xi_i = \Bigl(\sum_{j \geqslant 1} a_{ij} X_0^{(j-1)}\Bigr) dX_0 ,
  \qquad
  d\xi_1 = \Bigl(\sum_{j \geqslant 1} a_{1j} X_0^{(j-1)}\Bigr) dX_0 ,
\]

\[
  \xi_{i-1} = \sum_{j \geqslant 1} a_{i-1,j} X_0^{(j)}
            + \Bigl(\sum_{j \geqslant 0} b_{i-1,j} X_0^{(j)}\Bigr) dX_0 ,
\]

\[
  \xi_{i-1}\,d\xi_1
  = \Bigl(\sum_{k \geqslant 1} a_{1k} X_0^{(k-1)}\Bigr)
    \Bigl(\sum_{k \geqslant 1} a_{i-1,k} X_0^{(k)}\Bigr) dX_0 .
\]

\[
  \varphi : A \to A \ \text{endom.\ de $S$-algèbre}
  \quad [\varphi(x+y) = \varphi(x) + \varphi(y),\ \varphi(xy) = \varphi(x)\varphi(y),\ \varphi(1) = 1],
\]

\[
  \varphi\, d = d\, \varphi , \qquad \alpha\,\varphi = \alpha .
\]

\[
  \varphi(X_0^{(i)}) = \xi_i \in A \quad (i \geqslant 1),
  \qquad \xi_i = A_i + B_i\,dX_0 , \quad A_i, B_i \in S\{X_0\} \ (i \geqslant 1)
\]

\[
  \begin{cases}
    A_i = A_1^{(i)} \\
    \alpha(A_1) = 0 , \ \text{i.e. } A_1 \in S\{X_0\}^{+} .
  \end{cases}
\]

\[
  \boxed{\begin{array}{l} A_1 \in S\{X_0\}^{+} \\ B_i \in S\{X_0\} \ (i \geqslant 1) \end{array}}
\]

\[
  B_i = A_1^{(i-1)} B_1 .
\]

\[
  \beta : A_1 = S\{X_0, dX_0\} = k\{T\}\{X_0, dX_0\} \longrightarrow S ,
\]

\[
  A_1 \simeq k\{X_0, X_1, dX_0, dX_1\}/\bigl((X_0 + X_1 - T,\ dX_0 + dX_1)\bigr)_{\mathrm{pd}} ,
\]

\[
  \begin{cases}
    \beta(X_0^{(i)}) = T^{(i)} \\
    \beta(dX_0) = 0
  \end{cases}
\]

\[
  \beta(A_1)^{(i)} = T^{(i)} \qquad \forall i \geqslant 1 ,
\]

\[
  \boxed{\beta(A_1) = T} \qquad \text{i.e. } A_1(T) = T .
\]

\[
  A_1 - X_0 = \varphi^0_0(X_0) - X_0 = -\bigl(\varphi^0_0(X_1) - X_1\bigr) = f \in A_1^0
\]

\[
  S\{X_0\} \simeq k\{T\}\{X_0, X_1\}/(X_0 + X_1 - T)_{\mathrm{pd}}
\]

\[
  A_1 = \underbrace{S\{X_0\}}_{A_1^0} + \underbrace{S\{X_0\}\,dX_0}_{A_1^1}
      \simeq S\{X_1\} + S\{X_1\}\,dX_1 \ ).
\]

\[
  f \in (X_0)_{\mathrm{pd}} \cap (X_1)_{\mathrm{pd}} ,
\]

\[
  \boxed{f = \sum_{i, j \geqslant 1} f_{ij}\, X_0^{(i)} X_1^{(j)} , \quad f_{ij} \in k}
\]

\[
  A_1 \simeq S\{X_0, X_1, dX_0, dX_1\}/\!/(X_0 + X_1 - T,\ dX_0 + dX_1)_{\mathrm{pd}} ,
\]

\[
  f \in \operatorname{Ker}\bigl(k\{X_0, X_1\} \to k\{X_0\}\bigr)
    \cap \operatorname{Ker}\bigl(k\{X_0, X_1\} \to k\{X_1\}\bigr) ,
\]

\[
  \text{i.e.}\quad \boxed{f = \sum_{i, j \geqslant 1} c_{ij}\, X_0^{(i)} X_1^{(j)}}
\]

\[
  \boxed{B_i \in A_1^0 = k\{X_0, X_1\} \qquad (i \geqslant 0)}
\]

\[
  \begin{cases}
    \varphi(X_0^{(i)}) = (X_0 + f)^{(i)} + B_i\,dX_0 \\
    \varphi(dX_0) = 1 + df
  \end{cases}
\]

\[
  \varphi^1(X_1^{(i)}) = \varphi^1\bigl((T - X_0)^{(i)}\bigr)
  = \varphi^1\Bigl(\sum_{0 \leqslant j \leqslant i} T^{(i-j)} X_0^{(j)} (-1)^j\Bigr)
  = \Bigl(\sum_{0 \leqslant j \leqslant i} T^{(i-j)} (-1)^j B_j\Bigr) dX_0
\]

\[
  = \Bigl(\underbrace{\sum_{0 \leqslant j \leqslant i} (-1)^{j+1} T^{(i-j)} B_j}_{C_i}\Bigr) dX_1 .
\]

\[
  \varphi^1\bigl((X_0 + X_1)^{(i)}\bigr) = \varphi^1(T^{(i)}) = 0 ,
  \qquad
  (X_0 + X_1)^{(i)} = \sum_{j+h=i} X_0^{(j)} X_1^{(h)} ,
\]

\[
  \varphi^1\bigl((X_0 + X_1)^{(i)}\bigr)
  = \text{comp.\ de degré 1 de }\sum_{j+h=i} \varphi(X_0^{(j)})\,\varphi(X_1^{(h)}) ,
\]

\[
  \varphi(X_0^{(j)}) = (X_0 + f)^{(j)} + B_j\,dX_0 ,
  \qquad
  \varphi(X_1^{(h)}) = (X_1 - f)^{(h)} + C_h\,dX_1 , \quad dX_1 = -dX_0 ,
\]

\[
  \sum_{j+k=i} \Bigl[ -(X_0 + f)^{(j)} C_k + B_j (X_1 - f)^{(k)} \Bigr] = 0 .
\]

\[
  A_1 = k\{X_0, X_1\}[dX_0, dX_1]/(dX_0 + dX_1)\ \struck{\ill{}} ,
\]

\[
  \struck{\ill{}} \quad \varphi_n\, A_*(\sigma) = A_*(\sigma)\, \varphi_1
\]

\[
  \varphi_{n'}\, A_*(\tau) = A_*(\tau)\, \varphi_n .
\]

\[
  \varphi_{n'}\, \underbrace{A_*(\tau)\, A_*(\sigma)}_{A_*(\sigma\tau)}
  = A_*(\tau)\, \underbrace{\varphi_n\, A_*(\sigma)}_{A_*(\sigma)\,\varphi \ (\text{par hyp.\ sur } \varphi_n)}
  \qquad \forall \sigma : \Delta_n \to \Delta_1 ,
\]

\[
  \underbrace{\varphi_{n'}\, A_*(\sigma\tau)}_{A_*(\sigma\tau)\,\varphi \ (\text{par hyp.\ sur } \varphi_{n'})}
  \qquad\text{et}\qquad A_*(\tau)A_*(\sigma)\varphi = A_*(\sigma\tau)\,\varphi .
  \qquad \text{OK.}
\]

\[
  \sigma_r(0) = 0, \ \ldots, \ \sigma_r(r) = 0, \quad
  \sigma_r(r+1) = 1, \ \ldots, \ \sigma_r(n) = 1 ,
\]

\[
  \begin{cases}
    A_*(\sigma_r)(X_0) = X_0 + \cdots + X_r \overset{\mathrm{df}}{=} U_r \\
    A_*(\sigma_r)(X_1) = X_{r+1} + \cdots + X_n
  \end{cases}
\]

\[
  A_*(\sigma_r)(X_0^{(i)}) = U_r^{(i)} .
\]

\[
  A_n \simeq S\{U_0, \ldots, U_{n-1}\}[dU_0, \ldots, dU_{n-1}] .
\]

\[
  X_0 = U_0, \quad X_1 = U_1 - U_0, \quad \ldots, \quad X_{n-1} = U_{n-1} - U_{n-2}
\]

\[
  \varphi_n(\underbrace{U_{r,i}}_{U_r^{(i)}})
  = \varphi_n\bigl(\underbrace{\sigma_r^{*}}_{\overset{\mathrm{df}}{=} A_*(\sigma_r)}(X_0^{(i)})\bigr)
  = \sigma_r^{*}\bigl(\varphi(X_0^{(i)})\bigr)
  = \sigma_r^{*}(\xi_i) ,
\]

\[
  \varphi_n(U_r^{(i)}) = \sigma_r^{*}(\xi_i)
  = \sum_j a_{ij}\,U_r^{(j)} + \Bigl(\sum_j b_{ij}\,U_r^{(j)}\Bigr) dU_r .
\]

\[
  \varphi_n^0(U_r^{(i)}) = \sigma_r^{*}(\xi_i) \overset{\mathrm{df}}{=} \xi_{r,i} ;
\]

\[
  \xi_{r,i}\,\xi_{r,j} = c_{ij}\,\xi_{r,i+j} ,
\]

\[
  \sigma_r^{*}(\xi_i)\,\sigma_r^{*}(\xi_j) = c_{ij}\,\sigma_r^{*}(\xi_{i+j}) ;
\]

\[
  \varphi_n(dU_r) = d\varphi_n(U_r) = d\sigma_r^{*}(\xi_1) = \sigma_r^{*}(d\xi_1)
  \qquad
  \Bigl(= d\Bigl(\sum_{j \geqslant 0} a_{1j}\,U_r^{(j)}\Bigr)
  = \Bigl(\sum_{j \geqslant 1} a_{1j}\,U_r^{(j-1)}\Bigr) dU_r\Bigr) ,
\]

\[
  d\xi_1 = \sum_{j \geqslant 1} a_{1j}\,X_0^{(j-1)}\,dX_0 .
\]

\[
  \xi_{s,i} = \sigma_s^{*}(X_0^{(i)})
  = \sum_{j \geqslant 0} a_{ij}\,U_s^{(j)} + \Bigl(\sum_{j \geqslant 0} b_{ij}\,U_s^{(j)}\Bigr) dU_s .
\]

\[
  A = S\{U_0, \ldots, U_m\}[dU_0, \ldots, dU_m]
\]

\[
  X_0 + \sum_{i,j \geq 1} f_{ij}\, X_0^{(i)} \sum_{0 \leq k \leq j} T^{(j-k)} (-1)^k X_0^{k}
\]

\[
  \sum_{\substack{i,j,k \\ 0 \leq k \leq j\,;\ i,j \geq 1}} f_{ij}\, T^{(j-k)} (-1)^k X_0^{(i+k)}
\]

\[
  \begin{cases}
    i + k = \nu \geq 1 \\
    0 \leq k \leq \nu - 1 \\
    j \geq \operatorname{Sup}(k,1)
  \end{cases}
  \qquad i = \nu - k \geq 1, \quad j = k + \delta, \quad \delta = j - k \geq 0 ,
\]

\[
  \text{\struck{$\displaystyle \sum_{\nu} X_0^{\nu} \sum_{0 \leq k \leq \nu-1} (-1)^k f_{\nu,k+\delta}$}}
\]

\[
  \sum_{\nu \geq 1} X_0^{\nu} \underbrace{\sum_{\substack{0 \leq k \leq \nu - 1 \\ \delta \geq 0 \\ k + \delta \geq 1}} (-1)^k f_{\nu-k,\,k+\delta}\, T^{\delta}}_{a_{1,\nu}}
\]

\[
  A_0 = X_0 \bigl( \underbrace{1 + \sum_{\delta \geq 1} f_{1,\delta}\, T^{(\delta)}}_{a_{1,1}} \bigr) + \sum
  \qquad (k = 0,\ \nu = 1)
\]

\[
  \mathcal{A}_n(S, S^+, t) = S\{X_0, \ldots, X_n\}[dX_0, \ldots, dX_n] \big/ \bigl( \textstyle\sum X_i - t,\ \sum dX_i \bigr)_{\mathrm{pd}} .
\]

\[
  \mathcal{A}_*(u, \lambda) : \mathcal{A}_*(S, S^+, t) \longrightarrow \mathcal{A}_*(S, S'^+, t'),
\]

\[
  X_0 \longmapsto \lambda X_0 \qquad
  \Bigl(\text{donc } \textstyle\sum X_0 - t \longmapsto \sum \lambda X_0 - u(t) = \sum \lambda X_0 - \lambda t' = \lambda \bigl(\sum X_0 - t'\bigr)\Bigr),
\]

\[
  dX_0 \longmapsto \lambda\, dX_0 .
\]

\[
  \sum F_{i_0 \ldots i_p}\, dX_{i_0} \wedge \ldots \wedge dX_{i_p}, \qquad
  F_{i_0 \ldots i_p} \in S\{X_0, \ldots, X_n\}
\]

\[
  \mathcal{A}_*(k) \overset{\text{déf}}{=} \mathcal{A}_*(S_k, S_k^+, T), \qquad S_k = k\{T\},
\]

\[
  (\lambda' * \lambda)\{T\} = \lambda'\{T\} \; \text{\ill{}} \; \lambda'\{\lambda\} ;
\]

\[
  G_k,\ H_k \subset \Omega_k
\]

\[
  G_k \cap H_k = \bigl\{ u_\lambda \bigm| \lambda \in S_k,\ \lambda T = T \text{ i.e. } (\lambda\{T\} - 1)\, T = 0 \bigr\}
\]

\[
  \boxed{\lambda_0 = 1} \qquad \boxed{(i+1)\, \lambda_i = 0 \quad \forall i \geq 1}
\]

\[
  G_k \cap H_k \simeq \Bigl\{ u_\lambda \Bigm| \lambda = 1 + \sum_{i \geq 1} \lambda_i T^{(i)},\ (i+1)\, \lambda_i = 0 \Bigr\}
\]

\[
  \varphi_0(T^{(i)}) = \tau_i \in S_k \qquad i \geq 1
\]

\[
  \boxed{\tau_i \tau_j = c_{ij}\, \tau_{i+j}}
\]

\[
  \varphi_1 : \underbrace{S\{X_0\}[dX_0]}_{\mathcal{A}_S} \longrightarrow \underbrace{S'\{X_0\}[dX_0]}_{\mathcal{A}_{S'}}
\]

\[
  \xi_i \in \mathcal{A}_{S'}^{+} = \operatorname{Ker}(\alpha_{S'} : \mathcal{A}_{S'} \to S'),
  \qquad \xi_i = A_i + B_i\, dX_0, \quad A_i \in S'\{X_0\}^+,\ B_i \in S'\{X_0\}
\]

\[
  A_i = A_1^{(i)}
\]

\[
  \varphi_1(X_0^{(i)}) = \xi_i, \qquad \varphi_1(dX_0) = d\xi_1 .
\]

\[
  \varphi_1(X_0^{(i)}) = \xi_i, \quad i \geq 1 \qquad\qquad
  \boxed{\xi_i \xi_j = c_{ij}\, \xi_{i+j}} \quad (i, j \geq 1)
\]

\[
  \varphi_1(dX_0) = d\xi_1 \qquad\qquad
  \boxed{d\xi_i = \xi_{i-1}\, d\xi_1} \qquad
  \boxed{\xi_i \in S'\{X_0\}^+}
\]

\[
  \xi_i = A_i + B_i\, dX_0 \qquad A_i, B_i \in S'\{X_0\}, \quad A_i \in S'\{X_0\}^+
\]

\[
  dA_i = A_{i-1} A'_1
\]

\[
  A_i = \xi_1^{(i)}
\]

\[
  \beta_{S'}(\xi_i) = \varphi_0(T^{(i)}) \text{ i.e. } \quad
  \beta_{S'}(A_i) = \varphi_0(T^{(i)})
\]

\[
  \begin{cases}
    \beta_{S'}(A_1) = \varphi_0(T) \\
    \varphi_0(T^{(i)}) = \varphi_0(T)^{(i)}
  \end{cases}
\]

\[
  A_1 \text{\struck{$(X_0)$}} = X_0 + f \text{\struck{$(X_0)$}}
\]

\[
  f = \sum_{i \geq 1,\, j \geq 1} X_0^{(i)} X_1^{(j)}
\]

\[
  \operatorname{Hom}_{S\text{-alg à p.d.}}\bigl( (A(M^*), J(M^*)),\ (B^*, K^*) \bigr)
  \longrightarrow \operatorname{Hom}_{S\text{-mod grad}}(M^*, K^*)
\]

\[
  A = A^0 + A^1 \supset J^0 + J^1
\]

\[
  J^0 A^1 \subset J^1 \subset \overline{\varphi}^{-1}\bigl( \operatorname{Hom}(A^1, J^0) \bigr)
  \qquad \bigl(\operatorname{Hom}(A^1, J^0) \subset \operatorname{Hom}(A^1, A^0)\bigr)
\]

\[
  (A) \;
  \begin{cases}
    (0)\ x^{(i)} \ (i \geq 0) \quad x^{(0)} = 1,\ x^{(1)} = x,\ x^{(i)} \in J^0 \ (i \geq 1) \\
    (1)\ (x + y)^{(i)} = \sum_{j+k=i} x^{(j)} y^{(k)} & x, y \in J^0,\ i \geq 0 \\
    (2)\ (\lambda x)^{(i)} = \lambda^i x^{(i)} & \lambda \in A^0,\ x \in J^0 \\
    (3)\ x^{(i)} x^{(j)} = c_{ij}\, x^{(i+j)} & x \in J^0,\ i, j \geq 0, \quad c_{ij} = \frac{(i+j)!}{i!\,j!} \\
    (4)\ (x^{(i)})^{(j)} = d_{ij}\, x^{(ij)} & x \in J^0,\ i, j \geq 1, \quad d_{ij} = \frac{(ij)!}{(i!)^j\, j!}
  \end{cases}
\]

\[
  (*) \qquad (x + u)^{(i)} = x^{(i)} + x^{(i-1)} u \qquad x \in J^0,\ u \in J^1,\ i \geq 0
\]

\[
  H^i_{\mathrm{DR}}\bigl( \operatorname{Sym}^*(M) \otimes \overset{*}{\Lambda}(M) \bigr) \simeq
  \begin{cases}
    0 & \text{si } i \neq 0 \\
    A = \operatorname{Sym}^0 \otimes \overset{0}{\Lambda} & \text{si } i = 0
  \end{cases}
\]

\[
  H_i^{\mathrm{Kos}}\bigl( \operatorname{Sym}^* M \otimes \overset{*}{\Lambda} M \bigr) =
  \begin{cases}
    0 & \text{si } i \neq 0 \\
    \simeq A = \operatorname{Sym}^0 \otimes \Lambda^0 & \text{si } i = 0
  \end{cases}
\]

\[
  d(x^{(i)}) = x^{(i-1)}\, dx ,
\]

\[
  H^i_{\mathrm{DR\,p.d.}}\bigl( \Gamma^*(M) \otimes \overset{*}{\Lambda} M \bigr) \simeq
  \begin{cases}
    0 & \text{si } i \neq 0 \\
    A = \Gamma^0 \otimes \Lambda^0 & \text{si } i = 0
  \end{cases}
\]

\[
  H_i^{\mathrm{Kos\,p.d}}\bigl( \Gamma^* M \otimes \overset{*}{\Lambda} M \bigr) \simeq
  \begin{cases}
    0 & \text{si } i \neq 0 \\
    A = \Gamma^0 \otimes \Lambda^0 & \text{si } i = 0
  \end{cases}
\]

\[
  \begin{array}{ccc}
    \operatorname{Sym}^* M \otimes \overset{*}{\Lambda} M & \longrightarrow & \Gamma^* M \otimes \overset{*}{\Lambda} M \\
    \| & & \| \\
    C^*_{\mathrm{DR}} \operatorname{Sym}^* M & \longrightarrow & C^*_{\mathrm{DR\,p.d.}}(\Gamma^* M)
  \end{array}
\]

\[
  \operatorname{Sym}^* M \otimes \overset{*}{\Lambda} M \longrightarrow \Gamma^* M \otimes \overset{*}{\Lambda} M
\]

\[ \theta = d\delta + \delta d \]

\[ (d\delta + \delta d)x = \theta x = nx \qquad \text{$x$ de degré total $n$,} \]

\[ \text{i.e. } x \in \textstyle\sum_{p+q=n} S^p \otimes \Lambda^q . \]

\[ \mathrm{Sym}^kM \xrightarrow{\ d\ } (\mathrm{Sym}^{k-1}M) \otimes M \]

\[ M \otimes M' \to A ; \]

\[ \mathrm{Sym}^*M \times \Gamma^*M' \to A, \qquad \dot\Lambda M \times \dot\Lambda M' \to A \]

\[ \text{\struck{$\mathrm{Sym}$}}\ S^*\Lambda^*(M) \times \Gamma^*\Lambda^*(M') \to A . \]

\[ ND_\bullet : \mathfrak{B}^*_* \to \mathfrak{B}_\bullet \]

\[ ND_i(L_*) \simeq \mathrm{Hom}_{k,*}(\underbrace{\dot\Lambda^i\mathfrak{L}_*}_{\mathfrak{L}^i_*}, L_*) \]

\[ ND_\bullet(L_*) \simeq \mathrm{Hom}_{k,*}(\underbrace{\dot\Lambda\mathfrak{L}_*}_{\mathfrak{L}^\bullet_*}, L_*) ; \]

\[ \mathrm{Hom}_{k,*}(\dot\Lambda^i\mathfrak{L}_*, L_*) \simeq \mathrm{Hom}_{k_\bullet}(\mathfrak{L}(i)_\bullet, L_\bullet) . \]

\[ \begin{cases} \mathrm{Hom}_{k_\bullet}(C_\bullet^{i-1}, L_\bullet) \simeq L_i & (i \geq 1) \\
\mathrm{Hom}_{k_\bullet}(k[0]_\bullet, L_\bullet) \simeq L_0 \end{cases} \]

\[ ND^\bullet(L^*) \simeq \dot\Lambda\mathfrak{L}_* \otimes_{k,*} L^* \]

\[ b \longmapsto \mathrm{Hom}_{\mathrm{Ab}_{k_*}}(M_*, \mathrm{Hom}_{\mathfrak{B}}(L^*, b)) \ ] \]

\[ F(x) \simeq \mathrm{Hom}^*_k(\alpha^*(x), C^*_F), \qquad C^*_F \overset{\mathrm{dfn}}{=} F(C^*) \]

\[ DP : \mathrm{Hom}^\bullet_k(K^\bullet, C^\bullet) \xrightarrow{\ \sim\ } \mathrm{Hom}^*_k(K^*, C^*), \]

\[ k_* \to \mathfrak{L}_* \]

\[ \Psi_* = \mathfrak{L}_*/k_* : I \mapsto k^I/k . \]

\[ \mathfrak{L}^!_n = \{ x \in \mathfrak{L}_n \mid \partial_1(x) = \dots = \partial_n(x) = 0 \} \]

\[ \partial = \partial_0 : \mathfrak{L}^!_1 \to \mathfrak{L}^!_0 \qquad (\mathfrak{L}^!_1 \simeq k,\ \mathfrak{L}^!_0 \simeq k) \]

\[ \dot\Lambda^i_k\mathfrak{L}_* \text{ est homotope à } \dot\Lambda^i_k(0) = \begin{cases} 0 & \text{si } i \neq 0 \\ k_* & \text{si } i = 0 \end{cases} \]

\[ \pi_i(\Psi_*) \simeq \pi_{i-1}(k_*) = \begin{cases} 0 & \text{si } i \neq 1 \\ k & \text{si } i = 1 \end{cases} \]

\[ \pi_j(\dot\Lambda^i(\mathfrak{L}_*/k_*)) = \begin{cases} 0 & \text{si } j \neq i \\ k & \text{si } j = i \end{cases} \]

\[ (L\dot\Lambda^i)(k[1]) \simeq \dot\Lambda^i k[1] = k[\text{-}i] \]

\[ (L\dot\Lambda)(L_\bullet) \xrightarrow{\ \sim\ } \dot\Lambda(L_\bullet) . \]

\[ 0 \to k_* \to \mathfrak{L}_* \to \Psi_* \to 0 \qquad (\Psi_* \overset{\mathrm{dfn}}{=} \mathfrak{L}_*/k_*) \]

\[ 0 \to \underbrace{\dot\Lambda^{i-1}\Psi_* \otimes k_*}_{\dot\Lambda^{i-1}\Psi_*} \to \underbrace{\dot\Lambda^i\mathfrak{L}_*}_{\text{flasque}} \to \dot\Lambda^i\Psi_* \to 0 \]

\[ 0 \to (\dot\Lambda^{i-1}\Psi_*)^! \to \underbrace{(\dot\Lambda^i\mathfrak{L}_*)^!}_{\text{flasque}} \to \dot\Lambda^i\Psi_* \to 0 \]

\[ \pi_\ell(A_* \otimes \dot\Lambda^i\Psi_*) \simeq \pi_{\ell-i}(A_*) \]

\[ \underbrace{H_\ell(A^!_* \otimes B^!_*)}_{\pi_\ell(A_* \otimes B_*)} \simeq \underbrace{H_{\ell-i}(A^!_*)}_{\pi_{\ell-i}(A_* \otimes B_*)} \]

\[ A^{**}_* \simeq \Gamma^*(\mathfrak{L}_*) \otimes_k \dot\Lambda\Psi_* \qquad \text{(degré total, degré ext)} \]

\[ \pi_\ell(A^{*i}_*) \simeq \pi_{\ell-i}(\underbrace{A^{*0}_*}_{\Gamma^*(\mathfrak{L}_*)}) \rightleftarrows \pi_{\ell-i}(\underbrace{A^{00}_*}_{\Gamma^*(0) = k_*}) = \begin{cases} 0 & \text{si } \ell \neq i \\ k & \text{si } \ell = i \end{cases} \]

\[ \pi_\ell(\underbrace{A^{p,i}_*}_{\Gamma^{p-i}_k\mathfrak{L}_* \otimes \dot\Lambda^i\Psi_*}) \simeq \pi_{\ell-i}(\underbrace{A^{p-i,0}_*}_{\Gamma^{p-i}_*\mathfrak{L}_*}) \simeq \begin{cases} 0 & \text{si } \ell \neq i \text{ ou } p \neq i \\ k & \text{si } i = \ell = p \end{cases} \]

\[ \mathfrak{L}^{pi}_* = \begin{cases} \widetilde{\mathfrak{L}}^{p,i}_* & \text{si } i < p \\ Z^p(\widetilde{\mathfrak{L}}^{p,*}_*) & \text{si } i = p \\ 0 & \text{si } i > p \end{cases} \]

\[ \underbrace{\mathrm{Hom}(\mathcal{X}, \mathfrak{L}^{p*}_*)}_{C^{p*}_{\mathrm{DR\,pd}}(\mathcal{X}, k)} \simeq \mathrm{Hom}(X, \tau^{\leq p}\widetilde{\mathfrak{L}}^{p*}_*) \simeq \tau^{\leq p}\Gamma(X, \widetilde{\mathfrak{L}}^{p*}_*) \]

\[ \simeq \tau^{\leq p}\mathbb{R}\Gamma(\mathcal{X}, k) \]

\[ H^{p*}_{\mathrm{DR\,pd}}(\mathcal{X}, k) \to H^*(\mathcal{X}, k) \]

\[ H^{p,i}_{\mathrm{DR\,pd}}(\mathcal{X}, k) \begin{cases} \simeq H^i(\mathcal{X}, k) & \text{si } i \leq p \\ = 0 & \text{si } i > p . \end{cases} \]

\[ S^{[i]} = \text{idéal engendré par les } t^{[j]} \ (j \geq i) \]

\[ \widehat{A}^*_*(S, \gamma^*, t) \overset{\mathrm{dfn}}{=} \widehat{A}^{**}_* \]

\[ \widehat{A}^*_n \simeq S\{\{X_0, \dots, X_n\}\}/(\text{idéal à p.d.\ engendré par } \textstyle\sum X_i - t) \otimes_S \dot\Lambda[dX_0, \dots, dX_n]/\text{idéal}(\textstyle\sum dX_i) \]

\[ (\simeq S\{\{Y_0, \dots, Y_n\}\}[dY_0, \dots, dY_n]) \]

\[ \widehat{A}^*_* \simeq \widehat{A}^0_* \otimes_S \dot\Lambda\Psi^S_* \]

\[ \pi_\ell(\widehat{A}^i_*) \simeq \pi_\ell(\widehat{A}^0_* \otimes \dot\Lambda^i\Psi^S_*) \simeq \pi_{\ell-i}(\widehat{A}^0_*) \]

\[ \mathrm{Fil}^0(\widehat{A}^0_n) \simeq \widehat{A}^0_n \]

\begin{align*}
\mathrm{Fil}^p(\widehat{A}^0_n) &\simeq \Bigl\{ \textstyle\sum a_{i_0 \dots i_n} X_0^{[i_0]} \cdots X_n^{[i_n]} \Bigm| a_{i_0 \dots i_n} \in S^{[p - \sum_0^n i_\alpha]} \Bigr\} \\
&= \Bigl\{ \textstyle\sum a_{i_0, \dots, i_{n-1}} Y_0^{[i_0]} \cdots Y_{n-1}^{[i_{n-1}]} \Bigm| a_{i_0 \dots i_{n-1}} \in S^{[p - \sum_0 i_\alpha]} \Bigr\}
\end{align*}

\[ k = S/S^{[1]} \]

\[ \mathrm{Gr}^p(\widehat{A}^0_*) \simeq \Gamma^p_k(\mathfrak{L}^k_*) \]

\[ \pi_i(\mathrm{Gr}^p(\widehat{A}^0_*)) = \begin{cases} 0 & \text{si } p \neq 0,\ i \neq 0 \\ k & \text{si } i, p = 0 \end{cases} \]

\[ \& \begin{cases} \mathrm{Fil}^p(\widehat{A}^0_*) \text{ flasque pour tt } p \geq 1 \\
\pi_i(\widehat{A}^0_*) = \begin{cases} 0 & \text{si } j \neq 0 \\ k & \text{si } i = 0 \end{cases} \quad \text{si } p = 0 \end{cases} \]

\[ \mathrm{Fil}^p\widehat{A}^*_* = \sum_{0 \leq i \leq p} \mathrm{Fil}^{p-i}(\widehat{A}^0_*) \otimes_S \dot\Lambda^i\Psi_* \]

\begin{align*}
\pi_\ell(\mathrm{Fil}^p(\widehat{A}^i_*)) &= \pi_\ell(\mathrm{Fil}^{p-i}(\widehat{A}^0_*) \otimes_S \dot\Lambda^i\Psi_*) \\
&= \pi_{\ell-i}(\mathrm{Fil}^{p-i}(\widehat{A}^0_*)) \\
&= \begin{cases} 0 & \text{si } p \leq i \text{ et } \ell \neq i \\ k & \text{si } p \leq i = \ell \end{cases}
\end{align*}

\[ k\{T\} \to \mathrm{Gr}^*(S) \]

\[ t = 0, \quad \text{donc } S \xrightarrow{\ \sim\ } k \]

\[ \mathrm{Gr}^p\widehat{A}^0_* \simeq \Gamma^p_k(\Psi_*), \]

\[ \mathrm{Gr}^p(\widehat{A}^0_*) \simeq K_*(k, p) \]

\[ v_p(n!) = \frac{1}{p-1}(n - \mathrm{chiff}_p\,n) \]

\[ v_p(p^{(n)}) = v_p\Bigl(\frac{p^n}{n!}\Bigr) = \frac{1}{p-1}\bigl((p-2)n + \underline{\mathrm{chiff}_p\,n}\bigr) \]

\[ \bigl(= \mathrm{chiff}_p(n) \text{ si } p = 2\bigr) \]

\[ \gamma_{p,i} = \operatorname*{Inf}_{n \geq i} v_p(p^{(n)}) = \operatorname*{Inf}_{n \geq i} \frac{1}{p-1}\bigl(\underbrace{(p-2)n + \mathrm{chiff}_p(n)}_{\text{cette fonction n'est ni croissante ni décroissante…}}\bigr) \]

\[ = \begin{cases} 1 & \text{si } p = 2 \\ > \frac{p-2}{p-1}\,i \xrightarrow[i \to +\infty]{} +\infty & \text{si } p \neq 2 \end{cases} \]

\[ i = i_0 + i_1p + \dots + i_rp^r \]

\[ = \underbrace{\underbrace{(p-1)}_{i_0} + \dots + \underbrace{(p-1)p^{s-1}}_{i_{s-1}}}_{p^s - 1} + \underset{< p-1}{i_s}\,p^s + \dots + i_rp^r \]

\[ i + 1 = (i_s+1)p^s + \dots + i_rp^r \]

\[ (p-2)i + (p-1)s + \sum_{\alpha \geq s} i_\alpha \leq (p-2)(i+1) + \sum_{\alpha \geq s} i'_\alpha \]

\begin{align*}
[v_p(p^{(i)}) \leq v_p(p^{(i+1)})] &\Leftrightarrow [(p-1)s \leq (p-2)] \Leftrightarrow [s = 0] \Leftrightarrow [i_0 \neq p-1] \\
&\Leftrightarrow i + 1 \not\equiv 0 \ (p)
\end{align*}

\[
  \mathcal{M}_{N'} \overset{i_{N,N'}}{\hookrightarrow} \mathcal{M}_N \qquad N' \geq N
\]

\[
  \tau_{N';N} : \mathcal{M}_N \longrightarrow \mathcal{M}_{N'}
\]

\[
  \tau_{N';N}(M)_i =
  \begin{cases}
    0 & \text{si } i < N' \\
    M_i & \text{si } i \geq N'
  \end{cases}
\]

\[
  \begin{array}{ccc}
    \mathrm{Hom}_{\mathcal{M}_N}(i_{N,N'}(P), M) & \xleftarrow{\ \sim\ } & \mathrm{Hom}_{\mathcal{M}_{N'}}(P, \tau_{N';N}(M)) \\
    \| & & \| \\
    \mathrm{Hom}_{\mathcal{M}}(P, M) & \xleftarrow{\ \sim\ } & \mathrm{Hom}(P, \tau_{N';N}(M))
  \end{array}
\]

\[
  \mathrm{Hom}_{\mathcal{M}_N}(M, i_{N,N'}(P)) = \mathrm{Hom}_{\mathcal{M}}(M, P)
\]

\[
  = \mathrm{Hom}_{\mathcal{M}}\bigl(M/\text{ss-module engendré par les } M_i,\ i<N',\ P\bigr)
\]

\[
  \begin{array}{ccccccccc}
    0 & 0 & M_N & M_{N+1} & M_{N+2} & \cdots & M_{N'-1} & M_{N'} & M_{N'+1} \ \cdots \\
    0 & 0 & 0 & 0 & 0 & \cdots & 0 & P_{N'} & P_{N'+1} \ \cdots
  \end{array}
\]

\[
  q_{N';N}(M) = M/\text{ss-module de } M \text{ engendré par les } M_i,\ i < N'
\]

\[
  A^* \qquad \bullet \qquad d \qquad J \qquad \gamma \ \text{puiss.}
\]

\[
  \Bigl(\sum_{i \text{ pair}} x_i + \sum_{j \text{ impair}} x_j\Bigr)^{\alpha}
\]

\[
  (x + y)^{(\alpha)} = x^{(\alpha)} + x^{(\alpha-1)} y + \text{\struck{$x^{(\alpha-2)} y^{(2)}$}}
  \qquad (x \in J^0,\ y \in A^1)
\]

\[
  z^{(\alpha)} = (x + y)^{(\alpha)} = x^{(\alpha)} + x^{(\alpha-1)} y
\]

\[
  z'^{(\alpha)} = (x' + y')^{(\alpha)} = x'^{(\alpha)} + x'^{(\alpha-1)} y'
\]

\begin{align*}
  (z + z')^{(\alpha)} &= \bigl((x + x') + (y + y')\bigr)^{(\alpha)}
    = (x + x')^{(\alpha)} + (x + x')^{(\alpha-1)} (y + y') \\
  &= \sum_{i+j=\alpha} x^{(i)} x'^{(j)} + \Bigl(\sum_{i+j=\alpha-1} x^{(i)} x'^{(j)}\Bigr)(y + y')
\end{align*}

\begin{align*}
  \sum_{i+j=\alpha} z^{(i)} z'^{(j)}
  &= \sum_{i+j=\alpha} (x^{(i)} + x^{(i-1)} y)(x'^{(j)} + x'^{(j-1)} y') \\
  &= \sum_{i+j=\alpha} x^{(i)} x'^{(j)} + \Bigl(\sum_{i+j=\alpha-1} x^{(i)} x'^{(j)}\Bigr)(y + y') \\
  &\quad + \Bigl(\sum_{i+j=\alpha-2} x^{(i)} x'^{(j)}\Bigr) y y'
\end{align*}

\[
  \boxed{\text{OK si } yy' = 0}
\]

\[
  \mathrm{Hom}_{\mathcal{M}_{N'}}(\tau_{N';N}(M), P) \overset{?}{\xrightarrow{\ \sim\ }} \mathrm{Hom}_{\mathcal{M}_N}(M, \rho_{N,N'} P)
\]

\[
  \begin{array}{c}
    \| \\
    \mathrm{Hom}_{\mathcal{M}}(\tau_{N';N}(M), P) \\
    \cap \\
    \prod_{i \geq N'} \mathrm{Hom}_k(M_i, P_i)
  \end{array}
\]

\[
  \begin{array}{cccccccccc}
    0 & 0 & M_N & M_{N+1} & \cdots & M_{N'-1} & M_{N'} & M_{N'+1} & \cdots \\
      &   &     &         &        &          & \downarrow & \downarrow & \\
    0 & 0 & 0 & 0 &  & 0 & P_{N'} & P_{N'+1} & \cdots
  \end{array}
\]

\[
  \rho_{N,N'}(P)_i =
  \begin{cases}
    0 & \text{si } i < N \\
    \mathrm{Hom}^i_{\mathcal{M}}(\tau_{N'-i}\, S, P) \simeq \mathrm{Hom}_{\mathcal{M}_N}(T^{-i} \tau_{N'-i}\, S, P) & \text{si } i \geq N
  \end{cases}
\]

\[
  (\lambda\cdot z)^{(\alpha)} \overset{?}{=} \lambda^{\alpha} z^{(\alpha)}
  \qquad \lambda \in \hat{A},\ z \in J
\]

\[
  \textstyle\Bigl(\sum_{i \in I} x_i\Bigr)\Bigl(\sum_{j \in J} y_j\Bigr) = \sum_{i,j} x_i y_j
\]

\[
  \lambda = \xi + \eta \quad (\xi \in A^0,\ \eta \in A') \qquad z = x + y
\]

\[
  \lambda z = (\xi x + \eta y) + (\xi y + x\eta) \qquad (\xi x + \eta y \in A^0)
\]

\begin{align*}
  (\lambda z)^{(\alpha)} &= (\xi x + \eta y)^{(\alpha)} + (\xi x + \eta y)^{(\alpha-1)}(\xi y + x\eta) \\
  &= \sum_{i+j=\alpha} (\xi x)^{(i)} (\eta y)^{(j)}
     + \Bigl(\sum_{i+j=\alpha-1} (\xi x)^{(i)} (\eta y)^{(j)}\Bigr)(\xi y + x\eta)
\end{align*}

\begin{align*}
  \lambda^{\alpha} z^{(\alpha)} &= (\xi^{\alpha} + \alpha\,\xi^{\alpha-1}\eta)(x^{(\alpha)} + x^{(\alpha-1)} y) \\
  &= (\xi^{\alpha} x^{(\alpha)} + \alpha\,\xi^{\alpha-1} x^{(\alpha-1)}\,\eta y)
     + (\xi^{\alpha} x^{(\alpha-1)} y + \alpha\,\xi^{\alpha-1} x^{(\alpha)} \eta) \\
  &= \bigl((\xi x)^{(\alpha)} + \alpha\,(\xi x)^{(\alpha-1)}(\eta y)\bigr)
     + \bigl(\xi (\xi x)^{(\alpha-1)} y + x\,(x\xi)^{(\alpha-1)} \eta\bigr)
\end{align*}

\[
  (z^{(\alpha)})^{(\beta)} \overset{?}{=} z^{(\alpha\beta)}\, \frac{(\alpha\beta)!}{\beta!\,(\alpha!)^{\beta}}
  \qquad \Bigl(\frac{(\alpha\beta)!}{\beta!\,(\alpha!)^{\beta}} = c_{\alpha,\beta}\Bigr)
\]

\[
  \frac{1}{\beta!}\Bigl(\frac{z^{\alpha}}{\alpha!}\Bigr)^{\beta} = \frac{z^{\alpha\beta}}{\beta!\,(\alpha!)^{\beta}}
\]

\[
  = x^{(\alpha\beta)} c_{\alpha\beta} + x^{(\alpha\beta-1)} y\, c_{\alpha\beta}
\]

\begin{align*}
  (z^{(\alpha)})^{(\beta)} &= (x^{(\alpha)} + x^{(\alpha-1)} y)^{(\beta)} \\
  &= (x^{(\alpha)})^{(\beta)} + (x^{(\alpha)})^{(\beta-1)} x^{(\alpha-1)} y \\
  &= \frac{(\alpha\beta)!}{\beta!\,(\alpha!)^{\beta}}\, x^{(\alpha\beta)}
     + \frac{(\alpha(\beta-1))!}{(\beta-1)!\,(\alpha!)^{\beta-1}}\, x^{(\alpha(\beta-1))} x^{(\alpha-1)} y
\end{align*}

\[
  x^{(\alpha(\beta-1))} x^{(\alpha-1)} = x^{(\alpha\beta-1)}\, \frac{(\alpha\beta-1)!}{(\alpha(\beta-1))!\,(\alpha-1)!}
\]

\[
  = \ldots + \frac{(\alpha\beta-1)!}{(\alpha-1)!\,(\beta-1)!\,(\alpha!)^{\beta-1}}\, x^{(\alpha\beta-1)} y
\]

\[
  \frac{(\alpha\beta)!}{\alpha!\,\beta!\,(\alpha!)^{\beta-1}} = \frac{(\alpha\beta)!}{\beta!\,(\alpha!)^{\beta}}
  \qquad \text{OK}
\]

\[
  x^{(p)} x^{(q)} = \frac{(p+q)!}{p!\,q!}\, x^{(p+q)}
\]

\begin{align*}
  z^{(p)} z^{(q)} &\overset{?}{=} \frac{(p+q)!}{p!\,q!}\, z^{(p+q)} \\
  z^{(p)} z^{(q)} &= (x^{(p)} + x^{(p-1)} y)(x^{(q)} + x^{(q-1)} y) \\
  &= x^{(p)} x^{(q)} + (x^{(p)} x^{(q-1)} + x^{(p-1)} x^{(q)})\, y \\
  &= \frac{(p+q)!}{p!\,q!}\, x^{(p+q)}
     + \Bigl(\frac{(p+q-1)!}{p!\,(q-1)!} + \frac{(p+q-1)!}{(p-1)!\,q!}\Bigr) x^{(p+q-1)} y
\end{align*}

\[
  (p+q-1)!\,\frac{p+q}{p!\,q!} = \frac{(p+q)!}{p!\,q!} \qquad \text{OK}
\]

\[
  \mathcal{M}_{N'} \overset{i_{N'}}{\hookrightarrow} \mathcal{M}
\]

\[
  q_{N'} \quad i_{N'} \quad \tau_{N'} \quad \rho_{N'}
\]

\[
  \left\{
  \begin{array}{l}
    \tau_{N'}(M) \hookrightarrow M \\
    \tau_{N'}(M)_i = \begin{cases} 0 & \text{si } i < N' \\ M_i & \text{si } i \geq N' \end{cases}
  \end{array}
  \right.
\]

\[
  q_{N'}(M) = M/\text{ss-module engendré par les } M_i,\ i < N'
\]

\[
  \rho_{N'}(M)_i = \mathrm{Hom}^i(\tau_{N'-i}S, M) = \mathrm{Hom}_{\mathcal{M}_{N'}}(T^{-i}\tau_{N'-i}S, M)
\]

\[
  \mathcal{M}_{N'} \overset{i_{N,N'}}{\hookrightarrow} \mathcal{M}_N \hookrightarrow \mathcal{M}
\]

\[
  q_{N';N},\ i_{N,N'},\ \tau_{N';N},\ \rho_{N,N'}
\]

\[
  \left\{
  \begin{array}{l}
    q_{N';N} = q_{N'} | \mathcal{M}_N \\
    \tau_{N';N} = \tau_{N'} | \mathcal{M}_N
  \end{array}
  \right.
\]

\begin{align*}
  \mathrm{Hom}_{\mathcal{M}_{N'}}(\tau_{N';N} M, P) &\simeq \mathrm{Hom}_{\mathcal{M}_N}(M, \rho_{N,N'} P) \\
  \| \qquad & \\
  \mathrm{Hom}_{\mathcal{M}_{N'}}(\tau_{N'} M, P) & \\
  \wr \qquad & \\
  \mathrm{Hom}_{\mathcal{M}}(M, \rho_{N'} P) &\simeq \mathrm{Hom}_{\mathcal{M}}(i_N M, \rho_{N'} P) \\
  &\simeq \mathrm{Hom}_{\mathcal{M}_N}(M, \tau_N \rho_{N'} P)
\end{align*}

\[
  \hat{\Gamma}^*\Bigl(\frac{\mathbb{Z} \times \mathbb{Z}^I}{\mathbb{Z}}\Bigr) \otimes_{\mathbb{Z}} \overset{*}{\Lambda}\, \mathbb{Z}^I/\mathbb{Z}
\]

\[
  0 \longrightarrow \mathbb{Z} \xrightarrow{\ (-\mathrm{id},\, \mathrm{diag})\ } \mathbb{Z} \times \mathbb{Z}^I
\]

\[
  \mathbb{Z}\{(X_i)_{i \in I}\}[(dX_i)_{i \in I}]\big/\Bigl(\textstyle\sum dX_i = 0\Bigr)
\]

\[
  \Gamma^*(\mathbb{Z}^I) \otimes_{\mathbb{Z}} \overset{*}{\Lambda}(\mathbb{Z}^I/\mathbb{Z})
\]

\[
  \mathrm{Sym}^*(\mathbb{Z}^{(I)}) \otimes_{\mathbb{Z}} \overset{*}{\Lambda}\, \mathbb{Z}^{(I)},
\]

\[
  \begin{array}{ccc}
    A & \xrightarrow{\ \varphi\ } & A' \\
    \pi & & \pi'
  \end{array}
  \qquad
  (\varphi, \lambda) \quad
  \begin{array}{l}
    \varphi(\pi) = \lambda \pi' \\
    \varphi(\pi^{(\alpha)}) = \lambda^{\alpha} \pi'^{(\alpha)}
  \end{array}
\]

\[
  \begin{array}{rl}
    X_i,\ dX_i & A\{(X_i, dX_i)\}\big/\bigl\{\textstyle\sum X_i - \pi,\ \sum dX_i\bigr\} \\
    \downarrow \quad \downarrow & \\
    \lambda X_i,\ \lambda dX_i & A'\{(X_i, dX_i)\}\big/\bigl(\textstyle\sum X_i - \pi',\ \sum dX_i\bigr)
  \end{array}
\]

\[
  \Bigl(\sum \lambda_{ijk}\, dX_i\, dX_j\, dX_k\Bigr)^2
  = \sum_{\substack{i,j,k \\ i',j',k'}} \lambda_{ijk} \lambda_{i'j'k'}\, dX_i\, dX_j\, dX_k\, dX_{i'}\, dX_{j'}\, dX_{k'}
\]

\[
  \Bigl(\sum \lambda_{\alpha} \xi_{\alpha}\Bigr)\Bigl(\sum \lambda_{\beta} \xi_{\beta}\Bigr) = \sum_{\alpha,\beta} \lambda_{\alpha} \lambda_{\beta}\, \xi_{\alpha} \xi_{\beta}
\]

\[
  \frac{x^n}{n!}
\]

\[
  (A^*, \bullet, d, J, \gamma)
\]

\[
  \mathcal{N} = \mathrm{Mod}(k) \overset{i}{\longrightarrow} \mathcal{M}_0 = \mathcal{M}^+
\]

\[
  k \circ \rho_{0N} = k \circ \tau_0 \circ \rho_N = k \circ \rho_N
\]

\[
      \Bigl[\, = (\rho_N \tau_N(P \otimes_k S))_0 = (\rho_N(P \otimes_k \tau_N S))_0 \,\Bigr]
    \]

\[
      P \otimes S_{\alpha} \xrightarrow{\ \sim\ } \mathrm{Hom}^{\alpha}(\tau_{N-\alpha} S, P \otimes_k S)
    \]

\[
      \Bigl[\ \mathrm{Hom}^{\alpha}(S, P \otimes_k S) \ \Bigr]
    \]

\[
      \mathrm{Hom}^{\alpha}(S/\tau_{N-\alpha} S, P \otimes_k S) = 0
    \]

\[
  A\{(X_i)_{i \in I}, (Y_i)_{i \in I}\} \big/ \Bigl(\textstyle\sum X_i - \pi,\ \sum Y_i - \omega,\ \sum dX_i,\ \sum dY_i\Bigr)
\]

\[
  A \longrightarrow A^{(I)}
\]

\[
  \Gamma^*_{(A,\pi)}(A^I \otimes_A J)
\]

\[
  X_1 \otimes \pi,\ X_2 \otimes \pi,\ \ldots,\ X_n \otimes \pi
\]

\[
  \textstyle\sum X_i \otimes \pi - \pi
\]

\[
  J^I/J
\]

\[
  \Gamma^*_{(A,\pi)}(J^I)/\delta J
\]

\[
  \delta\pi = -\pi + \pi \qquad dX_i
\]

\[
  \Gamma^*_A \qquad A\{(X_i)_{i \in I}\} \big/ \Bigl(\textstyle\sum \pi X_i - \pi,\ \pi \sum dX_i\Bigr)
\]

\[
  \Gamma^{*+}_A J \longrightarrow J
\]

\[
  H^*(X, \Gamma^{*+} J) \longrightarrow H^*(X, J)
\]

\[
  H^{2i}(X, M) \xrightarrow{\ \gamma^{\alpha}\ } H^{2i\alpha}(X, \Gamma^{\alpha} M) \longrightarrow H^{2i\alpha}(X, \mathrm{Sym}^{\alpha} M)
\]

\[
  H^{2i}(X, J) \longrightarrow H^{2i\alpha}(X, \Gamma^{\alpha} J) \longrightarrow H^{2i\alpha}(X, J)
\]

\[
  A \xrightarrow{\ 2i\ } M, \qquad A \xrightarrow{\ 2i\alpha\ } \Gamma^{\alpha} M
\]

\[
  \Gamma^i M \rightleftarrows \mathrm{Sym}^i(M) \qquad x^{(i)} \mapsto x^i, \quad i!\,x^{(i)} = x^i
\]

\[
  \mathcal{M}/\mathrm{Nég}_{\infty} \simeq \varinjlim \mathcal{M}_N
\]

\[
  \overset{\text{déf}}{\Longleftrightarrow} \tau_N M = 0 \Longleftrightarrow M_i = 0 \text{ si } i \geq N
\]

\[
  \Longleftrightarrow \exists N \in \mathbb{Z} \text{ tel que } M\ N\text{-négligeable}
  \Longleftrightarrow M_i = 0 \text{ si } i \text{ assez grand}
\]

\[
  \mathrm{Nég}_N \subset \mathrm{Nég}_{N'} \subset \mathrm{Nég}_{\infty} \subset \mathrm{Nég} \qquad N \leq N'
\]

\[
  \begin{array}{ccc}
    \mathcal{M}/\mathrm{Nég}_N & \simeq & \mathcal{M}_N \\
    \downarrow & & \downarrow{\scriptstyle \tau_{N';N}} \\
    \mathcal{M}/\mathrm{Nég}_{N'} & \longrightarrow & \mathcal{M}_{N'}
  \end{array}
\]

\[
  \varinjlim \varphi_N \tau_N P \longrightarrow \varphi_\omega \tau_\omega P
\]

\[
  \Updownarrow
\]

\[
  \mathrm{Hom}_{\mathcal{M}}(M, \varinjlim \varphi_N \tau_N P) \longrightarrow \mathrm{Hom}_{\mathcal{M}}(M, \varphi_\omega \tau_\omega P) \quad \text{bij.}
\]

\begin{align*}
  \mathrm{Hom}_{\mathcal{M}}(M, \varinjlim \varphi_N \tau_N P)
  &\simeq \varinjlim \mathrm{Hom}_{\mathcal{M}}(M, \varphi_N \tau_N P) \\
  &\simeq \varinjlim \mathrm{Hom}_{\mathcal{M}_N}(\tau_N M, \tau_N P) \\
  &\simeq \varinjlim \mathrm{Hom}_{\mathcal{M}}(\tau_N M, P)
\end{align*}

\begin{align*}
  \mathrm{Hom}_{\mathcal{M}}(M, \varphi_\omega \tau_\omega P)
  &\simeq \mathrm{Hom}_{\mathcal{M}_\omega}(\tau_\omega M, \tau_\omega P) \\
  &\simeq \varinjlim_{\substack{M' \subset M \\ M/M' \text{ négl.}}} \mathrm{Hom}(M', P)
\end{align*}

\[
  \varinjlim \varphi_N \tau_N P \xrightarrow{\ \sim\ } \varphi_\omega \tau_\omega P
\]

\begin{align*}
  (\varphi_\omega \tau_\omega P)_i
  &\simeq \mathrm{Hom}^i(S, \varphi_\omega \tau_\omega P) \\
  &\simeq \mathrm{Hom}(S[-i], \varphi_\omega \tau_\omega P) \\
  &\simeq \mathrm{Hom}(\tau_\omega S[-i], \tau_\omega P) \\
  &\simeq \varinjlim_{\substack{M' \subset S[-i] \\ \text{t.q. } S[-i]/M' \text{ négl.}}} \mathrm{Hom}(M', P) \\
  &\simeq \varinjlim_{\substack{M' \subset S \\ \text{t.q. } S/M' \text{ négl.}}} \mathrm{Hom}^i(M', P) \\
  &\simeq \varinjlim_N \mathrm{Hom}^i(\tau_N S, P) \\
  &\simeq \varinjlim_N \mathrm{Hom}(\tau_N(S[-i]), \tau_N P)
\end{align*}

\[
  \textstyle\sum T^{(i)} \overset{T^{(\ell)}}{\longmapsto} \sum c_{i,\ell}\, T^{(i+\ell)}
\]

\[
  \Updownarrow
\]

\[
  P \xrightarrow{\ \sim\ } \varphi_\omega \tau_\omega P
\]

\[
  \Updownarrow
\]

\[
  \mathrm{Hom}^i(S, P) \xrightarrow{\ \sim\ } \varinjlim_N \mathrm{Hom}^i(\tau_N S, P)
\]

\[
  \mathrm{Hom}_{\mathcal{M}}(M, \varphi_N \text{\struck{\ill{}}} P) \simeq \mathrm{Hom}_{\mathcal{M}_N}(\tau_N M, \text{\struck{\ill{}}} P)
  \qquad (P \in \mathcal{M}_N)
\]

\begin{align*}
  (\varphi_N \tau_N P)_i
  &= \mathrm{Hom}^i_{\mathcal{M}}(S, \varphi_N \tau_N P) = \mathrm{Hom}_{\mathcal{M}}(S[-i], \varphi_N P) \\
  &= \mathrm{Hom}_{\mathcal{M}_N}(\tau_N(S[-i]), P) \\
  &\simeq \mathrm{Hom}_{\mathcal{M}_N}((\tau_{N-i}(S))(-i), P) \\
  &\simeq \mathrm{Hom}^i_{\mathcal{M}}(\tau_{N-i}(S), P)
\end{align*}

\begin{align*}
  \mathrm{Hom}_{\mathcal{M}}(M, \varphi_N \tau_N P) &= \mathrm{Hom}_{\mathcal{M}_N}(\tau_N M, \tau_N P) \\
  &\simeq \mathrm{Hom}_{\mathcal{M}}(i_N \tau_N M, P)
\end{align*}

\[
  h_M \varphi_N \tau_N = h_{i_N \tau_N}
\]

\[
  R^*(h_M \varphi_N \tau_N) \simeq R^* h_{i_N \tau_N} = \mathbb{R}\mathrm{Hom}(i_N \tau_N M, -)
\]

\[
  \| \qquad\qquad
\]

\[
  \mathbb{R}^*(h_M \varphi_N)\, \tau_N
  \qquad (R^* \tau_N = \tau_N \text{ car } \tau_N \text{ exact})
\]

\[
  h_M (R^* \varphi_N) \circ \tau_N \simeq \mathbb{R}\mathrm{Hom}(\tau_N M, -)
\]

\[
  \boxed{\mathrm{Hom}(M, R^* \varphi_N(\tau_N P)) \simeq \mathbb{R}\mathrm{Hom}(\tau_N M, P)}
\]

\[
  \mathrm{Hom}(M, (R^n \varphi_N)(\tau_N P)) \simeq \mathrm{Ext}^n(\tau_N M, P)
  \quad \text{si } M \text{ projectif} \in \mathcal{M}
\]

\[
  L_i\,\mathrm{Sym}^i(M,n) \simeq L_{i+n}\,\Lambda^i(M,
  \text{\struck{$n+1$}})
\]

\[
  L_i\,\Lambda^i(M,n) \simeq L_{i+n}\,\Gamma^i(M,n+1)
\]

\[
  M \to M \qquad
  K(M,n) \longrightarrow E(M,n+1) \longrightarrow K(M,n+1)
\]

\[
  \mathrm{Sym}\,K(M,n) \otimes \Lambda E(M,n+1)
\]

\[
  \Lambda^{n+1}\Psi \otimes_{\mathbb{Z}} M
\]

\[
  0 \to \mathrm{Sym}^i K(M,n) \to \mathrm{Sym}^{i-1}K(M,n)\otimes\Lambda^1E(M,n+1)
  \to \mathrm{Sym}^{i-2}K(M,n)\otimes\Lambda^2E(M,n+1)
\]

\[
  \to \cdots \to \Lambda^iE(M,n+1) \to \Lambda^iK(M,n+1) \to 0
\]

\[
  0 \to \Lambda^iK(M,n) \to \Lambda^{i-1}K(M,n)\otimes\mathrm{Sym}^1E(M,n+1)
  \to \cdots
\]

\[
  \cdots \to \mathrm{Sym}^iE(M,n+1) \to \mathrm{Sym}^iK(M,n+1) \to 0
\]

\[
  0 \to \Gamma^iK(M,n) \to \Gamma^{i-1}K(M,n)\otimes\Lambda^1E(\ \cdots)
\]

\[
  L_r\,\mathrm{Sym}^i K(M,n+1) \simeq L_{r-i}\,\Lambda^iK(M,n)
\]

\begin{align*}
  L_{\ast}\,\mathrm{Sym}^iX[1] &\simeq L_{\ast}\,\Lambda^iX[-i] \\
  L_{\ast}\,\Lambda^i(X[1]) &\simeq L_{\ast}\,\Gamma^iX[-i] \\
  L_{\ast}\,\mathrm{Sym}^iX[2] &\simeq L_{\ast}\,\Gamma^iX[-2i]
\end{align*}

\[
  L_r\,\mathrm{Sym}^{\ast}_{\uncertain{e}} \simeq L_r\,\mathbb{Z}[\quad]
\]

\[
  L_i\,\mathbb{Z}M_{\ast} = H_i(M_{\ast})
\]

\[
  L_i(\mathrm{Sym})M_{\ast}
\]

\[
  X \mapsto \pi_i\,\mathrm{SYM}^{\infty}X \simeq H_iX
  \qquad \mathbb{Z}X \qquad \pi_i\,\mathbb{Z}X
\]

\[
  G = \mathbb{Z}^{k} \times \mathbb{Z}
\]

\[
  K(G,n) \sim \mathrm{Sym}^{\infty}M(G,n) \qquad G \to M(G)
\]

\[
  H_i(G,n) = \pi_i\,\mathbb{Z}K(G,n) \overset{\text{Dold-Thom}}{=}
  \pi_i\,\mathbb{Z}\,\mathrm{Sym}^{\infty}M(G,n)
\]

\[
  \pi_i\,\mathrm{Sym}(\mathbb{Z}M(G,n)) = L_i\,\mathrm{Sym}(G)
\]

\[
  H_{n+r}\,K(M,n) \overset{S}{\simeq} H_{n+r+1}\,K(M,n+1)
  \qquad \text{\struck{$n \leq r \leq 2n$}}\ \ r < n.
\]

\[
  H_{\ast}(K(\mathbb{F}_p,n),\mathbb{F}_p)^{\ast} \simeq
  H^{n+r}(K(\mathbb{F}_p,n),\mathbb{F}_p)
\]

\[
  K(\mathbb{Z},n) \to K(\mathbb{Z},n) \to K(\mathbb{F}_p,n)
\]

\[
  H^n(X,\mathbb{F}_p) \longrightarrow H^{n+r}(X,\mathbb{F}_p)
\]

\[
  H^{n+r}(K(\mathbb{F}_p,n),\mathbb{F}_p) \simeq
  H^{n+r+1}(K(\mathbb{F}_p,n+1),\mathbb{F}_p)
\]

\[
  P_n \qquad\qquad P_{n+1}
\]

\[
  H^{n+r-1}(K(\mathbb{F}_p,n-1),\mathbb{F}_p)
\]

\[
  p \neq 2 \qquad
  \mathbb{F}_p[P^0, P^1, \ldots, P^n, \ldots, \beta]\,/\,\text{Relations d'Adem}
\]

\[
  P^0 = 1 \qquad \deg P^i = 2i(p-1) \qquad \deg\beta = 1
\]

\[
  \mathbb{F}_2[\mathrm{Sq}^i]\,/\,\mathrm{Sq}^0 = 1 \quad (\mathrm{Sq}^1 = \beta) \quad \text{rel d'Adem}
  \qquad i \geq 0 \qquad \deg \mathrm{Sq}^i = i
\]

\[
  P^i(x \times y) = \sum_{j \geq 0} P^j(x)\,P^{i-j}(y)
\]

\[
  P^I = \beta^{\varepsilon_0}P^{\eta_1}\beta^{\varepsilon_1}P^{\eta_2}\cdots
  \qquad I = (\varepsilon_0,\eta_1,\varepsilon_1,\eta_2,\ldots)
  \qquad \varepsilon_i = 0, 1 \qquad \eta_i \geq 0
\]

\[
  \eta_i \geq p\,\eta_{i+1} + \varepsilon_i
\]

\[
  A \longrightarrow H^{\ast}(K(\mathbb{F}_p,n),\mathbb{F}_p)
  \qquad x \mapsto x\,\iota_n
\]

\[
  E_2 = \mathrm{Tor}^{H_{\ast}(K(\mathbb{F}_p,n),\mathbb{F}_p)}(\mathbb{F}_p,\mathbb{F}_p)
  \Longrightarrow H_{\ast}(K(\mathbb{F}_p,n+1),\mathbb{F}_p)
\]

\[
  K(\mathbb{F}_p,n+1) = B_{K(\mathbb{F}_p,n)}
\]

\[
  \cdots\ \mathbb{Z}(G\times G) \rightrightarrows \mathbb{Z}G \longrightarrow \mathbb{Z}
  \qquad \mathbb{Z}(G\times G) = \mathbb{Z}G\otimes\mathbb{Z}G
\]

\[
  \cdots\ H_{\ast}(K(\mathbb{F}_p,n))^{\otimes 2} \rightrightarrows
  H_{\ast}K(\mathbb{F}_p,n) \to M
\]

\[
  A = \mathbb{Z}(G) \qquad \mathbb{Z}(EG) = {\ast} \qquad \mathbb{Z}(BG)
\]

\[
  H^{\ast}(K(\mathbb{F}_p,n+1),\mathbb{F}_p)\otimes H^{\ast}(K(\mathbb{F}_p,n),\mathbb{F}_p)
  \Longrightarrow H^{\ast}(\ast)
\]

\[
  \Delta P^i = \sum_{j+k=i} P^j \otimes P^k .
\]

\[
  A_{\ast} = \mathrm{Sym}(\xi_i) \otimes \Lambda(\tau_j)
  \qquad \text{dual de } A
\]

\[
  \Delta\xi_k = \sum_i \xi_{k-i}^{p^i}\otimes\xi_i
  \qquad
  \Delta\tau_k = \tau_k\otimes 1 + \sum_{i=0}^{k} \xi_{k-i}^{p^i}\otimes\tau_i
\]

\[
  A \otimes B \longrightarrow C \qquad \mathbb{F}_p\text{-modules}
\]

\[
  K(A,i) \wedge K(B,j) \longrightarrow K(C,i+j)
\]

\[
  K(\mathbb{F}_p,n)\wedge K(\mathbb{F}_p,m) \longrightarrow K(\mathbb{F}_p,m+n)
\]

\[
  H^{\ast}(K(\mathbb{F}_p,n))\otimes H^{\ast}(K(\mathbb{F}_p,m)) \longleftarrow
  H^{\ast}(K(\mathbb{F}_p,m+n),\mathbb{F}_p)
\]

\[
  A \otimes A \longleftarrow A
\]

\[
  L\,\mathrm{Sym}^r = 0 \qquad r \neq p^s \qquad s \geq 0
\]

\[
  L\,\mathrm{Sym}^{p^s}K(\mathbb{F}_p,n)
\]

\[
  L_{n+i}\,\mathrm{Sym}\,K(\mathbb{F}_p,n) \xrightarrow[+1]{S}
  L_{n+i+1}\,\mathrm{Sym}\,K(\mathbb{F}_p,n+1)
\]

\[
  L^{\mathrm{st}}_{\ast}\mathrm{Sym}\,\mathbb{F}_p = A_{\ast}
\]

\[
  L^{\mathrm{st}}_{\ast}\,\text{\struck{$\mathbb{F}_p$}}\,[\mathbb{F}_p] =
\]

\[
  A \xrightarrow{\ \varepsilon\ } \mathbb{F}_p
\]

\[
  B(A) \qquad \overline{B}(A)
\]

\[
  A\otimes A\otimes A \rightrightarrows A\otimes A
  \underset{1\otimes\varepsilon}{\overset{\mu}{\rightrightarrows}} A
  \qquad \to \mathbb{Z}EG
\]

\[
  A\otimes A \rightrightarrows A \rightrightarrows \mathbb{F}_p
  \qquad \mathbb{Z}BG
\]

\[
  \cdots\ \mathbb{Z}G^3 \rightrightarrows \mathbb{Z}G^2 \rightrightarrows \mathbb{Z}G
  \qquad \mathbb{Z}G^2 = \mathbb{Z}G\otimes\mathbb{Z}G
\]

\[
  A = \mathbb{Z}K(G,n) \qquad \overline{B}(A) \sim \mathbb{Z}BK(G,n)
\]

\[
  \overline{B}(A)\otimes\overline{B}(A) \rightleftarrows \overline{B}(A\otimes A)
  \longrightarrow \overline{B}(A)
\]

\[
  E(R) \subset GL(R) \longrightarrow K_1(R) \to 0
\]

\[
  E(R) \hookrightarrow GL(R) \xrightarrow{\ \det\ } R^{\ast} \to 0
\]

\[
  0 \to K_2(R) \to St(R) \to E(R) \hookrightarrow Gl(R) \longrightarrow K_1(R) \to 0
\]

\[
  \pi_1(E(R)) = K_2(R)
\]

\[
  BGl(R)^{+}
\]

\[
  \mathrm{Ext}^2(\mathbb{Z},G_m)
\]

\[
  H^i_{\mathrm{Zar}}(X,\underline{K}_j) \Longrightarrow K_{j-i}(X)
\]

\[
  H^i(X,\underline{K}_i) = i\text{-cycles}/\text{équiv rat}.
\]

\[
  0 \to G_m \to i_{\ast}G_{m,K} \to \bigoplus_{\text{pts de codim } 1}\mathbb{Z} \to 0
\]

\[
  0 \to \underline{K}_1 \to i_{\ast}\underline{K}_1 \to \bigoplus_{\text{pts } 1}\underline{K}_0 \to 0
\]

\[
  0 \to \underline{K}_i \to i_{\ast}\underline{K}_i \to \bigoplus_{\text{codim } 1} i_{x\ast}K_{i-1}
  \to \cdots \to \bigoplus_{\text{pts fermés}} i_{x\ast}\mathbb{Z} \to 0
\]

\[
  H^0(X,\ \cdots)\,/\,H^0(\quad)
\]

\[
  A = \mathbb{F}_p[P^i,\beta]\,/\,P^0 = 1,\ \text{Adem}
  \qquad \deg P^i = 2i(p-1) \qquad \deg\beta = 1
\]

\[
  A = \mathbb{F}_2[\mathrm{Sq}^i]\,/\,\mathrm{Sq}^0 = 1,\ \text{Adem en car } 2
  \qquad \deg\mathrm{Sq}^i = i
\]

\[
  \mathrm{Sq}^a\mathrm{Sq}^b = \sum_{j=0}^{a<2b} (\quad)\,\mathrm{Sq}^{a+b-j}\mathrm{Sq}^j
\]

\[
  K(M\times N,n) \sim K(M,n)\times K(N,n)
\]

\[
  G \in \mathcal{R} \text{ des car } p \qquad
  H^{\ast}(K(G,n),\mathcal{R})
\]

\[
  \mathbb{Z} \xrightarrow{\ p\ } \mathbb{Z} \to \mathbb{Z}/p
\]

\[
  \xrightarrow{0 = p} H_{\ast}(K(\mathbb{F}_p,n),\mathbb{Z}) \to H_{\ast}(K(\mathbb{F}_p),\mathbb{F}_p)
  \to H_{\ast-1}(K(\mathbb{F}_p),\mathbb{Z}) \xrightarrow{\ p=0\ } 0
\]

\[
  K(\mathbb{Z},n) \to K(\mathbb{Z},n) \to K(\mathbb{F}_p,n)
\]

\[
  0 \to H_{\ast}(K(\mathbb{Z},n),\mathbb{F}_p) \to H_{\ast}(K(\mathbb{F}_p,n),\mathbb{F}_p)
  \to H_{\ast-1}(K(\mathbb{Z},n),\mathbb{F}_p) \to 0
\]

\[
  \mathrm{Sym}(\xi_i)\otimes\Lambda(\tau_i)
\]

\[
  \tau_0 \to 1 \qquad
  \begin{cases} \tau_i \to 0 & i\neq 0 \\ \xi_j \to 0 & j>0 \end{cases}
\]

\[
  \begin{cases} \tau_j \to \xi_j \\ \xi_j \to 0 \end{cases}
\]

\[
  \tau_j \longmapsto \xi_j \qquad \xi_j \longmapsto 0
\]

\[
  \mathrm{Im}\,\beta_2 = \ker\beta_2 = H_{\ast}(K(\mathbb{Z},n),\mathbb{Z})_{(p)} .
\]

\[
  H_{\ast}(K(\mathbb{Z},n),\mathbb{F}_p) \xrightarrow{\ \sim\ } H_{\ast}(K(\mathbb{F}_p,n),\mathbb{Z})
\]

\[
  (\ast) \qquad H_{\ast}(K(\mathbb{F}_p,n),\mathbb{F}_p) = A_{\ast} \longleftarrow
\]

\[
  H_{\ast}(K(G,n),H) \simeq H_{\ast}(K(H,n),G) \quad \text{additif.}
\]

\[
  \pi_i(\mathbb{Z}X) \simeq H_i(X)
\]

\[
  K(\mathbb{Z},n)\wedge X \xrightarrow{\ A(X)\ } \mathbb{Z}^{+}(S^nX)
  \qquad \mathbb{Z}^{+}(X,x) = \mathbb{Z}X/\mathbb{Z}x
\]

\[
  K(\mathbb{Z},n) = \mathbb{Z}^{+}[S^n]
\]

\[
  \left.\begin{aligned}
    \mathbb{Z}(X\times Y) &\simeq \mathbb{Z}X\otimes\mathbb{Z}Y \\
    \mathbb{Z}^{+}(X\wedge Y) &\simeq \mathbb{Z}^{+}X\otimes\mathbb{Z}^{+}Y
  \end{aligned}\right\}
  \ \begin{array}{l} \text{EZ} \\ \text{EZ pointé.} \end{array}
\]

\[
  K(\mathbb{Z},n)\wedge X \to K(\mathbb{Z},n)\wedge\mathbb{Z}^{+}X
\]

\[
  \mathbb{Z}^{+}(S^n)\wedge X \to \mathbb{Z}^{+}(S^n)\wedge\mathbb{Z}^{+}X
  \to \mathbb{Z}^{+}(S^n)\otimes\mathbb{Z}^{+}X \xrightarrow{\ \mathrm{EZ}\ } \mathbb{Z}^{+}(S^nX)
\]

\[
  \pi_{n+r}(K(\mathbb{Z},n)\wedge X) = \pi_{n+r}(\mathbb{Z}^{+}(S^nX))
  = \widetilde{H}_{n+r}(S^nX) = \widetilde{H}_r(X)
\]

\[
  \pi_i(K(\mathbb{Z},n)\wedge S^m) \qquad
  \mathbb{Z}^{+}(S^n\wedge S^m) = \mathbb{Z}^{+}(S^{n+m})
\]

\[
  S^m\wedge K(\mathbb{Z},n) \xrightarrow{\ \mathrm{qis}\ } K(\mathbb{Z},n+m)
\]

\[
  S^m \to \mathbb{Z}^{+}(S^m) = K(\mathbb{Z},m)
\]

\[
  S^m\wedge K(\mathbb{Z},n) \to K(\mathbb{Z},m)\wedge K(\mathbb{Z},n)
  \xrightarrow{\ \cup\ } K(\mathbb{Z},n+m).
\]

\[
  Y \hookrightarrow X \to X/Y
\]

\[
  h_r(Y) \to h_r(X) \to h_r(X/Y) \to h_{r-1}(Y) \to \cdots
\]

\[
  \begin{array}{ccc}
    K(\mathbb{Z},n)\wedge K(\mathbb{F}_p,m) & \longrightarrow & K(\mathbb{F}_p,m)\wedge K(\mathbb{Z},n) \\
    \big\downarrow & & \big\downarrow \\
    K(\mathbb{F}_p,n)\wedge K(\mathbb{F}_p,m) & \longrightarrow & K(\mathbb{F}_p,n)\wedge K(\mathbb{F}_p,m)
  \end{array}
\]

\[
  \pi_{i+m}(K(H,m)\wedge K(G,n)) \simeq \pi_{i+n}(K(G,n)\wedge K(H,m))
\]

\[
  X_{\ast}(G) \longrightarrow G \qquad X_n(G) = \mathbb{Z}(G^n\times\mathbb{Z}^{+})
\]

\[
  \mathbb{Z}^{\mathbb{Z}^G} \rightrightarrows \mathbb{Z}^G \qquad G
  \qquad\qquad G \to G\,! \qquad H
\]

\[
  \mathbb{Z}^{\mathbb{Z}K(G,n)} \rightrightarrows \mathbb{Z}^{K(G,n)}
  \qquad K(G,n) = G\otimes_{\mathbb{Z}}\Lambda^n\Psi
\]

\[
  K(\mathbb{Z},n)\wedge K(G,p) \xrightarrow{\ \sim\ } \mathbb{Z}(S^n\wedge K(G,p))
  \longrightarrow \mathbb{Z}K(G,n+p)
\]

\[
  S^n\wedge K(G,p) \to K(G,n+p)
\]

\[
  K(\mathbb{Z},p_1)\wedge K(\mathbb{Z},p_2)\wedge\cdots\wedge K(\mathbb{Z},p_r)\wedge K(G,p_{r+1})
  \to \mathbb{Z}^{r}K(G,\ p)
\]

\[
  \sum_{i=1}^{r+1} p_i = p
\]

\[
  r\text{ qis} \qquad K(\mathbb{Z},n)\wedge K(G,p) \longrightarrow \mathbb{Z}K(G,n+p)
\]

\[
  r = \min(n+2p-1,\ 2n+p-1)
\]

\[
  K(\mathbb{Z},n)\wedge X \to \mathbb{Z}[S^n\wedge X]
\]

\[
  K(\mathbb{Z},n)\wedge K(\mathbb{Z},m)\wedge K(G,p) \to \mathbb{Z}[S^n\wedge K(\mathbb{Z},m)\wedge K(G,p)]
\]

\[
  \downarrow
\]

\[
  \mathbb{Z}[K(\mathbb{Z},n+m)\wedge K(G,p)]
\]

\[
  \downarrow
\]

\[
  \mathbb{Z}[\mathbb{Z}[S^{n+m}\wedge K(G,p)]]
\]

\[
  \downarrow
\]

\[
  \mathbb{Z}[\mathbb{Z}K(G,n+m+p)]
\]

\[
  H^q(X,R) \simeq H^q(X,\mathbb{F}_p)\otimes_{\mathbb{F}_p} R
\]

\[
  P^i_R = P^i\otimes 1 \qquad i>0 \qquad 2i(p-1)
\]

\[
  P^0_R = P^0\otimes F \qquad (P^0 = 1)
\]

\[
  H^{\ast}(K(G_a,n),G_a) = \mathcal{A}\cdot\text{les op}
\]

\[
  \mathbb{H}^{\ast}(\mathbb{Z}[K(G_a,n)]^{\sim},G_a)
\]

\[
  \mathbb{H}^{\ast}(X,F) = [X,K(F,n)]
  \qquad X' \to K(F,n),\ X' \to X
\]

\[
  \mathcal{E}xt^p(\Lambda(X),F) = \varinjlim_{\substack{X'\to X\\ \text{qis}\\ \text{fibrant}}} [X',K(F,p)]
\]

\[
  \overset{\text{déf}}{=} \mathbb{H}^p(X,F)
\]

\[
  \mathcal{E}xt^p(\Lambda(-),F) \to \mathrm{Ext}^q(\Lambda(\ ),G)
\]

\[
  \mathbb{Z}X \xrightarrow{\ f\ } K(A,q)
\]

\[
  \mathbb{H}^n(K(G_a,m),G_a) = R^n\,\mathrm{Sym}(\mathbb{F}_p[m])
  = \text{dual de } L^n\Gamma\,(\mathbb{F}_p[m])
\]

\[
  K(G_a,n-1) \to \ast \to K(G_a,n)
\]

\[
  G_a^3 \qquad G_a\times G_a \qquad G_a \qquad\qquad
  k[x,y,z] \leftleftarrows k[x,y] \leftleftarrows k[x]
\]

\[
  \iota_1 \in H^1(K(G_a,1),G_a)
\]

\[
  \text{Witt} \leftarrow \beta \in H^2(K(G_a,1),G_a)
\]

\[
  0 \to G_a \to W_2 \to G_a \to 0 \qquad
  0 \to \mathbb{Z}/p \to \mathbb{Z}/p^2 \to \mathbb{Z}/p \to 0
\]

\[
  \Bigl|\ H^{\ast}(K(\mathbb{F}_{p^h},m),\mathbb{F}_{p^h})
\]

\[
  H^{\ast}(K(R,n),R) = {}
\]

\[
  \bigoplus P^If\ \ i_n \qquad f\in\mathrm{Hom}(R,R)
\]

\[
  H^{\ast}(X,\mathcal{O}) \to H^{\ast}(X,\mathcal{O}) \qquad x \mapsto P^Ix
\]

\[
  x\in H^{\ast}(K(\mathcal{O},m),\mathcal{O})
  \qquad K(\mathcal{O},m) \longrightarrow K(\mathcal{O},n)
  \qquad \forall\, R/\mathbb{F}_p
\]

\[
  K(R,m) \longrightarrow K(R,n)
\]

\[
  x\in H^{\ast}(K(\mathcal{O},n),\mathcal{O})_h \xrightarrow{\ \text{injective}\ }
  \mathbb{H}^{\ast}(K(\mathbb{F}_{p^h},n),\mathbb{F}_{p^h})
\]

\[
  G_a^{\mathrm{parf}} = \varinjlim_{F} G_a
\]

\[
  \mathrm{Ext}^i(G_a^{\mathrm{parf}},G_a^{\mathrm{parf}}) = 0 \qquad i>1.
\]

\[
  \forall \text{ fais simpl } X : \quad
  E_2^{i,j} = \mathrm{Ext}^i_{\mathbb{F}_p}(\underline{H}_j(X),F) \Longrightarrow \mathbb{H}^{i+j}(X,F)
\]

\[
  \mathrm{Hom}(G_a,G_a) = k[F] \qquad \mathrm{Ext}^1(G_a,G_a) = 0
\]

\[
  E \qquad \mathrm{Ext}^i(H_j(K(G_a,n)),G_a) \Longrightarrow \mathbb{H}^{\ast}(K(G_a,n),G_a)
\]

\[
  E_2^{i,n} = \mathrm{Ext}^i(H_n(K(G_a,n)),G_a) = \mathrm{Ext}^i(G_a,G_a)
\]

\[
  \underline{H}_j(K(G_a,n)) = \bigoplus G_a \qquad G_a \xrightarrow{\ p=0\ } G_a
\]

\[
  \mathrm{Ext}^i(H_j(K(G_a,n)),G_a) \Longrightarrow \mathbb{H}^{i+j}(K(G_a,n),G_a)
\]

\[
  \mathrm{Ext}^i(H_n(K(G_a,n)),G_a) \to H^{n+i}(K(G_a,n),G_a)
  \to \mathrm{Hom}(H_{n+i}(K(G_a,n)),G_a)
\]

\[
  \to \mathrm{Ext}^{i+1}(G_a,G_a) \xrightarrow{\ \partial\ } H^{n+i+1}(K(G_a,n),G_a)
\]

\[
  \mathrm{Ext}^{i+1}(\mathbb{F}_{p^h},\mathbb{F}_{p^h}) \xrightarrow{\ \partial\ }
  H^{n+i+1}(K(\mathbb{F}_{p^h},n),\mathbb{F}_{p^h})
  \qquad \mathrm{Ext}^{i+1}(\mathbb{F}_{p^h},\mathbb{F}_{p^h}) = 0
\]

\[
  \simeq \mathrm{Hom}(G_a\otimes H_{n+i}(K(\mathbb{F}_p,n)),G_a)
  = H^{n+i}(K(\mathbb{F}_p,n),\mathbb{F}_p)\otimes\mathrm{Hom}(G_a,G_a)
\]

\[
  P^I\iota_n \qquad \beta P^{i_1}\beta\cdots P^{i_k}\iota_n \qquad \text{longueur } n+1
\]

\[
  P^d\iota_n \longrightarrow P^d\iota_n\otimes F
\]

\[
  \mathrm{Ext}^{\ast}_{\mathbb{F}_p}(G_a,G_a) \qquad
  \mathrm{Ext}^{2i+1}(G_a,G_a) = 0 \quad \forall i \quad i>0
\]

\[
  \mathrm{Ext}^{2i}(G_a,G_a) = k[F]/(F^d) \qquad d = v_p(i).
\]

\[
  \mathcal{A} \to A \qquad P^i \mapsto P^i \qquad P^0 \mapsto 1
\]

\[
  \mathcal{A} \to \mathcal{A}\otimes\mathcal{A} \qquad P^i \mapsto \sum_{j+k=i} P^j\otimes P^k
\]

\[
  \mathbb{Z}^iK(G_a,n) \cdots \mathbb{Z}\mathbb{Z}K(G_a,n) \rightrightarrows \mathbb{Z}K(G_a,n)
\]

\[
  E_1 \qquad \cdots\ \mathcal{A}^{\otimes 2}\otimes\mathcal{A} \leftleftarrows \mathcal{A}\otimes\mathcal{A} \leftleftarrows \mathcal{A}
\]

\[
  \cdots\ \mathcal{A}_{\ast}^{\otimes 2}\otimes\mathcal{A}_{\ast} \rightrightarrows \mathcal{A}_{\ast}\otimes\mathcal{A}_{\ast} \rightrightarrows \mathcal{A}_{\ast}
\]

\[
  E^2 = \mathrm{Tor}^{A_{\ast}}(\mathcal{A}_{\ast},\mathbb{F}_p) \Longrightarrow E^2 \text{ par Koszul}
\]

\[
  \mathrm{Ext}^i(\mathbb{Z}^jK(G_a,n),G_a) \Longrightarrow \mathrm{Ext}^{i+j}(G_a,G_a)
\]

\[
  \mathbb{Z}[\mathbb{Z}^{j-1}K(G_a,n)] \sim \mathbb{Z}[K(\mathbb{Z})\wedge K(\mathbb{Z})\wedge\cdots\wedge K(G_a,n)]
\]

\[
  E_2^{i,j} = \widetilde{\mathbb{H}}^{i}(K(\mathbb{Z})^{\wedge j-1}\wedge K(G_a),G_a)
  \overset{\text{Künneth}}{\sim}
  \widetilde{\mathbb{H}}^{\ast}(K(\mathbb{Z}))^{\otimes j-1}\otimes\widetilde{\mathbb{H}}^{\ast}(K(G_a),G_a)
\]

\[
  \mathbb{H}^{\ast}(K(\mathbb{F}_p),\mathbb{F}_p) = A \qquad
  \mathbb{H}^{\ast}(K(G_a),G_a) = \mathcal{A}
\]