Cote n° 137 · pages 3–85
· 234 displayed formulas · Dévissage des complexes ½ simpliciaux ∂ - parfaits et structures multiplicatives… (1975 ou 1976) : notes manuscrites (s.d.).
Inventory dating : [vers 1975-1976]
Édition de démonstration
\[\operatorname{Hom}_{\uncertain{\mathrm{Zab}}}\bigl(L_{\bullet} \otimes_{k}
\Lambda^{q}\Psi_{*k},\ \Lambda^{q'}\Psi_{*k} \otimes_{k} M\bigr) = 0\]
LaTeX source
\[
\operatorname{Hom}_{\uncertain{\mathrm{Zab}}}\bigl(L_{\bullet} \otimes_{k}
\Lambda^{q}\Psi_{*k},\ \Lambda^{q'}\Psi_{*k} \otimes_{k} M\bigr) = 0
\]\[\cdots \longrightarrow L'_{q+1} \longrightarrow L'_{q} \longrightarrow 0
\longrightarrow \cdots\]
LaTeX source
\[
\cdots \longrightarrow L'_{q+1} \longrightarrow L'_{q} \longrightarrow 0
\longrightarrow \cdots
\]\[\Bigl(\prod_{i \geqslant 1} \Lambda^{i}\Phi_{*k} \otimes_{k}
M_{i}\Bigr) \times \prod_{j \geqslant 0} \bigl(\Lambda^{j}\Psi_{*k}\bigr)
\otimes_{k} N_{j} .\]
LaTeX source
\[
\Bigl(\prod_{i \geqslant 1} \Lambda^{i}\Phi_{*k} \otimes_{k}
M_{i}\Bigr) \times \prod_{j \geqslant 0} \bigl(\Lambda^{j}\Psi_{*k}\bigr)
\otimes_{k} N_{j} .
\]\[\prod_{i \geqslant 1} \bigl(0 \to M_{i} \to M_{i} \to 0 \to
\cdots\bigr) \times \prod_{j \geqslant 0} \bigl(0 \to N_{j} \to 0 \to
\cdots\bigr)\]
LaTeX source
\[
\prod_{i \geqslant 1} \bigl(0 \to M_{i} \to M_{i} \to 0 \to
\cdots\bigr) \times \prod_{j \geqslant 0} \bigl(0 \to N_{j} \to 0 \to
\cdots\bigr)
\]\[L_{n} = \prod_{1 \leqslant i \leqslant n+1} \Lambda^{i}\Phi_{n} \otimes_{k}
M_{i} \times \prod_{0 \leqslant j \leqslant n} \Lambda^{j}\Psi_{n}
\otimes N_{j}\]
LaTeX source
\[
L_{n} = \prod_{1 \leqslant i \leqslant n+1} \Lambda^{i}\Phi_{n} \otimes_{k}
M_{i} \times \prod_{0 \leqslant j \leqslant n} \Lambda^{j}\Psi_{n}
\otimes N_{j}
\]\[\rho(n) \overset{\mathrm{df}}{=} \operatorname{rg}_{k} L_{n} =
\sum_{1 \leqslant i \leqslant n+1} \mu_{i} \binom{n+1}{i} +
\sum_{0 \leqslant j \leqslant n} \nu_{j} \binom{n}{j}\]
LaTeX source
\[
\rho(n) \overset{\mathrm{df}}{=} \operatorname{rg}_{k} L_{n} =
\sum_{1 \leqslant i \leqslant n+1} \mu_{i} \binom{n+1}{i} +
\sum_{0 \leqslant j \leqslant n} \nu_{j} \binom{n}{j}
\]\[\sum_{i \geqslant 1} \mu_{i} \binom{n+1}{i} = f_{L_{\bullet}}(n) -
\sum_{j \geqslant 0} \nu_{j} \binom{n}{j} \qquad (n \geqslant -1)\]
LaTeX source
\[
\sum_{i \geqslant 1} \mu_{i} \binom{n+1}{i} = f_{L_{\bullet}}(n) -
\sum_{j \geqslant 0} \nu_{j} \binom{n}{j} \qquad (n \geqslant -1)
\]\[f_{L_{\bullet}}(n-1) - \sum_{j \geqslant 0} \nu_{j} P_{j}(n-1) =
\sum_{i \geqslant 1} \mu_{i} P_{i}(n)\]
LaTeX source
\[
f_{L_{\bullet}}(n-1) - \sum_{j \geqslant 0} \nu_{j} P_{j}(n-1) =
\sum_{i \geqslant 1} \mu_{i} P_{i}(n)
\]\[(\mu_{0} = 0) \iff \Bigl(f(-1) = \sum (-1)^{j}\nu_{j}\Bigr)
\quad \bigl[= \chi((\nu_{j}))\bigr].\]
LaTeX source
\[
(\mu_{0} = 0) \iff \Bigl(f(-1) = \sum (-1)^{j}\nu_{j}\Bigr)
\quad \bigl[= \chi((\nu_{j}))\bigr].
\]\[L_{\bullet} \simeq \prod_{i \geqslant 1} M_{i} \otimes_{k}
\Lambda^{i}\Phi_{\bullet} \times \prod_{j \geqslant 0} N_{j} \otimes_{k}
\Lambda^{j}\Psi_{*} \times \prod_{\substack{i \geqslant 1\\ \ell \in
\mathfrak{B}}} \Phi_{i,\ell}^{\rho_{i,\ell}}\]
LaTeX source
\[
L_{\bullet} \simeq \prod_{i \geqslant 1} M_{i} \otimes_{k}
\Lambda^{i}\Phi_{\bullet} \times \prod_{j \geqslant 0} N_{j} \otimes_{k}
\Lambda^{j}\Psi_{*} \times \prod_{\substack{i \geqslant 1\\ \ell \in
\mathfrak{B}}} \Phi_{i,\ell}^{\rho_{i,\ell}}
\]\[\begin{cases}
\operatorname{rg} Z_{i}(L^{!}) = \mu_{i+1} + \rho_{i+1} + \nu_{i} \\
\operatorname{rg} L^{!}_{i} = \mu_{i+1} + \rho_{i+1} + \rho_{i} +
\nu_{i}
\end{cases}\]
LaTeX source
\[
\begin{cases}
\operatorname{rg} Z_{i}(L^{!}) = \mu_{i+1} + \rho_{i+1} + \nu_{i} \\
\operatorname{rg} L^{!}_{i} = \mu_{i+1} + \rho_{i+1} + \rho_{i} +
\nu_{i}
\end{cases}
\]\[L^{!} \simeq \prod_{i} K_{i}, \qquad
K_{i} : \quad \cdots \to 0 \to P_{i} \hookrightarrow Q_{i} \to 0 \to
\cdots \quad \text{(en degrés $i$ et $i-1$)}\]
LaTeX source
\[
L^{!} \simeq \prod_{i} K_{i}, \qquad
K_{i} : \quad \cdots \to 0 \to P_{i} \hookrightarrow Q_{i} \to 0 \to
\cdots \quad \text{(en degrés $i$ et $i-1$)}
\]\[\overline{K}_{i} : \ \cdots \to 0 \to P_{i} \hookrightarrow
\overline{P}_{i} \to 0 \to 0, \qquad
\cdots \to 0 \to N_{i-1} \to 0\]
LaTeX source
\[
\overline{K}_{i} : \ \cdots \to 0 \to P_{i} \hookrightarrow
\overline{P}_{i} \to 0 \to 0, \qquad
\cdots \to 0 \to N_{i-1} \to 0
\]\[\cdots \to k \xrightarrow{\ m_{i,\alpha}\ } k \to 0 \qquad
(1 \leqslant \alpha \leqslant N_{i},\ N_{i} = \operatorname{rg} P_{i} =
\operatorname{rg} \overline{P}_{i})\]
LaTeX source
\[
\cdots \to k \xrightarrow{\ m_{i,\alpha}\ } k \to 0 \qquad
(1 \leqslant \alpha \leqslant N_{i},\ N_{i} = \operatorname{rg} P_{i} =
\operatorname{rg} \overline{P}_{i})
\]\[\cdots \to 0 \to k \xrightarrow{\ n\ } k \to 0 \to \cdots \qquad
\text{(en degrés $i$ et $i-1$)}\]
LaTeX source
\[
\cdots \to 0 \to k \xrightarrow{\ n\ } k \to 0 \to \cdots \qquad
\text{(en degrés $i$ et $i-1$)}
\]\[0 \to \Lambda^{i-1}\Psi \to \Phi(i,n) \to \Lambda^{i}\Psi \to 0\]
LaTeX source
\[
0 \to \Lambda^{i-1}\Psi \to \Phi(i,n) \to \Lambda^{i}\Psi \to 0
\]\[\pi_{j}(\Phi(i,n)) =
\begin{cases}
0 & \text{si } j \neq i-1 \\
k/\operatorname{div} n & \text{si } j = i-1 .
\end{cases}\]
LaTeX source
\[
\pi_{j}(\Phi(i,n)) =
\begin{cases}
0 & \text{si } j \neq i-1 \\
k/\operatorname{div} n & \text{si } j = i-1 .
\end{cases}
\]\[\operatorname{rg} L_{n} = \sum_{i \leqslant n+1} (\mu_{i} + \rho_{i})
\binom{n+1}{i} + \sum_{0 \leqslant j \leqslant n} \nu_{j} \binom{n}{j} .\]
LaTeX source
\[
\operatorname{rg} L_{n} = \sum_{i \leqslant n+1} (\mu_{i} + \rho_{i})
\binom{n+1}{i} + \sum_{0 \leqslant j \leqslant n} \nu_{j} \binom{n}{j} .
\]\[\operatorname{Hom}_{k}(\Phi(i,n), L_{\bullet}) \simeq
d_{L_{\bullet},i}^{-1}(n L_{i-1}) \subset L_{i}\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Phi(i,n), L_{\bullet}) \simeq
d_{L_{\bullet},i}^{-1}(n L_{i-1}) \subset L_{i}
\]\[\operatorname{Hom}(\Phi(i,n), \Phi(j,m)_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i,\ i+1 \\
\{\xi \in k \mid m\xi \in nk\} = n'k \simeq k & \text{si } j = i \\
k & \text{si } j = i+1
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}(\Phi(i,n), \Phi(j,m)_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i,\ i+1 \\
\{\xi \in k \mid m\xi \in nk\} = n'k \simeq k & \text{si } j = i \\
k & \text{si } j = i+1
\end{cases}
\]\[\operatorname{Hom}_{k}(\Phi(i,n), \Lambda^{j}\Psi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Phi(i,n), \Lambda^{j}\Psi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i
\end{cases}
\]\[\operatorname{Hom}_{k}(\Lambda^{i}\Phi_{*}, \Lambda^{j}\Psi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Lambda^{i}\Phi_{*}, \Lambda^{j}\Psi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i
\end{cases}
\]\[\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, L_{\bullet}) \simeq
Z_{i}(L_{*}) \simeq k^{\mu_{i+1} + \rho_{i+1} + \nu_{i}}\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, L_{\bullet}) \simeq
Z_{i}(L_{*}) \simeq k^{\mu_{i+1} + \rho_{i+1} + \nu_{i}}
\]\[\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{j}\Psi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{j}\Psi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i
\end{cases}
\]\[\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{i}\Psi_{*}) = k\,\mathrm{id}_{\Lambda^{i}\Psi_{*}} \simeq k ,\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{i}\Psi_{*}) = k\,\mathrm{id}_{\Lambda^{i}\Psi_{*}} \simeq k ,
\]\[\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Phi_{\bullet}(j,n)) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \\
k & \text{si } j = i+1
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Phi_{\bullet}(j,n)) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \\
k & \text{si } j = i+1
\end{cases}
\]\[\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{j}\Phi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \\
k & \text{si } j = i+1
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{j}\Phi_{*}) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \\
k & \text{si } j = i+1
\end{cases}
\]\[\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{i+1}\Phi_{*}) = k\,\partial \simeq k\]
LaTeX source
\[
\operatorname{Hom}_{k}(\Lambda^{i}\Psi_{*}, \Lambda^{i+1}\Phi_{*}) = k\,\partial \simeq k
\]\[0 \to \Phi^{0} \to \Phi^{1} \to \Phi^{2} \to \cdots \to \Phi^{i} \xrightarrow{\ \partial_i\ } \Phi^{i+1} \to \cdots\]
LaTeX source
\[
0 \to \Phi^{0} \to \Phi^{1} \to \Phi^{2} \to \cdots \to \Phi^{i} \xrightarrow{\ \partial_i\ } \Phi^{i+1} \to \cdots
\]\[\operatorname{Hom}(\Phi^{i}, \Phi^{j}) \simeq
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
\simeq k \ (\text{engendré par } \mathrm{id}) & \text{si } j = i \\
\simeq k \ (\text{engendré par } \partial_i) & \text{si } j = i+1
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}(\Phi^{i}, \Phi^{j}) \simeq
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
\simeq k \ (\text{engendré par } \mathrm{id}) & \text{si } j = i \\
\simeq k \ (\text{engendré par } \partial_i) & \text{si } j = i+1
\end{cases}
\]\[\Psi^{0} \simeq \Phi^{0}\]
LaTeX source
\[
\Psi^{0} \simeq \Phi^{0}
\]\[(*) \qquad 0 \to \Psi^{i-1} \xrightarrow{\ \partial_i\ } \Phi^{i} \xrightarrow{\ u_i\ } \Psi^{i} \to 0
\qquad \text{ext.\ can.}\]
LaTeX source
\[
(*) \qquad 0 \to \Psi^{i-1} \xrightarrow{\ \partial_i\ } \Phi^{i} \xrightarrow{\ u_i\ } \Psi^{i} \to 0
\qquad \text{ext.\ can.}
\]\[\begin{aligned}
&\operatorname{Hom}(\Psi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } \mathrm{id})
\end{cases} \\
&\operatorname{Hom}(\Phi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } u_i)
\end{cases} \\
&\operatorname{Hom}(\Psi^{i}, \Phi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j})) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \ (i \neq 0) \\
k & \text{si } j = i+1 \ (\text{base } \partial_i) \\
k & \text{si } j = i = 0 \ (\text{base } \mathrm{id})
\end{cases}
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
&\operatorname{Hom}(\Psi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } \mathrm{id})
\end{cases} \\
&\operatorname{Hom}(\Phi^{i}, \Psi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j+1})) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
k & \text{si } j = i \ (\text{base } u_i)
\end{cases} \\
&\operatorname{Hom}(\Psi^{i}, \Phi^{j}) \ (\subset \operatorname{Hom}(\Phi^{i}, \Phi^{j})) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \ (i \neq 0) \\
k & \text{si } j = i+1 \ (\text{base } \partial_i) \\
k & \text{si } j = i = 0 \ (\text{base } \mathrm{id})
\end{cases}
\end{aligned}
\]\[\operatorname{Hom}(\Phi(i,-), \Phi(j,-)) \to \operatorname{Hom}(\Psi^{i-1}, \Psi^{j})\]
LaTeX source
\[
\operatorname{Hom}(\Phi(i,-), \Phi(j,-)) \to \operatorname{Hom}(\Psi^{i-1}, \Psi^{j})
\]\[\operatorname{Hom}(\Phi(i,n), \Phi(j,m)) \xleftarrow{\ \approx\ } \operatorname{Hom}(\Psi^{i}, \Psi^{j-1})
\simeq
\begin{cases}
k & \text{cas } j = i+1 \\
0 & \text{si } j \neq i+1
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}(\Phi(i,n), \Phi(j,m)) \xleftarrow{\ \approx\ } \operatorname{Hom}(\Psi^{i}, \Psi^{j-1})
\simeq
\begin{cases}
k & \text{cas } j = i+1 \\
0 & \text{si } j \neq i+1
\end{cases}
\]\[q^{*}(e_{i,m}) = m\,e_i , \qquad p_{*}(e_{i,n}) = n\,e_i\]
LaTeX source
\[
q^{*}(e_{i,m}) = m\,e_i , \qquad p_{*}(e_{i,n}) = n\,e_i
\]\[(q\,m - p\,n)\,e_i = 0 .\]
LaTeX source
\[ (q\,m - p\,n)\,e_i = 0 . \]
\[\text{(c)} \qquad e_i \text{ est une base de } \operatorname{Ext}^{1}_{\mathcal{C}_0}(\Psi^{i}, \Psi^{i-1})\]
LaTeX source
\[
\text{(c)} \qquad e_i \text{ est une base de } \operatorname{Ext}^{1}_{\mathcal{C}_0}(\Psi^{i}, \Psi^{i-1})
\]\[q\,m = p\,n\]
LaTeX source
\[ q\,m = p\,n \]
\[\frac{q}{p} = \frac{n}{m} = \frac{n'}{m'}\]
LaTeX source
\[
\frac{q}{p} = \frac{n}{m} = \frac{n'}{m'}
\]\[p = \struck{\ill{}}\ t\,m' , \qquad q = t\,n' \qquad (t \in k)\]
LaTeX source
\[
p = \struck{\ill{}}\ t\,m' , \qquad q = t\,n' \qquad (t \in k)
\]\[\operatorname{Hom}(\Phi(i,n), \Phi(i,m)) \simeq k .\]
LaTeX source
\[
\operatorname{Hom}(\Phi(i,n), \Phi(i,m)) \simeq k .
\]\[\underline{\mathrm{Hom}}(\Phi(i,-), \Phi(j,-)) \simeq
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
k & \text{si } j = i+1 \\
k & \text{si } j = i
\end{cases}\]
LaTeX source
\[
\underline{\mathrm{Hom}}(\Phi(i,-), \Phi(j,-)) \simeq
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
k & \text{si } j = i+1 \\
k & \text{si } j = i
\end{cases}
\]\[n' = \frac{n}{(m,n)}, \qquad m' = \frac{m}{(m,n)}\]
LaTeX source
\[
n' = \frac{n}{(m,n)}, \qquad m' = \frac{m}{(m,n)}
\]\[\underline{\mathrm{Hom}}(\Phi(i,-), \Psi^{j}) \subset \underline{\mathrm{Hom}}(\Phi(i,-), \Phi^{j+1})\]
LaTeX source
\[
\underline{\mathrm{Hom}}(\Phi(i,-), \Psi^{j}) \subset \underline{\mathrm{Hom}}(\Phi(i,-), \Phi^{j+1})
\]\[\underline{\mathrm{Hom}}(\Psi^{i-1}, \Psi^{j}) = 0 \quad \text{si } j \neq i-1 .\]
LaTeX source
\[
\underline{\mathrm{Hom}}(\Psi^{i-1}, \Psi^{j}) = 0 \quad \text{si } j \neq i-1 .
\]\[\underline{\mathrm{Hom}}(\Phi(i,-), \Psi^{j}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
\simeq k & \text{si } j = i
\end{cases}\]
LaTeX source
\[
\underline{\mathrm{Hom}}(\Phi(i,-), \Psi^{j}) \simeq
\begin{cases}
0 & \text{si } j \neq i \\
\simeq k & \text{si } j = i
\end{cases}
\]\[\underline{\mathrm{Hom}}(\Psi^{i}, \Phi(j,-)) \longrightarrow \underline{\mathrm{Hom}}(\Psi^{i}, \Psi^{j})\]
LaTeX source
\[
\underline{\mathrm{Hom}}(\Psi^{i}, \Phi(j,-)) \longrightarrow \underline{\mathrm{Hom}}(\Psi^{i}, \Psi^{j})
\]\[\underline{\mathrm{Hom}}(\Psi^{i}, \Phi(j,-)) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \\
k & \text{si } j = i+1
\end{cases}\]
LaTeX source
\[
\underline{\mathrm{Hom}}(\Psi^{i}, \Phi(j,-)) \simeq
\begin{cases}
0 & \text{si } j \neq i+1 \\
k & \text{si } j = i+1
\end{cases}
\]\[C \xrightarrow{\text{can}} \widehat{C} \xrightarrow{\text{rest.}} \widehat{A} =
\underline{\mathrm{Hom}}(A^{\circ}, \mathrm{Ens})\]
LaTeX source
\[
C \xrightarrow{\text{can}} \widehat{C} \xrightarrow{\text{rest.}} \widehat{A} =
\underline{\mathrm{Hom}}(A^{\circ}, \mathrm{Ens})
\]\[r^{\circ} : C^{\circ} \longrightarrow \widehat{A^{\circ}} \quad (\text{resp. } \widehat{A^{\circ}}_{ab})\]
LaTeX source
\[
r^{\circ} : C^{\circ} \longrightarrow \widehat{A^{\circ}} \quad (\text{resp. } \widehat{A^{\circ}}_{ab})
\]\[\widehat{A}_{\mathcal{T}} \simeq (\widehat{A^{\circ}}_{\mathcal{T}'})^{\circ}\]
LaTeX source
\[
\widehat{A}_{\mathcal{T}} \simeq (\widehat{A^{\circ}}_{\mathcal{T}'})^{\circ}
\]\[\underline{\mathrm{Hom}}_{\mathcal{T}'}(C, C') \longrightarrow \underline{\mathrm{Hom}}(A, C')\]
LaTeX source
\[
\underline{\mathrm{Hom}}_{\mathcal{T}'}(C, C') \longrightarrow \underline{\mathrm{Hom}}(A, C')
\]\[\underline{\mathrm{Hom}}_{add}(C, C') \simeq \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') .\]
LaTeX source
\[
\underline{\mathrm{Hom}}_{add}(C, C') \simeq \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') .
\]\[C \xrightarrow{\text{incl}} \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A^{\circ}, \mathrm{Ab})\]
LaTeX source
\[
C \xrightarrow{\text{incl}} \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A^{\circ}, \mathrm{Ab})
\]\[\underline{\mathrm{Hom}}_{add}(D, C') \xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C')\]
LaTeX source
\[
\underline{\mathrm{Hom}}_{add}(D, C') \xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C')
\]\[\underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') \xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{add}(D, C')\]
LaTeX source
\[
\underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') \xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{add}(D, C')
\]\[\begin{array}{cl}
D & \times\ \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') \longrightarrow C' \\
\cap & \\
\underline{\mathrm{Hom}}_{\frac{1}{2}add}(A^{\circ}, \mathrm{Ab}) &
\end{array}\]
LaTeX source
\[
\begin{array}{cl}
D & \times\ \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') \longrightarrow C' \\
\cap & \\
\underline{\mathrm{Hom}}_{\frac{1}{2}add}(A^{\circ}, \mathrm{Ab}) &
\end{array}
\]\[(F, G) \longmapsto F \otimes_{C} G\]
LaTeX source
\[
(F, G) \longmapsto F \otimes_{C} G
\]\[C \xrightarrow{\ \approx\ } D'^{\circ} \subset \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, \mathrm{Ab})^{\circ} ,\]
LaTeX source
\[
C \xrightarrow{\ \approx\ } D'^{\circ} \subset \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, \mathrm{Ab})^{\circ} ,
\]\[\underline{\mathrm{Hom}}_{add}(D'^{\circ}, C') \xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C')\]
LaTeX source
\[
\underline{\mathrm{Hom}}_{add}(D'^{\circ}, C') \xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C')
\]\[\begin{array}{cl}
D'^{\circ} & \times\ \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') \longrightarrow C' \\
\cap & \\
\underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, \mathrm{Ab})^{\circ} &
\end{array}\]
LaTeX source
\[
\begin{array}{cl}
D'^{\circ} & \times\ \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, C') \longrightarrow C' \\
\cap & \\
\underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, \mathrm{Ab})^{\circ} &
\end{array}
\]\[\underline{\mathrm{Hom}}_{C}(F, G) .\]
LaTeX source
\[
\underline{\mathrm{Hom}}_{C}(F, G) .
\]\[\mathcal{S}^{\circ} \longrightarrow \struck{\ill{}}\ \underline{\mathrm{Hom}}_{add}(C, \mathrm{Ab})\]
LaTeX source
\[
\mathcal{S}^{\circ} \longrightarrow \struck{\ill{}}\ \underline{\mathrm{Hom}}_{add}(C, \mathrm{Ab})
\]\[\mathcal{S}^{\circ} \longrightarrow \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, \mathrm{Ab})\]
LaTeX source
\[
\mathcal{S}^{\circ} \longrightarrow \underline{\mathrm{Hom}}_{\frac{1}{2}add}(A, \mathrm{Ab})
\]\[\Phi^{i} = \dot{\Lambda}\Phi_{\bullet} \ (i \geq 1), \qquad \Psi^{j} = \Lambda T_{\bullet} \ (j \geq 0).\]
LaTeX source
\[
\Phi^{i} = \dot{\Lambda}\Phi_{\bullet} \ (i \geq 1), \qquad \Psi^{j} = \Lambda T_{\bullet} \ (j \geq 0).
\]\[M_{\bullet} \longmapsto F_{X_\bullet}(M_\bullet) = F(M_\bullet) = \underline{\mathrm{Hom}}_{\widehat{\Delta}}(X_\bullet, M_\bullet)\]
LaTeX source
\[
M_{\bullet} \longmapsto F_{X_\bullet}(M_\bullet) = F(M_\bullet) = \underline{\mathrm{Hom}}_{\widehat{\Delta}}(X_\bullet, M_\bullet)
\]\[\begin{cases}
F(M_\bullet) \otimes_{k} F(N_\bullet) \longrightarrow F(M_\bullet \otimes N_\bullet) \\
\Gamma^{i} F(M_\bullet) \longrightarrow F(\Gamma^{i}(M_\bullet))
\end{cases}\]
LaTeX source
\[
\begin{cases}
F(M_\bullet) \otimes_{k} F(N_\bullet) \longrightarrow F(M_\bullet \otimes N_\bullet) \\
\Gamma^{i} F(M_\bullet) \longrightarrow F(\Gamma^{i}(M_\bullet))
\end{cases}
\]\[0 \to \mathbb{C}^{0} \xrightarrow{d^{0}} \mathbb{C}^{1} \xrightarrow{d^{1}} \cdots \to \mathbb{C}^{i}
\xrightarrow{d^{i}} \mathbb{C}^{i+1} \to \cdots\]
LaTeX source
\[
0 \to \mathbb{C}^{0} \xrightarrow{d^{0}} \mathbb{C}^{1} \xrightarrow{d^{1}} \cdots \to \mathbb{C}^{i}
\xrightarrow{d^{i}} \mathbb{C}^{i+1} \to \cdots
\]\[\operatorname{Hom}(\mathbb{C}^{i}, \mathbb{C}^{j}) \simeq
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
k & \text{si } j = i, \text{ base } \mathrm{id}_{\mathbb{C}^{i}} \\
k & \text{si } j = i+1, \text{ base } d^{i}
\end{cases}\]
LaTeX source
\[
\operatorname{Hom}(\mathbb{C}^{i}, \mathbb{C}^{j}) \simeq
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
k & \text{si } j = i, \text{ base } \mathrm{id}_{\mathbb{C}^{i}} \\
k & \text{si } j = i+1, \text{ base } d^{i}
\end{cases}
\]\[\mathbb{C}^{i} = C^{i}_{\bullet} = \lbrace \cdots 0 \to 0 \to
\underset{\text{degré } i+1}{k} \xrightarrow{\ \mathrm{id}\ } \underset{\text{degré } i}{k} \to 0 \to \cdots \to 0 \rbrace\]
LaTeX source
\[
\mathbb{C}^{i} = C^{i}_{\bullet} = \lbrace \cdots 0 \to 0 \to
\underset{\text{degré } i+1}{k} \xrightarrow{\ \mathrm{id}\ } \underset{\text{degré } i}{k} \to 0 \to \cdots \to 0 \rbrace
\]\[\underline{\mathrm{Hom}}(C^{i}_{*}, C^{j}_{*}) =
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
k & \text{si } j = i, \text{ base } \mathrm{id}_{C^{i}_{*}} \\
k & \text{si } j = i+1, \text{ base } d^{i} = T_{*}\wedge
\end{cases}\]
LaTeX source
\[
\underline{\mathrm{Hom}}(C^{i}_{*}, C^{j}_{*}) =
\begin{cases}
0 & \text{si } j \neq i, i+1 \\
k & \text{si } j = i, \text{ base } \mathrm{id}_{C^{i}_{*}} \\
k & \text{si } j = i+1, \text{ base } d^{i} = T_{*}\wedge
\end{cases}
\]\[\underline{\mathrm{Hom}}(\mathcal{C}, \mathcal{B}) \xrightarrow[\ \approx\ ]{\text{rest}}
\underline{\mathrm{Hom}}_{k\text{-lin}}(\mathcal{A}, \mathcal{B})\]
LaTeX source
\[
\underline{\mathrm{Hom}}(\mathcal{C}, \mathcal{B}) \xrightarrow[\ \approx\ ]{\text{rest}}
\underline{\mathrm{Hom}}_{k\text{-lin}}(\mathcal{A}, \mathcal{B})
\]\[\mathcal{C} \longrightarrow \mathit{Ab}_k, \qquad M_* \longmapsto \operatorname{Hom}_{\mathcal{C}}(L_*, M_*) ;\]
LaTeX source
\[
\mathcal{C} \longrightarrow \mathit{Ab}_k, \qquad M_* \longmapsto \operatorname{Hom}_{\mathcal{C}}(L_*, M_*) ;
\]\[\mathcal{C}^{\circ} \hookrightarrow \underline{\mathrm{Hom}}_{k\text{-lin}}(\mathcal{C}, \mathit{Ab}_k)
\xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{\mathrm{ac}}(\mathcal{A}, \mathit{Ab}_k)
\xrightarrow{\ \approx\ } \mathit{Ab}_k^{\bullet}\]
LaTeX source
\[
\mathcal{C}^{\circ} \hookrightarrow \underline{\mathrm{Hom}}_{k\text{-lin}}(\mathcal{C}, \mathit{Ab}_k)
\xrightarrow{\ \approx\ } \underline{\mathrm{Hom}}_{\mathrm{ac}}(\mathcal{A}, \mathit{Ab}_k)
\xrightarrow{\ \approx\ } \mathit{Ab}_k^{\bullet}
\]\[\mathcal{C}^{\circ} \xhookrightarrow{\ \alpha^{\bullet}\ } \mathit{Ab}_k^{\bullet}\]
LaTeX source
\[
\mathcal{C}^{\circ} \xhookrightarrow{\ \alpha^{\bullet}\ } \mathit{Ab}_k^{\bullet}
\]\[0 \to \operatorname{Hom}(L_*, C^0_*) \to \operatorname{Hom}(L_*, C^1_*) \longrightarrow \cdots ,\]
LaTeX source
\[
0 \to \operatorname{Hom}(L_*, C^0_*) \to \operatorname{Hom}(L_*, C^1_*) \longrightarrow \cdots ,
\]\[\alpha^{\bullet}(C^i_*) = \bigl( 0 \to 0 \to \cdots 0 \to \underset{\text{degré } i}{k} \xrightarrow{\ \mathrm{id}\ } \underset{\text{degré } i+1}{k} \to 0 \to \cdots \bigr) = K^i_{\bullet}\]
LaTeX source
\[
\alpha^{\bullet}(C^i_*) = \bigl( 0 \to 0 \to \cdots 0 \to \underset{\text{degré } i}{k} \xrightarrow{\ \mathrm{id}\ } \underset{\text{degré } i+1}{k} \to 0 \to \cdots \bigr) = K^i_{\bullet}
\]\[\begin{array}{ccc}
\alpha^{\bullet}(C^i_*) & \xleftarrow{\ \alpha^{\bullet}(d^i)\ } & \alpha^{\bullet}(C^{i+1}_*) \\
\| & & \| \\
K^i_{\bullet} & \longrightarrow & K^{i+1}_{\bullet}
\end{array}
\qquad k \to 0\]
LaTeX source
\[
\begin{array}{ccc}
\alpha^{\bullet}(C^i_*) & \xleftarrow{\ \alpha^{\bullet}(d^i)\ } & \alpha^{\bullet}(C^{i+1}_*) \\
\| & & \| \\
K^i_{\bullet} & \longrightarrow & K^{i+1}_{\bullet}
\end{array}
\qquad k \to 0
\]\[\mathcal{C}^{\circ} \longrightarrow \mathit{Ab}_k^{*}, \qquad M_* \longmapsto \operatorname{Hom}_{\mathcal{C}}(M_*, L_*)\]
LaTeX source
\[
\mathcal{C}^{\circ} \longrightarrow \mathit{Ab}_k^{*}, \qquad M_* \longmapsto \operatorname{Hom}_{\mathcal{C}}(M_*, L_*)
\]\[\begin{array}{ccc}
\mathcal{C} \hookrightarrow \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{C}^{\circ}, \mathit{Ab}_k^{\bullet}) & \xrightarrow{\ \sim\ } & \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{C}, \mathit{Ab}_k^{\circ})^{\circ} \\
\downarrow{\scriptstyle \wr} & & \downarrow{\scriptstyle \wr} \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{A}^{\circ}, \mathit{Ab}_k^{\bullet}) & \xrightarrow{\ \sim\ } & \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{A}, \mathit{Ab}_k^{\circ})^{\circ} \\
\downarrow & & \downarrow{\scriptstyle \wr} \\
\lbrace \mathit{Ab}_k^{*} \rbrace^{\circ} & & (\mathit{Ab}_k^{\circ *})^{\circ}
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
\mathcal{C} \hookrightarrow \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{C}^{\circ}, \mathit{Ab}_k^{\bullet}) & \xrightarrow{\ \sim\ } & \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{C}, \mathit{Ab}_k^{\circ})^{\circ} \\
\downarrow{\scriptstyle \wr} & & \downarrow{\scriptstyle \wr} \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{A}^{\circ}, \mathit{Ab}_k^{\bullet}) & \xrightarrow{\ \sim\ } & \underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{A}, \mathit{Ab}_k^{\circ})^{\circ} \\
\downarrow & & \downarrow{\scriptstyle \wr} \\
\lbrace \mathit{Ab}_k^{*} \rbrace^{\circ} & & (\mathit{Ab}_k^{\circ *})^{\circ}
\end{array}
\]\[\mathcal{C}^{\circ} \xhookrightarrow{\ \beta^{\bullet}\ } \mathit{Ab}_{k*}\]
LaTeX source
\[
\mathcal{C}^{\circ} \xhookrightarrow{\ \beta^{\bullet}\ } \mathit{Ab}_{k*}
\]\[\beta_{\bullet}(\mathbb{C}^i) = \bigl( \cdots 0 \to \cdots 0 \to \underset{i-1}{k} \to \underset{i}{k} \to 0 \to \cdots \to 0 \bigr) \quad \text{pour } i \geqslant 1\]
LaTeX source
\[
\beta_{\bullet}(\mathbb{C}^i) = \bigl( \cdots 0 \to \cdots 0 \to \underset{i-1}{k} \to \underset{i}{k} \to 0 \to \cdots \to 0 \bigr) \quad \text{pour } i \geqslant 1
\]\[\beta_{\bullet}(C^0) = \bigl( 0 \to \underset{\text{degré } 0}{k} \to 0 \bigr)\]
LaTeX source
\[
\beta_{\bullet}(C^0) = \bigl( 0 \to \underset{\text{degré } 0}{k} \to 0 \bigr)
\]\[(d^{i+1}_{i}) = \mathrm{id}_k \qquad i \geqslant \uncertain{0}\]
LaTeX source
\[
(d^{i+1}_{i}) = \mathrm{id}_k \qquad i \geqslant \uncertain{0}
\]\[M \longmapsto \check{M}, \qquad \mathit{Ab}_k^{\circ} \longrightarrow \mathit{Ab}_k\]
LaTeX source
\[
M \longmapsto \check{M}, \qquad \mathit{Ab}_k^{\circ} \longrightarrow \mathit{Ab}_k
\]\[\begin{array}{ccc}
\operatorname{Hom}(\mathbb{C}^{n}, x) \times \operatorname{Hom}(x, \mathbb{C}^{n+1}) & \longrightarrow & \operatorname{Hom}(\mathbb{C}^{n}, \mathbb{C}^{n+1}) \simeq k \\
(u, v) & \longmapsto & vu
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
\operatorname{Hom}(\mathbb{C}^{n}, x) \times \operatorname{Hom}(x, \mathbb{C}^{n+1}) & \longrightarrow & \operatorname{Hom}(\mathbb{C}^{n}, \mathbb{C}^{n+1}) \simeq k \\
(u, v) & \longmapsto & vu
\end{array}
\]\[F_{C^{\bullet}} : \mathbb{K} \longrightarrow \mathcal{B}\]
LaTeX source
\[
F_{C^{\bullet}} : \mathbb{K} \longrightarrow \mathcal{B}
\]\[F(x) \xrightarrow{\ \sim\ } \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), F(\mathbb{C}^{\bullet}))
= \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), \mathcal{C}^{\bullet}_F),\]
LaTeX source
\[
F(x) \xrightarrow{\ \sim\ } \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), F(\mathbb{C}^{\bullet}))
= \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), \mathcal{C}^{\bullet}_F),
\]\[\operatorname{Hom}(b, \operatorname{Hom}^{\bullet}(K^{\bullet}, \mathcal{C}^{\bullet}))
\simeq \operatorname{Hom}_{\mathcal{B}^{\bullet}}(b \otimes_k K^{\bullet}; C^{\bullet}) ;\]
LaTeX source
\[
\operatorname{Hom}(b, \operatorname{Hom}^{\bullet}(K^{\bullet}, \mathcal{C}^{\bullet}))
\simeq \operatorname{Hom}_{\mathcal{B}^{\bullet}}(b \otimes_k K^{\bullet}; C^{\bullet}) ;
\]\[\prod_i \operatorname{Hom}_k(K^i, C^i) \xrightarrow{\ \delta\ } \prod_i \operatorname{Hom}_k(K^i, C^{i+1})\]
LaTeX source
\[
\prod_i \operatorname{Hom}_k(K^i, C^i) \xrightarrow{\ \delta\ } \prod_i \operatorname{Hom}_k(K^i, C^{i+1})
\]\[(1) \qquad F(x) \longrightarrow \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), C^{\bullet}_F)
= \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), F(\mathbb{C}^{\bullet}))\]
LaTeX source
\[
(1) \qquad F(x) \longrightarrow \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), C^{\bullet}_F)
= \operatorname{Hom}^{\bullet}_k(\alpha^{\bullet}(x), F(\mathbb{C}^{\bullet}))
\]\[(1') \qquad F(x) \otimes \alpha^{\bullet}(x) \longrightarrow C^{\bullet} = F(\mathbb{C}^{\bullet})\]
LaTeX source
\[
(1') \qquad F(x) \otimes \alpha^{\bullet}(x) \longrightarrow C^{\bullet} = F(\mathbb{C}^{\bullet})
\]\[(1'') \qquad \underset{\in \mathcal{B}}{F(x)} \otimes \underset{\in \mathit{Ab}_k^{\bullet}}{\operatorname{Hom}(x, \mathbb{C}^{\bullet})} \xrightarrow{\ \mathrm{can}\ } F(\mathbb{C}^{\bullet})\]
LaTeX source
\[
(1'') \qquad \underset{\in \mathcal{B}}{F(x)} \otimes \underset{\in \mathit{Ab}_k^{\bullet}}{\operatorname{Hom}(x, \mathbb{C}^{\bullet})} \xrightarrow{\ \mathrm{can}\ } F(\mathbb{C}^{\bullet})
\]\[F(x) \otimes \operatorname{Hom}(x, C) \longrightarrow F(C)\]
LaTeX source
\[
F(x) \otimes \operatorname{Hom}(x, C) \longrightarrow F(C)
\]\[(2) \qquad F(\mathbb{C}^i) \longrightarrow \operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i, F(\mathbb{C}^{\bullet}))\]
LaTeX source
\[
(2) \qquad F(\mathbb{C}^i) \longrightarrow \operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i, F(\mathbb{C}^{\bullet}))
\]\[\operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i, \underset{\in \operatorname{Ob} \mathcal{B}^{\bullet}}{\mathcal{C}^{\bullet}}) \simeq C^i\]
LaTeX source
\[
\operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i, \underset{\in \operatorname{Ob} \mathcal{B}^{\bullet}}{\mathcal{C}^{\bullet}}) \simeq C^i
\]\[\mathcal{B}_{\bullet} \underset{\mathrm{DP}^{*}}{\overset{\mathrm{ND}_{\bullet}}{\rightleftarrows}} \mathcal{B}_{*}
\qquad \text{donc aussi (en appliquant à } \mathcal{B}^{\circ}\text{)} \qquad
\mathcal{B}^{\bullet} \underset{\mathrm{DP}_{*}}{\overset{\mathrm{ND}^{\bullet}}{\rightleftarrows}} \mathcal{B}^{*}\]
LaTeX source
\[
\mathcal{B}_{\bullet} \underset{\mathrm{DP}^{*}}{\overset{\mathrm{ND}_{\bullet}}{\rightleftarrows}} \mathcal{B}_{*}
\qquad \text{donc aussi (en appliquant à } \mathcal{B}^{\circ}\text{)} \qquad
\mathcal{B}^{\bullet} \underset{\mathrm{DP}_{*}}{\overset{\mathrm{ND}^{\bullet}}{\rightleftarrows}} \mathcal{B}^{*}
\]\[\mathit{Ab}_{k\bullet} \xrightarrow{\ \approx\ } \mathit{Ab}_{k*}, \qquad \mathit{Ab}_k^{\bullet} \xrightarrow{\ \approx\ } \mathit{Ab}_k^{*}\]
LaTeX source
\[
\mathit{Ab}_{k\bullet} \xrightarrow{\ \approx\ } \mathit{Ab}_{k*}, \qquad \mathit{Ab}_k^{\bullet} \xrightarrow{\ \approx\ } \mathit{Ab}_k^{*}
\]\[\alpha_{*}(\mathbb{C}^i) \simeq \mathrm{DP}(C^{i}_{\bullet}) \simeq \overset{i+1}{\Lambda} \Phi_{*}\]
LaTeX source
\[
\alpha_{*}(\mathbb{C}^i) \simeq \mathrm{DP}(C^{i}_{\bullet}) \simeq \overset{i+1}{\Lambda} \Phi_{*}
\]\[\alpha^{*}(\mathbb{C}^i) \simeq \mathrm{DP}(C^{\bullet}_{i}) \simeq \overset{i+1}{\Lambda} \check{\Phi}_{*} \ )\]
LaTeX source
\[
\alpha^{*}(\mathbb{C}^i) \simeq \mathrm{DP}(C^{\bullet}_{i}) \simeq \overset{i+1}{\Lambda} \check{\Phi}_{*} \ )
\]\[\struck{\ill{}} \qquad \mathbb{C}^{*} = \mathrm{DP}(\mathbb{C}^{\bullet})\]
LaTeX source
\[
\struck{\ill{}} \qquad \mathbb{C}^{*} = \mathrm{DP}(\mathbb{C}^{\bullet})
\]\[\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathbb{K}, \mathcal{B}) \longrightarrow \mathcal{B}^{*}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathbb{K}, \mathcal{B}) \longrightarrow \mathcal{B}^{*}
\]\[0 \to L'_{\bullet} \to L_{\bullet} \to L''_{\bullet} \to 0\]
LaTeX source
\[
0 \to L'_{\bullet} \to L_{\bullet} \to L''_{\bullet} \to 0
\]\[0 \to L'_n \to L_n \to L''_n \to 0 \longrightarrow\]
LaTeX source
\[ 0 \to L'_n \to L_n \to L''_n \to 0 \longrightarrow \]
\[(*) \qquad 0 \to A' \xrightarrow{\ q\ } A \xrightarrow{\ p\ } A'' \to 0\]
LaTeX source
\[
(*) \qquad 0 \to A' \xrightarrow{\ q\ } A \xrightarrow{\ p\ } A'' \to 0
\]\[\left\lbrace
\begin{array}{l}
0 \to \operatorname{Hom}(B, A') \to \operatorname{Hom}(B, A) \to \operatorname{Hom}(B, A'') \\
0 \to \operatorname{Hom}(A'', B) \to \operatorname{Hom}(A, B) \to \operatorname{Hom}(A', B)
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
0 \to \operatorname{Hom}(B, A') \to \operatorname{Hom}(B, A) \to \operatorname{Hom}(B, A'') \\
0 \to \operatorname{Hom}(A'', B) \to \operatorname{Hom}(A, B) \to \operatorname{Hom}(A', B)
\end{array}
\right.
\]\[0 \to A \xrightarrow{\ \mathrm{id}\ } A \to 0 \to 0, \qquad 0 \to 0 \to A \xrightarrow{\ \mathrm{id}\ } A \to 0\]
LaTeX source
\[
0 \to A \xrightarrow{\ \mathrm{id}\ } A \to 0 \to 0, \qquad 0 \to 0 \to A \xrightarrow{\ \mathrm{id}\ } A \to 0
\]\[0 \to A \to B \to K \to 0 \qquad (\text{resp.\ } 0 \to P \to A \to B \to 0)\]
LaTeX source
\[
0 \to A \to B \to K \to 0 \qquad (\text{resp.\ } 0 \to P \to A \to B \to 0)
\]\[0 \to P \to A \to R \to 0, \qquad 0 \to R \to B \to Q \to 0\]
LaTeX source
\[ 0 \to P \to A \to R \to 0, \qquad 0 \to R \to B \to Q \to 0 \]
\[A \xrightarrow{\ u\ } B \xrightarrow{\ v\ } C\]
LaTeX source
\[
A \xrightarrow{\ u\ } B \xrightarrow{\ v\ } C
\]\[\begin{array}{l}
0 \to \operatorname{Ker} u \to A \to \operatorname{Coim} u \to 0 \\
\qquad\qquad \operatorname{Coim} u \simeq \operatorname{Im} u \\
\qquad 0 \to \operatorname{Im} u \to \operatorname{Ker} v \to H \to 0 \\
\qquad\qquad\qquad 0 \to \operatorname{Ker} v \to B \to \operatorname{Coim} v \to 0 \\
\qquad\qquad\qquad\qquad \operatorname{Coim} v \simeq \operatorname{Im} v \\
\qquad\qquad\qquad 0 \to \operatorname{Im} v \to C \to \operatorname{Coker} v \to 0
\end{array}\]
LaTeX source
\[
\begin{array}{l}
0 \to \operatorname{Ker} u \to A \to \operatorname{Coim} u \to 0 \\
\qquad\qquad \operatorname{Coim} u \simeq \operatorname{Im} u \\
\qquad 0 \to \operatorname{Im} u \to \operatorname{Ker} v \to H \to 0 \\
\qquad\qquad\qquad 0 \to \operatorname{Ker} v \to B \to \operatorname{Coim} v \to 0 \\
\qquad\qquad\qquad\qquad \operatorname{Coim} v \simeq \operatorname{Im} v \\
\qquad\qquad\qquad 0 \to \operatorname{Im} v \to C \to \operatorname{Coker} v \to 0
\end{array}
\]\[\underline{\operatorname{Hom}}_{k, \Sigma\text{-ex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \hookrightarrow \mathcal{B}^{\bullet}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{k, \Sigma\text{-ex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \hookrightarrow \mathcal{B}^{\bullet}
\]\[\underline{\operatorname{Hom}}_{k\text{-}\Sigma\text{ex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \hookrightarrow \mathcal{B}^{\bullet}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{k\text{-}\Sigma\text{ex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \hookrightarrow \mathcal{B}^{\bullet}
\]\[\to \Psi^{i}_{\bullet} = \struck{H^i(C^{\bullet})} \ k[i] \quad \text{placé en degré } i,\]
LaTeX source
\[
\to \Psi^{i}_{\bullet} = \struck{H^i(C^{\bullet})} \ k[i] \quad \text{placé en degré } i,
\]\[0 \to \Psi^0_{\bullet} \to C^0_{\bullet} \to \Psi^1_{\bullet} \to 0\]
LaTeX source
\[
0 \to \Psi^0_{\bullet} \to C^0_{\bullet} \to \Psi^1_{\bullet} \to 0
\]\[0 \to \Psi^1_{\bullet} \to C^1_{\bullet} \to \Psi^2_{\bullet} \to 0\]
LaTeX source
\[
0 \to \Psi^1_{\bullet} \to C^1_{\bullet} \to \Psi^2_{\bullet} \to 0
\]\[0 \to \Psi^2_{\bullet} \to C^2_{\bullet} \to \Psi^3_{\bullet} \to \cdots\]
LaTeX source
\[
0 \to \Psi^2_{\bullet} \to C^2_{\bullet} \to \Psi^3_{\bullet} \to \cdots
\]\[\operatorname{Fil}_i(L_{\bullet})_j = \left\lbrace
\begin{array}{ll}
L_j & \text{si } j \leqslant i \\
0 & \text{si } j > i
\end{array}
\right.\]
LaTeX source
\[
\operatorname{Fil}_i(L_{\bullet})_j = \left\lbrace
\begin{array}{ll}
L_j & \text{si } j \leqslant i \\
0 & \text{si } j > i
\end{array}
\right.
\]\[\struck{\operatorname{Fil}} \operatorname{Gr}_i(L_{\bullet}) \simeq L_i \ \text{placé en degré } i
\ \underset{\mathrm{can}}{\simeq} \ \Psi^i_{\bullet} \otimes L_i \ \underset{\text{non can}}{\simeq} \ (\Psi^i_{\bullet})^{f_i}\]
LaTeX source
\[
\struck{\operatorname{Fil}} \operatorname{Gr}_i(L_{\bullet}) \simeq L_i \ \text{placé en degré } i
\ \underset{\mathrm{can}}{\simeq} \ \Psi^i_{\bullet} \otimes L_i \ \underset{\text{non can}}{\simeq} \ (\Psi^i_{\bullet})^{f_i}
\]\[\begin{array}{ccc}
\mathcal{B}^{\bullet}_{\Sigma\text{-sex}} & \longrightarrow & \underline{\operatorname{Hom}}_{\Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \\
\cup & & \cup \\
\mathcal{B}^{\bullet}_{\Sigma\text{-ex}} & \longrightarrow & \underline{\operatorname{Hom}}_{\Sigma\text{-ex}}(\mathcal{P}_{\bullet}, \mathcal{B})
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
\mathcal{B}^{\bullet}_{\Sigma\text{-sex}} & \longrightarrow & \underline{\operatorname{Hom}}_{\Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \\
\cup & & \cup \\
\mathcal{B}^{\bullet}_{\Sigma\text{-ex}} & \longrightarrow & \underline{\operatorname{Hom}}_{\Sigma\text{-ex}}(\mathcal{P}_{\bullet}, \mathcal{B})
\end{array}
\]\[\begin{array}{ccc}
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \mathcal{B}) & \underset{\sigma}{\overset{\rho}{\rightleftarrows}} & \mathcal{B}^{\bullet} \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \widehat{\mathcal{B}}) & & \\
\downarrow & \nwarrow{\scriptstyle \sigma} & \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \mathcal{B}) & \xrightarrow{\ \rho\ } & \mathcal{B}^{\bullet}
\end{array}
\qquad \text{$\mathcal{B}$ cat.\ additive avec $\Sigma$ classe de suites exactes et $\Sigma$-noyaux}\]
LaTeX source
\[
\begin{array}{ccc}
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \mathcal{B}) & \underset{\sigma}{\overset{\rho}{\rightleftarrows}} & \mathcal{B}^{\bullet} \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \widehat{\mathcal{B}}) & & \\
\downarrow & \nwarrow{\scriptstyle \sigma} & \\
\underline{\operatorname{Hom}}_{k\text{-lin}}(\mathcal{P}_{\bullet}, \mathcal{B}) & \xrightarrow{\ \rho\ } & \mathcal{B}^{\bullet}
\end{array}
\qquad \text{$\mathcal{B}$ cat.\ additive avec $\Sigma$ classe de suites exactes et $\Sigma$-noyaux}
\]\[\underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B})
\underset{\sigma}{\overset{\rho}{\rightleftarrows}} \mathcal{B}^{\bullet}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B})
\underset{\sigma}{\overset{\rho}{\rightleftarrows}} \mathcal{B}^{\bullet}
\]\[\left\lbrace
\begin{array}{l}
\rho(F) = F(\mathbb{C}^{\bullet}) \\
\sigma(C^{\bullet})(L_{\bullet}) \simeq \operatorname{Hom}^{\bullet}_k(\check{L}_{\bullet}, C^{\bullet})
\end{array}
\right.\]
LaTeX source
\[
\left\lbrace
\begin{array}{l}
\rho(F) = F(\mathbb{C}^{\bullet}) \\
\sigma(C^{\bullet})(L_{\bullet}) \simeq \operatorname{Hom}^{\bullet}_k(\check{L}_{\bullet}, C^{\bullet})
\end{array}
\right.
\]\[\operatorname{Hom}(\rho(F), C^{\bullet}) \xrightarrow{\ \sim\ } \operatorname{Hom}(F, \sigma(C^{\bullet}))\]
LaTeX source
\[
\operatorname{Hom}(\rho(F), C^{\bullet}) \xrightarrow{\ \sim\ } \operatorname{Hom}(F, \sigma(C^{\bullet}))
\]\[\operatorname{Hom}(\rho(F), C^{\bullet}) \simeq \operatorname{Hom}(F(\mathbb{C}^{\bullet}), C^{\bullet})\]
LaTeX source
\[
\operatorname{Hom}(\rho(F), C^{\bullet}) \simeq \operatorname{Hom}(F(\mathbb{C}^{\bullet}), C^{\bullet})
\]\[F(L_{\bullet}) \longrightarrow \operatorname{Hom}^{\bullet}_k(\check{L}_{\bullet}, C^{\bullet})\]
LaTeX source
\[
F(L_{\bullet}) \longrightarrow \operatorname{Hom}^{\bullet}_k(\check{L}_{\bullet}, C^{\bullet})
\]\[\check{L}^{\bullet} \otimes_k F(L_{\bullet}) \longrightarrow C^{\bullet}\]
LaTeX source
\[
\check{L}^{\bullet} \otimes_k F(L_{\bullet}) \longrightarrow C^{\bullet}
\]\[F(L) \otimes L^i \simeq \operatorname{Hom}(L_{\bullet}, \mathbb{C}^i) \otimes F(L_{\bullet}) \longrightarrow F(\mathbb{C}^i)\]
LaTeX source
\[
F(L) \otimes L^i \simeq \operatorname{Hom}(L_{\bullet}, \mathbb{C}^i) \otimes F(L_{\bullet}) \longrightarrow F(\mathbb{C}^i)
\]\[F \xrightarrow{\ u\ } \sigma\rho F\]
LaTeX source
\[
F \xrightarrow{\ u\ } \sigma\rho F
\]\[\begin{array}{ccc}
\operatorname{Hom}(\rho F, C^{\bullet}) & \longrightarrow & \operatorname{Hom}(F, \sigma C^{\bullet}) \\
& \searrow & \nearrow{\scriptstyle \alpha \mapsto \alpha \circ u} \\
& & \operatorname{Hom}(\sigma\rho F, \sigma C^{\bullet})
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
\operatorname{Hom}(\rho F, C^{\bullet}) & \longrightarrow & \operatorname{Hom}(F, \sigma C^{\bullet}) \\
& \searrow & \nearrow{\scriptstyle \alpha \mapsto \alpha \circ u} \\
& & \operatorname{Hom}(\sigma\rho F, \sigma C^{\bullet})
\end{array}
\]\[\rho\sigma C^{\bullet} \longrightarrow C^{\bullet}\]
LaTeX source
\[
\rho\sigma C^{\bullet} \longrightarrow C^{\bullet}
\]\[\rho(\sigma(C^{\bullet})) = \sigma(C^{\bullet})(\mathbb{C}^{\bullet}) = \bigl(\underbrace{\sigma(C^{\bullet})(\mathbb{C}^i)}_{\operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i,\, C^{\bullet}) \,=\, C^i}\bigr)_i \simeq (C^i)_i \quad \text{ok.}\]
LaTeX source
\[
\rho(\sigma(C^{\bullet})) = \sigma(C^{\bullet})(\mathbb{C}^{\bullet}) = \bigl(\underbrace{\sigma(C^{\bullet})(\mathbb{C}^i)}_{\operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i,\, C^{\bullet}) \,=\, C^i}\bigr)_i \simeq (C^i)_i \quad \text{ok.}
\]\[\mathcal{B}^{\bullet} \xhookrightarrow{\ \sigma\ } \underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}} \subset \underline{\operatorname{Hom}}_{k\text{-lin}}\]
LaTeX source
\[
\mathcal{B}^{\bullet} \xhookrightarrow{\ \sigma\ } \underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}} \subset \underline{\operatorname{Hom}}_{k\text{-lin}}
\]\[F \xrightarrow{\ u\ } \sigma\rho(F)\]
LaTeX source
\[
F \xrightarrow{\ u\ } \sigma\rho(F)
\]\[F(\mathbb{C}^i_{\bullet}) \to \sigma(\rho(F))(\mathbb{C}^i_{\bullet}) = \operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i, \underset{= F(\mathbb{C}^{\bullet})}{\rho(F)}) \simeq F(\mathbb{C}^i_{\bullet})\]
LaTeX source
\[
F(\mathbb{C}^i_{\bullet}) \to \sigma(\rho(F))(\mathbb{C}^i_{\bullet}) = \operatorname{Hom}^{\bullet}_k(\mathbb{C}^{\bullet}_i, \underset{= F(\mathbb{C}^{\bullet})}{\rho(F)}) \simeq F(\mathbb{C}^i_{\bullet})
\]\[\begin{aligned}
F(\Psi^i_{\bullet}) \to \operatorname{Hom}^{\bullet}_k(\Psi^{\bullet}_i, F(\mathbb{C}^{\bullet})) &\simeq Z^i(F(\mathbb{C}^{\bullet})) \\
&\simeq \operatorname{Ker}(F(\mathbb{C}^i_{\bullet}) \to F(\mathbb{C}^{i+1}_{\bullet})) \\
&\simeq F(\underbrace{\operatorname{Ker}(\mathbb{C}^i \to \mathbb{C}^{i+1})}_{\Sigma\text{-Ker}\,!}) \simeq F(\Psi^i_{\bullet})
\end{aligned}\]
LaTeX source
\[
\begin{aligned}
F(\Psi^i_{\bullet}) \to \operatorname{Hom}^{\bullet}_k(\Psi^{\bullet}_i, F(\mathbb{C}^{\bullet})) &\simeq Z^i(F(\mathbb{C}^{\bullet})) \\
&\simeq \operatorname{Ker}(F(\mathbb{C}^i_{\bullet}) \to F(\mathbb{C}^{i+1}_{\bullet})) \\
&\simeq F(\underbrace{\operatorname{Ker}(\mathbb{C}^i \to \mathbb{C}^{i+1})}_{\Sigma\text{-Ker}\,!}) \simeq F(\Psi^i_{\bullet})
\end{aligned}
\]\[\left.
\begin{array}{c}
\cdots \to 0 \to L_p \xrightarrow{\ \mathrm{id}\ } L_p \to 0 \\
\uparrow{\scriptstyle d_{p+1}} \qquad \uparrow{\scriptstyle d_p} \\
\cdots \to L_{p+n} \to L_{p+n-1} \to \cdots \to L_{p+1} \to L_p \to 0 \to 0
\end{array}
\right\rbrace
\qquad
\begin{array}{c}
(\mathbb{C}^{p}_{\bullet})^{r} \\
\uparrow \\
L_{\bullet}
\end{array}\]
LaTeX source
\[
\left.
\begin{array}{c}
\cdots \to 0 \to L_p \xrightarrow{\ \mathrm{id}\ } L_p \to 0 \\
\uparrow{\scriptstyle d_{p+1}} \qquad \uparrow{\scriptstyle d_p} \\
\cdots \to L_{p+n} \to L_{p+n-1} \to \cdots \to L_{p+1} \to L_p \to 0 \to 0
\end{array}
\right\rbrace
\qquad
\begin{array}{c}
(\mathbb{C}^{p}_{\bullet})^{r} \\
\uparrow \\
L_{\bullet}
\end{array}
\]\[\begin{array}{ccccccccc}
0 \to & (\Psi^p_{\bullet})^r & \to & L_{\bullet} & \longrightarrow & L'_{\bullet} & \to 0 \\
& \| & & \downarrow & & \downarrow & \\
0 \to & (\Psi^p_{\bullet})^r & \to & (\mathbb{C}^p_{\bullet})^r & \longrightarrow & (\Psi^{p+1}_{\bullet})^r & \to 0
\end{array}\]
LaTeX source
\[
\begin{array}{ccccccccc}
0 \to & (\Psi^p_{\bullet})^r & \to & L_{\bullet} & \longrightarrow & L'_{\bullet} & \to 0 \\
& \| & & \downarrow & & \downarrow & \\
0 \to & (\Psi^p_{\bullet})^r & \to & (\mathbb{C}^p_{\bullet})^r & \longrightarrow & (\Psi^{p+1}_{\bullet})^r & \to 0
\end{array}
\]\[\underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \xrightarrow{\ \approx\ } \mathcal{B}^{\bullet}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \xrightarrow{\ \approx\ } \mathcal{B}^{\bullet}
\]\[\underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \xrightarrow[\rho]{\ \approx\ } \mathcal{B}^{\bullet}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{k, \Sigma\text{-sex}}(\mathcal{P}_{\bullet}, \mathcal{B}) \xrightarrow[\rho]{\ \approx\ } \mathcal{B}^{\bullet}
\]\[F(L_{\bullet}) \simeq \operatorname{Hom}^{\bullet}_k(L^{\bullet}, \underbrace{F(\mathbb{C}^{\bullet})}_{= C^{\bullet}})\]
LaTeX source
\[
F(L_{\bullet}) \simeq \operatorname{Hom}^{\bullet}_k(L^{\bullet}, \underbrace{F(\mathbb{C}^{\bullet})}_{= C^{\bullet}})
\]\[\operatorname{Hom}_{\bullet}(\lambda_{\bullet}, L_{\bullet}) \xrightarrow{\ \sim\ } \operatorname{Hom}(L^{\bullet}, \check{\lambda}_{\bullet})\]
LaTeX source
\[
\operatorname{Hom}_{\bullet}(\lambda_{\bullet}, L_{\bullet}) \xrightarrow{\ \sim\ } \operatorname{Hom}(L^{\bullet}, \check{\lambda}_{\bullet})
\]\[L \longmapsto \operatorname{Hom}(L, \mathbb{C}^0)^{\vee}\]
LaTeX source
\[
L \longmapsto \operatorname{Hom}(L, \mathbb{C}^0)^{\vee}
\]\[\begin{array}{ccc}
\operatorname{Hom}(\Psi^0, L) & \longleftarrow & \operatorname{Hom}(\mathbb{C}^0, L) \quad \bigl[\longleftarrow \operatorname{Hom}(\mathbb{C}^1, L)\bigr] \\
\wr & & \\
\operatorname{Hom}(L, \mathbb{C}^0)^{\vee} & &
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
\operatorname{Hom}(\Psi^0, L) & \longleftarrow & \operatorname{Hom}(\mathbb{C}^0, L) \quad \bigl[\longleftarrow \operatorname{Hom}(\mathbb{C}^1, L)\bigr] \\
\wr & & \\
\operatorname{Hom}(L, \mathbb{C}^0)^{\vee} & &
\end{array}
\]\[\operatorname{Hom}(\mathbb{C}^0, L) \times \operatorname{Hom}(L, \mathbb{C}^0) \longrightarrow \operatorname{Hom}(\mathbb{C}^0, \mathbb{C}^0) \simeq k, \qquad (u, v) \longmapsto vu\]
LaTeX source
\[
\operatorname{Hom}(\mathbb{C}^0, L) \times \operatorname{Hom}(L, \mathbb{C}^0) \longrightarrow \operatorname{Hom}(\mathbb{C}^0, \mathbb{C}^0) \simeq k, \qquad (u, v) \longmapsto vu
\]\[\operatorname{Hom}(\Psi^0, L) \longleftarrow \operatorname{Hom}(\mathbb{C}^0, L) \longleftarrow \operatorname{Hom}(\mathbb{C}^1, L)\]
LaTeX source
\[
\operatorname{Hom}(\Psi^0, L) \longleftarrow \operatorname{Hom}(\mathbb{C}^0, L) \longleftarrow \operatorname{Hom}(\mathbb{C}^1, L)
\]\[\Psi^0 \longrightarrow \mathbb{C}^0 \longrightarrow \mathbb{C}^1\]
LaTeX source
\[
\Psi^0 \longrightarrow \mathbb{C}^0 \longrightarrow \mathbb{C}^1
\]\[L + (\Psi^0)^r, \qquad r \in \mathbb{N},\ L \in \operatorname{Ob} \mathbb{K}\]
LaTeX source
\[
L + (\Psi^0)^r, \qquad r \in \mathbb{N},\ L \in \operatorname{Ob} \mathbb{K}
\]\[0 \to L' \to L \to L'' \qquad \bigl(\ldots L' \to L \to L''\bigr)\]
LaTeX source
\[ 0 \to L' \to L \to L'' \qquad \bigl(\ldots L' \to L \to L''\bigr) \]
\[\underline{\operatorname{Hom}}_{\Sigma\text{-sex}} \longrightarrow \mathcal{B}^{\bullet} \longrightarrow \underline{\operatorname{Hom}}_{\mathrm{sex}} \longrightarrow \underline{\operatorname{Hom}}_{\mathrm{sex}}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{\Sigma\text{-sex}} \longrightarrow \mathcal{B}^{\bullet} \longrightarrow \underline{\operatorname{Hom}}_{\mathrm{sex}} \longrightarrow \underline{\operatorname{Hom}}_{\mathrm{sex}}
\]\[\underline{\operatorname{Hom}}_{\Sigma\text{-ex}} \simeq \mathcal{B}^{\mathrm{rés}}\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{\Sigma\text{-ex}} \simeq \mathcal{B}^{\mathrm{rés}}
\]\[\operatorname{Ext}^1(L^{\bullet}, M) \Longleftarrow 0\]
LaTeX source
\[
\operatorname{Ext}^1(L^{\bullet}, M) \Longleftarrow 0
\]\[\underline{\operatorname{Hom}}_{\Sigma\text{-ex}}(\mathcal{P}, \mathcal{B}) \xrightarrow{\ \approx\ } \underline{\operatorname{Hom}}_{\Sigma\text{-ex}}(\mathcal{P}_0, \mathcal{B})\]
LaTeX source
\[
\underline{\operatorname{Hom}}_{\Sigma\text{-ex}}(\mathcal{P}, \mathcal{B}) \xrightarrow{\ \approx\ } \underline{\operatorname{Hom}}_{\Sigma\text{-ex}}(\mathcal{P}_0, \mathcal{B})
\]\[\begin{array}{l}
0 \to \Psi^0 \to \mathbb{C}^0 \to \Psi^1 \to 0 \\
\qquad 0 \to \Psi^1 \to \mathbb{C}^1 \to \Psi^2 \to 0 \\
\qquad\qquad 0 \to \Psi^2 \to \mathbb{C}^2 \to \Psi^3 \to 0 \\
\qquad\qquad\qquad \cdots
\end{array}\]
LaTeX source
\[
\begin{array}{l}
0 \to \Psi^0 \to \mathbb{C}^0 \to \Psi^1 \to 0 \\
\qquad 0 \to \Psi^1 \to \mathbb{C}^1 \to \Psi^2 \to 0 \\
\qquad\qquad 0 \to \Psi^2 \to \mathbb{C}^2 \to \Psi^3 \to 0 \\
\qquad\qquad\qquad \cdots
\end{array}
\]\[\Phi^i \to \Psi^j, \qquad \Psi^i \to \Phi^j, \qquad \Psi^i \to \Psi^j \qquad (\Phi^i \to \Phi^j \text{ déjà connus})\]
LaTeX source
\[
\Phi^i \to \Psi^j, \qquad \Psi^i \to \Phi^j, \qquad \Psi^i \to \Psi^j \qquad (\Phi^i \to \Phi^j \text{ déjà connus})
\]\[0 \to L^0 \to M \to N \to 0\]
LaTeX source
\[ 0 \to L^0 \to M \to N \to 0 \]
\[0 \to R \to L \to M \to 0\ )\]
LaTeX source
\[ 0 \to R \to L \to M \to 0\ ) \]
\[L \longmapsto L_{\bullet} = \operatorname{Hom}(\Phi^{\bullet}, L)\]
LaTeX source
\[
L \longmapsto L_{\bullet} = \operatorname{Hom}(\Phi^{\bullet}, L)
\]\[0 \to N \to A \to A' \to 0 ,\]
LaTeX source
\[ 0 \to N \to A \to A' \to 0 , \]
\[C^{\bullet} \longmapsto Z^{0}(C^{\bullet}) = H^{0}(C^{\bullet})
\quad \text{sur } \mathcal{B}^{\bullet},\]
LaTeX source
\[
C^{\bullet} \longmapsto Z^{0}(C^{\bullet}) = H^{0}(C^{\bullet})
\quad \text{sur } \mathcal{B}^{\bullet},
\]\[\mathrm{R}F(\Psi^0) \simeq F(\mathbb{C}^{\bullet})\]
LaTeX source
\[
\mathrm{R}F(\Psi^0) \simeq F(\mathbb{C}^{\bullet})
\]\[\mathcal{P}_{*} \supset \mathcal{Q}_{*} \supset \mathcal{P}_{0*} \supset
\mathbb{K}_{*}\]
LaTeX source
\[
\mathcal{P}_{*} \supset \mathcal{Q}_{*} \supset \mathcal{P}_{0*} \supset
\mathbb{K}_{*}
\]\[(L_{*} \otimes M_{*})_{[n]} = L_{[n]} \otimes M_{[n]} \ )\]
LaTeX source
\[
(L_{*} \otimes M_{*})_{[n]} = L_{[n]} \otimes M_{[n]} \ )
\]\[\underset{\mathbf{1}_{*}}{\Psi^{0}_{*}} \otimes
\underset{\mathbf{1}_{*}}{\Psi^{0}_{*}} \simeq
\underset{\mathbf{1}_{*}}{\Psi^{0}_{*}}
\quad (\Rightarrow \text{stabilité de } \mathcal{P}_0)\]
LaTeX source
\[
\underset{\mathbf{1}_{*}}{\Psi^{0}_{*}} \otimes
\underset{\mathbf{1}_{*}}{\Psi^{0}_{*}} \simeq
\underset{\mathbf{1}_{*}}{\Psi^{0}_{*}}
\quad (\Rightarrow \text{stabilité de } \mathcal{P}_0)
\]\[\Psi^{i}_{*} \otimes \Psi^{j}_{*} \simeq \Psi^{i+j}_{*} +
\text{élément de } \mathbb{K}
\quad (\Rightarrow \text{stabilité de } \mathcal{P}_{1*} = \mathcal{Q})\]
LaTeX source
\[
\Psi^{i}_{*} \otimes \Psi^{j}_{*} \simeq \Psi^{i+j}_{*} +
\text{élément de } \mathbb{K}
\quad (\Rightarrow \text{stabilité de } \mathcal{P}_{1*} = \mathcal{Q})
\]\[F : \mathcal{B}' \longrightarrow \mathcal{B}\]
LaTeX source
\[
F : \mathcal{B}' \longrightarrow \mathcal{B}
\]\[F(X) \otimes F(Y) \xrightarrow{\ u_{X,Y}\ } F(X \otimes Y)\]
LaTeX source
\[
F(X) \otimes F(Y) \xrightarrow{\ u_{X,Y}\ } F(X \otimes Y)
\]\[\mathbb{K} \xrightarrow{\ F\ } \mathcal{B}\]
LaTeX source
\[
\mathbb{K} \xrightarrow{\ F\ } \mathcal{B}
\]\[F(X) \otimes F(Y) \longrightarrow F(X \otimes Y)\]
LaTeX source
\[ F(X) \otimes F(Y) \longrightarrow F(X \otimes Y) \]
\[F(\mathbb{C}^{\bullet}) \otimes F(\mathbb{C}^{\bullet}) \longrightarrow
F(\mathbb{C}^{\bullet} * \mathbb{C}^{\bullet}) \quad \text{dans }
\mathcal{B}^{\bullet\bullet}\]
LaTeX source
\[
F(\mathbb{C}^{\bullet}) \otimes F(\mathbb{C}^{\bullet}) \longrightarrow
F(\mathbb{C}^{\bullet} * \mathbb{C}^{\bullet}) \quad \text{dans }
\mathcal{B}^{\bullet\bullet}
\]\[\mathbb{C}^{i}_{\bullet} * \mathbb{C}^{j}_{\bullet} =
\mathrm{N}_{\bullet}(\mathbb{C}^{i}_{*} \otimes
\mathbb{C}^{j}_{*})\]
LaTeX source
\[
\mathbb{C}^{i}_{\bullet} * \mathbb{C}^{j}_{\bullet} =
\mathrm{N}_{\bullet}(\mathbb{C}^{i}_{*} \otimes
\mathbb{C}^{j}_{*})
\]\[\mathbb{F}^{ij} \qquad F(\mathbb{C}^{i}) \otimes_{\mathcal{B}}
F(\mathbb{C}^{j}) \longrightarrow
\underline{\mathrm{Hom}}_{k}\bigl((\mathbb{C}^{i}_{\bullet} *
\mathbb{C}^{j}_{\bullet})^{\vee}, C^{\bullet}\bigr)\]
LaTeX source
\[
\mathbb{F}^{ij} \qquad F(\mathbb{C}^{i}) \otimes_{\mathcal{B}}
F(\mathbb{C}^{j}) \longrightarrow
\underline{\mathrm{Hom}}_{k}\bigl((\mathbb{C}^{i}_{\bullet} *
\mathbb{C}^{j}_{\bullet})^{\vee}, C^{\bullet}\bigr)
\]\[\mathbb{F}^{ij} \qquad (C^{i} \underset{\mathcal{B}}{\otimes} C^{j})
\underset{k}{\otimes}
\underbrace{(C^{\bullet}_{i} * C^{\bullet}_{j})}_{\in\, \mathbb{K}^{\bullet}}
\longrightarrow C^{\bullet}\]
LaTeX source
\[
\mathbb{F}^{ij} \qquad (C^{i} \underset{\mathcal{B}}{\otimes} C^{j})
\underset{k}{\otimes}
\underbrace{(C^{\bullet}_{i} * C^{\bullet}_{j})}_{\in\, \mathbb{K}^{\bullet}}
\longrightarrow C^{\bullet}
\]\[\begin{align*}
(1^{*}) \qquad & C^{i} \underset{\mathcal{B}}{\otimes} C^{j}
\longrightarrow \underline{\mathrm{Hom}}^{*}_{k}(C^{*}_{i} \otimes
C^{*}_{j},\ C^{*}) \\
\text{ou encore} \qquad
(2^{*}) \qquad & (C^{i} \otimes_{\mathcal{B}} C^{j}) \otimes_{k}
\underbrace{(\overset{i}{\Lambda}\Phi^{*} \otimes
\overset{j}{\Lambda}\Phi^{*})}_{C^{*}_{ij}}
\xrightarrow{\ u^{ij}\ } C^{*}
\end{align*}\]
LaTeX source
\begin{align*}
(1^{*}) \qquad & C^{i} \underset{\mathcal{B}}{\otimes} C^{j}
\longrightarrow \underline{\mathrm{Hom}}^{*}_{k}(C^{*}_{i} \otimes
C^{*}_{j},\ C^{*}) \\
\text{ou encore} \qquad
(2^{*}) \qquad & (C^{i} \otimes_{\mathcal{B}} C^{j}) \otimes_{k}
\underbrace{(\overset{i}{\Lambda}\Phi^{*} \otimes
\overset{j}{\Lambda}\Phi^{*})}_{C^{*}_{ij}}
\xrightarrow{\ u^{ij}\ } C^{*}
\end{align*}\[\begin{array}{l}
0 \to L \to I' \to L'' \to 0 \\
\qquad\qquad 0 \to L' \to J'' \to L'' \to 0
\end{array}\]
LaTeX source
\[
\begin{array}{l}
0 \to L \to I' \to L'' \to 0 \\
\qquad\qquad 0 \to L' \to J'' \to L'' \to 0
\end{array}
\]\[F(X) \simeq \mathrm{R}^{0}F(L) \simeq \mathrm{Ker}\bigl(F(I) \to F(J)\bigr)\]
LaTeX source
\[
F(X) \simeq \mathrm{R}^{0}F(L) \simeq \mathrm{Ker}\bigl(F(I) \to F(J)\bigr)
\]\[\underline{\mathbf{1}}_{\mathcal{B}'} \longrightarrow
F(\Psi_0) \simeq Z^{0}(C^{\bullet})\]
LaTeX source
\[
\underline{\mathbf{1}}_{\mathcal{B}'} \longrightarrow
F(\Psi_0) \simeq Z^{0}(C^{\bullet})
\]\[\mathrm{Tot}^{\bullet}(\mathbb{C}^{\bullet}_{*} * \mathbb{C}^{\bullet}_{*})
\longrightarrow \mathbb{C}^{\bullet}_{*}\]
LaTeX source
\[
\mathrm{Tot}^{\bullet}(\mathbb{C}^{\bullet}_{*} * \mathbb{C}^{\bullet}_{*})
\longrightarrow \mathbb{C}^{\bullet}_{*}
\]\[\begin{array}{ccc}
F_{C^{\bullet}}(\mathbb{C}^{\bullet}_{*} * \mathbb{C}^{\bullet}_{*}) &
\longrightarrow & F(\mathbb{C}^{\bullet}_{*}) \\
\uparrow & & \Vert \\
F_{C^{\bullet}}(\mathbb{C}^{\bullet}) \otimes_{\mathcal{B}}
F_{C^{\bullet}}(\mathbb{C}^{\bullet}) & & C^{\bullet} \\
\Vert & & \\
C^{\bullet} \otimes C^{\bullet} & &
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
F_{C^{\bullet}}(\mathbb{C}^{\bullet}_{*} * \mathbb{C}^{\bullet}_{*}) &
\longrightarrow & F(\mathbb{C}^{\bullet}_{*}) \\
\uparrow & & \Vert \\
F_{C^{\bullet}}(\mathbb{C}^{\bullet}) \otimes_{\mathcal{B}}
F_{C^{\bullet}}(\mathbb{C}^{\bullet}) & & C^{\bullet} \\
\Vert & & \\
C^{\bullet} \otimes C^{\bullet} & &
\end{array}
\]\[C^{\bullet} \otimes_{\mathcal{B}} C^{\bullet} \longrightarrow C^{\bullet}\]
LaTeX source
\[
C^{\bullet} \otimes_{\mathcal{B}} C^{\bullet} \longrightarrow C^{\bullet}
\]\[\boxed{(f + g) \circ h = f \circ h + g \circ h}\]
LaTeX source
\[
\boxed{(f + g) \circ h = f \circ h + g \circ h}
\]\[f \circ (g + h) \neq f \circ g + f \circ h \ !\]
LaTeX source
\[ f \circ (g + h) \neq f \circ g + f \circ h \ ! \]
\[\mathbb{Z}^{\times} \longrightarrow \mathrm{End}(\mathcal{P}_1)
\longleftarrow \mathrm{End}(\mathcal{P})\]
LaTeX source
\[
\mathbb{Z}^{\times} \longrightarrow \mathrm{End}(\mathcal{P}_1)
\longleftarrow \mathrm{End}(\mathcal{P})
\]\[\mathcal{P}(X, Y) = \bigoplus_{n \geq 0} \mathcal{P}_n(X, Y)\]
LaTeX source
\[
\mathcal{P}(X, Y) = \bigoplus_{n \geq 0} \mathcal{P}_n(X, Y)
\]\[f_n \circ (\lambda\,\mathrm{id}_X) = \lambda^{n} f_n\]
LaTeX source
\[
f_n \circ (\lambda\,\mathrm{id}_X) = \lambda^{n} f_n
\]\[(*) \qquad \mathcal{P}(X_1 \times \cdots \times X_r, Y) =
\bigoplus_{n = (n_1, \ldots, n_r) \in \mathbb{N}^{r}}
\mathcal{P}_n(X_1, \ldots, X_r; Y)\]
LaTeX source
\[
(*) \qquad \mathcal{P}(X_1 \times \cdots \times X_r, Y) =
\bigoplus_{n = (n_1, \ldots, n_r) \in \mathbb{N}^{r}}
\mathcal{P}_n(X_1, \ldots, X_r; Y)
\]\[f \circ (\lambda_1 \mathrm{id}_{X_1}, \ldots, \lambda_r \mathrm{id}_{X_r})
= \lambda_1^{n_1} \cdots \lambda_r^{n_r} f\]
LaTeX source
\[
f \circ (\lambda_1 \mathrm{id}_{X_1}, \ldots, \lambda_r \mathrm{id}_{X_r})
= \lambda_1^{n_1} \cdots \lambda_r^{n_r} f
\]\[P = \sum_{n = (n_1, \ldots, n_r) \in \mathbb{N}^{r}} P_n\]
LaTeX source
\[
P = \sum_{n = (n_1, \ldots, n_r) \in \mathbb{N}^{r}} P_n
\]\[u(\lambda) \cdot f = \lambda_1^{n_1} \cdots \lambda_r^{n_r} f\]
LaTeX source
\[
u(\lambda) \cdot f = \lambda_1^{n_1} \cdots \lambda_r^{n_r} f
\]\[P_n = \lbrace f \in P \mid u(\lambda) f = \lambda_1^{n_1} \cdots
\lambda_r^{n_r} f \quad \forall \lambda \in \mathbb{Z}^{r} \rbrace\]
LaTeX source
\[
P_n = \lbrace f \in P \mid u(\lambda) f = \lambda_1^{n_1} \cdots
\lambda_r^{n_r} f \quad \forall \lambda \in \mathbb{Z}^{r} \rbrace
\]\[(\lambda^{n} - \lambda^{n'})\, g_{n'} = 0 \qquad \forall \lambda \in
\mathbb{Z}^{r}\]
LaTeX source
\[
(\lambda^{n} - \lambda^{n'})\, g_{n'} = 0 \qquad \forall \lambda \in
\mathbb{Z}^{r}
\]\[\begin{array}{ccccc}
X_1 & \times \cdots \times & X_r & \xrightarrow{\ (n_1, \ldots, n_r)\ } &
Y \\
\uparrow {\scriptstyle m^{1}_{1}, \ldots, m^{1}_{s_1}} & &
\uparrow {\scriptstyle m^{r}_{1}, \ldots, m^{r}_{s_r}} & & \\
Z^{1}_{1} \times \cdots \times Z^{1}_{s_1} & & Z^{r}_{1}, \ldots,
Z^{r}_{s_r} & &
\end{array}\]
LaTeX source
\[
\begin{array}{ccccc}
X_1 & \times \cdots \times & X_r & \xrightarrow{\ (n_1, \ldots, n_r)\ } &
Y \\
\uparrow {\scriptstyle m^{1}_{1}, \ldots, m^{1}_{s_1}} & &
\uparrow {\scriptstyle m^{r}_{1}, \ldots, m^{r}_{s_r}} & & \\
Z^{1}_{1} \times \cdots \times Z^{1}_{s_1} & & Z^{r}_{1}, \ldots,
Z^{r}_{s_r} & &
\end{array}
\]\[\mathcal{P}_0(0, Y) \longrightarrow \mathcal{P}_0(X, Y)\]
LaTeX source
\[
\mathcal{P}_0(0, Y) \longrightarrow \mathcal{P}_0(X, Y)
\]\[k \longrightarrow \mathrm{End}(\mathcal{P}_1) \longleftarrow
\mathrm{End}(\mathcal{P})\]
LaTeX source
\[
k \longrightarrow \mathrm{End}(\mathcal{P}_1) \longleftarrow
\mathrm{End}(\mathcal{P})
\]\[\mathcal{P}_1^{\circ} \times \mathcal{P}_1^{\circ} \times \mathcal{P}_1
\longrightarrow \mathit{Ab}, \qquad
(X, Y; Z) \longmapsto \mathcal{P}_{(1,1)}(X, Y; Z)\]
LaTeX source
\[
\mathcal{P}_1^{\circ} \times \mathcal{P}_1^{\circ} \times \mathcal{P}_1
\longrightarrow \mathit{Ab}, \qquad
(X, Y; Z) \longmapsto \mathcal{P}_{(1,1)}(X, Y; Z)
\]\[Y \longmapsto \mathcal{P}_n(X, Y)\]
LaTeX source
\[
Y \longmapsto \mathcal{P}_n(X, Y)
\]\[\mathbf{1} \otimes X \simeq X \quad \text{i.e.} \quad
\mathcal{P}_{(1,1)}(\mathbf{1}, X; Y) \simeq \mathcal{P}_1(X, Y)
\; (= \mathrm{Hom}_{\mathcal{P}_1}(X, Y)) \ ?\]
LaTeX source
\[
\mathbf{1} \otimes X \simeq X \quad \text{i.e.} \quad
\mathcal{P}_{(1,1)}(\mathbf{1}, X; Y) \simeq \mathcal{P}_1(X, Y)
\; (= \mathrm{Hom}_{\mathcal{P}_1}(X, Y)) \ ?
\]\[\mathcal{P}_{1,1}(\mathbf{1}, \mathbf{1}; Y) \simeq
\mathcal{P}_1(\mathbf{1}, Y) \ ?\]
LaTeX source
\[
\mathcal{P}_{1,1}(\mathbf{1}, \mathbf{1}; Y) \simeq
\mathcal{P}_1(\mathbf{1}, Y) \ ?
\]\[X \times Y \times Z \longrightarrow T \times Z\]
LaTeX source
\[ X \times Y \times Z \longrightarrow T \times Z \]
\[\underset{1,n}{\mathcal{P}}(T, Z; M) \longrightarrow
\mathcal{P}_{1,1,n}(X, Y, Z; M)\]
LaTeX source
\[
\underset{1,n}{\mathcal{P}}(T, Z; M) \longrightarrow
\mathcal{P}_{1,1,n}(X, Y, Z; M)
\]\[\mathcal{P}_{1}(T; M) \simeq \mathcal{P}_{1,1}(X, Y; M)\]
LaTeX source
\[
\mathcal{P}_{1}(T; M) \simeq \mathcal{P}_{1,1}(X, Y; M)
\]\[\mathcal{P}_1(X_1 \otimes \cdots \otimes X_r; M) \simeq
\mathcal{P}_{1, \ldots, 1}(X_1, \ldots, X_r; M)\]
LaTeX source
\[
\mathcal{P}_1(X_1 \otimes \cdots \otimes X_r; M) \simeq
\mathcal{P}_{1, \ldots, 1}(X_1, \ldots, X_r; M)
\]\[\Gamma^{\nu_1}(X_1) \otimes \cdots \otimes \Gamma^{\nu_r}(X_r)
\longleftarrow \Gamma^{\nu}(X_1, \ldots, X_r)\]
LaTeX source
\[
\Gamma^{\nu_1}(X_1) \otimes \cdots \otimes \Gamma^{\nu_r}(X_r)
\longleftarrow \Gamma^{\nu}(X_1, \ldots, X_r)
\]\[\mathcal{P}^{\mathfrak{g}}_{\nu}(X_1, \ldots, X_r; M) \longleftarrow
\mathcal{P}_{1, \ldots, 1}(\Gamma^{\nu_1} X_1, \ldots,
\Gamma^{\nu_r} X_r; M)\]
LaTeX source
\[
\mathcal{P}^{\mathfrak{g}}_{\nu}(X_1, \ldots, X_r; M) \longleftarrow
\mathcal{P}_{1, \ldots, 1}(\Gamma^{\nu_1} X_1, \ldots,
\Gamma^{\nu_r} X_r; M)
\]\[X_1 \xrightarrow{\ \gamma_{\nu_1}\ } \Gamma^{\nu_1} X_1, \quad \ldots,
\quad X_r \xrightarrow{\ \gamma_{\nu_r}\ } \Gamma^{\nu_r}(X_r)\]
LaTeX source
\[
X_1 \xrightarrow{\ \gamma_{\nu_1}\ } \Gamma^{\nu_1} X_1, \quad \ldots,
\quad X_r \xrightarrow{\ \gamma_{\nu_r}\ } \Gamma^{\nu_r}(X_r)
\]\[\Gamma^{n}(X_1 \times \cdots \times X_r) \xrightarrow{\ \sim\ }
\sum_{\substack{\nu = (\nu_i) \in \mathbb{N}^{r} \\ \sum \nu_i = n}}
\underbrace{\Gamma^{\nu_1}(X_1) \otimes \cdots \otimes
\Gamma^{\nu_r}(X_r)}_{\Gamma^{\nu_1 \ldots \nu_r}(X_1, \ldots, X_r)}\]
LaTeX source
\[
\Gamma^{n}(X_1 \times \cdots \times X_r) \xrightarrow{\ \sim\ }
\sum_{\substack{\nu = (\nu_i) \in \mathbb{N}^{r} \\ \sum \nu_i = n}}
\underbrace{\Gamma^{\nu_1}(X_1) \otimes \cdots \otimes
\Gamma^{\nu_r}(X_r)}_{\Gamma^{\nu_1 \ldots \nu_r}(X_1, \ldots, X_r)}
\]\[X \otimes \mathbf{1} = \Gamma^{1}(X) \otimes \Gamma^{0}(0) \simeq
\Gamma^{1,0}(X, 0) \simeq X\]
LaTeX source
\[
X \otimes \mathbf{1} = \Gamma^{1}(X) \otimes \Gamma^{0}(0) \simeq
\Gamma^{1,0}(X, 0) \simeq X
\]\[X \longmapsto \Gamma^{*}(X) = (\Gamma^{i}(X))_{i \geq 0}\]
LaTeX source
\[
X \longmapsto \Gamma^{*}(X) = (\Gamma^{i}(X))_{i \geq 0}
\]\[\mathcal{P}_1 \longrightarrow \mathrm{Grad}(\mathcal{P}_1)\]
LaTeX source
\[
\mathcal{P}_1 \longrightarrow \mathrm{Grad}(\mathcal{P}_1)
\]\[(*) \qquad \Gamma^{p}(\Gamma^{q}(X)) \longleftarrow \Gamma^{pq}(X)\]
LaTeX source
\[
(*) \qquad \Gamma^{p}(\Gamma^{q}(X)) \longleftarrow \Gamma^{pq}(X)
\]\[X \longrightarrow \Gamma^{p}(\Gamma^{q}(X))\]
LaTeX source
\[
X \longrightarrow \Gamma^{p}(\Gamma^{q}(X))
\]\[X \xrightarrow[\text{hom.\ univ.\ de degré } q]{\ e^{X}_{q}\ }
\Gamma^{q}(X)
\xrightarrow[\text{hom.\ univ.\ de degré } p]{\ e^{\Gamma^{q}(X)}_{p}\ }
\Gamma^{p}(\Gamma^{q}(X))\]
LaTeX source
\[
X \xrightarrow[\text{hom.\ univ.\ de degré } q]{\ e^{X}_{q}\ }
\Gamma^{q}(X)
\xrightarrow[\text{hom.\ univ.\ de degré } p]{\ e^{\Gamma^{q}(X)}_{p}\ }
\Gamma^{p}(\Gamma^{q}(X))
\]\[\Gamma^{1} X = X, \qquad \Gamma^{0} X = \mathbf{1}\]
LaTeX source
\[
\Gamma^{1} X = X, \qquad \Gamma^{0} X = \mathbf{1}
\]\[\Gamma^{*} : (\mathcal{P}_1, \oplus) \longrightarrow
(\mathcal{P}_1^{\mathrm{grad.}}, \otimes)\]
LaTeX source
\[
\Gamma^{*} : (\mathcal{P}_1, \oplus) \longrightarrow
(\mathcal{P}_1^{\mathrm{grad.}}, \otimes)
\]\[\Gamma^{*}(0) = \mathbf{1} \quad \text{i.e.} \quad
\Gamma^{i}(0) = \begin{cases} \mathbf{1} & \text{si } i = 0 \\
0 & \text{si } i > 0 \end{cases}\]
LaTeX source
\[
\Gamma^{*}(0) = \mathbf{1} \quad \text{i.e.} \quad
\Gamma^{i}(0) = \begin{cases} \mathbf{1} & \text{si } i = 0 \\
0 & \text{si } i > 0 \end{cases}
\]\[\Gamma^{*}(X \oplus Y) \simeq \Gamma^{*}(X) \otimes \Gamma^{*}(Y)\]
LaTeX source
\[
\Gamma^{*}(X \oplus Y) \simeq \Gamma^{*}(X) \otimes \Gamma^{*}(Y)
\]\[\Gamma^{i}(M) = \begin{cases} 0 & \text{si } i < 0 \\
\mathbf{1} & \text{si } i = 0 \\
M & \text{si } i = 1 \end{cases}\]
LaTeX source
\[
\Gamma^{i}(M) = \begin{cases} 0 & \text{si } i < 0 \\
\mathbf{1} & \text{si } i = 0 \\
M & \text{si } i = 1 \end{cases}
\]\[\Gamma^{pq} \xrightarrow{\ \alpha_{pq}\ } \Gamma^{p} \circ \Gamma^{q}\]
LaTeX source
\[
\Gamma^{pq} \xrightarrow{\ \alpha_{pq}\ } \Gamma^{p} \circ \Gamma^{q}
\]\[\Gamma^{pq}(X) \xrightarrow{\ \alpha^{X}_{pq}\ } \Gamma^{p}(\Gamma^{q}(X))\ )\]
LaTeX source
\[
\Gamma^{pq}(X) \xrightarrow{\ \alpha^{X}_{pq}\ } \Gamma^{p}(\Gamma^{q}(X))\ )
\]\[\begin{align*}
(**) \qquad
& \Gamma^{q} \xrightarrow{\ \alpha_{1,q} = \mathrm{id}\ }
\Gamma^{1} \circ \Gamma^{q} = \mathrm{id} \circ \Gamma^{q} = \Gamma^{q} \\
& \Gamma^{q} \xrightarrow{\ \alpha_{q,1} = \mathrm{id}\ }
\Gamma^{q} \circ \Gamma^{1} = \Gamma^{q} \circ \mathrm{id} = \Gamma^{q} \\[1ex]
(***) \qquad
& \Gamma^{0} \xrightarrow{\ \alpha_{0,q} = \mathrm{id}\ }
\Gamma^{0} \circ \Gamma^{q} = \Gamma^{0} \\
& \Gamma^{0} \xrightarrow[\ \simeq\ ]{\ \alpha_{q,0}\ }
\Gamma^{q} \circ \Gamma^{0} .
\end{align*}\]
LaTeX source
\begin{align*}
(**) \qquad
& \Gamma^{q} \xrightarrow{\ \alpha_{1,q} = \mathrm{id}\ }
\Gamma^{1} \circ \Gamma^{q} = \mathrm{id} \circ \Gamma^{q} = \Gamma^{q} \\
& \Gamma^{q} \xrightarrow{\ \alpha_{q,1} = \mathrm{id}\ }
\Gamma^{q} \circ \Gamma^{1} = \Gamma^{q} \circ \mathrm{id} = \Gamma^{q} \\[1ex]
(***) \qquad
& \Gamma^{0} \xrightarrow{\ \alpha_{0,q} = \mathrm{id}\ }
\Gamma^{0} \circ \Gamma^{q} = \Gamma^{0} \\
& \Gamma^{0} \xrightarrow[\ \simeq\ ]{\ \alpha_{q,0}\ }
\Gamma^{q} \circ \Gamma^{0} .
\end{align*}\[\mathcal{P}_p(X, Y) = \mathrm{Hom}(\Gamma^{p} X, Y)\]
LaTeX source
\[
\mathcal{P}_p(X, Y) = \mathrm{Hom}(\Gamma^{p} X, Y)
\]\[\begin{cases}
\mathcal{P}_1(X, Y) = \mathrm{Hom}_{\mathcal{P}_1}(X, Y) \\
\mathcal{P}_0(X, Y) = \mathrm{Hom}_{\mathcal{P}_1}(\mathbf{1}, Y)
\end{cases}\]
LaTeX source
\[
\begin{cases}
\mathcal{P}_1(X, Y) = \mathrm{Hom}_{\mathcal{P}_1}(X, Y) \\
\mathcal{P}_0(X, Y) = \mathrm{Hom}_{\mathcal{P}_1}(\mathbf{1}, Y)
\end{cases}
\]\[\begin{array}{ccc}
\mathcal{P}_p(X, Y) \times \mathcal{P}_q(Y, Z) & \longrightarrow &
\mathcal{P}_{pq}(X, Z) \\
(f, g) & \longmapsto & g \circ f
\end{array}\]
LaTeX source
\[
\begin{array}{ccc}
\mathcal{P}_p(X, Y) \times \mathcal{P}_q(Y, Z) & \longrightarrow &
\mathcal{P}_{pq}(X, Z) \\
(f, g) & \longmapsto & g \circ f
\end{array}
\]\[\Gamma^{p} X \xrightarrow{\ f\ } Y \qquad \Gamma^{q} Y \xrightarrow{\ g\ } Z\]
LaTeX source
\[
\Gamma^{p} X \xrightarrow{\ f\ } Y \qquad \Gamma^{q} Y \xrightarrow{\ g\ } Z
\]\[g \circ f = g\, \Gamma^{q}(f)\, \alpha^{X}_{q,p}\]
LaTeX source
\[
g \circ f = g\, \Gamma^{q}(f)\, \alpha^{X}_{q,p}
\]\[g \circ f = g \quad \text{si } g \text{ de degré } 0\]
LaTeX source
\[
g \circ f = g \quad \text{si } g \text{ de degré } 0
\]\[\mathbf{1} \xrightarrow[\ \simeq\ ]{\ \alpha_{q,0}\ } \Gamma^{q}(\mathbf{1})\]
LaTeX source
\[
\mathbf{1} \xrightarrow[\ \simeq\ ]{\ \alpha_{q,0}\ } \Gamma^{q}(\mathbf{1})
\]\[k \to \mathrm{End}(\mathrm{id}_{\mathcal{P}_1}) \quad
(\text{ou } \mathrm{id}_{\mathcal{P}})\]
LaTeX source
\[
k \to \mathrm{End}(\mathrm{id}_{\mathcal{P}_1}) \quad
(\text{ou } \mathrm{id}_{\mathcal{P}})
\]\[f \circ (\mathrm{id}_{\lambda_1} \times \cdots \times \mathrm{id}_{\lambda_r})
= \lambda_1^{\nu_1} \cdots \lambda_r^{\nu_r} f\]
LaTeX source
\[
f \circ (\mathrm{id}_{\lambda_1} \times \cdots \times \mathrm{id}_{\lambda_r})
= \lambda_1^{\nu_1} \cdots \lambda_r^{\nu_r} f
\]\[\Gamma^{p}(\Gamma^{q}(M)) \longrightarrow \Gamma^{pq}(M)\]
LaTeX source
\[
\Gamma^{p}(\Gamma^{q}(M)) \longrightarrow \Gamma^{pq}(M)
\]\[\mathcal{P}(X, Y) \times \mathcal{P}(Y, Z) \longrightarrow \mathcal{P}(X, Z)\]
LaTeX source
\[
\mathcal{P}(X, Y) \times \mathcal{P}(Y, Z) \longrightarrow \mathcal{P}(X, Z)
\]\[\mathcal{P}(Y, Z) \longrightarrow
(\text{Appl.\ } k\text{-polynomiales})(\mathcal{P}(X, Y), \mathcal{P}(X, Z))\]
LaTeX source
\[
\mathcal{P}(Y, Z) \longrightarrow
(\text{Appl.\ } k\text{-polynomiales})(\mathcal{P}(X, Y), \mathcal{P}(X, Z))
\]\[f \longmapsto \Gamma^{p}(f) : \mathrm{Hom}_1(X, Y) \longrightarrow
\mathrm{Hom}_1(\Gamma^{p} X, \Gamma^{p} Y)\]
LaTeX source
\[
f \longmapsto \Gamma^{p}(f) : \mathrm{Hom}_1(X, Y) \longrightarrow
\mathrm{Hom}_1(\Gamma^{p} X, \Gamma^{p} Y)
\]