Cote n° 56 · pages 2–34
· 63 displayed formulas · Lefschetz-Hodge - Vanishing cycles : notes manuscrites (s.d.), tiré à part (1962).
Inventory dating : 1962
Édition de démonstration
\[\cdots \to H^i(X_0) \to H^i(\overline{X}_1) \to \Phi^{i+1} \to H^{i+1}(X_0)
\to H^{i+1}(\overline{X}_1) \to \cdots\]
LaTeX source
\[
\cdots \to H^i(X_0) \to H^i(\overline{X}_1) \to \Phi^{i+1} \to H^{i+1}(X_0)
\to H^{i+1}(\overline{X}_1) \to \cdots
\]\[0 \to \Phi^{2\nu} \to H^{2\nu}(X_0) \to H^{2\nu}(\overline{X}_1) \to
\Phi^{2\nu+1} \to 0\]
LaTeX source
\[
0 \to \Phi^{2\nu} \to H^{2\nu}(X_0) \to H^{2\nu}(\overline{X}_1) \to
\Phi^{2\nu+1} \to 0
\]\[\left.
\begin{array}{ll}
H^{*}(X_0) : & 1, \xi_0, \ldots, \xi_0^{\nu-1}, [\xi_0^{\nu}, \lambda'_{\nu}],
\tfrac{1}{2}\xi_0^{\nu+1}, \ldots, \tfrac{1}{2}\xi_0^{2\nu-1} \\[4pt]
H^{*}(\overline{X}_1) : & 1, \xi_1, \ldots, \xi_1^{\nu-1}, \tfrac{1}{2}\xi_1^{\nu},
\tfrac{1}{2}\xi_1^{\nu+1}, \ldots, \tfrac{1}{2}\xi_1^{2\nu-1}
\end{array}
\right]
\quad
\begin{array}{l}
\xi_0 \mapsto \xi_1 \\
\lambda'_{\nu} \mapsto 0
\end{array}\]
LaTeX source
\[
\left.
\begin{array}{ll}
H^{*}(X_0) : & 1, \xi_0, \ldots, \xi_0^{\nu-1}, [\xi_0^{\nu}, \lambda'_{\nu}],
\tfrac{1}{2}\xi_0^{\nu+1}, \ldots, \tfrac{1}{2}\xi_0^{2\nu-1} \\[4pt]
H^{*}(\overline{X}_1) : & 1, \xi_1, \ldots, \xi_1^{\nu-1}, \tfrac{1}{2}\xi_1^{\nu},
\tfrac{1}{2}\xi_1^{\nu+1}, \ldots, \tfrac{1}{2}\xi_1^{2\nu-1}
\end{array}
\right]
\quad
\begin{array}{l}
\xi_0 \mapsto \xi_1 \\
\lambda'_{\nu} \mapsto 0
\end{array}
\]\[\left.
\begin{array}{ll}
H^{*}(X_0) : & 1, \xi_0, \ldots, \xi_0^{\nu-1}, \xi_0^{\nu},
\tfrac{1}{2}\xi_0^{\nu+1}, \ldots, \tfrac{1}{2}\xi_0^{2\nu} \\[4pt]
H^{*}(\overline{X}_1) : & 1, \xi_1, \ldots, \xi_1^{\nu-1}, [\xi_1^{\nu}, \lambda_{\nu}],
\tfrac{1}{2}\xi_1^{\nu+1}, \ldots, \tfrac{1}{2}\xi_1^{2\nu}
\end{array}
\right]
\quad \xi_0 \mapsto \xi_1\]
LaTeX source
\[
\left.
\begin{array}{ll}
H^{*}(X_0) : & 1, \xi_0, \ldots, \xi_0^{\nu-1}, \xi_0^{\nu},
\tfrac{1}{2}\xi_0^{\nu+1}, \ldots, \tfrac{1}{2}\xi_0^{2\nu} \\[4pt]
H^{*}(\overline{X}_1) : & 1, \xi_1, \ldots, \xi_1^{\nu-1}, [\xi_1^{\nu}, \lambda_{\nu}],
\tfrac{1}{2}\xi_1^{\nu+1}, \ldots, \tfrac{1}{2}\xi_1^{2\nu}
\end{array}
\right]
\quad \xi_0 \mapsto \xi_1
\]\[\cdots \to H^i(U_0) \to H^i(\overline{U}_1) \to \Phi^{i+1} \to H^{i+1}(X_0)
\to H^{i+1}(\overline{X}_1) \to \cdots\]
LaTeX source
\[
\cdots \to H^i(U_0) \to H^i(\overline{U}_1) \to \Phi^{i+1} \to H^{i+1}(X_0)
\to H^{i+1}(\overline{X}_1) \to \cdots
\]\[H^i(\overline{U}_1) \simeq \Phi^{i+1}\]
LaTeX source
\[
H^i(\overline{U}_1) \simeq \Phi^{i+1}
\]\[\Phi^i = 0 \ \text{si}\ i \neq n, \qquad
\Phi^n = H^{n-1}(\overline{U}_1)\]
LaTeX source
\[
\Phi^i = 0 \ \text{si}\ i \neq n, \qquad
\Phi^n = H^{n-1}(\overline{U}_1)
\]\[\struck{H^{i-1}(X) \to H^i(\widehat{C}) \to H^i(\widehat{C}') \to H^i(X) \to
H^{i+1}(\widehat{C})}\]
LaTeX source
\[
\struck{H^{i-1}(X) \to H^i(\widehat{C}) \to H^i(\widehat{C}') \to H^i(X) \to
H^{i+1}(\widehat{C})}
\]\[\begin{array}{ccccccc}
\cdots \to H^{i-1}(X) & \to & H^i(\widehat{C}) & \to & H^i(\widehat{C}') & \to & H^i(X) \to \cdots \\
\downarrow & & \downarrow & & \downarrow & & \| \\
H^{i-1}(X) & \to & H^i(C) & \to & H^i(C') & \xrightarrow{\text{bij}} & H^i(X)
\end{array}\]
LaTeX source
\[
\begin{array}{ccccccc}
\cdots \to H^{i-1}(X) & \to & H^i(\widehat{C}) & \to & H^i(\widehat{C}') & \to & H^i(X) \to \cdots \\
\downarrow & & \downarrow & & \downarrow & & \| \\
H^{i-1}(X) & \to & H^i(C) & \to & H^i(C') & \xrightarrow{\text{bij}} & H^i(X)
\end{array}
\]\[\boxed{H^i(C) = 0 \quad \text{si}\ i \neq 0}\]
LaTeX source
\[
\boxed{H^i(C) = 0 \quad \text{si}\ i \neq 0}
\]\[0 \to H^i(\widehat{C}) \to H^i(\widehat{C}') \to H^i(X) \to 0\]
LaTeX source
\[
0 \to H^i(\widehat{C}) \to H^i(\widehat{C}') \to H^i(X) \to 0
\]\[\boxed{H^i(\widehat{C}) \xleftarrow{\ \sim\ } H^{i-2}(X)(-1) \qquad i \neq 0}\]
LaTeX source
\[
\boxed{H^i(\widehat{C}) \xleftarrow{\ \sim\ } H^{i-2}(X)(-1) \qquad i \neq 0}
\]\[\boxed{H^{*}(X) \xrightarrow{\ \sim\ } H^{*}(\widehat{C} - a)}, \qquad
\widehat{C} - a = \widehat{E}, \qquad \widehat{C} - a - X = E\]
LaTeX source
\[
\boxed{H^{*}(X) \xrightarrow{\ \sim\ } H^{*}(\widehat{C} - a)}, \qquad
\widehat{C} - a = \widehat{E}, \qquad \widehat{C} - a - X = E
\]\[H^{i-2}(X)(-1) \to H^i(\widehat{E}) \to \boxed{H^i(E)} \to H^{i-1}(X)(-1)
\to H^{i+1}(\widehat{E})\]
LaTeX source
\[
H^{i-2}(X)(-1) \to H^i(\widehat{E}) \to \boxed{H^i(E)} \to H^{i-1}(X)(-1)
\to H^{i+1}(\widehat{E})
\]\[\boxed{
\begin{array}{lll}
i \leqslant n & H^i(E) \simeq P^i(X) & \text{mod torsion} \\
i \geqslant n+1 & H^i(E) \simeq P^{\uncertain{i-2}}(X)(-1) & \text{mod. de torsion}
\end{array}}\]
LaTeX source
\[
\boxed{
\begin{array}{lll}
i \leqslant n & H^i(E) \simeq P^i(X) & \text{mod torsion} \\
i \geqslant n+1 & H^i(E) \simeq P^{\uncertain{i-2}}(X)(-1) & \text{mod. de torsion}
\end{array}}
\]\[\cdots \to H^{i-1}(C) \to H^{i-1}(E) \to H^i_a(C) \to H^i(C) \to H^i(E) \to
H^{i+1}_a(C) \to \cdots\]
LaTeX source
\[
\cdots \to H^{i-1}(C) \to H^{i-1}(E) \to H^i_a(C) \to H^i(C) \to H^i(E) \to
H^{i+1}_a(C) \to \cdots
\]\[\to H^0_a(C) \to H^0(C) \xrightarrow{\text{bij}} H^0(E) \to H^1_a(C) \to
H^1(C) \to H^1(E) \to H^2_a(C) \to H^2(C)\]
LaTeX source
\[
\to H^0_a(C) \to H^0(C) \xrightarrow{\text{bij}} H^0(E) \to H^1_a(C) \to
H^1(C) \to H^1(E) \to H^2_a(C) \to H^2(C)
\]\[\boxed{
\begin{array}{l}
H^0_a(C) = H^1_a(C) = 0 \\
H^i_a(C) \simeq H^{i-1}(E) \quad \text{si}\ i \geqslant 2
\end{array}}\]
LaTeX source
\[
\boxed{
\begin{array}{l}
H^0_a(C) = H^1_a(C) = 0 \\
H^i_a(C) \simeq H^{i-1}(E) \quad \text{si}\ i \geqslant 2
\end{array}}
\]\[\underline{H^i(Q'^{\,n+1}) \xleftarrow{\ \sim\ } H^{i-2}(Q^n)(-1)} \quad
\text{si}\ i \neq 0 \quad \text{donc :}\]
LaTeX source
\[
\underline{H^i(Q'^{\,n+1}) \xleftarrow{\ \sim\ } H^{i-2}(Q^n)(-1)} \quad
\text{si}\ i \neq 0 \quad \text{donc :}
\]\[\begin{array}{ll}
\left(1, \xi, \ldots, \xi^{\nu}, \tfrac{1}{2}\xi^{\nu+1}, \ldots,
\tfrac{1}{2}\xi^{2\nu}\right) & \text{si}\ \boxed{n+1 = 2\nu} \\[6pt]
\left(1, \xi, \ldots, \xi^{\nu}, [\xi^{\nu+1}, \lambda'_{\nu+1}],
\tfrac{1}{2}\xi^{\nu+2}, \ldots, \tfrac{1}{2}\xi^{2\nu+1}\right) &
\text{si}\ \boxed{n+1 = 2\nu+1}
\end{array}\]
LaTeX source
\[
\begin{array}{ll}
\left(1, \xi, \ldots, \xi^{\nu}, \tfrac{1}{2}\xi^{\nu+1}, \ldots,
\tfrac{1}{2}\xi^{2\nu}\right) & \text{si}\ \boxed{n+1 = 2\nu} \\[6pt]
\left(1, \xi, \ldots, \xi^{\nu}, [\xi^{\nu+1}, \lambda'_{\nu+1}],
\tfrac{1}{2}\xi^{\nu+2}, \ldots, \tfrac{1}{2}\xi^{2\nu+1}\right) &
\text{si}\ \boxed{n+1 = 2\nu+1}
\end{array}
\]\[\left\{
\begin{array}{ll}
H^i(E) = 0 & \text{si}\ i \neq 0, 2\nu - 2, 4\nu - 1 \\[4pt]
H^{2(\nu-1)}(E) \simeq
\left\{ \begin{array}{ll} 0, & \ell \neq 2 \\ \mathbf{Z}/2\mathbf{Z} & \text{si}\ \ell = 2 \end{array} \right. \\[10pt]
H^{4\nu-1}(E) \simeq \mu(-2\nu)
\end{array}
\right.
\qquad \boxed{n + 1 = 2\nu} \ \ n = 2\nu - 1\]
LaTeX source
\[
\left\{
\begin{array}{ll}
H^i(E) = 0 & \text{si}\ i \neq 0, 2\nu - 2, 4\nu - 1 \\[4pt]
H^{2(\nu-1)}(E) \simeq
\left\{ \begin{array}{ll} 0, & \ell \neq 2 \\ \mathbf{Z}/2\mathbf{Z} & \text{si}\ \ell = 2 \end{array} \right. \\[10pt]
H^{4\nu-1}(E) \simeq \mu(-2\nu)
\end{array}
\right.
\qquad \boxed{n + 1 = 2\nu} \ \ n = 2\nu - 1
\]\[\left\{
\begin{array}{l}
H^i(E) = 0 \quad \text{si}\ i \neq (0), 2\nu - 1, 2\nu + 1, 4\nu + 1 \\
H^0(E) = \mu(0) \\
H^{2\nu-1}(E) \simeq \mu(-\nu+1) \\
H^{2\nu+1}(E) \simeq \mu(-\nu-1) \\
H^{\uncertain{4\nu+1}}(E) \simeq \mu(-2\nu-1)
\end{array}
\right.
\qquad \boxed{n + 1 = 2\nu + 1} \ \ n = 2\nu\]
LaTeX source
\[
\left\{
\begin{array}{l}
H^i(E) = 0 \quad \text{si}\ i \neq (0), 2\nu - 1, 2\nu + 1, 4\nu + 1 \\
H^0(E) = \mu(0) \\
H^{2\nu-1}(E) \simeq \mu(-\nu+1) \\
H^{2\nu+1}(E) \simeq \mu(-\nu-1) \\
H^{\uncertain{4\nu+1}}(E) \simeq \mu(-2\nu-1)
\end{array}
\right.
\qquad \boxed{n + 1 = 2\nu + 1} \ \ n = 2\nu
\]\[\cdots \to H^i(\widehat{C}) \to H^i(X) \to H^{i+1}_c(C) \to H^{i+1}(\widehat{C})
\to H^{i+1}(X)\]
LaTeX source
\[
\cdots \to H^i(\widehat{C}) \to H^i(X) \to H^{i+1}_c(C) \to H^{i+1}(\widehat{C})
\to H^{i+1}(X)
\]\[\boxed{H^{*}_a(C) \simeq H^{*}_c(C) \quad ??}\]
LaTeX source
\[
\boxed{H^{*}_a(C) \simeq H^{*}_c(C) \quad ??}
\]\[H^i_c(U^n, A) \simeq
\left\{
\begin{array}{ll}
0 & \text{si}\ i \neq n, 2n \\
A \otimes \nu'_n & \text{si}\ i = n \\
A \otimes \mu_m(-n) & \text{si}\ i = 2n
\end{array}
\right.
\quad (\text{si}\ n \neq 0)\]
LaTeX source
\[
H^i_c(U^n, A) \simeq
\left\{
\begin{array}{ll}
0 & \text{si}\ i \neq n, 2n \\
A \otimes \nu'_n & \text{si}\ i = n \\
A \otimes \mu_m(-n) & \text{si}\ i = 2n
\end{array}
\right.
\quad (\text{si}\ n \neq 0)
\]\[\nu'_n = \check{\nu}_n \otimes \mu_m(-n) =
\left\{
\begin{array}{ll}
\mu_m(-\tfrac{n}{2}) & n\ \text{pair} \\
\mu_m(-\tfrac{n+1}{2}) & n\ \text{impair}
\end{array}
\right.\]
LaTeX source
\[
\nu'_n = \check{\nu}_n \otimes \mu_m(-n) =
\left\{
\begin{array}{ll}
\mu_m(-\tfrac{n}{2}) & n\ \text{pair} \\
\mu_m(-\tfrac{n+1}{2}) & n\ \text{impair}
\end{array}
\right.
\]\[\nu'_n = \check{\nu}_n = \mu_m(-\tfrac{n}{2}) \quad n\ \text{pair}, \qquad
\nu'_n = \nu_n \otimes \mu_m(1) \quad n\ \text{impair}\]
LaTeX source
\[
\nu'_n = \check{\nu}_n = \mu_m(-\tfrac{n}{2}) \quad n\ \text{pair}, \qquad
\nu'_n = \nu_n \otimes \mu_m(1) \quad n\ \text{impair}
\]\[H^i(U^n, A) \simeq
\left\{
\begin{array}{ll}
0 & \text{si}\ i \neq 0, n \\
A \otimes \nu_n & \text{si}\ i = 0 \\
A & \text{si}\ i = n .
\end{array}
\right.\]
LaTeX source
\[
H^i(U^n, A) \simeq
\left\{
\begin{array}{ll}
0 & \text{si}\ i \neq 0, n \\
A \otimes \nu_n & \text{si}\ i = 0 \\
A & \text{si}\ i = n .
\end{array}
\right.
\]\[\nu_n \simeq \mu_m(-\tfrac{n}{2}) \ \text{si}\ n\ \text{pair}, \qquad
\nu_n \simeq \mu_m(-\tfrac{n \pm 1}{2}) \ \text{si}\ n\ \text{impair}\]
LaTeX source
\[
\nu_n \simeq \mu_m(-\tfrac{n}{2}) \ \text{si}\ n\ \text{pair}, \qquad
\nu_n \simeq \mu_m(-\tfrac{n \pm 1}{2}) \ \text{si}\ n\ \text{impair}
\]\[H^{i-1}(U^n) \to H^{i-2}(Q^{n-1})(-1) \to H^i(Q^n) \to H^i(U^n) \to
H^{i-1}(Q^{n-1})(-1) \to \cdots\]
LaTeX source
\[
H^{i-1}(U^n) \to H^{i-2}(Q^{n-1})(-1) \to H^i(Q^n) \to H^i(U^n) \to
H^{i-1}(Q^{n-1})(-1) \to \cdots
\]\[H^{i-2}(Q^{n-1})(-1) \xrightarrow{\alpha_*} H^i(Q^n) \ \text{est}\
\left|
\begin{array}{ll}
\text{surjectif} & \text{si}\ i \neq 0, n \\
\text{injectif} & \text{si}\ i \neq 1, n+1
\end{array}
\right|
\ \text{bijectif si}\ i \neq 0, \struck{\ill{}}\, n, n+1\]
LaTeX source
\[
H^{i-2}(Q^{n-1})(-1) \xrightarrow{\alpha_*} H^i(Q^n) \ \text{est}\
\left|
\begin{array}{ll}
\text{surjectif} & \text{si}\ i \neq 0, n \\
\text{injectif} & \text{si}\ i \neq 1, n+1
\end{array}
\right|
\ \text{bijectif si}\ i \neq 0, \struck{\ill{}}\, n, n+1
\]\[0 \to H^{n-2}(Q^{n-1})(-1) \to H^n(Q^n) \to H^n(U^n) \to
H^{n-1}(Q^{n-1})(-1) \to H^{n+1}(Q^n) \to 0\]
LaTeX source
\[
0 \to H^{n-2}(Q^{n-1})(-1) \to H^n(Q^n) \to H^n(U^n) \to
H^{n-1}(Q^{n-1})(-1) \to H^{n+1}(Q^n) \to 0
\]\[0 \to H^{n-2}(Q^{n-1})(-1) \to H^n(Q^n) \to H^n(U^n) \to 0\]
LaTeX source
\[
0 \to H^{n-2}(Q^{n-1})(-1) \to H^n(Q^n) \to H^n(U^n) \to 0
\]\[0 \to H^{n}(U^{n}) \to H^{n-1}(Q^{n-1})(-1) \to H^{n+1}(Q^{n}) \to 0\]
LaTeX source
\[
0 \to H^{n}(U^{n}) \to H^{n-1}(Q^{n-1})(-1) \to H^{n+1}(Q^{n}) \to 0
\]\[\left|
\begin{array}{l}
\text{bijectif si}\ \struck{\ill{}}\ i \geqslant 2,\ i \neq n \\
\text{injectif pour tt}\ i
\end{array}
\right\}\ (n\ \text{pair})
\qquad
\left|
\begin{array}{l}
\text{bijectif si}\ i \geqslant 2,\ i \neq n+1 \\
\text{surjectif pour tt}\ i \geqslant 1
\end{array}
\right\}\ (n\ \text{impair})\]
LaTeX source
\[
\left|
\begin{array}{l}
\text{bijectif si}\ \struck{\ill{}}\ i \geqslant 2,\ i \neq n \\
\text{injectif pour tt}\ i
\end{array}
\right\}\ (n\ \text{pair})
\qquad
\left|
\begin{array}{l}
\text{bijectif si}\ i \geqslant 2,\ i \neq n+1 \\
\text{surjectif pour tt}\ i \geqslant 1
\end{array}
\right\}\ (n\ \text{impair})
\]\[\left|
\begin{array}{l}
\text{bijectif si}\ i \leqslant 2n-2,\ i \neq n \\
\text{injectif pour tt}\ i
\end{array}
\right\}\ n\ \text{pair}
\qquad
\left|
\begin{array}{l}
\text{bijectif si}\ i \leqslant 2n-2,\ i \neq n-1 \\
\text{injectif pour tt}\ i \leqslant 2n-\ldots
\end{array}
\right\}\ n\ \text{impair}\]
LaTeX source
\[
\left|
\begin{array}{l}
\text{bijectif si}\ i \leqslant 2n-2,\ i \neq n \\
\text{injectif pour tt}\ i
\end{array}
\right\}\ n\ \text{pair}
\qquad
\left|
\begin{array}{l}
\text{bijectif si}\ i \leqslant 2n-2,\ i \neq n-1 \\
\text{injectif pour tt}\ i \leqslant 2n-\ldots
\end{array}
\right\}\ n\ \text{impair}
\]\[n = 2\nu \qquad H^2(Q^n) \;\text{---}\; H^{2\nu-2}(Q^n)\]
LaTeX source
\[
n = 2\nu \qquad H^2(Q^n) \;\text{---}\; H^{2\nu-2}(Q^n)
\]\[\xi^0 \;-\; \xi^{\nu-1} \quad \xi^{\nu} \quad \xi^{\nu+1} \ \cdots \ \xi^{2\nu}\]
LaTeX source
\[
\xi^0 \;-\; \xi^{\nu-1} \quad \xi^{\nu} \quad \xi^{\nu+1} \ \cdots \ \xi^{2\nu}
\]\[Q^n \quad
\left\{
\begin{array}{l}
\xi^i \in H^{2i}(Q^n, \mu(i)) \qquad \struck{0 \leqslant i \leqslant 2n} \\
\struck{\lambda_{2\nu} \in H^{2\nu}(Q^{2\nu})}\ \ \lambda_{2\nu} \in H^{2\nu}(Q^{2\nu})
\end{array}
\right.\]
LaTeX source
\[
Q^n \quad
\left\{
\begin{array}{l}
\xi^i \in H^{2i}(Q^n, \mu(i)) \qquad \struck{0 \leqslant i \leqslant 2n} \\
\struck{\lambda_{2\nu} \in H^{2\nu}(Q^{2\nu})}\ \ \lambda_{2\nu} \in H^{2\nu}(Q^{2\nu})
\end{array}
\right.
\]\[Q^{n-1} \subset Q^n \qquad
\xi_i \longleftarrow \xi_i \qquad
\xi_i \rightsquigarrow \xi_{i+2}
\qquad
\begin{pmatrix} 1, & 1 \\ 1, & -1 \end{pmatrix}\]
LaTeX source
\[
Q^{n-1} \subset Q^n \qquad
\xi_i \longleftarrow \xi_i \qquad
\xi_i \rightsquigarrow \xi_{i+2}
\qquad
\begin{pmatrix} 1, & 1 \\ 1, & -1 \end{pmatrix}
\]\[\begin{array}{ccccc}
Q^0 \subset & Q^1 \subset & Q^2 \subset & Q^3 \subset & Q^4 \\[4pt]
(\lambda_0, \xi^0) & \xi^0 & \xi^0 & \xi^0 & \xi^0 \\
& \tfrac12 \xi & (\lambda_1, \xi) & \xi & \xi \\
& & \tfrac12 \xi^2 & \tfrac12 \xi^2 & (\lambda_2, \xi^2) \\
& & & \tfrac12 \xi^3 & \tfrac12 \xi^3 \\
& & & & \tfrac12 \xi^4
\end{array}\]
LaTeX source
\[
\begin{array}{ccccc}
Q^0 \subset & Q^1 \subset & Q^2 \subset & Q^3 \subset & Q^4 \\[4pt]
(\lambda_0, \xi^0) & \xi^0 & \xi^0 & \xi^0 & \xi^0 \\
& \tfrac12 \xi & (\lambda_1, \xi) & \xi & \xi \\
& & \tfrac12 \xi^2 & \tfrac12 \xi^2 & (\lambda_2, \xi^2) \\
& & & \tfrac12 \xi^3 & \tfrac12 \xi^3 \\
& & & & \tfrac12 \xi^4
\end{array}
\]\[\boxed{
\begin{array}{ll}
Q^{n = 2\nu} : & \xi^0, \ldots, \xi^{\nu-1}, [\xi^{\nu}, \lambda_{\nu}],
\tfrac12 \xi^{\nu+1}, \ldots, \tfrac12 \xi^{2\nu} \\[4pt]
Q^{n = 2\nu+1} : & \xi^0, \ldots, \xi^{\nu}, \tfrac12 \xi^{\nu+1}, \ldots,
\tfrac12 \xi^{2\nu+1}
\end{array}}
\qquad \lambda_{\nu}\]
LaTeX source
\[
\boxed{
\begin{array}{ll}
Q^{n = 2\nu} : & \xi^0, \ldots, \xi^{\nu-1}, [\xi^{\nu}, \lambda_{\nu}],
\tfrac12 \xi^{\nu+1}, \ldots, \tfrac12 \xi^{2\nu} \\[4pt]
Q^{n = 2\nu+1} : & \xi^0, \ldots, \xi^{\nu}, \tfrac12 \xi^{\nu+1}, \ldots,
\tfrac12 \xi^{2\nu+1}
\end{array}}
\qquad \lambda_{\nu}
\]\[Q^i \subset Q^j \qquad\qquad \xi\]
LaTeX source
\[ Q^i \subset Q^j \qquad\qquad \xi \]
\[\struck{\mathbf{Z}[T][T'] / T^{2\nu+2},\ 2T' - T^{\nu+1}}\]
LaTeX source
\[
\struck{\mathbf{Z}[T][T'] / T^{2\nu+2},\ 2T' - T^{\nu+1}}
\]\[\left\{
\begin{array}{l}
\xi^0, \ldots, \xi^{\nu-1}, [\xi^{\nu}, \lambda_{\nu}], \xi_{\nu+1}, \ldots, \xi_{2\nu} \\
\xi^0, \ldots, \xi^{\nu}, \xi_{\nu+1}, \ldots, \xi_{2\nu+1}
\end{array}
\right.
\qquad
\left[
\begin{array}{l}
2\xi_i = \xi^i \\
\xi^i \xi_j = \xi_{i+j} \\
\xi_i \xi_j = 0 \ \text{pr raisons de degré} \\
\lambda_{\nu}^2 = 2\xi_{2\nu} = \xi^{2\nu} \\
\lambda_{\nu} \xi = 0 \\
\lambda_{\nu} \xi_{\alpha} = 0 \quad \alpha \geqslant \nu+1 \ \text{raisons de degré}
\end{array}
\right.\]
LaTeX source
\[
\left\{
\begin{array}{l}
\xi^0, \ldots, \xi^{\nu-1}, [\xi^{\nu}, \lambda_{\nu}], \xi_{\nu+1}, \ldots, \xi_{2\nu} \\
\xi^0, \ldots, \xi^{\nu}, \xi_{\nu+1}, \ldots, \xi_{2\nu+1}
\end{array}
\right.
\qquad
\left[
\begin{array}{l}
2\xi_i = \xi^i \\
\xi^i \xi_j = \xi_{i+j} \\
\xi_i \xi_j = 0 \ \text{pr raisons de degré} \\
\lambda_{\nu}^2 = 2\xi_{2\nu} = \xi^{2\nu} \\
\lambda_{\nu} \xi = 0 \\
\lambda_{\nu} \xi_{\alpha} = 0 \quad \alpha \geqslant \nu+1 \ \text{raisons de degré}
\end{array}
\right.
\]\[Q^i \subset Q^j \quad (i \leqslant j)\]
LaTeX source
\[ Q^i \subset Q^j \quad (i \leqslant j) \]
\[\left.
\begin{array}{l}
\xi^{\alpha} \longleftarrow \xi^{\alpha} \\
\xi_{\alpha} \longleftarrow \xi_{\alpha} \\
0 \longleftarrow \lambda_{\nu} \\
\xi_{\nu} \longleftarrow \varphi_{\nu}
\end{array}
\right\}\ \text{si}\ i < j = 2\nu\]
LaTeX source
\[
\left.
\begin{array}{l}
\xi^{\alpha} \longleftarrow \xi^{\alpha} \\
\xi_{\alpha} \longleftarrow \xi_{\alpha} \\
0 \longleftarrow \lambda_{\nu} \\
\xi_{\nu} \longleftarrow \varphi_{\nu}
\end{array}
\right\}\ \text{si}\ i < j = 2\nu
\]\[\left.
\begin{array}{l}
\xi^{\alpha} \rightsquigarrow \xi^{\alpha + (j-i)} \\
\xi_{\alpha} \rightsquigarrow \xi_{\alpha + (j-i)} \\
\lambda_{\nu} \rightsquigarrow 0 \\
\varphi_{\nu} \rightsquigarrow \xi_{\nu + (j-i)}
\end{array}
\right\}\ \text{si}\ i = 2\nu < j\]
LaTeX source
\[
\left.
\begin{array}{l}
\xi^{\alpha} \rightsquigarrow \xi^{\alpha + (j-i)} \\
\xi_{\alpha} \rightsquigarrow \xi_{\alpha + (j-i)} \\
\lambda_{\nu} \rightsquigarrow 0 \\
\varphi_{\nu} \rightsquigarrow \xi_{\nu + (j-i)}
\end{array}
\right\}\ \text{si}\ i = 2\nu < j
\]\[\begin{bmatrix} 1, & 1 \\ 1, & -1 \end{bmatrix}\]
LaTeX source
\[
\begin{bmatrix} 1, & 1 \\ 1, & -1 \end{bmatrix}
\]\[\begin{array}{ccccccc}
Q^0 & Q^1 & Q^2 & Q^3 & Q^4 & Q^5 & Q^6 \\[4pt]
[\lambda_0, \xi^0] & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 \\
& \tfrac12 \xi & [\lambda_1, \xi] & \xi & \xi & \xi & \xi \\
& & \tfrac12 \xi^2 & \tfrac12 \xi^2 & [\lambda_2, \xi^2] & \xi^2 & \xi^2 \\
& & & \tfrac12 \xi^3 & \tfrac12 \xi^3 & \tfrac12 \xi^3 & [\lambda_3, \xi^3] \\
& & & & \tfrac12 \xi^4 & \tfrac12 \xi^4 & \tfrac12 \xi^4 \\
& & & & & \tfrac12 \xi^5 & \tfrac12 \xi^5 \\
& & & & & & \tfrac12 \xi^6
\end{array}\]
LaTeX source
\[
\begin{array}{ccccccc}
Q^0 & Q^1 & Q^2 & Q^3 & Q^4 & Q^5 & Q^6 \\[4pt]
[\lambda_0, \xi^0] & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 & \xi^0 = 1 \\
& \tfrac12 \xi & [\lambda_1, \xi] & \xi & \xi & \xi & \xi \\
& & \tfrac12 \xi^2 & \tfrac12 \xi^2 & [\lambda_2, \xi^2] & \xi^2 & \xi^2 \\
& & & \tfrac12 \xi^3 & \tfrac12 \xi^3 & \tfrac12 \xi^3 & [\lambda_3, \xi^3] \\
& & & & \tfrac12 \xi^4 & \tfrac12 \xi^4 & \tfrac12 \xi^4 \\
& & & & & \tfrac12 \xi^5 & \tfrac12 \xi^5 \\
& & & & & & \tfrac12 \xi^6
\end{array}
\]\[L_{\cdot}(F_{\cdot}) \otimes L_{\cdot}(G_{\cdot}) = F \mathbin{\underline{\otimes}} G\]
LaTeX source
\[
L_{\cdot}(F_{\cdot}) \otimes L_{\cdot}(G_{\cdot}) = F \mathbin{\underline{\otimes}} G
\]\[n = 2\nu\]
LaTeX source
\[ n = 2\nu \]
\[H^{n-1}(U^{n-1})(1) \xrightarrow[\partial]{\text{ins}} H^n(Q^n)
\xrightarrow{\text{res}} H^n(U^n)\]
LaTeX source
\[
H^{n-1}(U^{n-1})(1) \xrightarrow[\partial]{\text{ins}} H^n(Q^n)
\xrightarrow{\text{res}} H^n(U^n)
\]\[\begin{array}{lll}
\ell^{2\nu+1} & & \\
\ell^{2\nu+1} \rightsquigarrow \lambda_{\nu} &
[\lambda_{\nu}, \xi^{\nu}] \rightsquigarrow [2\ell^{2\nu}, 0] & \\
& \varphi_{\nu} = \tfrac12 (\lambda_{\nu} + \xi^{\nu}) \rightsquigarrow \ell^{2\nu} &
\end{array}\]
LaTeX source
\[
\begin{array}{lll}
\ell^{2\nu+1} & & \\
\ell^{2\nu+1} \rightsquigarrow \lambda_{\nu} &
[\lambda_{\nu}, \xi^{\nu}] \rightsquigarrow [2\ell^{2\nu}, 0] & \\
& \varphi_{\nu} = \tfrac12 (\lambda_{\nu} + \xi^{\nu}) \rightsquigarrow \ell^{2\nu} &
\end{array}
\]\[\lambda_{\nu} \longleftarrow \ell_{2\nu}\]
LaTeX source
\[
\lambda_{\nu} \longleftarrow \ell_{2\nu}
\]\[H^{n+1}_c(U^{n+1})(+1) \xleftarrow[\partial]{} H^n(Q^n)
\xleftarrow{i_{\partial}} H^n_c(U^n)\]
LaTeX source
\[
H^{n+1}_c(U^{n+1})(+1) \xleftarrow[\partial]{} H^n(Q^n)
\xleftarrow{i_{\partial}} H^n_c(U^n)
\]\[\begin{array}{ll}
[2\ell_{2\nu+1}, 0] \longleftarrow [\lambda_{\nu}, \xi^{\nu}] & \\
\ell_{2\nu+1} \longleftarrow \varphi_{\nu} & \\
2\ell_{2\nu+1} \longleftarrow \ell_{2\nu} &
\end{array}\]
LaTeX source
\[
\begin{array}{ll}
[2\ell_{2\nu+1}, 0] \longleftarrow [\lambda_{\nu}, \xi^{\nu}] & \\
\ell_{2\nu+1} \longleftarrow \varphi_{\nu} & \\
2\ell_{2\nu+1} \longleftarrow \ell_{2\nu} &
\end{array}
\]\[Q^{n+1} - Q^{n} = U^{n+1}\]
LaTeX source
\[
Q^{n+1} - Q^{n} = U^{n+1}
\]\[\alpha_{\nu} + \beta_{\nu} = \xi^{\nu}, \qquad
\alpha_{\nu} - \beta_{\nu} = \lambda_{\nu}, \qquad
\varphi_{\nu} = \alpha_{\nu} \ \ldots\]
LaTeX source
\[
\alpha_{\nu} + \beta_{\nu} = \xi^{\nu}, \qquad
\alpha_{\nu} - \beta_{\nu} = \lambda_{\nu}, \qquad
\varphi_{\nu} = \alpha_{\nu} \ \ldots
\]\[Q^{2\nu} : \qquad \xi : H^{2i}(Q^{2\nu}) \to H^{2i+2}(Q^{2\nu})(1)
\qquad \struck{\ill{}}\ (0 \leqslant i \leqslant 2\nu - 1)\]
LaTeX source
\[
Q^{2\nu} : \qquad \xi : H^{2i}(Q^{2\nu}) \to H^{2i+2}(Q^{2\nu})(1)
\qquad \struck{\ill{}}\ (0 \leqslant i \leqslant 2\nu - 1)
\]\[Q^{2\nu+1} \qquad \xi : H^{2i}(Q^{2\nu+1}) \to H^{2i+2}(Q^{2\nu+1})(1)
\qquad (0 \leqslant i \leqslant 2\nu)\]
LaTeX source
\[
Q^{2\nu+1} \qquad \xi : H^{2i}(Q^{2\nu+1}) \to H^{2i+2}(Q^{2\nu+1})(1)
\qquad (0 \leqslant i \leqslant 2\nu)
\]\[1, H, H^2, \ldots, H^{n-1}, \underbrace{S', S''}, HS' = HS'',
\ldots, H^n S' = H^n S''\]
LaTeX source
\[
1, H, H^2, \ldots, H^{n-1}, \underbrace{S', S''}, HS' = HS'',
\ldots, H^n S' = H^n S''
\]\[1, H, \ldots, H^n, \frac{H^{n+1}}{2}, \ldots, \frac{H^{2n+1}}{2}\]
LaTeX source
\[
1, H, \ldots, H^n, \frac{H^{n+1}}{2}, \ldots, \frac{H^{2n+1}}{2}
\]