Cote n° 84 · pages 5–189 · 63 commutative diagrams · [Formes quadratiques] : notes manuscrites (s.d.).
Inventory dating : [à partir de 1982-vers 1986]
Édition de démonstration

batch 1 · p. 5 — read it beside the facsimile1 / 63
LaTeX source
\begin{tikzcd}
  & SW/\mu_2 = W^*/G_m \arrow[r] & 1 & \\
  1 \arrow[r] & SW \arrow[r] \arrow[u] & W^* \arrow[r, "N"] & G_m \arrow[r] & 1 \\
  1 \arrow[r] & \mu_2 \arrow[r] \arrow[u] & G_m \arrow[r, "{(\ )^2}"] \arrow[u] & G_m \arrow[r] \arrow[u, no head] & 1
\end{tikzcd}
batch 1 · p. 13 — read it beside the facsimile2 / 63
LaTeX source
\begin{tikzcd}[column sep=small, nodes={font=\small}]
  0 \arrow[r] & L \otimes L^\vee \arrow[r] \arrow[d, no head, "\wr"] & (E \otimes E) \otimes L^\vee \arrow[r] \arrow[dd, "\wr"] & \underline{\mathrm{Sym}}^2(E) \otimes L^\vee \arrow[r] \arrow[dd, "\wr"] & 0 \\
  & \underline{O} \arrow[d, "{\mathrm{id}_{\underline{O}}}", "\wr"'] & & & \\
  0 \arrow[r] & \underline{O} \arrow[r] & \underline{\mathrm{End}}(E) \arrow[r] & \underline{\mathrm{End}}(E)/\underline{O} \arrow[r] & 0
\end{tikzcd}
batch 1 · p. 19 — read it beside the facsimile3 / 63
LaTeX source
\begin{tikzcd}
  0 \arrow[r] & \underline{O} \arrow[r] \arrow[d, "\mathrm{id}"] & A_Q \arrow[r] \arrow[d, "u^!_Q"] & \Lambda \arrow[r] \arrow[d, "u_Q"] & 0 \\
  0 \arrow[r] & \underline{O} \arrow[r] & \underline{\mathrm{End}}(E) \arrow[r] & \underline{\mathrm{End}}(E)/\underline{O} \arrow[r] & 0
\end{tikzcd}
batch 3 · p. 53 — read it beside the facsimile4 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\scriptsize}]
\mathrm{Tors}(\mathbb{Z}/2\mathbb{Z}) \arrow[r, "\mathrm{Ass}"] \arrow[d, "\wr"'] & \mathrm{Tors}(\mu_{2}) \arrow[r, "\mathrm{Oub}"] \arrow[d, "\wr"] & \mathrm{Rev\,quad}(X) \arrow[d] \\
\mathrm{Triples}(L, \delta, T_{0}) \arrow[r, "{\mathrm{Ass}'\,?}"] & \mathrm{Couples}(L, \theta) \arrow[r, "{\mathrm{Oub}'\,?}"] & \mathrm{Triples}(L, \delta, T_{0})
\end{tikzcd}
batch 3 · p. 54 — read it beside the facsimile5 / 63
LaTeX source
\begin{tikzcd}
\mathrm{Tors}((\mathbb{Z}/2\mathbb{Z})_{X}) \arrow[r] \arrow[d] & \mathrm{Tors}(\mu_{2X}) \\
\mathrm{Rev\,quad}(X) &
\end{tikzcd}
batch 3 · p. 56 — read it beside the facsimile6 / 63
LaTeX source
\begin{tikzcd}[column sep=normal, row sep=normal]
0 \arrow[r] & L \arrow[r] \arrow[d, "u \mapsto u^{2}"'] & E \arrow[r, "\psi"] \arrow[d, "x \mapsto x^{(2)}"] & \mathcal{O}_{X} \arrow[r] \arrow[d, "\lambda \mapsto \lambda^{2}"] & 0 \\
0 \arrow[r] & L^{\otimes 2} \arrow[r] & E^{(2)} \arrow[r, "\psi^{(2)}"] & \mathcal{O}_{X} \arrow[r] & 0
\end{tikzcd}
batch 4 · p. 62 — read it beside the facsimile7 / 63
LaTeX source
\begin{tikzcd}
P_{X_1} \times P_{X_2} \arrow[r] & P_{X_1 \wedge X_2} \\
X_1 \times X_2 \arrow[r] \arrow[u, hook] & X_1 \wedge X_2 \arrow[u, hook]
\end{tikzcd}
batch 4 · p. 67 — read it beside the facsimile8 / 63
LaTeX source
\begin{tikzcd}[column sep=small]
 & L_1 \otimes L_2 \arrow[dl] \arrow[dr] & \\
L_1 \otimes E_2 \arrow[dr] & & E_1 \otimes L_2 \arrow[dl] \\
 & E_1 \otimes E_2 &
\end{tikzcd}
batch 4 · p. 68 — read it beside the facsimile9 / 63
LaTeX source
\begin{tikzcd}[column sep=small]
 & L_1 \otimes L_2 \arrow[dl] \arrow[dr] & \\
L_1 \otimes E_2 \arrow[dr, "\alpha_1"'] & & E_1 \otimes L_2 \arrow[dl, "\alpha_2"] \\
 & L_1 \otimes L_2 &
\end{tikzcd}
batch 4 · p. 69 — read it beside the facsimile10 / 63
LaTeX source
\begin{tikzcd}[column sep=small]
0 \arrow[r] & E_1 *_0 E_2 \arrow[r, "i"] \arrow[d] & E_1 \otimes E_2 \arrow[r] \arrow[d] & \mathcal{O} \arrow[r] \arrow[d, "\mathrm{id}"] & 0 \\
0 \arrow[r] & L_1 \otimes L_2 \arrow[r, "i"] & E_1 * E_2 \arrow[r, "\psi"] & \mathcal{O} \arrow[r] & 0
\end{tikzcd}
batch 4 · p. 69 — read it beside the facsimile11 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & E_1 *_0 E_2 \arrow[dl, "\mathrm{inc}"'] \arrow[dr, "\mu"] & \\
E_1 \otimes E_2 \arrow[dr, dashed] \arrow[ddr, "{\rho = \pi}"'] & & L_1 \otimes L_2 \arrow[dl, dashed] \arrow[ddl, "{\sigma = \chi\,\mathrm{id}_{L_1 \otimes L_2}}"] \\
 & E_1 * E_2 \arrow[d, "\pi_T"] & \\
 & M &
\end{tikzcd}
batch 5 · p. 87 — read it beside the facsimile12 / 63
LaTeX source
\begin{tikzcd}
0 \arrow[r] & L_1 \times L_2 \arrow[r] \arrow[d] & {E_1 \overset{\leftrightarrows}{\otimes} E_2 = E(P_1 \times P_2)} \arrow[r] \arrow[d] & \mathcal{O} \arrow[r] \arrow[d, no head, "="] & 0 \\
0 \arrow[r] & L_1 \otimes L_2 \arrow[r] & E_1 * E_2 \arrow[r] & \mathcal{O} \arrow[r] & 0
\end{tikzcd}
batch 5 · p. 96 — read it beside the facsimile13 / 63
LaTeX source
\begin{tikzcd}
0 \arrow[r] & L \arrow[r] \arrow[d, "\mathrm{id}"] & E \arrow[r] \arrow[d, "g"', "\wr"] & \mathcal{O} \arrow[r] \arrow[d, "\mathrm{id}"] & 0 \\
0 \arrow[r] & L \arrow[r] & \mathcal{A} \otimes L \arrow[r, "\varphi \otimes \mathrm{id}_L"] & \mathcal{O} \arrow[r] & 0
\end{tikzcd}
batch 5 · p. 97 — read it beside the facsimile14 / 63
LaTeX source
\begin{tikzcd}
E \arrow[d] \arrow[r, no head, "="] & \underline{L} \arrow[d, "-\mathrm{id}_L"] & \oplus & \mathcal{O}.x_0 \arrow[d, "\mathrm{id}"] \\
\mathcal{A} \otimes L \arrow[r, no head, "\simeq"] & \underline{L} & \oplus & \mathcal{O} .
\end{tikzcd}
batch 5 · p. 98 — read it beside the facsimile15 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & E_1 \otimes E_2 \arrow[dr, "*"] & & \\
L_1 \otimes E_2 \oplus E_1 \otimes L_2 \arrow[ur, "\mathrm{inj}"] \arrow[r, no head] & L_1 \otimes L_2 \arrow[r, "\mathrm{inj}"] & E \arrow[r, "i"] & E_1 \otimes E_2
\end{tikzcd}
batch 5 · p. 100 — read it beside the facsimile16 / 63
LaTeX source
\begin{tikzcd}[column sep=small, nodes={font=\small}]
0 \arrow[r] & {\mathcal{A}_1 \underset{\mathcal{O}}{\oplus} \mathcal{A}_2} \arrow[r] & \mathcal{A}_1 \otimes \mathcal{A}_2 \arrow[r] \arrow[d] & L_1^{\vee} \otimes L_2^{\vee} \arrow[r] \arrow[d, no head, "\|"] & 0 \\
0 \arrow[r] & \mathcal{O} \arrow[r] \arrow[u, "\tau"'] & \mathcal{A} \arrow[r] & L_1^{\vee} \otimes L_2^{\vee} \arrow[r] & 0
\end{tikzcd}
batch 5 · p. 100 — read it beside the facsimile17 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & \mathcal{O} \\
\mathcal{A}_1 \otimes_0 \mathcal{A}_2 = \mathcal{A}_1 \underset{\mathcal{O}}{\oplus} \mathcal{A}_2 \arrow[ur] \arrow[dr] & \\
 & \mathcal{A}_1 \otimes \mathcal{A}_2
\end{tikzcd}
batch 6 · p. 109 — read it beside the facsimile18 / 63
LaTeX source
\begin{tikzcd}
  k \arrow[r, "\alpha"] & A \arrow[l, bend left=40, "t"] \arrow[l, bend left=70, "n"]
\end{tikzcd}
batch 6 · p. 110 — read it beside the facsimile19 / 63
LaTeX source
\begin{tikzcd}
  A_i \arrow[r, bend left=20, "t_i"] \arrow[r, bend right=20, "n_i"'] & k
\end{tikzcd}
batch 6 · p. 110 — read it beside the facsimile20 / 63
LaTeX source
\begin{tikzcd}
  A_1 \arrow[dr, "\alpha_1"'] & A_2 \arrow[d, "\alpha_2"] & A_3 \arrow[dl, "\alpha_3"] \\
  & A &
\end{tikzcd}
batch 6 · p. 110 — read it beside the facsimile21 / 63
LaTeX source
\begin{tikzcd}[column sep=large, row sep=large]
  & A \arrow[dl, bend right=15, "T_1"'] \arrow[dl, bend left=15, "N_1"] \arrow[d, bend right=15, "T_2"'] \arrow[d, bend left=15, "N_2"] \arrow[dr, bend right=15, "T_3"'] \arrow[dr, bend left=15, "N_3"] & \\
  A_1 & A_2 & A_3
\end{tikzcd}
batch 6 · p. 111 — read it beside the facsimile22 / 63
LaTeX source
\begin{tikzcd}
  A_1 \arrow[r, no head, "\alpha_1"] \arrow[d, no head] & A \arrow[d, no head, "\alpha_2"] \\
  k \arrow[r, no head] & A_2
\end{tikzcd}
batch 6 · p. 111 — read it beside the facsimile23 / 63
LaTeX source
\begin{tikzcd}
  A_1 \otimes_k A_2 \arrow[r] \arrow[d, "t_1 \otimes \mathrm{id}_{A_2}"'] & A \arrow[d, "T_1"] \\
  A_2 \arrow[r, no head] & A_2
\end{tikzcd}
batch 6 · p. 112 — read it beside the facsimile24 / 63
LaTeX source
\begin{tikzcd}
  A_1 \otimes_k A_2 \arrow[r] \arrow[d, "{n_1 \otimes_k \mathrm{id}_{A_2} = N_{12}}"'] & A \arrow[d, "N_2"] \\
  A_2 \arrow[r, no head] & A_2
\end{tikzcd}
batch 6 · p. 117 — read it beside the facsimile25 / 63
LaTeX source
\begin{tikzcd}
  B \arrow[rr, "f"] \arrow[dr] & & A \\
  & (\bigotimes A_i)^{\Gamma} \arrow[ur] &
\end{tikzcd}
batch 6 · p. 120 — read it beside the facsimile26 / 63
LaTeX source
\begin{tikzcd}[row sep=small]
  & X_1 \\
  X_1 \times X_2 \arrow[ur] \arrow[r] \arrow[dr] & X_2 \\
  & X_3 = X_1 \wedge X_2
\end{tikzcd}
batch 7 · p. 121 — read it beside the facsimile27 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & X_{1\,\mathfrak{X}_0} \arrow[dr] & \\
  X_1 \times X_2 \times X_3 \arrow[ur, "{(\mathrm{pr}_1,\wedge)}"] \arrow[r, "{(\mathrm{pr}_2,\wedge)}"] \arrow[dr, "{(\mathrm{pr}_3,\wedge)}"'] & X_{2\,\mathfrak{X}_0} \arrow[r] & \mathfrak{X}_0 \\
  & X_{3\,\mathfrak{X}_0} \arrow[ur] &
\end{tikzcd}
batch 7 · p. 123 — read it beside the facsimile28 / 63
LaTeX source
\begin{tikzcd}
  0 \arrow[r] & L \arrow[r] \arrow[d, "\alpha"] & E \arrow[r] \arrow[d, "\beta"] \arrow[l, bend right=40, "\pi"'] & M \arrow[r] \arrow[d, "\gamma"] \arrow[l, bend right=40, "\varpi"'] & 0 \\
  0 \arrow[r] & L' \arrow[r] & E' \arrow[r] \arrow[l, bend left=40, "\pi'"] & M' \arrow[r] \arrow[l, bend left=40, "\varpi'"] & 0
\end{tikzcd}
batch 7 · p. 123 — read it beside the facsimile29 / 63
LaTeX source
\begin{tikzcd}
  E_1 \arrow[d] & E_2 \arrow[d] \\
  E_1' & E_2'
\end{tikzcd}
batch 7 · p. 123 — read it beside the facsimile30 / 63
LaTeX source
\begin{tikzcd}
  E_1 \wedge E_2 \arrow[r] & E_1'
\end{tikzcd}
batch 7 · p. 123 — read it beside the facsimile31 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & E_1 \otimes E_2 \arrow[rr] & & E_1' \otimes E_2' \arrow[dr] & \\
  E_1 \otimes L_2,\ L_1 \otimes E_2 \arrow[ur] \arrow[dr, hook] & & & & E_1' \wedge E_2' \\
  & L_1 \otimes L_2 \arrow[rr] & & L_1' \otimes L_2' \arrow[ur] &
\end{tikzcd}
batch 7 · p. 125 — read it beside the facsimile32 / 63
LaTeX source
\begin{tikzcd}
  \mathcal{E}_{(p_1,q_1)(p_2,q_2)} \arrow[r, "T_u"] \arrow[d, "T_{-\sigma b}"'] & \mathcal{E}_{(p_1',q_1')(p_2',q_2')} \arrow[d, "T_{-\sigma b'}"] \\
  \mathcal{E} \arrow[r, "T_{-u}"'] & \mathcal{E}
\end{tikzcd}
batch 7 · p. 125 — read it beside the facsimile33 / 63
LaTeX source
\begin{tikzcd}
  L_1 \otimes L_2 \arrow[r, hook] \arrow[d, Rightarrow] & E \arrow[r, two heads] & M_1 \otimes M_2 \arrow[ld, no head] \\
  L_1' \otimes L_2' & M_1' \otimes M_2' &
\end{tikzcd}
batch 7 · p. 127 — read it beside the facsimile34 / 63
LaTeX source
\begin{tikzcd}
  \mathcal{E}_{(p_1,q_1)(p_2,q_2)} = M_1 \otimes M_2 \oplus L_1 \otimes L_2 \arrow[d, "\wr"', "T_{v(p_1,q_1)(p_2,q_2)}"] \\
  \mathcal{E}_{(\tilde p_1,\tilde q_1)(\tilde p_2,\tilde q_2)} = M_1 \otimes M_2 \oplus L_1 \otimes L_2
\end{tikzcd}
batch 7 · p. 131 — read it beside the facsimile35 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & & E_1 \vee E_2 \arrow[dll, "\Theta_{p_1 * p_2}"', "\sim"] \arrow[d, "\Theta_{p_1' * p_2'}", "\wr"'] \arrow[drr, "\Theta_{p_1'' * p_2''}", "\sim"'] & & \\
  M \oplus L \arrow[rr, "{\Theta_{(p_1,p_2)(p_1',p_2')}}", "{(u_1,u_2,u)}"'] & & M \oplus L \arrow[rr, "{\Theta_{(p_1',p_2')(p_1'',p_2'')}}", "{(u_1',u_2',u')}"'] & & M \oplus L
\end{tikzcd}
batch 7 · p. 133 — read it beside the facsimile36 / 63
LaTeX source
\begin{tikzcd}[column sep=small, nodes={font=\scriptsize}]
  & E_1 \vee E_2 \arrow[dl, "\Theta_{p_1 * p_2}"'] \arrow[dr, "\Theta_{p_1' * p_2'}"] & \\
  (M_1 \otimes M_2) \oplus (L_1 \otimes L_2) \arrow[rr, "\Lambda_u"'] & & (M_1 \otimes M_2) \oplus (L_1 \otimes L_2)
\end{tikzcd}
batch 7 · p. 133 — read it beside the facsimile37 / 63
LaTeX source
\begin{tikzcd}
  0 \arrow[r] & \mathcal{H} \arrow[r, "\alpha"] \arrow[d, "-\mathrm{id}"'] & \mathcal{E} \arrow[r, "\psi"] \arrow[d, "\Phi"] & \underline{O} \arrow[r] \arrow[d, no head, "\Vert" description] & 0 \\
  0 \arrow[r] & \mathcal{H} \arrow[r] & \mathcal{H}om^{!}(\mathcal{E}, \mathcal{H}) \arrow[r] & \underline{O} \arrow[r] & 0
\end{tikzcd}
batch 7 · p. 135 — read it beside the facsimile38 / 63
LaTeX source
\begin{tikzcd}
  0 \arrow[r] & L \arrow[r] \arrow[d, "-\mathrm{id}_L"'] & E \arrow[r, "\psi"] \arrow[d, "\wr"] & \underline{O} \arrow[r] \arrow[d, no head, "\Vert" description] & 0 \\
  0 \arrow[r] & L \arrow[r] & \mathcal{H}om^{!}(E, L) \arrow[r] & \underline{O} \arrow[r] & 0
\end{tikzcd}
batch 7 · p. 139 — read it beside the facsimile39 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  (E_1 \otimes L_2) \oplus (L_1 \otimes E_2) \arrow[r, Rightarrow] \arrow[dr, "\pi"'] & E_1 \otimes E_2 \arrow[r] \arrow[d, "x_1 \otimes x_2 \mapsto x_1 * x_2"] & M_1 \otimes M_2 \arrow[r] \arrow[d, no head, "\wr"] & 0 \\
  & L_1 \otimes L_2 \arrow[r] & E \arrow[r] \arrow[l, bend left=40, "p"] & M_1 \otimes M_2 \arrow[l, bend left=30, "{m_1 \otimes m_2 \mapsto x_1 * x_2}"]
\end{tikzcd}
batch 8 · p. 141 — read it beside the facsimile40 / 63
LaTeX source
\begin{tikzcd}[column sep=large]
E_1\otimes_0 E_2 \arrow[r, hook, "\mu"] \arrow[dr, dashed, "\pi"'] & E_1\otimes E_2 \arrow[r, "F_{11}"] & E_1\otimes E_2 \\
 & L_1\otimes L_2 \arrow[ur, dashed, "\alpha_1\otimes\alpha_2"'] &
\end{tikzcd}
batch 8 · p. 141 — read it beside the facsimile41 / 63
LaTeX source
\begin{tikzcd}[column sep=large]
E_1\otimes E_2 \arrow[r, "F_{11}"] \arrow[dr, "\psi_1\otimes\psi_2"'] & E_1\otimes E_2 \arrow[r, "\nu"] & E_1\otimes^{0} E_2 \\
 & M_1\otimes M_2 \arrow[ur, "\varpi"'] &
\end{tikzcd}
batch 8 · p. 143 — read it beside the facsimile42 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & E_1\otimes E_2 & \\
x_1\otimes u_2 \arrow[r] \arrow[dr] & x_1\otimes u_2 \arrow[r] & f_1(x_1)\circledast f_2(u_2) \\
 & \pi_1(x_1)\otimes u_2 \arrow[r] & f_1^{!}(\pi_1(x_1))\otimes f_2^{!}(u_2)
\end{tikzcd}
batch 8 · p. 143 — read it beside the facsimile43 / 63
LaTeX source
\begin{tikzcd}[column sep=small]
 & E_1\otimes E_2 \arrow[r] & L_1\otimes L_2 \\
E_1\otimes L_2 \arrow[ur] \arrow[r] & L_1\otimes L_2 \arrow[ur] &
\end{tikzcd}
batch 8 · p. 153 — read it beside the facsimile44 / 63
LaTeX source
\begin{tikzcd}[column sep=large]
E_1\otimes_0 E_2 \arrow[r, "F_{11}^{!}"] \arrow[d] \arrow[dr, "\pi"] & E_1\otimes E_2 \arrow[r] & E_1\otimes^0 E_2 \\
E_1\otimes E_2 \arrow[dr, "\psi_1\otimes\psi_2"'] & L_1\otimes L_2 \arrow[u, "\alpha_1\otimes\alpha_2"'] & \\
 & M_1\otimes M_2 \arrow[uur, bend right=20] &
\end{tikzcd}
batch 9 · p. 161 — read it beside the facsimile45 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
E_1 \otimes E_2 \arrow[rr, "?"] \arrow[dr] \arrow[d] & & E_1 \otimes E_2 \arrow[d] \\
E_1 \otimes L_2 \oplus_{L_1 \otimes L_2} L_1 \otimes E_2 \arrow[rr, "{\pi_1, \pi_2}"'] \arrow[drr] & E \arrow[ur] & L_1 \otimes L_2 \\
 & & \overline{L_1' \otimes L_2'}
\end{tikzcd}
batch 9 · p. 161 — read it beside the facsimile46 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & E_1 \otimes_0 E_2 \arrow[r] \arrow[d, "\pi_2"] & E_1 \otimes E_2 \arrow[r] \arrow[d] & L_1' \otimes L_2' \arrow[r] \arrow[d, no head, "\|" description] & 0 \\
0 \arrow[r] & E_1 \otimes L_2 \arrow[r] \arrow[d, "\text{inc. can.}"] & E \arrow[r] \arrow[d, "?"] & L_1' \otimes L_2' \arrow[r] \arrow[d, "\chi\,\mathrm{id}"] & 0 \\
0 \arrow[r] & E_1 \otimes_0 E_2 \arrow[r] & E_1 \otimes E_2 \arrow[r] & L_1' \otimes L_2' \arrow[r] & {}
\end{tikzcd}
batch 9 · p. 165 — read it beside the facsimile47 / 63
LaTeX source
\begin{tikzcd}[column sep=tiny, row sep=small, nodes={font=\scriptsize}]
 & \Sigma = (E_1 \otimes L_2) \oplus (L_1 \otimes E_2) \arrow[dl, "{\mathrm{id}_1 \otimes \alpha_2,\ \alpha_1 \otimes \mathrm{id}_2}"'] \arrow[dr, "{\pi_1 \otimes \mathrm{id}_2,\ \mathrm{id}_1 \otimes \pi_2}"] & \\
E_1 \otimes E_2 \arrow[d, "F_{11}"'] \arrow[drr, "F_{12}"] & & L_1 \otimes L_2 \arrow[dll, "F_{21}"'] \arrow[d, "F_{22}"] \\
E_1 \otimes E_2 \arrow[dr, "{\mathrm{id}_1 \otimes \psi_2,\ \psi_1 \otimes \mathrm{id}_2}"'] & & M_1 \otimes M_2 \arrow[dl, "{\varpi_1 \otimes \mathrm{id}_2,\ \mathrm{id}_1 \otimes \varpi_2}"] \\
 & (E_1 \otimes M_2) \times (M_1 \otimes E_2) = \Pi &
\end{tikzcd}
batch 9 · p. 165 — read it beside the facsimile48 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & \Sigma \arrow[dl] \arrow[dr] & \\
E_1 \otimes E_2 \arrow[drr, "{0,\ F_{12} = \psi_1 \otimes \psi_2}"'] & & L_1 \otimes L_2 \arrow[d, "F_{22} = 0"] \\
 & & M_1 \otimes M_2
\end{tikzcd}
batch 9 · p. 165 — read it beside the facsimile49 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
L_1 \otimes L_2 \arrow[d, "F_{21}"'] \arrow[r, "F_{22} = 0"] & M_1 \otimes M_2 \arrow[dl] \\
E_1 \otimes E_2 \arrow[r] & \Pi
\end{tikzcd}
batch 9 · p. 167 — read it beside the facsimile50 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & L_1 \otimes L_2 \arrow[dl] \arrow[dr] & \\
E_1 \otimes L_2 \arrow[dr] & \text{cocart.} & L_1 \otimes E_2 \arrow[dl] \\
 & E_1 \otimes_0 E_2 &
\end{tikzcd}
batch 9 · p. 167 — read it beside the facsimile51 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=normal]
0 \arrow[r] & E_1 \otimes_0 E_2 \arrow[r, "\mu"] \arrow[d, "{\pi_1 \otimes \mathrm{id},\ \mathrm{id} \otimes \pi_2 \ (\text{cocart.})}"'] & E_1 \otimes E_2 \arrow[r] \arrow[d] & M_1 \otimes M_2 \arrow[r] \arrow[d, no head, "\|" description] & 0 \\
0 \arrow[r] & L_1 \otimes L_2 \arrow[r] \arrow[d, no head, "\|" description] & E_1 \vee E_2 \arrow[r] \arrow[d, "?"] & M_1 \otimes M_2 \arrow[r] \arrow[d, no head, "\|" description] & 0 \\
0 \arrow[r] & L_1 \otimes L_2 \arrow[r] \arrow[d, no head, "\|" description] & E_1 \wedge E_2 \arrow[r] \arrow[d, "\text{cart.}"] & M_1 \otimes M_2 \arrow[r] \arrow[d, "{\varpi_1 \otimes \mathrm{id},\ \mathrm{id} \otimes \varpi_2}"] & 0 \\
0 \arrow[r] & L_1 \otimes L_2 \arrow[r] & E_1 \otimes E_2 \arrow[r] & E_1 \otimes^{\circ} E_2 \arrow[r] & 0
\end{tikzcd}
batch 9 · p. 167 — read it beside the facsimile52 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & E_1 \otimes^{\circ} E_2 \arrow[dl] \arrow[dr] & \\
E_1 \otimes M_2 \arrow[dr] & \text{cart.} & M_1 \otimes E_2 \arrow[dl] \\
 & M_1 \otimes M_2 &
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=normal, row sep=small]
0 \arrow[r] & L_1 \arrow[r, "\alpha_1"] & E_1 \arrow[r, "\psi_1"] \arrow[l, bend right=40, dashed, "\pi_1"'] & M_1 \arrow[r] \arrow[l, bend right=40, dashed, "\varpi_1"'] & 0 \\
0 \arrow[r] & L_2 \arrow[r, "\alpha_2"] & E_2 \arrow[r, "\psi_2"] \arrow[l, bend left=40, dashed, "\pi_2"] & M_2 \arrow[r] \arrow[l, bend left=40, dashed, "\varpi_2"] & 0
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & E_1 \otimes_0 E_2 \arrow[r] \arrow[d, "{\pi_1 \otimes \mathrm{id},\ \mathrm{id} \otimes \pi_2}"'] & E_1 \otimes E_2 \arrow[r] \arrow[d] & M_1 \otimes M_2 \arrow[d, no head, "\|" description] \\
0 \arrow[r] & L_1 \otimes L_2 \arrow[r] & E_1 \vee E_2 \arrow[r] & M_1 \otimes M_2
\end{tikzcd}
batch 9 · p. 169 — read it beside the facsimile55 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & L_1 \otimes L_2 \arrow[r] \arrow[d, no head, "\|" description] & E_1 \wedge E_2 \arrow[r] \arrow[d] & M_1 \otimes M_2 \arrow[d] \\
0 \arrow[r] & L_1 \otimes L_2 \arrow[r] & E_1 \otimes E_2 \arrow[r] & E_1 \otimes^{\circ} E_2
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=normal, row sep=small]
\mathcal{A}_1 \otimes \mathcal{A}_2 \arrow[r, "\text{com. par div. à } \mathcal{A}"] \arrow[rr, bend left=30, "\sigma"] & \mathcal{A}_1 \wedge \mathcal{A}_2 = \mathcal{A} \arrow[r] & \mathcal{A}_1 \otimes \mathcal{A}_2 \\
E_1 \otimes E_2 & E = E_1 \wedge E_2 \supset L_1 \otimes L_2 \arrow[l] \arrow[d] & E_1 \otimes E_2 \arrow[l, "\text{com. diff. à } E"'] \arrow[d] \arrow[ll, bend left=30, "\sigma"'] \\
 & 0 & 0
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=normal, row sep=small]
\mathcal{A}_1 \otimes_0 \mathcal{A}_2 \arrow[r, "\mathrm{Tr}"] \arrow[d] & \underline{O} \arrow[r, "\mathrm{id}"] \arrow[d] & \underline{O} \arrow[d] \\
\mathcal{A}_1 \otimes \mathcal{A}_2 \arrow[r] \arrow[d] & \mathcal{A} \arrow[r] \arrow[d] & \mathcal{A}_1 \otimes \mathcal{A}_2 \arrow[d] \\
L_1^{\vee} \otimes L_2^{\vee} \arrow[r, "\mathrm{id}"] \arrow[d] & L_1^{\vee} \otimes L_2^{\vee} \arrow[r, "2"] \arrow[d] & L_1^{\vee} \otimes L_2^{\vee} \arrow[d] \\
0 & 0 & 0
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=normal, row sep=small]
\mathcal{A}_1, \mathcal{A}_2 \arrow[r, "\rightrightarrows" description] & \mathcal{A}_1 \otimes_0 \mathcal{A}_2 \arrow[r, "\mathrm{Tr}"] \arrow[dr, hook] & \underline{O} \arrow[r] & \mathcal{A} \\
 & & \mathcal{A}_1 \otimes \mathcal{A}_2 \arrow[ur] &
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=normal, row sep=small]
{E_1 \otimes L_2,\ L_1 \otimes E_2} \arrow[r, "\rightrightarrows" description] & E_1 \otimes_0 E_2 \arrow[r, "\pi"] \arrow[dr, hook] & L_1 \otimes L_2 \arrow[r] & E \\
 & & E_1 \otimes E_2 \arrow[ur] &
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=normal, row sep=small]
 & & \mathcal{A}_1 \otimes \mathcal{A}_2 \\
\mathcal{A}_1 \otimes_0 \mathcal{A}_1 = \mathcal{A}_1 \otimes_{\underline{O}} \mathcal{A}_2 \arrow[urr, hook] \arrow[drr, "{\mathrm{Tr}_1, \mathrm{Tr}_2}"'] & & \\
 & & \underline{O}
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
E_1 \otimes_0 E_2 \arrow[r, hook] \arrow[d, no head, "\|" description] & E_1 \otimes E_2 \\
E_1 \otimes L_2 \oplus_{L_1 \otimes L_2} L_1 \otimes E_2 \arrow[r, "{(\pi_1 \otimes \mathrm{id},\ \mathrm{id} \otimes \pi_2)}"] & L_1 \otimes L_2
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=small, row sep=large, nodes={font=\small}]
E_1\otimes E_2 \arrow[r] \arrow[rr, bend left=20] & E_1\wedge E_2 \arrow[r, Rightarrow, "?"] & E_1\otimes E_2 \arrow[d, leftrightarrow, "\text{can}"] \\
E_1\otimes_0 E_2 \arrow[u, "\text{can}"] \arrow[r, "\pi"'] & L_1\otimes L_2 \arrow[u] \arrow[ur] \arrow[r, "{\mathrm{id}}"'] & L_1\otimes L_2
\end{tikzcd}
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LaTeX source
\begin{tikzcd}[column sep=large, row sep=small, nodes={font=\small}]
A_1\otimes A_2 \arrow[r, "*"] \arrow[rr, bend left=20] & A \arrow[r, "\xi"] & A_1\otimes A_2 \\
E_1\otimes E_2 \arrow[d] & E \arrow[l, "\xi"'] \arrow[d] & E_1\otimes E_2 \arrow[l, "(*)"'] \arrow[d] \\
\underline{O} & \underline{O} & \underline{O}
\end{tikzcd}