Cote n° 57 · pages 26–303 · 33 commutative diagrams · Picard : tapuscrit annoté (s.d.), lettres (s.d., 1962).
Inventory dating : 1962-[vers 1968]
Édition de démonstration

batch 2 · p. 26 — read it beside the facsimile1 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & Z' \arrow[dr, "g''"] & \\
X \arrow[ur, "g'"] \arrow[rr, "g"] \arrow[d, "f"'] & & Z \arrow[dll, "h"] \\
Y & &
\end{tikzcd}
batch 2 · p. 32 — read it beside the facsimile2 / 33
LaTeX source
\begin{tikzcd}
X \times X \arrow[r] \arrow[d, "{u \times u}"'] & X \arrow[d, "u"] \\
G \times G \arrow[r] & G
\end{tikzcd}
batch 3 · p. 59 — read it beside the facsimile3 / 33
LaTeX source
\begin{tikzcd}
P \arrow[d] & P_X \arrow[l] \arrow[d] \\
S & X \arrow[l] \arrow[ul, dashed, "\varphi"']
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile4 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & 1 \arrow[dl] \arrow[dr, Rightarrow] & 2 \arrow[d, Rightarrow] \\
\cdot \arrow[dd, dashed] & & 3 \arrow[d, Rightarrow] \\
 & & 4 \arrow[dll, dashed] \arrow[d, Rightarrow] \\
6 \arrow[drr, Rightarrow] & & 5 \arrow[d, Rightarrow] \\
 & & 7 \arrow[d, Rightarrow] \\
 & & 8 \arrow[d, Rightarrow] \\
 & & 9 \arrow[r] & 10
\end{tikzcd}
batch 4 · p. 80 — read it beside the facsimile5 / 33
LaTeX source
\begin{tikzcd}
X \arrow[d, "f"'] \\
S
\end{tikzcd}
batch 5 · p. 82 — read it beside the facsimile6 / 33
LaTeX source
\begin{tikzcd}[column sep=small]
 & & \underline{P}_{X/S} \arrow[dl, "j"'] \arrow[d] & \\
0 \arrow[r] & f_*(\underline{E}_{X/S}) \arrow[r] & f_*(\underline{\mathrm{Div}}_{X/S})^{w} \arrow[r] & 0
\end{tikzcd}
batch 5 · p. 84 — read it beside the facsimile7 / 33
LaTeX source
\begin{tikzcd}[column sep=small]
0 \arrow[r] & \underline{\omega}_{X/S} \arrow[r] \arrow[d, "\mathrm{can}"] & f_*(\underline{E}_{X/S}) \arrow[r] \arrow[d, "\alpha"] & \underline{\mathrm{Pic}}^{w}_{X/S} \arrow[r] \arrow[d, "\mathrm{can}"] & 0 \\
0 \arrow[r] & \underline{\omega}_{X/S}/\mathrm{Im}\, f_*(\mathcal{O}_X^*) \arrow[r] & \mathcal{M}^{\#}_{X/S} \arrow[r] & \underline{\mathrm{Pic}}^{w}_{X/S} \arrow[r] & 0
\end{tikzcd}
batch 5 · p. 87 — read it beside the facsimile8 / 33
LaTeX source
\begin{tikzcd}
X \arrow[r] \arrow[d, "f"'] & A \arrow[dl] \\
S &
\end{tikzcd}
batch 5 · p. 90 — read it beside the facsimile9 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & \underline{\omega}^{*}_{X/S} \arrow[r] & \mathcal{M}_{X/S} \arrow[r] & \underline{\mathrm{Pic}}^{w}_{X/S} \arrow[r] & 0 \\
0 \arrow[r] & \underline{\omega}_{A/S} \arrow[r] \arrow[u] & \mathcal{M}_{A/S} \arrow[r] \arrow[u] & \underline{\mathrm{Pic}}^{w}_{A/S} \arrow[r] \arrow[u] & 0 \\
0 \arrow[r] & \underline{\omega}_{A/S} \arrow[r] \arrow[u, no head, "\Vert" description] & \mathcal{M}^{0}_{A/S} \arrow[r] \arrow[u] & B \arrow[r] \arrow[u, hook] & 0
\end{tikzcd}
batch 6 · p. 101 — read it beside the facsimile10 / 33
LaTeX source
\begin{tikzcd}
P(X) \arrow[r] \arrow[d] & P(X') \arrow[d] \\
P(U) \arrow[r] & P(U')
\end{tikzcd}
batch 6 · p. 101 — read it beside the facsimile11 / 33
LaTeX source
\begin{tikzcd}
\mathrm{Pic}(X) \arrow[r] \arrow[d] & \mathrm{Pic}(X') \arrow[d] \\
\mathrm{Pic}(U) \arrow[r] & \mathrm{Pic}(U')
\end{tikzcd}
batch 6 · p. 106 — read it beside the facsimile12 / 33
LaTeX source
\begin{tikzcd}
X \arrow[d] & X' \arrow[l] \arrow[d] \\
S & S' \arrow[l]
\end{tikzcd}
batch 6 · p. 117 — read it beside the facsimile13 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
0 \arrow[r] & A^* \arrow[r] & A_f^* \arrow[r] & \mathbb{Z}^I \arrow[r] & \mathbb{Z}^{(I)} \\
& K^*/A^* \arrow[r] \arrow[d] & K^*/A_f^* \arrow[d] & & \\
& \mathbb{Z} \arrow[r] & \mathbb{Z}^I & &
\end{tikzcd}
batch 6 · p. 117 — read it beside the facsimile14 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A_f^* \arrow[r] & \mathbb{Z}^I \\
A^*\times\mathbb{Z} \arrow[r] \arrow[u] & \mathbb{Z}^{?} \arrow[u]
\end{tikzcd}
batch 6 · p. 120 — read it beside the facsimile15 / 33
LaTeX source
\begin{tikzcd}
\Gamma(X_f,\mathcal{O}^*_{X_f}) \arrow[r] & \mathbb{Z}^I \\
\Gamma(X,\mathcal{O}_X^*)\times\mathbb{Z} \arrow[u] \arrow[r] & \mathbb{Z} \arrow[u]
\end{tikzcd}
batch 7 · p. 126 — read it beside the facsimile16 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X & X' \arrow[l, "p"'] & X'' \arrow[l, "{q_1, q_2}"'] & X''' \arrow[l, "{r_1, r_2, r_3}"'] \\
Z \arrow[u, "i"] & Z' \arrow[u, "{i'}"] \arrow[l, "\bar{p}"] & Z'' \arrow[u, "{i''}"] \arrow[l, "{\bar{q}_1, \bar{q}_2}"'] & \\
\Sigma \arrow[u, "g"] & \Sigma' \arrow[u, "{\bar{g}'}"'] & &
\end{tikzcd}
batch 7 · p. 132 — read it beside the facsimile17 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\underline{\mathrm{Pic}}_{X/k} \arrow[d, no head] & \underline{\mathrm{Pic}}_{Y/k} \arrow[l] \arrow[d, "\text{affine}"] \\
\underline{\mathrm{Pic}}_{X_{\mathrm{red}}} & \underline{\mathrm{Pic}}_{Y_{\mathrm{red}}} \arrow[l, "?"']
\end{tikzcd}
batch 9 · p. 163 — read it beside the facsimile18 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \times \mathrm{Pic} \arrow[d] & \\
\mathrm{Pic} & \mathrm{Div} \arrow[l, "d"]
\end{tikzcd}
batch 9 · p. 180 — read it beside the facsimile19 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\underline{\underline{\mathrm{Pic}}} \arrow[dr, "\varphi"'] & & \underline{\underline{\mathrm{Div}}} \arrow[ll, "p"'] \arrow[dl, "\psi"] \\
& S &
\end{tikzcd}
batch 10 · p. 186 — read it beside the facsimile20 / 33
LaTeX source
\begin{tikzcd}
E \arrow[d, "f"'] & E' \arrow[l, "h'"'] \arrow[d, "f'"'] \\
X & X' \arrow[l, "h"] \arrow[u, bend right=40, "q"']
\end{tikzcd}
batch 10 · p. 188 — read it beside the facsimile21 / 33
LaTeX source
\begin{tikzcd}
\tilde{C} \arrow[d] & \tilde{C} \times_{C} C_a = \tilde{S} \arrow[l] \arrow[d] \\
C & C_a = S \arrow[l]
\end{tikzcd}
batch 10 · p. 188 — read it beside the facsimile22 / 33
LaTeX source
\begin{tikzcd}[column sep=small]
\mathrm{Pic}(C_a - a) \arrow[d, "\wr"'] & \mathrm{Pic}(C - a) \arrow[l, "\sim"'] \arrow[d, "\wr"] \\
\mathrm{Pic}(\tilde{S} - X) & \mathrm{Pic}(\tilde{C} - X) \arrow[l, "\sim"']
\end{tikzcd}
batch 12 · p. 221 — read it beside the facsimile23 / 33
LaTeX source
\begin{tikzcd}
X^{(n)}_{a} \arrow[d] & \mathcal{S} \arrow[d] \\
S^{(0)}_{e} = \operatorname{Spec}(\mathbf{Z}[t]) & S \arrow[l]
\end{tikzcd}
batch 13 · p. 244 — read it beside the facsimile24 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\operatorname{Pic}(X_m) \arrow[r] & \underline{\operatorname{Pic}}_{X_m/k}(k) \\
\operatorname{Pic}(X_{n'}) \arrow[r] \arrow[u] & \underline{\operatorname{Pic}}_{X_{n'}/k}(k) \arrow[u] \\
\operatorname{Pic}(X_n) \arrow[r] \arrow[u] & \underline{\operatorname{Pic}}_{X_n/k}(k) \arrow[u]
\end{tikzcd}
batch 13 · p. 255 — read it beside the facsimile25 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
H^i(X_0, \mathbb{Z}) \arrow[r] \arrow[d, no head, "="] & R^if_*(\mathcal{O}_X)_y \arrow[r] \arrow[d, "u^i"] & R^if_*(\mathcal{O}^*_X)_y \arrow[r, "\partial^i"] \arrow[d, "w^i"] & H^{i+1}(X_0, \mathbb{Z}) \arrow[r, "\alpha^{i+1}"] \arrow[d, no head, "="] & R^{i+1}f_*(\mathcal{O}_X)_y \arrow[d, "u^{i+1}"] \\
H^i(X_0, \mathbb{Z}) \arrow[r] & R^if_*(\mathcal{O}_X)_y^{\wedge} \arrow[r] & \varprojlim_n H^i(X_n, \mathcal{O}^*_{X_n}) \arrow[r, "\hat{\partial}^i"] & H^{i+1}(X_0, \mathbb{Z}) \arrow[r, "\hat{\alpha}^{i+1}"] & R^{i+1}f_*(\mathcal{O}_X)_y^{\wedge}
\end{tikzcd}
batch 14 · p. 274 — read it beside the facsimile26 / 33
LaTeX source
\begin{tikzcd}
X \arrow[d] & X' \arrow[l] \arrow[d] \\
S & S' \arrow[l]
\end{tikzcd}
batch 15 · p. 282 — read it beside the facsimile27 / 33
LaTeX source
\begin{tikzcd}[row sep=small]
X \arrow[d, "g"] \\
Y \arrow[d, "f"] \\
S
\end{tikzcd}
batch 15 · p. 283 — read it beside the facsimile28 / 33
LaTeX source
\begin{tikzcd}
X \arrow[d, "f"'] & A \times_S B \arrow[l, "\Gamma"'] \arrow[d, "\mathrm{pr}_1"] \\
Y \arrow[d, "g"'] & B \arrow[l, "\Gamma"'] \\
S &
\end{tikzcd}
batch 15 · p. 286 — read it beside the facsimile29 / 33
LaTeX source
\begin{tikzcd}
X \arrow[d, "p"'] & A \times_S B \arrow[l, "\Gamma"'] \arrow[dl, "q"] \\
S &
\end{tikzcd}
batch 15 · p. 288 — read it beside the facsimile30 / 33
LaTeX source
\begin{tikzcd}
& X \arrow[dl, "\alpha"'] \arrow[dr, "p\beta"] & \\
A \arrow[rr, "u"] & & B
\end{tikzcd}
batch 15 · p. 297 — read it beside the facsimile31 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X_1 \amalg_Y X_2 & \\
X_1 \arrow[ur] & & X_2 \arrow[ul] \\
& Y \arrow[ul, "\varphi_1"] \arrow[ur, "\varphi_2"'] &
\end{tikzcd}
batch 16 · p. 303 — read it beside the facsimile32 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[d, "p"] & Y \arrow[l, "i"'] \arrow[dl, "{pi = \varepsilon f i = q}"] \\
X' \arrow[d, "f'"] & \\
S \arrow[u, bend right=40, "\varepsilon"'] &
\end{tikzcd}
batch 16 · p. 303 — read it beside the facsimile33 / 33
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
0 \arrow[r] & p_*(\underline{\mathcal{O}}_X^{*}) \arrow[r] & p_*(\underline{R}^{*}_{X/S}) \arrow[r] & p_*(\underline{\mathcal{D}}_{X/S}) \arrow[r] & 0 \\
0 \arrow[r] & \underline{\mathcal{O}}_{X'}^{*} \arrow[r] \arrow[u, "\alpha"] & \underline{R}^{*}_{X'/S} \arrow[r] \arrow[u, "\beta"] & \underline{\mathcal{D}}_{X'/S} \arrow[r] \arrow[u, "\gamma"] & 0
\end{tikzcd}