Cote n° 130 · pages 4–46 · 19 diagrammes commutatifs · Hauteurs et symboles locaux : notes manuscrites (s.d.).
Datation de l’inventaire : [vers 1971]
Édition de démonstration

batch 1 · p. 4 — lire la page1 / 19
LaTeX source
\begin{tikzcd}[column sep=small]
  D_a(A_K)/D_{\ell}(A_K) \times Z_0^{*}(A_K)/Z_{\ell}^{*}(A_K) \arrow[r] \arrow[d, no head, "\wr" description] & \mathbb{Z}/m\mathbb{Z} \arrow[d, no head, "\wr" description] \\
  A^{*}(K) & {}_{m}(\mathbb{Q}/\mathbb{Z})
\end{tikzcd}
batch 1 · p. 9 — lire la page2 / 19
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\begin{tikzcd}
  X & \tilde{X} \arrow[l, "\varphi"'] \arrow[d] & \tilde{Y}_{\bar{t}} \\
  & \mathbb{P}^{1} &
\end{tikzcd}
batch 1 · p. 9 — lire la page3 / 19
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\begin{tikzcd}
  \mathrm{Pic}(X) \arrow[r, hook] \arrow[dr] & \mathrm{Pic}(\tilde{X}) \arrow[d] \\
  & \mathrm{Pic}(\tilde{Y}_{\bar{\eta}})
\end{tikzcd}
batch 1 · p. 9 — lire la page4 / 19
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\begin{tikzcd}
  \mathrm{Pic}(X^{*})' \arrow[r, hook] \arrow[dr] & \mathrm{Pic}(\tilde{X})' \arrow[d] \\
  & \mathrm{Pic}^{0}(\tilde{X}_{\bar{\eta}}) = J^{0}(\tilde{X}_{\bar{\eta}})
\end{tikzcd}
batch 1 · p. 11 — lire la page5 / 19
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\begin{tikzcd}
  \mathrm{Pic}^{0}(X) \arrow[r, "\sim"] & \mathrm{Pic}^{0}(\tilde{X}) \arrow[r, "\sim"] & J(Y_{\bar{\eta}})^{\mathrm{fixe}}(k) \arrow[d, no head, "\wr" description] \\
  & & J(Y_{\bar{\eta}})^{\mathrm{fixe}}(k(\eta))
\end{tikzcd}
batch 1 · p. 11 — lire la page6 / 19
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\begin{tikzcd}
  NS(X)' \arrow[r, hook] & NS(\tilde{X})' \arrow[d] \\
  & J'(k(\eta))
\end{tikzcd}
batch 2 · p. 21 — lire la page7 / 19
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\begin{tikzcd}
X \arrow[d, "f"] \\
S
\end{tikzcd}
batch 2 · p. 23 — lire la page8 / 19
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\begin{tikzcd}
& \underline{T} \\
\underline{A}^{\circ} & \underline{B} \arrow[l]
\end{tikzcd}
batch 2 · p. 23 — lire la page9 / 19
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\begin{tikzcd}
A_\eta & B_\eta \arrow[l] \\
& T_\eta \arrow[u]
\end{tikzcd}
batch 2 · p. 23 — lire la page10 / 19
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\begin{tikzcd}
\underline{A} \arrow[r, bend left=20] & \underline{B_1 \times B_2} \arrow[l, bend left=20] \\
B_1 \times B_2 \arrow[u]
\end{tikzcd}
batch 2 · p. 23 — lire la page11 / 19
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\begin{tikzcd}
B_\eta & A_\eta \arrow[l, "\beta"'] & B_\eta \arrow[l, "\alpha"'] \arrow[ll, bend right=40, "N"'] \\
\underline{B} & \underline{A} \arrow[l] \arrow[u] & \underline{B} \arrow[l]
\end{tikzcd}
batch 2 · p. 23 — lire la page12 / 19
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\begin{tikzcd}
0 \arrow[d, leftrightarrow] \\
\underline{A} \\
\underline{B} \arrow[u] \\
C \arrow[u]
\end{tikzcd}
batch 2 · p. 23 — lire la page13 / 19
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\begin{tikzcd}
0 \\
A \arrow[d] \\
B \arrow[d] \\
C
\end{tikzcd}
batch 2 · p. 25 — lire la page14 / 19
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\begin{tikzcd}
X_\eta & X \arrow[d, no head] & X' \arrow[r, hook] \arrow[d, no head] & J \\
\eta & S & S
\end{tikzcd}
batch 2 · p. 25 — lire la page15 / 19
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\begin{tikzcd}
X' \arrow[r, hook] & \widetilde{X''} \arrow[r, "\mathrm{birat.}"] & X'' \arrow[r, hook] & J
\end{tikzcd}
batch 2 · p. 27 — lire la page16 / 19
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\begin{tikzcd}
X \arrow[rr, "\alpha"] \arrow[dr, no head] & & A \arrow[dl, no head] \\
& S &
\end{tikzcd}
batch 2 · p. 31 — lire la page17 / 19
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\begin{tikzcd}
X' \arrow[rr, "\alpha"] \arrow[dr, no head] & & A \arrow[dl, no head] \\
& S &
\end{tikzcd}
batch 3 · p. 44 — lire la page18 / 19
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\begin{tikzcd}[column sep=small, row sep=small]
 & & 0 & & \\
0 \arrow[r, dashed] & \underline{\Phi}_X / \mathrm{Im}\, \underline{\mathcal{O}}^*_X \arrow[r, dashed, "\simeq"] & \underline{\Phi}_X / \mathrm{Im}\, \underline{\mathcal{O}}^*_X \arrow[u] \arrow[r, dashed] & 0 & \\
0 \arrow[r] & \underline{\Phi}_X \arrow[r] \arrow[u, dashed] & \underline{QF}_X \arrow[r] \arrow[u] & \underline{\mathrm{Div}}_X \arrow[u, dashed, "\wr"] & \\
0 \arrow[r] & \underline{\mathcal{O}}^*_X / \underline{U}_X \arrow[r] \arrow[u] & \underline{R}^*_X / \underline{U}_X \arrow[r, dashed] \arrow[u] & \underline{\mathrm{Div}}_X \arrow[r, dashed] \arrow[u, dashed, "\wr"] & 0 \\
 & 0 \arrow[u] & 0 \arrow[u] & 0 \arrow[u, dashed, "\wr"] &
\end{tikzcd}
batch 3 · p. 46 — lire la page19 / 19
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & 0 & 0 & 0 & 0 \\
0 \arrow[r] & \Phi(X) / (\mathbf{G}_m(X)/U) \arrow[r] \arrow[u] & \Gamma(X, \underline{\Phi}_X / (\underline{\mathcal{O}}^*_X / \underline{U}_X)) \arrow[r] \arrow[u] & \mathrm{Pic}(X) \arrow[r] \arrow[u] & H^1(X, \underline{\Phi}_X) \arrow[u] \\
0 \arrow[r] & \Phi(X) \arrow[r] \arrow[u] & QF(X) \arrow[r] \arrow[u] & \mathrm{Div}(X) \arrow[r] \arrow[u] & H^1(X, \underline{\Phi}_X) \arrow[u, no head, "\Vert" description] \\
0 \arrow[r] & \mathbf{G}_m(X)/U \arrow[r] \arrow[u] & R(X)^*/U \arrow[r] \arrow[u] & \mathrm{Div}_{\ell}(X) \arrow[r] \arrow[u] & 0 \arrow[u] \\
 & 0 \arrow[u] & 0 \arrow[u] & 0 \arrow[u] &
\end{tikzcd}