Cote n° 49 · pages 2–8 · 17 displayed formulas · Picard des schémas abéliens : notes manuscrites (s.d.).
Inventory dating : s.d.
Édition de démonstration

batch 1 · p. 2 — read it beside the facsimile1 / 17 · 11 distinct symbols, 16 written
\[\varphi_{s} : A \longrightarrow A'\struck{^{0}} = \mathrm{Pic}^{\tau}_{X/S}\]
LaTeX source
\[
  \varphi_{s} : A \longrightarrow A'\struck{^{0}} = \mathrm{Pic}^{\tau}_{X/S}
\]
batch 1 · p. 2 — read it beside the facsimile2 / 17 · 11 distinct symbols, 24 written
\[\varphi_{s}(t) = \text{classe de } \tau_{t}^{*}(L)\, L^{-1}\]
LaTeX source
\[
  \varphi_{s}(t) = \text{classe de } \tau_{t}^{*}(L)\, L^{-1}
\]
batch 1 · p. 2 — read it beside the facsimile3 / 17 · 9 distinct symbols, 19 written
\[\dim A'_{0} \;\leqslant\; \dim H^{1}(A_{0}, \mathcal{O}_{A_{0}}) \;\leqslant\; \dim A_{0}\]
LaTeX source
\[
  \dim A'_{0} \;\leqslant\; \dim H^{1}(A_{0}, \mathcal{O}_{A_{0}})
  \;\leqslant\; \dim A_{0}
\]
batch 1 · p. 3 — read it beside the facsimile4 / 17 · 10 distinct symbols, 13 written
\[\dim H^{1}(A_{0}, \mathcal{O}_{A_{0}}) = n ,\]
LaTeX source
\[
  \dim H^{1}(A_{0}, \mathcal{O}_{A_{0}}) = n ,
\]
batch 1 · p. 4 — read it beside the facsimile5 / 17 · 10 distinct symbols, 26 written
\[R^{*}f_{*}(\mathcal{O}_{A}) \;\simeq\; \Lambda^{*}\!\left(R^{1}f_{*}(\mathcal{O}_{A})\right)\]
LaTeX source
\[
  R^{*}f_{*}(\mathcal{O}_{A}) \;\simeq\;
  \Lambda^{*}\!\left(R^{1}f_{*}(\mathcal{O}_{A})\right)
\]
batch 1 · p. 4 — read it beside the facsimile6 / 17 · 14 distinct symbols, 35 written
\[R^{1}f_{*}(\mathcal{O}_{A}) \;=\; \mathrm{Hom}\!\left(\struck{\varepsilon^{*}(\Omega^{1}_{A/S})}\; \Omega^{1}_{\varepsilon},\; \mathcal{O}_{S}\right)\]
LaTeX source
\[
  R^{1}f_{*}(\mathcal{O}_{A}) \;=\;
  \mathrm{Hom}\!\left(\struck{\varepsilon^{*}(\Omega^{1}_{A/S})}\;
  \Omega^{1}_{\varepsilon},\; \mathcal{O}_{S}\right)
\]
batch 1 · p. 4 — read it beside the facsimile7 / 17 · 11 distinct symbols, 21 written
\[R^{*}f_{*}(\mathcal{O}_{A})_{y} \longrightarrow H^{*}(X_{y}, \mathcal{O}_{X_{y}})\]
LaTeX source
\[
  R^{*}f_{*}(\mathcal{O}_{A})_{y} \longrightarrow H^{*}(X_{y}, \mathcal{O}_{X_{y}})
\]
batch 1 · p. 6 — read it beside the facsimile8 / 17 · 9 distinct symbols, 25 written
\[(*) \qquad \mathrm{Pic}_{A/S} \longrightarrow \mathrm{Hom}_{S\text{-gr}}(A, A')\]
LaTeX source
\[
  (*) \qquad \mathrm{Pic}_{A/S} \longrightarrow \mathrm{Hom}_{S\text{-gr}}(A, A')
\]
batch 1 · p. 6 — read it beside the facsimile9 / 17 · 9 distinct symbols, 17 written
\[(**) \qquad \mathrm{Pic}_{A/S} \times_{S} A \longrightarrow A'\]
LaTeX source
\[
  (**) \qquad \mathrm{Pic}_{A/S} \times_{S} A \longrightarrow A'
\]
batch 1 · p. 6 — read it beside the facsimile10 / 17 · 7 distinct symbols, 12 written
\[(\xi, t) \;\rightsquigarrow\; t_{*}(\xi) - \xi .\]
LaTeX source
\[
  (\xi, t) \;\rightsquigarrow\; t_{*}(\xi) - \xi .
\]
batch 1 · p. 6 — read it beside the facsimile11 / 17 · 10 distinct symbols, 23 written
\[\pi^{*}(\xi) \;=\; \mathrm{pr}_{1}^{*}(\xi) + \mathrm{pr}_{2}^{*}(\xi)\]
LaTeX source
\[
  \pi^{*}(\xi) \;=\; \mathrm{pr}_{1}^{*}(\xi) + \mathrm{pr}_{2}^{*}(\xi)
\]
batch 1 · p. 6 — read it beside the facsimile12 / 17 · 7 distinct symbols, 23 written
\[\text{Nér-Sév}_{A/S} \longrightarrow \mathrm{Hom}_{S\text{-gr}}(A, A')\]
LaTeX source
\[
  \text{Nér-Sév}_{A/S} \longrightarrow
  \mathrm{Hom}_{S\text{-gr}}(A, A')
\]
batch 1 · p. 7 — read it beside the facsimile13 / 17 · 7 distinct symbols, 23 written
\[\text{Nér-Sév}_{A/S} \longrightarrow \mathrm{Hom}_{S\text{-gr}}(A, A')\]
LaTeX source
\[
  \text{Nér-Sév}_{A/S} \longrightarrow
  \mathrm{Hom}_{S\text{-gr}}(A, A')
\]
batch 1 · p. 8 — read it beside the facsimile14 / 17 · 12 distinct symbols, 35 written
\[\varphi_{\xi} = 0 \quad \text{ssi} \quad \pi^{*}(\xi) - \mathrm{pr}_{1}^{*}(\xi) - \mathrm{pr}_{2}^{*}(\xi) = 0 .\]
LaTeX source
\[
  \varphi_{\xi} = 0 \quad \text{ssi} \quad
  \pi^{*}(\xi) - \mathrm{pr}_{1}^{*}(\xi) - \mathrm{pr}_{2}^{*}(\xi) = 0 .
\]
batch 1 · p. 8 — read it beside the facsimile15 / 17 · 9 distinct symbols, 25 written
\[t_{1}^{*}(\xi_{1}) = \xi_{1} \qquad (\text{ici } \xi_{1} = \xi \times_{S} S_{1}) .\]
LaTeX source
\[
  t_{1}^{*}(\xi_{1}) = \xi_{1} \qquad (\text{ici } \xi_{1} = \xi \times_{S} S_{1}) .
\]
batch 1 · p. 8 — read it beside the facsimile16 / 17 · 11 distinct symbols, 41 written
\[t_{1}^{*}\!\left(\mathrm{pr}_{2}^{*}(L)\right) = \left(\mathrm{pr}_{2} \circ \mathrm{transl}(t_{1})\right)^{*}(L)\]
LaTeX source
\[
  t_{1}^{*}\!\left(\mathrm{pr}_{2}^{*}(L)\right) =
  \left(\mathrm{pr}_{2} \circ \mathrm{transl}(t_{1})\right)^{*}(L)
\]
batch 1 · p. 8 — read it beside the facsimile17 / 17 · 13 distinct symbols, 44 written
\[i_{1}^{*}(\cdots) = (\pi i_{1})^{*}(L) - (\mathrm{pr}_{1} i_{1})^{*}(L) - (\mathrm{pr}_{2} i_{1})^{*}(L) \;=\; 0\]
LaTeX source
\[
  i_{1}^{*}(\cdots) = (\pi i_{1})^{*}(L) - (\mathrm{pr}_{1} i_{1})^{*}(L)
  - (\mathrm{pr}_{2} i_{1})^{*}(L) \;=\; 0
\]