Cote n° 91 · pages 15–135 · 32 commutative diagrams · Autour de Néron / Greenberg-Néron. Foncteurs Hom (méthodes non-projectives) : notes manuscrites (s.d.), lettre (1967).
Inventory dating : [à partir de 1964-vers 1970]
Édition de démonstration

batch 1 · p. 15 — read it beside the facsimile1 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
Z \arrow[r, "\varphi"] \arrow[dr, "h"'] & X \arrow[d, no head, "f"] & Y \arrow[dl, no head, "g"] \\
 & S &
\end{tikzcd}
batch 1 · p. 15 — read it beside the facsimile2 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
{Z'} \arrow[r, "{\varphi'}"] \arrow[rr, bend left=20, "v"] \arrow[dr, "{h'}"'] & {X'} \arrow[r, "w"] \arrow[d, "{f'}"] & {Y'} \arrow[dl, "{g'}"] \\
 & {S'} &
\end{tikzcd}
batch 1 · p. 17 — read it beside the facsimile3 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
\text{(1)} & P(u_0,S) \arrow[r] \arrow[d] & P(u_0,S_\eta)\ast \arrow[d] \\
 & \ast P(v_0,S) \arrow[r] & P(v_0,S_\eta)
\end{tikzcd}
batch 1 · p. 17 — read it beside the facsimile4 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
\text{(2)} & \mathrm{Hom}(M_{u_0}^{*},J) \arrow[r] \arrow[d] & \mathrm{Hom}(M_{u_0},J_\eta) \arrow[d] \\
 & \mathrm{Hom}(M_{v_0},J) \arrow[r] & \mathrm{Hom}(M_{v_0},J_\eta)
\end{tikzcd}
batch 2 · p. 22 — read it beside the facsimile5 / 32
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\begin{tikzcd}[column sep=small]
0 \arrow[r] & \mathcal{O}_{\Gamma_0}\otimes_{S_0} J \arrow[r] \arrow[d] & E \arrow[r] \arrow[d] & \underline{I}_{\Gamma_0} \arrow[r] \arrow[d] & 0 \\
0 \arrow[r] & \mathcal{O}_{\Gamma'_0}\otimes_{S_0} J \arrow[r] & E' \arrow[r] & \underline{I}_{\Gamma'_0} \arrow[r] & 0
\end{tikzcd}
batch 2 · p. 22 — read it beside the facsimile6 / 32
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\begin{tikzcd}[row sep=small]
 & P_0 \arrow[d] \\
Z \arrow[r] & X
\end{tikzcd}
batch 2 · p. 23 — read it beside the facsimile7 / 32
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\begin{tikzcd}
 & L \arrow[r, "i"] & M \arrow[r, "j"] & N \arrow[r] & 0 \\
0 \arrow[r] & L' \arrow[r, "i'"] \arrow[u, leftrightarrow, "\alpha"'] & M' \arrow[r, "j'"] \arrow[u, leftrightarrow, "\beta"'] & N' \arrow[r] \arrow[u, "\gamma"'] & 0
\end{tikzcd}
batch 2 · p. 23 — read it beside the facsimile8 / 32
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\begin{tikzcd}[row sep=small]
 & & L \arrow[dll, "\sigma"'] \\
J \arrow[d, dashed, "\lambda"'] & & L' \arrow[u, "\alpha"'] \\
G & &
\end{tikzcd}
batch 2 · p. 24 — read it beside the facsimile9 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
\operatorname{Ker} i \arrow[rr, dashed] \arrow[d] & & L \arrow[dll, dashed] & \\
J & & L' \arrow[ll, dashed] \arrow[u, dashed, "?"'] & M' \arrow[l, dashed] \\
 & & & \operatorname{Ker}\beta \arrow[u, dashed] \arrow[ulll, bend left=10]
\end{tikzcd}
batch 2 · p. 24 — read it beside the facsimile10 / 32
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\begin{tikzcd}
L + M' \arrow[r, two heads, "{(i,\beta)}"] \arrow[d, "{(\sigma,\, \sigma\alpha\pi') = \rho}"'] & M \arrow[ddl, dashed, "\mu"] \\
J \arrow[d, "\lambda"'] & \\
G &
\end{tikzcd}
batch 2 · p. 34 — read it beside the facsimile11 / 32
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\begin{tikzcd}
 & X \arrow[d] \arrow[dl, dashed, "\varphi"'] & Y \arrow[dl] & \\
S & T \arrow[l, "f"'] & T \times_S T \arrow[l, bend right=15, "\mathrm{pr}_1"'] \arrow[l, bend left=15, "\mathrm{pr}_2"] \arrow[ll, bend left=40, "g"] &
\end{tikzcd}
batch 2 · p. 38 — read it beside the facsimile12 / 32
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\begin{tikzcd}
X \arrow[d, "f"] \\
Y = \operatorname{Spec}(\mathcal{O})
\end{tikzcd}
batch 2 · p. 39 — read it beside the facsimile13 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
 & X \arrow[d] & \\
S & T \arrow[l, "f"'] & X \times_S S' \arrow[ul] \arrow[dl] \\
S' \arrow[u] & S' \times_S T = T' \arrow[l, "f'"] \arrow[u] \arrow[ur, bend right=20] &
\end{tikzcd}
batch 2 · p. 39 — read it beside the facsimile14 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
 & X_s \arrow[d] & \\
s & T_s \arrow[l, no head] & \\
 & & X'_{s'} \arrow[uul] \arrow[dl] \\
s' & T'_{s'} \arrow[l, no head] \arrow[uu] &
\end{tikzcd}
batch 2 · p. 40 — read it beside the facsimile15 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
 & X \arrow[d] & \\
S & T \arrow[l] & \\
H \arrow[u] & H \times_S T = \bigcup H_i \times_S T \arrow[l] \arrow[u] \arrow[uu, bend right=30] &
\end{tikzcd}
batch 3 · p. 43 — read it beside the facsimile16 / 32
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\begin{tikzcd}
  & X \arrow[d] \\
  S & T \arrow[l, "f"']
\end{tikzcd}
batch 3 · p. 49 — read it beside the facsimile17 / 32
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\begin{tikzcd}
  f_{*}(X/T) \arrow[r] & f_{*}(Y/T) \\
  S'_1 \arrow[u] \arrow[r, hook] & S' \arrow[u] \arrow[ul, dashed]
\end{tikzcd}
batch 3 · p. 49 — read it beside the facsimile18 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
  & T & & Y & \\
  S & & T' \arrow[ul] \arrow[ur] \arrow[rr, dashed] \arrow[dl] & & X \arrow[ul] \\
  & S' \arrow[ul] & & T'_1 \arrow[ul] \arrow[ur] \arrow[dl] & \\
  & & S'_1 \arrow[ul] & &
\end{tikzcd}
batch 3 · p. 50 — read it beside the facsimile19 / 32
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\begin{tikzcd}
  & f_{*}(X) \arrow[d, leftrightarrow] \\
  \operatorname{Spec}(k') \arrow[r] & f_{*}(Y)
\end{tikzcd}
batch 3 · p. 53 — read it beside the facsimile20 / 32
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\begin{tikzcd}[column sep=small]
  X_s \arrow[r] & Y_s & & X'_s \arrow[r] & Y'_s \\
  k(s)=k \quad T_s & & k' & T'_s \arrow[u] \arrow[ur] &
\end{tikzcd}
batch 5 · p. 86 — read it beside the facsimile21 / 32
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\begin{tikzcd}
\text{Préschémas atomiques} \arrow[d, "{\text{foncteur pleinement fidèle}}"] \\
\text{Foncteurs covariants}\ (\mathrm{Corps}) \to (\mathrm{Ens})
\end{tikzcd}
batch 5 · p. 86 — read it beside the facsimile22 / 32
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\begin{tikzcd}
\text{Préschémas atomiques sur } k\ [\text{i.e. sur } \operatorname{Spec}(k)] \arrow[d, "{\text{foncteur pleinement fidèle}}"] \\
\text{Foncteurs covariants}\ (\text{Extensions de } k) \to (\mathrm{Ens})
\end{tikzcd}
batch 5 · p. 88 — read it beside the facsimile23 / 32
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\begin{tikzcd}[column sep=small, nodes={font=\scriptsize}]
(\mathrm{Sch}) \arrow[r] & (\mathrm{Sch}) \arrow[r, "\mathrm{red}"', "\text{fidèle}"] & (\text{ens. } \mathrm{Sch}) \arrow[r, "\text{fidèle}"'] & (\text{Sch. atom.}) \arrow[r, "\text{pl.\ fid.}"'] & {\underline{\operatorname{Hom}}((\mathrm{Corps}), \mathrm{Ens})}
\end{tikzcd}
batch 5 · p. 90 — read it beside the facsimile24 / 32
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\begin{tikzcd}
X \arrow[r, bend left=20, "f"] & Y \arrow[l, bend left=20, "g"]
\end{tikzcd}
batch 5 · p. 100 — read it beside the facsimile25 / 32
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\begin{tikzcd}[column sep=large]
(\text{Ens. élémentaires sur } k) \arrow[r, "\text{pl.\ fidèle}"'] & {\underline{\operatorname{Hom}}\bigl((\text{Ens. élém.}/k)^{\circ}, (\mathrm{Ens})\bigr)}
\end{tikzcd}
batch 5 · p. 100 — read it beside the facsimile26 / 32
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\begin{tikzcd}
& X & \\
S \arrow[ur, dashed] & S' \arrow[l] \arrow[u] & S'' \arrow[l, bend left=12] \arrow[l, bend right=12]
\end{tikzcd}
batch 5 · p. 100 — read it beside the facsimile27 / 32
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\begin{tikzcd}[column sep=small]
X \times_Y X \arrow[r, bend left=12] \arrow[r, bend right=12] & X \arrow[r] \arrow[dr] & Y \\
& & X/\mathrm{Rel}
\end{tikzcd}
batch 6 · p. 113 — read it beside the facsimile28 / 32
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\begin{tikzcd}[column sep=small, row sep=large, nodes={font=\scriptsize}]
(\mathrm{Cons\,Aff}) \arrow[r] \arrow[rr, bend right=25, "\text{canonique}"'] \arrow[rrr, bend right=35, "\mathrm{can}"'] & (\mathrm{Complexes}(X,E)) \arrow[r] & \mathrm{Ind}(\mathrm{Cons\,Aff})' \arrow[r] \arrow[r, bend right=40, "\mathrm{can}"'] & \underline{\mathrm{Hom}}((\mathrm{Cons\,Aff})^{\circ},\mathrm{Ens})
\end{tikzcd}
batch 6 · p. 117 — read it beside the facsimile29 / 32
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\begin{tikzcd}[column sep=large, row sep=large, nodes={font=\scriptsize}]
X \arrow[r, "\alpha"] \arrow[rr, bend left=30, "{\gamma\ [\mathrm{Cons\ cpct}]}"] & Y \arrow[r, "\beta=\varprojlim"] & K \arrow[l, bend left=35, "{\beta'\ \text{pl. fid.}}"] \arrow[ll, bend left=55, "{\gamma'\ \text{pl. fid.}}"]
\end{tikzcd}
batch 6 · p. 119 — read it beside the facsimile30 / 32
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\begin{tikzcd}[column sep=large, row sep=large, nodes={font=\scriptsize}]
(\text{Préschémas quasi-séparés}) \arrow[r, "\overline{\alpha}"] \arrow[rr, bend left=30, "\overline{\gamma}"] & \mathrm{Ind}(\mathrm{Cons\,cpct})\text{ flasques} \arrow[r, "\overline{\beta}", "\text{surjectif}"'] & (\text{schémas loc.\ cpcts réduits}) \arrow[l, bend left=35, "{\overline{\beta}'\ \text{pl. fid.}}"] \arrow[ll, bend left=50, "{\overline{\gamma}'\ \text{pl. fid.}}"]
\end{tikzcd}
batch 6 · p. 119 — read it beside the facsimile31 / 32
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\begin{tikzcd}[column sep=large]
(\mathrm{Cons}) \arrow[r, "\text{pl. fid.}"] & \underline{\mathrm{Hom}}((\mathrm{Cons\,cpct})^{\circ},(\mathrm{Ens}))
\end{tikzcd}
batch 7 · p. 135 — read it beside the facsimile32 / 32
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\begin{tikzcd}[column sep=small, row sep=large, nodes={font=\scriptsize}]
 & \operatorname{Pro}(\mathrm{Sch}_{/S}) \\
 (\mathrm{Sch}_{/S}) & \operatorname{Pro}'(\mathrm{Sch}_{/S}) \arrow[l, "\varprojlim"'] \arrow[u, hook, "\text{sous-catégorie pleine}"'] \\
 & \operatorname{Pro}''(\mathrm{Sch}_{/S}) \arrow[u, hook, "\text{sous-catégorie pleine}"'] \arrow[ul, "\text{foncteur pleinement fidèle}"]
\end{tikzcd}