Cote n° 33 · pages 2–161 · 32 commutative diagrams · SGA 1965. MS [Manuscrit] brouillon : notes manuscrites (s.d.).
Inventory dating : 1965-[vers 1971]
Édition de démonstration

batch 1 · p. 2 — read it beside the facsimile1 / 32
LaTeX source
\begin{tikzcd}
Y \arrow[d, "f"'] & Y' \arrow[l, "h"'] \arrow[d, "{f'}"] \\
X & X' \arrow[l, "g"]
\end{tikzcd}
batch 1 · p. 11 — read it beside the facsimile2 / 32
LaTeX source
\begin{tikzcd}
Y \arrow[rr, hook] \arrow[dr, "\text{lisse}"'] & & X \arrow[dl, "\text{lisse}"] \\
& k &
\end{tikzcd}
batch 2 · p. 29 — read it beside the facsimile3 / 32
LaTeX source
\begin{tikzcd}[column sep=small]
  \Gamma\bigl(Y, \underline{H}^{2d}_Y(\mu_{n,X}^{\otimes d})\bigr) \arrow[d, "\wr"] & \\
  H^{2d}_Y(X, \mu_{n,X}^{\otimes d}) \arrow[r] & \Gamma\bigl(S, R^{2d}_{!} f_{*}(\mu_{n,X}^{\otimes d})\bigr)
\end{tikzcd}
batch 2 · p. 35 — read it beside the facsimile4 / 32
LaTeX source
\begin{tikzcd}
  Y \arrow[r, hook, "i", "(d)"'] \arrow[d, "g"', "(r)"] & X \arrow[dl, "f", "(r+d)"'] \\
  S &
\end{tikzcd}
batch 2 · p. 35 — read it beside the facsimile5 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  & R^{2r+2d}_{\text{supp.\ propres}/S,\ \subset Y} f_{*}(T_{X/S}) \arrow[d] \\
  (\mathbb{Z}/n\mathbb{Z})_S & R^{2r+2d}_{!} g_{*X}(T_{X/S}) \arrow[l]
\end{tikzcd}
batch 3 · p. 41 — read it beside the facsimile6 / 32
LaTeX source
\begin{tikzcd}
X \arrow[r, hook, "i"] \arrow[d, "f"'] & \overline{X} \arrow[dl, "\overline{f}"] \\
Y &
\end{tikzcd}
batch 3 · p. 43 — read it beside the facsimile7 / 32
LaTeX source
\begin{tikzcd}
X \arrow[d, "f"'] & X' \arrow[l, "h"'] \arrow[d, "{f'}"] \\
Y & Y' \arrow[l, "g"]
\end{tikzcd}
batch 3 · p. 55 — read it beside the facsimile8 / 32
LaTeX source
\begin{tikzcd}
Y \arrow[r, hook, "i"] \arrow[dr, "g"'] & X \arrow[d, "f"] \\
& S
\end{tikzcd}
batch 3 · p. 55 — read it beside the facsimile9 / 32
LaTeX source
\begin{tikzcd}
Y \arrow[r, hook] \arrow[d] & X \arrow[d] \\
Y' \arrow[r] \arrow[dr, "{g'}"'] & X' = S[t_1, \ldots, t_d] \arrow[d, "{f'}"] \\
& S
\end{tikzcd}
batch 3 · p. 57 — read it beside the facsimile10 / 32
LaTeX source
\begin{tikzcd}
Y \arrow[d, "i"'] & Y' \arrow[l, "g"'] \arrow[d, "{i'}"] \\
X & X' \arrow[l, "f"]
\end{tikzcd}
batch 4 · p. 61 — read it beside the facsimile11 / 32
LaTeX source
\begin{tikzcd}
X \arrow[r, hook, "i"] \arrow[d, "f"'] & X' \arrow[dl, "{f'}"] \\
Y &
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile12 / 32
LaTeX source
\begin{tikzcd}[column sep=small]
 & X''' \arrow[ddl] \arrow[ddr] & \\
 & X \arrow[u] \arrow[dl] \arrow[dr] & \\
X' & & X''
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile13 / 32
LaTeX source
\begin{tikzcd}
 & X'' \arrow[r, "g"] \arrow[dr, "{f''}"'] & X' \arrow[d, "{f'}"] \\
X \arrow[ur, "{i''}"] \arrow[urr, "{i'}" description] \arrow[rr, "f"'] & & Y
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile14 / 32
LaTeX source
\begin{tikzcd}
X \arrow[r, hook, "j"] \arrow[d, "g"'] & X' \arrow[d, "{g'}"] \\
Y \arrow[r, hook, "i"'] & Y' \arrow[u, bend right=40, "\sigma"']
\end{tikzcd}
batch 4 · p. 73 — read it beside the facsimile15 / 32
LaTeX source
\begin{tikzcd}
Y \arrow[rr, hook, "i"] \arrow[dr, "g"'] & & X \arrow[dl, "f"] \\
 & S &
\end{tikzcd}
batch 4 · p. 75 — read it beside the facsimile16 / 32
LaTeX source
\begin{tikzcd}[column sep=small, nodes={font=\scriptsize}]
\mathbb{R}_!g_*\bigl(\mathbb{R}g^!(G^\bullet)\bigr) \arrow[rr] \arrow[dr, "{\mathrm{Tr}_g(G^\bullet)}"'] & & \mathbb{R}_!f_*\,\mathbb{R}f^!(G^\bullet) \arrow[dl, "{\mathrm{Tr}_f(G^\bullet)}"] \\
 & G^\bullet &
\end{tikzcd}
batch 4 · p. 77 — read it beside the facsimile17 / 32
LaTeX source
\begin{tikzcd}[column sep=small, nodes={font=\scriptsize}]
\mathrm{Ext}^i\bigl(Y; F^\bullet, \mathbb{R}g^!(G^\bullet)\bigr) \arrow[r, "\Theta_g"] \arrow[d, "{\Theta_i}"'] & \mathrm{Ext}^i\bigl(S; \mathbb{R}_!g_*(F^\bullet), G^\bullet\bigr) \arrow[d, "{\text{transitivité}}"] \\
\mathrm{Ext}^i\bigl(X; \mathbb{R}_!i_*(F^\bullet), \mathbb{R}f^!(G^\bullet)\bigr) \arrow[r, "\Theta_f"] & \mathrm{Ext}^i\bigl(S; \mathbb{R}_!f_*(\mathbb{R}_!i_*(F^\bullet)), G^\bullet\bigr)
\end{tikzcd}
batch 5 · p. 83 — read it beside the facsimile18 / 32
LaTeX source
\begin{tikzcd}
X \arrow[r, "f"] \arrow[dr, "h"'] & Y \arrow[d, "g"] \\
& Z
\end{tikzcd}
batch 5 · p. 85 — read it beside the facsimile19 / 32
LaTeX source
\begin{tikzcd}
X \arrow[r, hook] \arrow[d] & X'_{Y} \arrow[r, hook] \arrow[dl] & X' \arrow[dl] \\
Y \arrow[r, hook] \arrow[d] & Y' \arrow[dl, "\text{lisse}"] & \\
Z & &
\end{tikzcd}
batch 5 · p. 87 — read it beside the facsimile20 / 32
LaTeX source
\begin{tikzcd}
X \arrow[d, "f"] & F^{\cdot} \\
Y & G^{\cdot}
\end{tikzcd}
batch 5 · p. 91 — read it beside the facsimile21 / 32
LaTeX source
\begin{tikzcd}
X \arrow[d, "f"] & X' \arrow[l, "g'"'] \arrow[d, "f'"] \\
Y & Y' \arrow[l, "g"]
\end{tikzcd}
batch 5 · p. 91 — read it beside the facsimile22 / 32
LaTeX source
\begin{tikzcd}
X \arrow[d, "f"] & \\
Y & F,G
\end{tikzcd}
batch 5 · p. 95 — read it beside the facsimile23 / 32
LaTeX source
\begin{tikzcd}
X' \arrow[r, hook, "j"] \arrow[d, "f'"'] & X \arrow[d, "f"] \\
Y' \arrow[r, hook, "i"'] & Y
\end{tikzcd}
batch 6 · p. 105 — read it beside the facsimile24 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
  X \arrow[d, "f"'] & K^{\bullet} \\
  Y & L_{\bullet}
\end{tikzcd}
batch 6 · p. 105 — read it beside the facsimile25 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
  X' \arrow[r, "j'"] \arrow[d, "f'"'] & X \arrow[d, "f"] \\
  Y' \arrow[r, "i"'] & Y
\end{tikzcd}
batch 6 · p. 113 — read it beside the facsimile26 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
  X' \arrow[r, "j"] \arrow[d, "f'"'] & X \arrow[d, "f"] \\
  Y' \arrow[r, "i"'] & Y
\end{tikzcd}
batch 6 · p. 115 — read it beside the facsimile27 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
  & Z = X \times_S Y \arrow[dl, "g'"'] \arrow[dr, "f'"] \arrow[dd, "h"] & \\
  X \arrow[dr, "f"'] & & Y \arrow[dl, "g"] \\
  & S &
\end{tikzcd}
batch 8 · p. 143 — read it beside the facsimile28 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & X\times_S Y \arrow[dl, "\mathrm{pr}_1"'] \arrow[dr, "\mathrm{pr}_2"] \arrow[dd, "h"] & \\
F_\bullet\;\; X \arrow[dr, "f"'] & & Y\;\; G_\bullet \arrow[dl, "g"] \\
 & S &
\end{tikzcd}
batch 8 · p. 145 — read it beside the facsimile29 / 32
LaTeX source
\begin{tikzcd}
A_X \arrow[r, "\varepsilon"] \arrow[drr, "\tau_{\mathcal{L}_\bullet}"'] & \mathbb{R}\mathcal{H}om(\mathcal{L}_\bullet, \mathcal{L}_\bullet) & \check{\mathcal{L}}_\bullet \overset{L}{\otimes} \mathcal{L}_\bullet \arrow[l, "\simeq"'] \arrow[d, "\tau"] \\
 & & A_X
\end{tikzcd}
batch 8 · p. 157 — read it beside the facsimile30 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathbb{R}\mathrm{Hom}^{\bullet}\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) \times \mathbb{R}\mathrm{Hom}^{\bullet}\bigl(\mathrm{pr}_2^*(G_\bullet), \mathrm{pr}_1^!(F_\bullet)\bigr) \arrow[r] \arrow[d, no head, "\wr"] & \mathbb{R}K^\bullet_{X\times_S Y} \arrow[d] \\
\mathbb{R}\mathrm{Hom}^{\bullet}\bigl(f_*(F_\bullet), g_*(G_\bullet)\bigr) \times \mathbb{R}\mathrm{Hom}^{\bullet}\bigl(g_*(G_\bullet), f_*(F_\bullet)\bigr) \arrow[r] & A_S
\end{tikzcd}
batch 8 · p. 159 — read it beside the facsimile31 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathbb{R}u_*\bigl(\mathbb{R}\mathcal{H}om(u^*\mathrm{pr}_1^*(F_\bullet), u^!(\mathrm{pr}_2^!(G_\bullet)))\bigr) \arrow[r, "\sim"] \arrow[d, no head, "\wr"] & \mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), u_*u^!(\mathrm{pr}_2^!(G_\bullet))\bigr) \arrow[d, "\varepsilon"] \\
\mathbb{R}\mathcal{H}om\bigl(u_*u^*\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr) \arrow[r] & \mathbb{R}\mathcal{H}om\bigl(\mathrm{pr}_1^*(F_\bullet), \mathrm{pr}_2^!(G_\bullet)\bigr)
\end{tikzcd}
batch 9 · p. 161 — read it beside the facsimile32 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & T \times_{Z} T' \arrow[dl] \arrow[dr] & \\
T \arrow[dr] & & T' \arrow[dl] \\
 & X \times_{S} Y = Z &
\end{tikzcd}