Cote n° 34 · pages 11–206 · 39 commutative diagrams · SGA 7 : notes manuscrites (s.d.).
Inventory dating : [vers 1967-1973]
Édition de démonstration

batch 1 · p. 11 — read it beside the facsimile1 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
K' \arrow[r, no head] & \widetilde{K}' \arrow[r, no head] & \overline{K'} \\
K \arrow[u, no head] \arrow[r, no head] & \widetilde{K} \arrow[u, no head] \arrow[r, no head] & \overline{K} \arrow[u, no head]
\end{tikzcd}
batch 1 · p. 11 — read it beside the facsimile2 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
V' \arrow[r, no head] & \widetilde{V}' & k' \arrow[r, no head] & \bar{k}' \\
V \arrow[u, no head] \arrow[r, no head] & \widetilde{V} \arrow[u, no head] & k \arrow[u, no head] \arrow[r, no head] & \bar{k} \arrow[u, no head]
\end{tikzcd}
batch 1 · p. 11 — read it beside the facsimile3 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
1 \arrow[r] & I' \arrow[r] \arrow[d] & \pi' \arrow[r] \arrow[d, no head] & \pi'_0 \arrow[r] \arrow[d, no head] & 1 \\
1 \arrow[r] & I \arrow[r] & \pi \arrow[r] & \pi_0 \arrow[r] & 1
\end{tikzcd}
batch 1 · p. 11 — read it beside the facsimile4 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
1 \arrow[r] & P' \arrow[r] \arrow[d] & I' \arrow[r] \arrow[d] & I'_t \arrow[r] \arrow[d, no head] & 1 \\
1 \arrow[r] & P \arrow[r] & I \arrow[r] & I_t \arrow[r] & 1
\end{tikzcd}
batch 2 · p. 26 — read it beside the facsimile5 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
1 \arrow[r] & U/N \arrow[r] \arrow[d, hook] & \pi/N \arrow[r] \arrow[d, hook] & \pi/U \arrow[r] \arrow[d, "\wr"] & 1 \\
1 \arrow[r] & G^{0}(R) \arrow[r] & G(R) \arrow[r] & G(R)/G^{0}(R) \arrow[r] & 1
\end{tikzcd}
batch 3 · p. 55 — read it beside the facsimile6 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & V \arrow[r, hook] \arrow[d] \arrow[dl] & \pi \arrow[d] \\
\operatorname{Im} V \arrow[r, hook] \arrow[dr] & U \arrow[r, hook] \arrow[d, "\wr"] \arrow[dr] & \pi^{\mathrm{ab}}(\ell) \arrow[d] \\
 & U \arrow[r, hook] & \pi_0^{\mathrm{ab}}(\ell)
\end{tikzcd}
batch 3 · p. 56 — read it beside the facsimile7 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
A_0 \arrow[r, no head] \arrow[d, no head] & A \arrow[d, no head] \\
X_0 \arrow[r, no head] \arrow[d, no head] & X \arrow[d, no head] \\
S_0 \arrow[r, no head] & S
\end{tikzcd}
batch 3 · p. 59 — read it beside the facsimile8 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[r, no head] \arrow[d, no head] & X' \arrow[d, no head] \\
k \arrow[r, no head] & k'
\end{tikzcd}
batch 3 · p. 60 — read it beside the facsimile9 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\operatorname{Gal}(\bar K'/K') \arrow[r] \arrow[d] & \operatorname{Gal}(\bar k'/k') \\
\operatorname{Gal}(\bar K/K) &
\end{tikzcd}
batch 4 · p. 69 — read it beside the facsimile10 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
1 \arrow[r] & \pi_1(\overline{U},\xi) \arrow[r] \arrow[d, "\alpha"] & \pi_1(U,\xi) \arrow[r] \arrow[d, "\beta"] & \pi_1(S,\xi) \arrow[r] \arrow[d, "\wr"] & 1 \\
1 \arrow[r] & \pi_1(\overline{X},\xi) \arrow[r] & \pi_1(X,\xi) \arrow[r] & \pi_1(S,\xi) \arrow[r] & 1
\end{tikzcd}
batch 4 · p. 74 — read it beside the facsimile11 / 39
LaTeX source
\begin{tikzcd}
C \arrow[r, hook] \arrow[d] & C' = L_t \cap X'_K \arrow[d, no head] \\
X_K \arrow[r, hook] & X'_K
\end{tikzcd}
batch 4 · p. 76 — read it beside the facsimile12 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
1 \arrow[r] & \pi_1(\overline{C},\xi_1) \arrow[r] \arrow[d, "\alpha"] & \pi_1(C,\xi_1) \arrow[r] \arrow[d, "\beta"] & \pi_1(\mathrm{Spec}\,K,\xi_1) \arrow[r] \arrow[d, "\gamma"] & 1 \\
1 \arrow[r] & \pi_1(\overline{X},\xi) \arrow[r] & \pi_1(X,\xi) \arrow[r] & \pi_1(\mathrm{Spec}\,k,\xi) \arrow[r] & 1
\end{tikzcd}
batch 5 · p. 83 — read it beside the facsimile13 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[d, no head] & X_K \arrow[d, no head] & X_{\bar K} \arrow[d, no head] \\
S \arrow[dr, no head] & K \arrow[dr, no head] & \bar K \\
 & \tilde S \arrow[r, no head] & \tilde K
\end{tikzcd}
batch 5 · p. 84 — read it beside the facsimile14 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[r, no head] \arrow[d, no head] & X_\alpha \arrow[r, no head] \arrow[d, no head] & X_\beta \arrow[d, no head] \\
S \arrow[r, no head] & S_\alpha \arrow[r, no head] & S_\beta
\end{tikzcd}
batch 5 · p. 84 — read it beside the facsimile15 / 39
LaTeX source
\begin{tikzcd}
H^*(X) \arrow[r] \arrow[d, "\wr"] & H^*(X_\alpha) \arrow[d, "\wr"] \\
H^*(X_s) \arrow[r] & H^*(X_{\alpha, s_\alpha})
\end{tikzcd}
batch 5 · p. 92 — read it beside the facsimile16 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\tilde U & \bar U \arrow[l] \\
\tilde X \arrow[u, hook] & \bar X \arrow[l] \arrow[u, hook]
\end{tikzcd}
batch 5 · p. 92 — read it beside the facsimile17 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
U \arrow[d, hook] & \tilde U \arrow[d, hook] & \bar U = U_{\bar\eta} \arrow[d, hook] \\
X \arrow[d, no head] & \tilde X \arrow[l] \arrow[d, no head] & \bar X = X_{\bar\eta} \arrow[l, "g"'] \arrow[d, no head] \\
S & \tilde S \arrow[l] & \bar S \arrow[l, no head]
\end{tikzcd}
batch 6 · p. 114 — read it beside the facsimile18 / 39
LaTeX source
\begin{tikzcd}[column sep=large]
p^*\bigl(R^1 f_*(\Lambda_U)\bigr) \arrow[r, "{(4.6.1)}"] \arrow[d, "\simeq"'] & R^1 f'_*(\Lambda_{U'}) \arrow[dd, "\simeq"] \\
p^*\Bigl(\coprod_i \Lambda_{D_i}(-1)\Bigr) \arrow[d, "\simeq"'] & \\
\coprod_i \Lambda_{\Delta_i}(-1) \arrow[r, "m.\mathrm{id}"] & \coprod_i \Lambda_{\Delta_i}(-1)
\end{tikzcd}
batch 6 · p. 114 — read it beside the facsimile19 / 39
LaTeX source
\begin{tikzcd}[column sep=large]
p^*\bigl(R^p f_*(\Lambda_U)\bigr) \arrow[r, "{(4.6.1)}"] \arrow[d, "\simeq"'] & R^p f'_*(\Lambda_{U'}) \arrow[dd, "\simeq"] \\
p^* \bigwedge^p\Bigl(\coprod_i \Lambda_{D_i}(-1)\Bigr) \arrow[d, "\simeq"'] & \\
\bigwedge^p \coprod_i \Lambda_{\Delta_i}(-1) \arrow[r, "m^p.\mathrm{id}"] & \bigwedge^p \coprod_i \Lambda_{\Delta_i}(-1)
\end{tikzcd}
batch 7 · p. 122 — read it beside the facsimile20 / 39
LaTeX source
\begin{tikzcd}
U' \arrow[r] \arrow[rr, bend left=30, "f"] & U \arrow[r] & X .
\end{tikzcd}
batch 7 · p. 125 — read it beside the facsimile21 / 39
LaTeX source
\begin{tikzcd}
X' \arrow[d, "p"'] & U \arrow[l, hook'] \arrow[d, "\wr"] \\
X \arrow[d, "f"'] & U \arrow[l, hook'] \\
S &
\end{tikzcd}
batch 7 · p. 126 — read it beside the facsimile22 / 39
LaTeX source
\begin{tikzcd}
X' \arrow[d, "p"'] & \\
X & U_{\overline{K}_t} \arrow[l, "g"] \arrow[lu, "g'"']
\end{tikzcd}
batch 7 · p. 131 — read it beside the facsimile23 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & X' \arrow[r, hook'] \arrow[dl] & X'_{\overline{\eta}} \arrow[dl] & U' \arrow[l, hook'] \arrow[dl] & & & \overline{U}' \arrow[lll] \arrow[dl] \\
X \arrow[d] & \widetilde{X} \arrow[l] \arrow[d] & \widetilde{X}_{\widetilde{\eta}} \arrow[l, hook'] \arrow[d] & \widetilde{U} \arrow[l, hook'] & & & \overline{U} \arrow[lll] \arrow[d] \arrow[llll, bend left=20, "g"'] & \\
S & \widetilde{S} \arrow[l] & \widetilde{\eta} \arrow[l] & & & & \overline{\eta} \arrow[llll] &
\end{tikzcd}
batch 7 · p. 132 — read it beside the facsimile24 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & X' \arrow[r, hook'] \arrow[dl] & X'_\eta \arrow[l] \arrow[dl] & U' \arrow[l, hook'] \arrow[dl] \arrow[r, no head] & U'_{\overline{\eta}} \arrow[dl] \\
X \arrow[d, no head] & X_\eta \arrow[l, hook'] \arrow[d] & U \arrow[l] \arrow[d] \arrow[r, no head] & U_{\overline{\eta}} \arrow[d, no head] & \\
S & \eta \arrow[l] & \eta \arrow[l] \arrow[r, no head] & \overline{\eta} &
\end{tikzcd}
batch 7 · p. 134 — read it beside the facsimile25 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & X' \arrow[r, hook'] \arrow[d] & U' \arrow[d, no head] & U'_{\overline{\eta}} \arrow[l] \arrow[r, no head] \arrow[dd, "\varphi"] & U' \times_{S_1} \overline{\lambda} \arrow[dd] \\
X & X_1 \arrow[l] \arrow[d] & U_1 \arrow[l, hook'] \arrow[d] & & \\
Y = S[T_1, \ldots, T_r] \arrow[d] & S_1 \arrow[l] \arrow[r, no head, "="] & S_1 \arrow[d] & S_{1\overline{\eta}} \arrow[l] \arrow[r, no head] \arrow[d, no head] & \overline{\lambda} \\
S & & S & \overline{\eta} \arrow[l] &
\end{tikzcd}
batch 7 · p. 139 — read it beside the facsimile26 / 39
LaTeX source
\begin{tikzcd}
H^i(X_s, \Lambda) \arrow[r, "\varphi^i_X"] \arrow[d] & H^i(X_{\overline{\eta}}, \Lambda) \arrow[d] \\
H^i(Y_s, \Lambda) \arrow[r, "\varphi^i_Y", "\sim"'] & H^i(Y_{\overline{\eta}}, \Lambda)
\end{tikzcd}
batch 8 · p. 144 — read it beside the facsimile27 / 39
LaTeX source
\begin{tikzcd}
V_s \arrow[r, hook, "i"] & V & V_\eta \arrow[l, hook', "j"'] \\
V_{\bar s} \arrow[u, "h_s"', "\wr"] \arrow[r, hook, "\bar i"] & \bar V \arrow[u, "h"] & V_{\bar\eta} \arrow[l, hook', "\bar j"'] \arrow[u, "h_\eta"']
\end{tikzcd}
batch 8 · p. 145 — read it beside the facsimile28 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
 & \mathbb{R}(\Gamma, I)(V_{\bar\eta}, \Lambda_\nu) \arrow[dl] & \\
\mathbb{R}(\Gamma, I)(V_{\bar s}, \underline{\Phi}^\bullet_{V_s}) \arrow[rr] & & \mathbb{R}(\Gamma, I)(\bar V, \Lambda_\nu) \arrow[ul, "\bar j^*"']
\end{tikzcd}
batch 8 · p. 146 — read it beside the facsimile29 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
 & \mathbb{R}\Gamma(V_{\bar\eta}, \Lambda_\nu) \arrow[dl] & \\
\mathbb{R}\Gamma(V_s, \underline{\Phi}^\bullet_\nu) \arrow[rr] & & \mathbb{R}\Gamma(\bar V, \Lambda_\nu) \arrow[ul, "\bar j^*"']
\end{tikzcd}
batch 8 · p. 147 — read it beside the facsimile30 / 39
LaTeX source
\begin{tikzcd}[column sep=tiny, nodes={font=\small}]
 & \mathbb{R}(\Gamma, I)(V_{\bar s}, \Lambda_\nu) \simeq \mathbb{R}\Gamma(V_s, \Lambda_\nu)^{\mathrm{triv}} \arrow[dl] & \\
\mathbb{R}(\Gamma, I)(\bar V, (\Lambda_\nu)_{V_{\bar\eta}, \bar V}) \arrow[rr] & & \mathbb{R}(\Gamma, I)(\bar V, \Lambda_\nu) \arrow[ul, "\bar i^*"']
\end{tikzcd}
batch 8 · p. 147 — read it beside the facsimile31 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
 & \mathbb{R}\Gamma(V_s, \Lambda_\nu) \arrow[dl] & \\
\mathbb{R}\Gamma(\bar V, (\Lambda_\nu)_{V_{\bar\eta}, \bar V}) \arrow[rr] & & \mathbb{R}\Gamma(\bar V, \Lambda_\nu) \arrow[ul, "\bar i^*"']
\end{tikzcd}
batch 8 · p. 153 — read it beside the facsimile32 / 39
LaTeX source
\begin{tikzcd}
V \arrow[r, no head] \arrow[d, no head, "\cap" description] & \bar V \arrow[dd, no head] \\
X' \arrow[d, no head] & \\
S & \bar S \arrow[l]
\end{tikzcd}
batch 8 · p. 156 — read it beside the facsimile33 / 39
LaTeX source
\begin{tikzcd}[column sep=tiny, nodes={font=\small}]
 & \mathbb{R}(\Gamma, \pi)(V_{\bar s}, \Lambda_\nu) \simeq \mathbb{R}\Gamma(V_s, \Lambda_\nu)^{\mathrm{triv}} = L^{\bullet\,\mathrm{triv}} \arrow[dl] & \\
R^\bullet = \mathbb{R}(\Gamma, \pi)(\bar V, (\Lambda_\nu)_{V_{\bar\eta}, \bar V}) \arrow[rr] & & \mathbb{R}(\Gamma, \pi)(V_{\bar\eta}, \Lambda_\nu) = K^\bullet \arrow[ul]
\end{tikzcd}
batch 8 · p. 157 — read it beside the facsimile34 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
 & \mathbb{R}\Gamma(V_s, \Lambda_\nu) = L^\bullet \arrow[dl] & \\
R^{\bullet\,\mathrm{ab}} = \mathbb{R}\Gamma(\bar V, (\Lambda_\nu)_{V_{\bar\eta}, \bar V}) \arrow[rr] & & \mathbb{R}\Gamma(V_{\bar\eta}, \Lambda_\nu) = K^{\bullet\,\mathrm{ab}} \arrow[ul, "k^*"']
\end{tikzcd}
batch 8 · p. 157 — read it beside the facsimile35 / 39
LaTeX source
\begin{tikzcd}[column sep=tiny, nodes={font=\small}]
 & \mathbb{R}\Gamma(V_s, \Lambda_\nu) \overset{\mathbb{L}}{\otimes}_{\Lambda_\nu} \mathbb{R}\Gamma^\pi(\Lambda_\nu) \simeq L^\bullet \oplus L^\bullet[-1] \arrow[dl] & \\
\mathbb{R}\Gamma^\pi(R^\bullet) \arrow[rr] & & \mathbb{R}\Gamma(V_\eta, \Lambda_\nu) \simeq \mathbb{R}\Gamma^\pi(K^\bullet) \arrow[ul, "\mathbb{R}\Gamma^\pi k^*"']
\end{tikzcd}
batch 8 · p. 158 — read it beside the facsimile36 / 39
LaTeX source
\begin{tikzcd}[column sep=small]
 & \mathbb{R}\Gamma(V_\eta, \Lambda_\nu) \arrow[dl, "\partial"'] & \\
\mathbb{R}\Gamma(V_s, \Lambda_\nu)(-1)[-2] \simeq \mathbb{R}\Gamma_{V_s}(\Lambda_{\nu\,V}) \arrow[rr, "i_*"] & & \mathbb{R}\Gamma(V, \Lambda_\nu) \arrow[ul, "j^*"']
\end{tikzcd}
batch 8 · p. 159 — read it beside the facsimile37 / 39
LaTeX source
\begin{tikzcd}
V_{\bar s} \arrow[r, "\bar i"] \arrow[d, "g_0"'] & \bar V \arrow[d, "g"] & V_{\bar\eta} \arrow[l, hook', "\bar j"'] \arrow[d, "h"] \\
V_s \arrow[r, "i"'] & V & V_\eta \arrow[l, "j"]
\end{tikzcd}
batch 11 · p. 204 — read it beside the facsimile38 / 39
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
  E_{2}^{pq} \arrow[r] \arrow[d, no head, "{=}" description] & E_{2}^{p+2,q-1} \arrow[d, no head, "{=}" description] \\
  \coprod_{\alpha_{1}<\dots<\alpha_{q}} H^{p}(Y_{\alpha_{1}\dots\alpha_{q}})(-q) \arrow[r] \arrow[d, no head, "\text{dual}"'] & \coprod_{\alpha_{1}<\dots<\alpha_{q-1}} H^{p+2}(Y_{\alpha_{1}\dots\alpha_{q-1}})(-(q-1)) \arrow[d, no head, "\text{dual}"] \\
  H^{2n-2-p-2q}(Y_{\alpha_{1}\dots\alpha_{q}})(n-1) & H^{2n-2q-p-2}(Y_{\alpha_{1}\dots\alpha_{q-1}})(n-1)
\end{tikzcd}
batch 11 · p. 206 — read it beside the facsimile39 / 39
LaTeX source
\begin{tikzcd}
  X \arrow[d, "{f\ \text{(propre)}}"'] & U \arrow[l, hook'] \arrow[ld, no head, "g" description] \arrow[d, no head, "f_{U}"] \\
  S & \eta \arrow[l, hook']
\end{tikzcd}