Cote n° 46 · pages 5–86
· 21 diagrammes commutatifs · Schémas en groupes et fibrés principau[x] (Divers) : notes manuscrites (s.d.), tapuscrit (s.d.).
Datation de l’inventaire : [vers 1966-vers 1972]
Édition de démonstration
LaTeX source
\begin{tikzcd}
G \arrow[r, "\varphi_a"] \arrow[d, "\rho_u"'] & G \arrow[d, "\rho_u"] \\
G \arrow[r, "\varphi_b"'] & G
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
& X'^{(p/X)} \arrow[dl] \arrow[ddr] & \\
X' \arrow[d, "p"'] & & X' \arrow[ll, "\mathrm{Fr}_{X'}"'] \arrow[ul, "\mathrm{Fr}_{X'/X}"'] \arrow[d, "p"] \\
X & & X \arrow[ll, "\mathrm{Fr}_X"]
\end{tikzcd}LaTeX source
\begin{tikzcd}
X \arrow[r, "\mathrm{Frob}(X)"] \arrow[d, "f"'] & X \arrow[d, "f"] \\
S \arrow[r, "\mathrm{Frob}(S)"'] & S
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=large]
X \arrow[r, "\mathrm{Frob}_S(X)"] \arrow[rr, bend left=30, "\mathrm{Frob}(X)"] & P_S(X) \arrow[r, "\mathrm{pr}_1"] \arrow[d] & X \arrow[d] \\
& S \arrow[r, "\mathrm{Frob}(S)"'] & S
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& P_S(X') \arrow[r] \arrow[d] & X' \arrow[d] \\
X \arrow[r] \arrow[dr, "f"'] & P_S(X) \arrow[r] \arrow[d] & X \arrow[d, "f"] \\
& S \arrow[r, "\mathrm{Frob}(S)"'] & S
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X' \arrow[r, "\mathrm{Frob}_X(X')"] \arrow[rr, bend left=30, "\mathrm{Frob}_S(X')"] \arrow[dr, "g"'] & P_X(X') \arrow[r, "r"] \arrow[d] & P_S(X') \arrow[r] \arrow[d] & X' \arrow[d, "g"] \\
& X \arrow[r, "\mathrm{Frob}_S(X)"] \arrow[rr, dashed, bend right=30, "\mathrm{Frob}(X)"'] \arrow[dr, "f"'] & P_S(X) \arrow[r] \arrow[d] & X \arrow[d, "f"] \\
& & S \arrow[r, "\mathrm{Frob}(S)"'] & S
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[d] & X' \arrow[l] \arrow[d] \\
S & S' \arrow[l]
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[d, "\mathrm{Fr}_X"'] & X' \arrow[l, "g"'] \arrow[d, "\mathrm{Fr}_{X'}"] \\
X & X' \arrow[l, "g"]
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\mathrm{Fr}_X^{*}(G) \arrow[d] & G \arrow[l] \arrow[d, "\text{épim}"] \\
\mathrm{Fr}_X^{*}(F) & F \arrow[l]
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X' \arrow[d, "p"'] & X'^{(p/X)} \arrow[l] \arrow[d] & X' \arrow[l, "\mathrm{Fr}_{X'/X}"'] \arrow[dl, "p"] \arrow[ll, bend right=30, "\mathbf{F}_{X'}"'] \\
X & X \arrow[l, "\mathbf{F}_X"] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X' \arrow[d, "\mathbf{F}_{X'}"'] & X' \arrow[l, "\mathbf{F}_{X'}"'] \arrow[d, "p"] \\
X & X \arrow[l, "\mathbf{F}_X"]
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X' \arrow[d, "p"'] & X'^{(p/S)} \arrow[l] \arrow[d] & X' \arrow[l, "\sim"'] \arrow[dl, "p"] \\
X & X \arrow[l, "\mathbf{F}_X"] &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
X \arrow[r, no head] \arrow[d, no head] & X \times_S \check{P} \arrow[d, no head] \\
S \arrow[r, no head] & \check{P}
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathrm{Progr}^{!}(C'_0) \arrow[r, "\alpha"] \arrow[d, hook, "i"'] & \mathrm{Gr}(C) \arrow[r, "\simeq", "\beta"'] & \mathrm{Gr}(C') \arrow[r, "\sigma"] \arrow[d, hook, "j"] & \mathrm{Gr\,qu.cpt}(\mathrm{Sch}_{/k}) \\
\mathrm{Progr}(C'_0) \arrow[rr, "k"'] & & \mathrm{Gr\,Pro}(C'_0) &
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
G_j \times G_j \arrow[r] & G_i \times G_i \arrow[r] \arrow[d] & H \times H \arrow[d] \\
& G_i \arrow[r, no head] & H
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
(G_i) \arrow[r] \arrow[d, no head] & G_{i_0} \arrow[d] \\
(X_j) \arrow[r, no head] & X_0
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
(G_i) \arrow[r] \arrow[d, leftrightarrow] & G_{i_1} \arrow[r] & G_{i_0} \arrow[d] \\
X = (X_j) \arrow[r] & X_{j_1} \arrow[r] \arrow[u] & X_0
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
e \arrow[r] & \mathbb{C}^{*} \arrow[r] & \mathrm{Gl}(n) \arrow[r] & \mathrm{GP}(n-1) \arrow[r] & e \\
e \arrow[r] & \mathbb{Z}/n\mathbb{Z} \arrow[r] \arrow[u, hook] & \mathrm{Sl}(n) \arrow[r] \arrow[u] & \mathrm{GP}(n-1) \arrow[r] \arrow[u, no head] & e
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\cdots \arrow[r] & H^1(X, \underline{\mathbb{C}}^{*}) \arrow[r] & H^1(X, \underline{\mathrm{Gl}}(n)) \arrow[r] & H^1(X, \mathrm{GP}(n-1)) \arrow[r] & H^2(X, \underline{\mathbb{C}}^{*}) \\
\cdots \arrow[r] & H^1(X, \mathbb{Z}/n\mathbb{Z}) \arrow[r] \arrow[u] & H^1(X, \underline{\mathrm{Sl}}(n)) \arrow[r] \arrow[u] & H^1(X, \mathrm{GP}(n-1)) \arrow[r] \arrow[u, no head] & H^2(X, \mathbb{Z}/n\mathbb{Z}) \arrow[u]
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
H^1(\Gamma, A^{*}) \arrow[r] \arrow[d] & H^1(\Gamma_d, A_{\mathfrak{m}}^{*}) \arrow[d] \\
H^1(\Gamma, \hat{A}^{*}) \arrow[r, "\sim"] & H^1(\Gamma_d, \hat{A}_{\mathfrak{m}}^{*})
\end{tikzcd}LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
H^1(\Gamma, A^{*}) \arrow[r] \arrow[d, "\varphi"'] & H^1(\Gamma_i, A^{*}) \arrow[d, "\varphi_i"] \\
H^1(\Gamma, A/\mathfrak{m}^{*}) \arrow[r, "\psi"] & H^1(\Gamma_i, A/\mathfrak{m}^{*})
\end{tikzcd}