Cote n° 136 · pages 24–363 · 63 commutative diagrams · Complexe de De Rham à puissance divisée [conférence de 1976 à l’IHÉS] : notes manuscrites (s.d.).
Inventory dating : [à partir de 1975-1976]
Édition de démonstration

batch 2 · p. 24 — read it beside the facsimile1 / 63
LaTeX source
\begin{tikzcd}
M\, T^{(i)} \arrow[r, "{\cdot T^{(j)}}"] \arrow[d, "{u_i \otimes T^{(i)}}"'] & M\, T^{(i+j)} \arrow[d, "{u_{i+j} \otimes T^{(i+j)}}"] \\
M'\, T^{(i)} \arrow[r, "{\cdot T^{(j)}}"'] & M'\, T^{(i+j)}
\end{tikzcd}
batch 3 · p. 55 — read it beside the facsimile2 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & & & 0 & \\
 & & & 0 & A_3^{3} \arrow[u] \arrow[l] & \cdots \arrow[l] \\
 & & 0 & A_2^{2} \arrow[u] \arrow[l] & A_3^{2} \arrow[u, "d"] \arrow[l, "\partial"'] & \cdots \arrow[l] \\
 & 0 & A_1^{1} \arrow[u] \arrow[l] & A_2^{1} \arrow[u, "d"] \arrow[l, "\partial"'] & A_3^{1} \arrow[u, "d"] \arrow[l, "\partial"'] & \cdots \arrow[l] \\
0 & A_0 \arrow[l] & A_1^{0} \arrow[u, "d"] \arrow[l, "\partial"'] & A_2^{0} \arrow[u, "d"] \arrow[l, "\partial"'] & A_3^{0} \arrow[u, "d"] \arrow[l, "\partial"'] & \cdots \arrow[l] \\
0 \arrow[r] & A_0 \arrow[u, "\varepsilon_0"] & A_0 \arrow[u, "\varepsilon_1"] \arrow[l, "0"'] & A_0 \arrow[u, "\varepsilon_2"] \arrow[l, "\mathrm{id}"'] & A_0 \arrow[u, "\varepsilon_3"] \arrow[l, "0"'] & \cdots \arrow[l, "\mathrm{id}"']
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile3 / 63
LaTeX source
\begin{tikzcd}
A^n_{n+1} \arrow[r, "\partial_{n+1}"] \arrow[d, "d^n"'] & A^n_n \arrow[d, "k_n"] \\
A^{n+1}_{n+1} \arrow[r, "k_{n+1}"'] & M
\end{tikzcd}
batch 4 · p. 65 — read it beside the facsimile4 / 63
LaTeX source
\begin{tikzcd}
\Gamma(X_*, A^*_{X_*}) \arrow[r] \arrow[d] & {C^*(X_*, M) = \Gamma(X_*, C^*_M|X_*)} \arrow[d, "\wr"] \\
R\Gamma(X_*, A^*_{X_*}) \arrow[r] & R\Gamma(X_*, M)
\end{tikzcd}
batch 4 · p. 66 — read it beside the facsimile5 / 63
LaTeX source
\begin{tikzcd}
\Gamma(X_*, A^*_{X_*}) \arrow[r] & R\Gamma(X_*, A^*_{X_*}) \\
 & R\Gamma(X_*, M) \arrow[u, "\wr"']
\end{tikzcd}
batch 4 · p. 68 — read it beside the facsimile6 / 63
LaTeX source
\begin{tikzcd}
A^n_* \arrow[r, "f"] \arrow[d, "\sigma^*_A"'] & C^n_* \arrow[d, "\sigma^*_C"] \\
A^m_* \arrow[r] & C^m_*
\end{tikzcd}
batch 4 · p. 69 — read it beside the facsimile7 / 63
LaTeX source
\begin{tikzcd}
\varinjlim_{U \supset \operatorname{Supp}\sigma} \Omega^*(U) \arrow[r] \arrow[d, "\text{restriction}"'] & A^*_m \arrow[d] \\
\varinjlim_{V \supset \operatorname{Supp}\alpha^*(\sigma)} \Omega^*(V) \arrow[r] & A^*_n
\end{tikzcd}
batch 4 · p. 71 — read it beside the facsimile8 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
R^*\Gamma(X, k) \arrow[r, "\varepsilon_{S^*_X}", "\sim"'] \arrow[d, "\wr"] & R^*\Gamma(X, S^*_X) \arrow[d, "\wr"] & \Gamma(X, S^*_X) \arrow[l, "\sim"'] \arrow[d, "\wr"] \\
R^*\Gamma(X, \Omega^*) \arrow[r, "\sim"] & R^*\Gamma(X, S'^*_X) & \Gamma(X, S'^*_X) \arrow[l, "\sim"'] \\
C^*(X, \Omega^*) \arrow[u] & &
\end{tikzcd}
batch 4 · p. 71 — read it beside the facsimile9 / 63
LaTeX source
\begin{tikzcd}
\Omega^* \arrow[r] & S'^*_X \\
k \arrow[u] \arrow[r] & S^*_X \arrow[u]
\end{tikzcd}
batch 4 · p. 71 — read it beside the facsimile10 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
C^*(X, \Omega^*) \arrow[r] \arrow[dd, "\wr"'] & {C^*(X, S'^*_X) \overset{\text{déf}}{=} C(S'_X, k)} \arrow[dd, "\wr"] & \\
 {[C(X, k)]} \arrow[rr, dashed] \arrow[u] & & {C^*(X, S^*_X) = C(S_X, k)} \arrow[ul, "\text{hypoth. } S'_X"'] \arrow[dd, "\wr"] \\
R^*\Gamma(X, \Omega^*) \arrow[r, "\sim"] & R^*\Gamma(X, S'^*_X) & \\
R^*\Gamma(X, k) \arrow[u, "\wr"] \arrow[rr, "\sim"] & & R^*\Gamma(X, S^*_X) \arrow[ul, "\wr"']
\end{tikzcd}
batch 5 · p. 81 — read it beside the facsimile11 / 63
LaTeX source
\begin{tikzcd}
 & L \arrow[dl, "u"'] \arrow[dr, "{u'}"] & \\
\Phi \arrow[rr, "\lambda"'] & & {\Phi'}
\end{tikzcd}
batch 5 · p. 84 — read it beside the facsimile12 / 63
LaTeX source
\begin{tikzcd}
L \arrow[r, "\lambda_{L}"] \arrow[d] & {L'} \arrow[d] \\
\Phi \arrow[r] & {\Phi'}
\end{tikzcd}
batch 5 · p. 88 — read it beside the facsimile13 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & \Phi \arrow[d, "\lambda"] \\
k \arrow[ur, "u"] \arrow[r, "{u'}"'] & {\Phi'}
\end{tikzcd}
batch 5 · p. 88 — read it beside the facsimile14 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & {\Phi'} \arrow[dr, hook] & & \\
k \arrow[ur, hook, "u"] \arrow[dr, hook, "{u'}"'] & & S \arrow[r, hook] & N \\
 & \Phi \arrow[ur, hook] & &
\end{tikzcd}
batch 5 · p. 88 — read it beside the facsimile15 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & N \arrow[dr, hook] & & \\
S \arrow[ur, hook] \arrow[dr, hook] & & T \arrow[r, hook] & {N''} \\
 & {N'} \arrow[ur, hook] & &
\end{tikzcd}
batch 5 · p. 88 — read it beside the facsimile16 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
\mathrm{DR}(u) \arrow[dr] & & \mathrm{DR}(N) \arrow[dr, "\sim"] & \\
 & \mathrm{DR}(S) \arrow[ur] \arrow[dr] & & \mathrm{DR}(N'') \\
\mathrm{DR}(u') \arrow[ur] & & \mathrm{DR}(N') \arrow[ur, "\sim"'] &
\end{tikzcd}
batch 5 · p. 90 — read it beside the facsimile17 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
 & \Phi \arrow[dr] & \\
k \arrow[ur] \arrow[r] \arrow[dr] & {\Phi'} \arrow[r] & {N''} \\
 & {\Phi''} \arrow[ur] &
\end{tikzcd}
batch 5 · p. 95 — read it beside the facsimile18 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
0 & 0 & A^{22} & \cdots \\
0 & A^{11} & A^{21} \arrow[u] & \cdots \\
k & A^{10} \arrow[u] & A^{20} \arrow[u] & \cdots
\end{tikzcd}
batch 5 · p. 95 — read it beside the facsimile19 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & k \arrow[r] & \Phi \arrow[r] \arrow[d, no head, "\Vert" description] & \Psi \arrow[r] \arrow[d] & 0 \\
 & & A^{10} & A^{20} &
\end{tikzcd}
batch 5 · p. 95 — read it beside the facsimile20 / 63
LaTeX source
\begin{tikzcd}
\mathrm{DR}(\Phi,T) \arrow[r] \arrow[d] & A^{**} \arrow[d] \\
\mathrm{DR}(\Phi',T') \arrow[r] & {A'^{**}}
\end{tikzcd}
batch 6 · p. 101 — read it beside the facsimile21 / 63
LaTeX source
\begin{tikzcd}
A_{*} \arrow[r] & \mathfrak{G}_{*} \otimes_{\mathbb{Z}} A_0 \\
k_{*} \arrow[u] \arrow[r, hook] & \mathfrak{G}_{*k} \;(= \mathfrak{G}_{*} \otimes_{\mathbb{Z}} k) \arrow[u, "{\mathfrak{G}_{*} \otimes_{\mathbb{Z}} u_0}"']
\end{tikzcd}
batch 6 · p. 101 — read it beside the facsimile22 / 63
LaTeX source
\begin{tikzcd}
\mathrm{DR}(A^{10}_{*}, T) \arrow[r] \arrow[d] & A^{**}_{*} \\
\mathrm{DR}(A^{10}_{0} \otimes_{\mathbb{Z}} \mathfrak{G}_{*}, T) & \mathrm{DR}(\mathfrak{G}_{*k}, T) \arrow[l]
\end{tikzcd}
batch 6 · p. 103 — read it beside the facsimile23 / 63
LaTeX source
\begin{tikzcd}
\mathrm{DRsp}(\mathcal{X}, k) \arrow[r, "\text{surj. sur ob.}"] \arrow[d, "\text{surjectif sur obj.}"'] & \mathrm{DRsp}(\mathcal{X}, k)_0 \\
\mathrm{DRsp}'(\mathcal{X}, k) \arrow[r, "\text{surj. sur objets}"'] & \mathrm{DRsp}'(\mathcal{X}, k)_0 \\
\mathrm{DRspfl}(\mathcal{X}, k) \arrow[r, "\text{surj. sur ob.}"] \arrow[d, "\text{surjectif sur ob.}"'] & \mathrm{DRspfl}(\mathcal{X}, k)_0 \arrow[d] \\
\mathrm{DRspfl}'(\mathcal{X}, k) \arrow[r, "\text{surj. sur ob.}"'] & \mathrm{DRspfl}'(\mathcal{X}, k)_0
\end{tikzcd}
batch 6 · p. 105 — read it beside the facsimile24 / 63
LaTeX source
\begin{tikzcd}
\widehat{A}^{**}_{*} \arrow[r, "u"] & \widehat{A}'^{**}_{*} \\
\widehat{\mathrm{DR}}(A^{10}_{*}, T) \arrow[u] \arrow[ur, dashed]
\end{tikzcd}
batch 6 · p. 105 — read it beside the facsimile25 / 63
LaTeX source
\begin{tikzcd}
k \arrow[r, hook] \arrow[d, no head, "\Vert" description] & A \arrow[d, "{v(1,0)}"] \\
k \arrow[r, hook] & A'^{1,0}
\end{tikzcd}
batch 6 · p. 105 — read it beside the facsimile26 / 63
LaTeX source
\begin{tikzcd}
 & & \mathfrak{G}_{*} \otimes A \\
k \arrow[r, hook] & A \arrow[r] \arrow[ur, no head] \arrow[dr] & \mathfrak{B} \arrow[d] \\
 & & A'^{pq}
\end{tikzcd}
batch 7 · p. 136 — read it beside the facsimile27 / 63
LaTeX source
\begin{tikzcd}
  s \arrow[r, "\kappa_n"] \arrow[d, "\pi_{n-1}"'] & (\mathcal{A}^{*})_n \arrow[d, "d_{\mathcal{A}^*}"] \\
  (\mathcal{A}^{*})_{n-1} \arrow[r, "\partial_{ss}"'] & (\mathcal{A}^{*})_{n-1}
\end{tikzcd}
batch 8 · p. 152 — read it beside the facsimile28 / 63
LaTeX source
\begin{tikzcd}
C \arrow[r, "\Delta_C"] \arrow[rr, bend right=30, "u_C"'] & C \otimes s \arrow[r, "\mathrm{id}_C \otimes u"] & C \otimes k \simeq C
\end{tikzcd}
batch 10 · p. 183 — read it beside the facsimile29 / 63
LaTeX source
\begin{tikzcd}
A_1 \arrow[r, bend left=20, "\alpha"] \arrow[r, bend right=20, "\beta"'] & A_0 = k\{T\}
\end{tikzcd}
batch 11 · p. 203 — read it beside the facsimile30 / 63
LaTeX source
\begin{tikzcd}
  S\{X_0, \ldots, X_n\}[dX_0, \ldots, dX_n] / \ldots \arrow[d] \\
  S'\{X_0, \ldots, X_n\}[dX_0, \ldots, dX_n] / \ldots
\end{tikzcd}
batch 11 · p. 207 — read it beside the facsimile31 / 63
LaTeX source
\begin{tikzcd}
  (1 \arrow[r] & H'_k \arrow[r] & \Omega_k \arrow[r] & G'_k = \operatorname{End}_{k\text{-alg}}(k\{T\}) = S_k) \\
  & & & G_k \arrow[u, "\simeq"'] \arrow[ull, bend left=20]
\end{tikzcd}
batch 11 · p. 207 — read it beside the facsimile32 / 63
LaTeX source
\begin{tikzcd}
  1 \arrow[r] & H_k \arrow[r] & \Omega_k \arrow[r] & G'_k = \operatorname{End}_{k\text{-alg augm.}}(k\{T\}) \\
  & & & G_k \arrow[u] \arrow[ull, bend left=20] \\
  & & & G_k \cap H_k \arrow[u]
\end{tikzcd}
batch 12 · p. 236 — read it beside the facsimile33 / 63
LaTeX source
\begin{tikzcd}[column sep=small]
H^i(\underbrace{\mathrm{Fil}^pC^*_{\mathrm{DR}}(\mathcal{X}, \widehat{A}^*_*)}_{C^*_{\mathrm{DR}}(\mathcal{X}, \mathrm{Fil}\widehat{A}^*_*)}) \arrow[r, "\simeq"] \arrow[d] & H^i(\mathcal{X}, \underbrace{S^{[p]}}_{\mathrm{Fil}^pS}) \arrow[d] \\
H^i(C^*_{\mathrm{DR}}(\mathcal{X}, \widehat{A}^*_*)) \arrow[r] & H^i(\mathcal{X}, S)
\end{tikzcd}
batch 12 · p. 238 — read it beside the facsimile34 / 63
LaTeX source
\begin{tikzcd}[column sep=small]
H^i(\mathrm{Fil}^pC^*_{\mathrm{DR}}(\mathcal{X}; S, t, \gamma^*)) \arrow[r, "\sim"] \arrow[d] & H^i(\mathcal{X}, \mathrm{Fil}^pS) \arrow[d] \\
H^i(\mathrm{Fil}^qC^*_{\mathrm{DR}}(\mathcal{X}; S, t, \gamma^*)) \arrow[r] & H^i(\mathcal{X}, \mathrm{Fil}^qS)
\end{tikzcd}
batch 13 · p. 258 — read it beside the facsimile35 / 63
LaTeX source
\begin{tikzcd}
  \tau_N S \arrow[r, "d^\circ 0"] \arrow[d, "T^{(\ell)}"'] & P \\
  \tau_{N+\ell}(S) \arrow[ur, "d^\circ \ell"']
\end{tikzcd}
batch 14 · p. 272 — read it beside the facsimile36 / 63
LaTeX source
\begin{tikzcd}
D^{+}(C) \arrow[r, bend left=20, "R\sigma"] & D^{+}(\mathcal{M}_\omega) \arrow[l, bend left=20, "R\psi"]
\end{tikzcd}
batch 14 · p. 272 — read it beside the facsimile37 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=normal, nodes={font=\small}]
 & & {\operatorname{Hom}(R\sigma K,\, R\sigma R\psi(L))} \arrow[dll] \\
{\operatorname{Hom}(R\sigma(K),\, L)} \arrow[rr, "{?\ \sim}"] \arrow[dr] & & {\operatorname{Hom}(K,\, R\psi(L))} \arrow[u] \arrow[dl, no head, "\wr" description] \\
 & {\operatorname{Hom}(R\psi R\sigma K,\, R\psi L)} &
\end{tikzcd}
batch 14 · p. 272 — read it beside the facsimile38 / 63
LaTeX source
\begin{tikzcd}
{R\sigma R\psi R\sigma K} \arrow[d] & {R\sigma R\psi L} \arrow[d] \\
R\sigma K & L
\end{tikzcd}
batch 14 · p. 276 — read it beside the facsimile39 / 63
LaTeX source
\begin{tikzcd}[column sep=small]
T(Z) \arrow[r] & T(Q) \\
T(P) \arrow[u] &
\end{tikzcd}
batch 15 · p. 282 — read it beside the facsimile40 / 63
LaTeX source
\begin{tikzcd}
V(P) & V(C\otimes C') \arrow[l] \arrow[dl] & V(C)\times V(C')\times V(S) \arrow[l, bend left=15] \arrow[l, bend right=15] \\
V(C'') & &
\end{tikzcd}
batch 15 · p. 282 — read it beside the facsimile41 / 63
LaTeX source
\begin{tikzcd}
P \arrow[r, hook, "i"] \arrow[d, "?"'] & C\otimes C' \arrow[d, bend right=20, "\alpha"'] \arrow[d, bend left=20, "\beta"] \\
P\otimes S \arrow[r, hook, "{i\otimes\mathrm{id}_S}"] & C\otimes C'\otimes S
\end{tikzcd}
batch 15 · p. 282 — read it beside the facsimile42 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & C'' \arrow[dl] \arrow[d] \arrow[dr] & \\
\cdots P_3 \arrow[r, hook] \arrow[d] & P_2 \arrow[r, hook] \arrow[d] & P_1 \arrow[r, hook] \arrow[dr] & C\otimes C' \arrow[d, bend right=20, "\alpha"'] \arrow[d, bend left=20, "\beta"] \\
P_2\otimes S \arrow[r, hook] & P_1\otimes S \arrow[rr, hook] & & C\otimes C'\otimes S
\end{tikzcd}
batch 15 · p. 284 — read it beside the facsimile43 / 63
LaTeX source
\begin{tikzcd}
S \arrow[r, "\Delta_S"] & S\otimes S & \\
C \arrow[u, "u"] \arrow[r, "\Delta_C"] & C\otimes C \arrow[u] \arrow[r, "{\mathrm{id}_C\otimes u}"] & C\otimes S
\end{tikzcd}
batch 15 · p. 284 — read it beside the facsimile44 / 63
LaTeX source
\begin{tikzcd}
V(C)\times V(S)\times V(C') \arrow[r, "{\mathrm{id}_{V(C)}\times\omega_{V(C')}}"] \arrow[d] & V(C)\times V(C') \arrow[d] \\
V(C'')\times V(S) \arrow[r] & V(C'')
\end{tikzcd}
batch 15 · p. 284 — read it beside the facsimile45 / 63
LaTeX source
\begin{tikzcd}
C \arrow[d] & C'' \arrow[l] \arrow[d] \\
C\otimes S & C''\otimes S \arrow[l]
\end{tikzcd}
batch 15 · p. 284 — read it beside the facsimile46 / 63
LaTeX source
\begin{tikzcd}
C'' \arrow[r] \arrow[d] & C\otimes A \arrow[d] \\
C''\otimes S \arrow[r] & C\otimes S\otimes A
\end{tikzcd}
batch 15 · p. 284 — read it beside the facsimile47 / 63
LaTeX source
\begin{tikzcd}
C'' \arrow[r, "\alpha"] \arrow[d, "\Delta_{C''}"'] & C\otimes C' \arrow[d, "{\Delta_C\otimes\mathrm{id}_{C'}}"] \\
C''\otimes S \arrow[r, "{\alpha\otimes\mathrm{id}_S}"'] & C\otimes S\otimes C'
\end{tikzcd}
batch 16 · p. 319 — read it beside the facsimile48 / 63
LaTeX source
\begin{tikzcd}
  A \arrow[r, "u_{pq}"] \arrow[d, "u_{p}"'] & \Gamma^{pq}(A) \arrow[r, "\Gamma^{pq}(f)"] & \Gamma^{pq}(X) \arrow[d, "\gamma^{X}_{pq}"] \\
  \Gamma^{p}(A) \arrow[r, "\Gamma^{p}(u_{q})"'] & \Gamma^{p}(\Gamma^{q}(A)) \arrow[r, "{\Gamma^{p}(\Gamma^{q}(f))}"'] & \Gamma^{p}(\Gamma^{q}(X))
\end{tikzcd}
batch 16 · p. 319 — read it beside the facsimile49 / 63
LaTeX source
\begin{tikzcd}
  A \arrow[r, "u_{pq}"] \arrow[d, "u_{p}"'] & \Gamma^{pq}(A) \arrow[d, "\gamma^{A}_{pq}"] \\
  \Gamma^{p}(A) \arrow[r, "\Gamma^{p}(u_{q})"'] & \Gamma^{p}(\Gamma^{q}(A))
\end{tikzcd}
batch 16 · p. 319 — read it beside the facsimile50 / 63
LaTeX source
\begin{tikzcd}
  A \otimes A \arrow[r, "u_{p} \otimes u_{q}"] & \Gamma^{p}(A) \otimes \Gamma^{q}(A) \arrow[d, "\mu^{A}_{p,q}"] \\
  A \arrow[u, "\Delta"] \arrow[r, "c_{pq}\, u_{p+q}"'] & \Gamma^{p+q}(A)
\end{tikzcd}
batch 17 · p. 321 — read it beside the facsimile51 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \arrow[r, "\Delta"] \arrow[d] & A\otimes A \arrow[r, "{(\Gamma^p f\circ u_p)\otimes(\Gamma^q g\circ u_q)}"] & \Gamma^p(X)\otimes\Gamma^q(X) \arrow[d] \\
{} & {} & \Gamma^{p+q}(X)
\end{tikzcd}
batch 17 · p. 321 — read it beside the facsimile52 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Gamma^p(F_A(X))\otimes\Gamma^q(F_A(X)) \arrow[r] \arrow[d] & F_A\Gamma^p(X)\otimes F_A\Gamma^q(X) \arrow[d] \\
\Gamma^{p+q}F_A(X) \arrow[dr] & F_A(\Gamma^p(X)\otimes\Gamma^q(X)) \arrow[d] \\
{} & F_A(\Gamma^{p+q}(X))
\end{tikzcd}
batch 17 · p. 321 — read it beside the facsimile53 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \arrow[r, "\Delta"] & A\otimes A \arrow[r, "u_p\otimes u_q"] & \Gamma^p(A)\otimes\Gamma^q(A) \arrow[d] \\
{} & {} & \Gamma^{p+q}(A)
\end{tikzcd}
batch 17 · p. 322 — read it beside the facsimile54 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \arrow[r, "u_p"] \arrow[d, "\Delta_A^{(p)}"'] & \Gamma^p(A) \arrow[r, "\Gamma^p(\Delta)"] & \Gamma^p(A\times A) \arrow[d, "\simeq"] \\
A\times\cdots\times A \arrow[rr] & {} & \sum_{i+j=p}\Gamma^iA\otimes\Gamma^jA
\end{tikzcd}
batch 17 · p. 328 — read it beside the facsimile55 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \arrow[r, "u_{pq}"] \arrow[d, "u_p"'] & \Gamma^{pq}(A) \arrow[d, "\gamma^A_{pq}"] \\
\Gamma^p(A) \arrow[r, "\Gamma^p(u_q)"'] & \Gamma^p(\Gamma^q(A))
\end{tikzcd}
batch 17 · p. 328 — read it beside the facsimile56 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \arrow[r, "f"] \arrow[d, "u_p"'] & \Gamma^*M & {} \\
\Gamma^pA \arrow[r, "\Gamma^pf"'] & \Gamma^p(\Gamma^*M) \arrow[r] & \Gamma^*M
\end{tikzcd}
batch 17 · p. 328 — read it beside the facsimile57 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
A \arrow[r, "f"] \arrow[d, "u_p"'] & J & {} \\
\Gamma^p(A) \arrow[r, "\Gamma^p(f)"'] & \Gamma^p(J) \arrow[r] & J
\end{tikzcd}
batch 17 · p. 331 — read it beside the facsimile58 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Gamma^pF(X) \arrow[r, "u_p"] \arrow[d, "\xi\mapsto\xi^{(q)}"'] & F(\Gamma^p(X)) \arrow[d, "F(\eta\mapsto\eta^{(q)})"] \\
\Gamma^{pq}(F(X)) \arrow[r, "u_{pq}"'] & F(\Gamma^{pq}(X))
\end{tikzcd}
batch 17 · p. 334 — read it beside the facsimile59 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\Gamma^pF(X) \arrow[r, "u^{F(X)}_p"] \arrow[d, "\gamma^q_{\Gamma^pF(X)}"'] & F\Gamma^p(X) \arrow[d, "F(\gamma^q_{\Gamma^p(X)})"] \\
\Gamma^q\Gamma^pF(X) \arrow[d] & F(\Gamma^q\Gamma^p(X)) \\
\Gamma^{qp}F(X) & {}
\end{tikzcd}
batch 17 · p. 338 — read it beside the facsimile60 / 63
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X_{[n]} \simeq \mathrm{Hom}(\Delta^n_*,X_*) \arrow[d] \arrow[drr] & {} & {} \\
k[X_{[n]}] \arrow[d, "\text{iso si } X_n \text{ fini}"'] & {} & {} \\
k[X_{[n]}]^{\vee\vee} & {} & \mathrm{Hom}(F_{X_*},F_{\Delta^n_*}) \arrow[ll, "\sim"']
\end{tikzcd}
batch 18 · p. 356 — read it beside the facsimile61 / 63
LaTeX source
\begin{tikzcd}
A_{**} \arrow[r, "N"] \arrow[d, "\Delta"'] & A_{\bullet\bullet} \arrow[d, "\int"] \\
A_* \arrow[r, "N"] & A_\bullet
\end{tikzcd}
batch 18 · p. 359 — read it beside the facsimile62 / 63
LaTeX source
\begin{tikzcd}
X_{***} \arrow[r, "N_{**}"] \arrow[d, "\Delta_{12}"'] & X_{\bullet\bullet\bullet} \arrow[r, "\int_{12}"] & X_{\bullet\bullet} \arrow[d, "\int"] \\
X_{**} \arrow[r, "N"] & X_{\bullet\bullet} & X_\bullet
\end{tikzcd}
batch 19 · p. 363 — read it beside the facsimile63 / 63
LaTeX source
\begin{tikzcd}
  \widetilde{H}^{n+1}(SX,\mathbb{F}_p) \arrow[r, "P_{n+1}"] &
  \widetilde{H}^{n+1+r}(SX,\mathbb{F}_p) \\
  \widetilde{H}^{n}(X,\mathbb{F}_p) \arrow[r, "P_n"] \arrow[u, "\wr"] &
  \widetilde{H}^{n+r}(X,\mathbb{F}_p) \arrow[u, "\wr"']
\end{tikzcd}