Cote n° 4 · pages 1–10 · 25 displayed formulas · Torsion en cohomologie cristalline : notes manuscrites (s.d.), copies de tapuscrit annoté (s.d.).
Inventory dating : [à partir de 1958]
Édition de démonstration

batch 1 · p. 1 — read it beside the facsimile1 / 25 · 11 distinct symbols, 25 written
\[H^{n}(X, \underline{O}_{X})_{\mathrm{ss}} \simeq (H^{n}(X) \otimes_{W} k)_{\mathrm{ss}}\]
LaTeX source
\[
H^{n}(X, \underline{O}_{X})_{\mathrm{ss}} \simeq (H^{n}(X) \otimes_{W} k)_{\mathrm{ss}}
\]
batch 1 · p. 1 — read it beside the facsimile2 / 25 · 14 distinct symbols, 53 written
\[(*) \qquad 0 \to (\mathrm{Tors}^{n}(X) \otimes k)_{\mathrm{ss}} \to H^{n}(X, \underline{O}_{X})_{\mathrm{ss}} \to (H^{n}_{\mathrm{lib}}(X) \otimes k)_{\mathrm{ss}} \to 0\]
LaTeX source
\[
(*) \qquad 0 \to (\mathrm{Tors}^{n}(X) \otimes k)_{\mathrm{ss}}
\to H^{n}(X, \underline{O}_{X})_{\mathrm{ss}}
\to (H^{n}_{\mathrm{lib}}(X) \otimes k)_{\mathrm{ss}} \to 0
\]
batch 1 · p. 1 — read it beside the facsimile3 / 25 · 17 distinct symbols, 53 written
\[(**) \qquad 0 \to (H^{n}_{\mathrm{lib}}(X) \otimes k)^{\mathrm{ss}} \to H^{0}(X, \Omega^{n}_{X})^{\mathrm{ss}} \to {}_{p}\mathrm{Tors}^{n+1}(X)^{\mathrm{ss}} \to 0\]
LaTeX source
\[
(**) \qquad 0 \to (H^{n}_{\mathrm{lib}}(X) \otimes k)^{\mathrm{ss}}
\to H^{0}(X, \Omega^{n}_{X})^{\mathrm{ss}}
\to {}_{p}\mathrm{Tors}^{n+1}(X)^{\mathrm{ss}} \to 0
\]
batch 1 · p. 2 — read it beside the facsimile4 / 25 · 12 distinct symbols, 22 written
\[(8.1.1) \qquad H_{\mathrm{DR}}(X) = H(X) \otimes^{L}_{W} k\]
LaTeX source
\[
(8.1.1) \qquad H_{\mathrm{DR}}(X) = H(X) \otimes^{L}_{W} k
\]
batch 1 · p. 2 — read it beside the facsimile5 / 25 · 16 distinct symbols, 44 written
\[(8.1.2) \qquad 0 \to H^{i}(X) \otimes_{W} k \to H^{i}_{\mathrm{DR}}(X) \to \mathrm{Tor}^{W}_{1}(H^{i+1}(X), k) \to 0 .\]
LaTeX source
\[
(8.1.2) \qquad 0 \to H^{i}(X) \otimes_{W} k \to H^{i}_{\mathrm{DR}}(X)
\to \mathrm{Tor}^{W}_{1}(H^{i+1}(X), k) \to 0 .
\]
batch 1 · p. 2 — read it beside the facsimile6 / 25 · 19 distinct symbols, 69 written
\[(8.1.3) \qquad \bigl(0 \to \mathrm{Ker}\, u_{0} \to \mathrm{Ker}\, u_{1} \to \mathrm{Ker}\, u_{2} \to\bigr)\; E(Y) \otimes_{W} k \to \underbrace{E_{\mathrm{DR}}(Y)}_{=\,\mathrm{Coker}\, u_{1}} \to \mathrm{Coker}\, u_{2} \to 0 ,\]
LaTeX source
\[
(8.1.3) \qquad \bigl(0 \to \mathrm{Ker}\, u_{0} \to \mathrm{Ker}\, u_{1} \to
\mathrm{Ker}\, u_{2} \to\bigr)\; E(Y) \otimes_{W} k \to
\underbrace{E_{\mathrm{DR}}(Y)}_{=\,\mathrm{Coker}\, u_{1}}
\to \mathrm{Coker}\, u_{2} \to 0 ,
\]
batch 1 · p. 2 — read it beside the facsimile7 / 25 · 15 distinct symbols, 74 written
\[E(Y) = \mathrm{Coker}\bigl(H^{n-1}(X) \xrightarrow{u_{0}} H^{n-1}(Y)\bigr), \qquad E_{\mathrm{DR}}(Y) = \mathrm{Coker}\bigl(H^{n-1}_{\mathrm{DR}}(X) \xrightarrow{u_{1}} H^{n-1}_{\mathrm{DR}}(Y)\bigr),\]
LaTeX source
\[
E(Y) = \mathrm{Coker}\bigl(H^{n-1}(X) \xrightarrow{u_{0}} H^{n-1}(Y)\bigr),
\qquad
E_{\mathrm{DR}}(Y) = \mathrm{Coker}\bigl(H^{n-1}_{\mathrm{DR}}(X)
\xrightarrow{u_{1}} H^{n-1}_{\mathrm{DR}}(Y)\bigr),
\]
batch 1 · p. 2 — read it beside the facsimile8 / 25 · 15 distinct symbols, 37 written
\[(8.1.4) \qquad u_{2} : \mathrm{Tor}^{W}_{1}(H^{n}(X), k) \to \mathrm{Tor}^{W}_{1}(H^{n}(Y), k).\]
LaTeX source
\[
(8.1.4) \qquad u_{2} : \mathrm{Tor}^{W}_{1}(H^{n}(X), k) \to
\mathrm{Tor}^{W}_{1}(H^{n}(Y), k).
\]
batch 1 · p. 3 — read it beside the facsimile9 / 25 · 16 distinct symbols, 37 written
\[(8.2.1) \qquad u'_{2} : \mathrm{Tors}^{n-1}(Y) \otimes_{W} k \to \mathrm{Tors}^{n+1}(X) \otimes_{W} k\]
LaTeX source
\[
(8.2.1) \qquad u'_{2} : \mathrm{Tors}^{n-1}(Y) \otimes_{W} k \to
\mathrm{Tors}^{n+1}(X) \otimes_{W} k
\]
batch 1 · p. 3 — read it beside the facsimile10 / 25 · 11 distinct symbols, 29 written
\[(8.3.1) \qquad \mathrm{Tors}^{n-1}(X) \simeq \mathrm{Tors}^{n-1}(Y).\]
LaTeX source
\[
(8.3.1) \qquad \mathrm{Tors}^{n-1}(X) \simeq \mathrm{Tors}^{n-1}(Y).
\]
batch 1 · p. 3 — read it beside the facsimile11 / 25 · 16 distinct symbols, 37 written
\[(8.3.2) \qquad u''_{2} : \mathrm{Tors}^{n-1}(X) \otimes_{W} k \to \mathrm{Tors}^{n+1}(X) \otimes_{W} k\]
LaTeX source
\[
(8.3.2) \qquad u''_{2} : \mathrm{Tors}^{n-1}(X) \otimes_{W} k \to
\mathrm{Tors}^{n+1}(X) \otimes_{W} k
\]
batch 1 · p. 4 — read it beside the facsimile12 / 25 · 8 distinct symbols, 14 written
\[(8.3.3) \qquad \mathrm{Ker}\, u_{0} = 0 .\]
LaTeX source
\[
(8.3.3) \qquad \mathrm{Ker}\, u_{0} = 0 .
\]
batch 1 · p. 4 — read it beside the facsimile13 / 25 · 18 distinct symbols, 71 written
\[(8.4.1) \qquad 0 \to (\mathrm{Ker}\, u_{1})_{\mathrm{ss}} \to (\mathrm{Ker}\, u_{2})_{\mathrm{ss}} \to (E(Y) \otimes_{W} k)_{\mathrm{ss}} \to E_{\mathrm{DR}}(Y)_{\mathrm{ss}} \to (\mathrm{Coker}\, u_{2})_{\mathrm{ss}} \to 0 .\]
LaTeX source
\[
(8.4.1) \qquad 0 \to (\mathrm{Ker}\, u_{1})_{\mathrm{ss}} \to
(\mathrm{Ker}\, u_{2})_{\mathrm{ss}} \to (E(Y) \otimes_{W} k)_{\mathrm{ss}}
\to E_{\mathrm{DR}}(Y)_{\mathrm{ss}} \to (\mathrm{Coker}\, u_{2})_{\mathrm{ss}}
\to 0 .
\]
batch 1 · p. 5 — read it beside the facsimile14 / 25 · 8 distinct symbols, 17 written
\[H^{i}_{\mathrm{DR}}(X) \to H^{i}(X, \underline{O}_{X})\]
LaTeX source
\[
H^{i}_{\mathrm{DR}}(X) \to H^{i}(X, \underline{O}_{X})
\]
batch 1 · p. 5 — read it beside the facsimile15 / 25 · 15 distinct symbols, 25 written
\[E_{1}^{a,b} = H^{b}(X, \underline{\Omega}^{a}_{X/k}) \Longrightarrow H^{\bullet}_{\mathrm{DR}}(X),\]
LaTeX source
\[
E_{1}^{a,b} = H^{b}(X, \underline{\Omega}^{a}_{X/k}) \Longrightarrow
H^{\bullet}_{\mathrm{DR}}(X),
\]
batch 1 · p. 6 — read it beside the facsimile16 / 25 · 26 distinct symbols, 66 written
\[(8.5.1) \qquad 0 \to \mathrm{Ker}\, v \xrightarrow{w} ({}_{p}\mathrm{Tors}^{n}(X))_{\mathrm{ss}} \to (E(Y) \otimes_{W} k)_{\mathrm{ss}} \xrightarrow{\varphi} \underbrace{E(Y, \underline{O}_{Y})_{\mathrm{ss}}}_{\simeq\, \mathrm{Coker}\, v} \to 0 ,\]
LaTeX source
\[
(8.5.1) \qquad 0 \to \mathrm{Ker}\, v \xrightarrow{w}
({}_{p}\mathrm{Tors}^{n}(X))_{\mathrm{ss}} \to
(E(Y) \otimes_{W} k)_{\mathrm{ss}} \xrightarrow{\varphi}
\underbrace{E(Y, \underline{O}_{Y})_{\mathrm{ss}}}_{\simeq\, \mathrm{Coker}\, v}
\to 0 ,
\]
batch 1 · p. 6 — read it beside the facsimile17 / 25 · 15 distinct symbols, 34 written
\[(8.5.2) \qquad v : H^{n-1}(X, \underline{O}_{X})_{\mathrm{ss}} \to H^{n-1}(Y, \underline{O}_{Y})_{\mathrm{ss}} ,\]
LaTeX source
\[
(8.5.2) \qquad v : H^{n-1}(X, \underline{O}_{X})_{\mathrm{ss}} \to
H^{n-1}(Y, \underline{O}_{Y})_{\mathrm{ss}} ,
\]
batch 1 · p. 6 — read it beside the facsimile18 / 25 · 20 distinct symbols, 49 written
\[(8.5.3) \qquad w : (\mathrm{Tors}^{n-1}(X) \otimes_{W} k)_{\mathrm{ss}} \xrightarrow{u_{d\xi}} (\mathrm{Tors}^{n+1}(X) \otimes_{W} k)_{\mathrm{ss}}\]
LaTeX source
\[
(8.5.3) \qquad w : (\mathrm{Tors}^{n-1}(X) \otimes_{W} k)_{\mathrm{ss}}
\xrightarrow{u_{d\xi}} (\mathrm{Tors}^{n+1}(X) \otimes_{W} k)_{\mathrm{ss}}
\]
batch 1 · p. 6 — read it beside the facsimile19 / 25 · 16 distinct symbols, 41 written
\[(8.5.4) \qquad E(Y) \overset{\mathrm{dfn}}{=} \mathrm{Coker}\bigl(H^{n-1}(X) \to H^{n-1}(Y)\bigr)\]
LaTeX source
\[
(8.5.4) \qquad E(Y) \overset{\mathrm{dfn}}{=}
\mathrm{Coker}\bigl(H^{n-1}(X) \to H^{n-1}(Y)\bigr)
\]
batch 1 · p. 6 — read it beside the facsimile20 / 25 · 21 distinct symbols, 59 written
\[(8.5.5) \qquad \dim (E(Y) \otimes_{W} k)_{\mathrm{ss}} = \dim E(Y, \underline{O}_{Y})_{\mathrm{ss}} + \bigl(\dim ({}_{p}\mathrm{Tors}^{n}(X))_{\mathrm{ss}} - \dim \mathrm{Ker}\, v\bigr) ,\]
LaTeX source
\[
(8.5.5) \qquad \dim (E(Y) \otimes_{W} k)_{\mathrm{ss}} =
\dim E(Y, \underline{O}_{Y})_{\mathrm{ss}} +
\bigl(\dim ({}_{p}\mathrm{Tors}^{n}(X))_{\mathrm{ss}}
- \dim \mathrm{Ker}\, v\bigr) ,
\]
batch 1 · p. 7 — read it beside the facsimile21 / 25 · 14 distinct symbols, 31 written
\[(8.5.6) \qquad \dim (E(Y) \otimes_{W} k)_{\mathrm{ss}} \geq \dim E(Y, \underline{O}_{Y})_{\mathrm{ss}}\]
LaTeX source
\[
(8.5.6) \qquad \dim (E(Y) \otimes_{W} k)_{\mathrm{ss}} \geq
\dim E(Y, \underline{O}_{Y})_{\mathrm{ss}}
\]
batch 1 · p. 7 — read it beside the facsimile22 / 25 · 11 distinct symbols, 16 written
\[H^{n-1}(X, \underline{O}_{X}(-d)) = 0 .\]
LaTeX source
\[
H^{n-1}(X, \underline{O}_{X}(-d)) = 0 .
\]
batch 1 · p. 8 — read it beside the facsimile23 / 25 · 13 distinct symbols, 24 written
\[\dim E(Y, \underline{O}_{Y})_{\mathrm{ss}} > 0 \qquad \text{pour} \quad d \to +\infty ,\]
LaTeX source
\[
\dim E(Y, \underline{O}_{Y})_{\mathrm{ss}} > 0 \qquad \text{pour} \quad
d \to +\infty ,
\]
batch 1 · p. 10 — read it beside the facsimile24 / 25 · 28 distinct symbols, 89 written
\[\begin{aligned} (8.5.11.1) \qquad \dim_{k} H^{n-1}(X, \underline{O})_{\mathrm{ss}} &= \dim_{\mathbf{F}_{p}} H^{n-1}(\overline{X}, \mathbf{Z}/p\mathbf{Z}) \\ &\geq \dim\bigl((H^{n-1}(X)/\mathrm{Tors}^{n-1}(X)) \otimes k\bigr)_{\mathrm{ss}} \end{aligned}\]
LaTeX source
\[
\begin{aligned}
(8.5.11.1) \qquad \dim_{k} H^{n-1}(X, \underline{O})_{\mathrm{ss}} &=
\dim_{\mathbf{F}_{p}} H^{n-1}(\overline{X}, \mathbf{Z}/p\mathbf{Z}) \\
&\geq
\dim\bigl((H^{n-1}(X)/\mathrm{Tors}^{n-1}(X)) \otimes k\bigr)_{\mathrm{ss}}
\end{aligned}
\]
batch 1 · p. 10 — read it beside the facsimile25 / 25 · 17 distinct symbols, 35 written
\[({}_{p}\mathrm{Tors}^{2}(X))_{\mathrm{ss}} \simeq H^{1}(X, \mathbf{Z}/p\mathbf{Z}) \otimes k = \mathrm{Ker}\, \varphi\]
LaTeX source
\[
({}_{p}\mathrm{Tors}^{2}(X))_{\mathrm{ss}} \simeq
H^{1}(X, \mathbf{Z}/p\mathbf{Z}) \otimes k = \mathrm{Ker}\, \varphi
\]