## Tag `07MM`

Chapter 51: Crystalline Cohomology > Section 51.24: Some further results

Remark 51.24.3 (Čech complex). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a separated

^{1}scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.Choose an affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$. Write $U_\lambda = \mathop{\rm Spec}(C_\lambda)$. Choose a polynomial algebra $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$. Having fixed these choices we can construct a Čech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

Given $n \geq 0$ and $\lambda_0, \ldots, \lambda_n \in \Lambda$ write $U_{\lambda_0 \ldots \lambda_n} = U_{\lambda_0} \cap \ldots \cap U_{\lambda_n}$. This is an affine scheme by assumption. Write $U_{\lambda_0 \ldots \lambda_n} = \mathop{\rm Spec}(C_{\lambda_0 \ldots \lambda_n})$. Set $$ P_{\lambda_0 \ldots \lambda_n} = P_{\lambda_0} \otimes_A \ldots \otimes_A P_{\lambda_n} $$ which comes with a canonical surjection onto $C_{\lambda_0 \ldots \lambda_n}$. Denote the kernel $J_{\lambda_0 \ldots \lambda_n}$ and set $D_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed divided power envelope of $J_{\lambda_0 \ldots \lambda_n}$ in $P_{\lambda_0 \ldots \lambda_n}$ relative to $\gamma$. Let $M_{\lambda_0 \ldots \lambda_n}$ be the $P_{\lambda_0 \ldots \lambda_n}$-module corresponding to the restriction of $\mathcal{F}$ to $\text{Cris}(U_{\lambda_0 \ldots \lambda_n}/S)$ via Proposition 51.17.4. By construction we obtain a cosimplicial divided power ring $D(*)$ having in degree $n$ the ring $$ D(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} D_{\lambda_0 \ldots \lambda_n} $$ (use that divided power envelopes are functorial and the trivial cosimplicial structure on the ring $P(*)$ defined similarly). Since $M_{\lambda_0 \ldots \lambda_n}$ is the ''value'' of $\mathcal{F}$ on the objects $\mathop{\rm Spec}(D_{\lambda_0 \ldots \lambda_n})$ we see that $M(*)$ defined by the rule $$ M(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} M_{\lambda_0 \ldots \lambda_n} $$ forms a cosimplicial $D(*)$-module. Now we claim that we have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = s(M(*)) $$ Here $s(-)$ denotes the cochain complex associated to a cosimplicial module (see Simplicial, Section 14.25).

Hints: The proof of this is similar to the proof of Proposition 51.21.1 (in particular the result holds for any module satisfying the assumptions of that proposition).

- This assumption is not strictly necessary, as using hypercoverings the construction of the remark can be extended to the general case. ↑

The code snippet corresponding to this tag is a part of the file `crystalline.tex` and is located in lines 4472–4539 (see updates for more information).

```
\begin{remark}[{\v C}ech complex]
\label{remark-cech-complex}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \Spec(A)$ and
$S_0 = \Spec(A/I)$. Let $X$ be a separated\footnote{This assumption is
not strictly necessary, as using hypercoverings the construction of the
remark can be extended to the general case.} scheme over
$S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a
crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.
\medskip\noindent
Choose an affine open covering
$X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$.
Write $U_\lambda = \Spec(C_\lambda)$. Choose a polynomial algebra
$P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$.
Having fixed these choices we can construct a {\v C}ech complex which
computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.
\medskip\noindent
Given $n \geq 0$ and $\lambda_0, \ldots, \lambda_n \in \Lambda$
write $U_{\lambda_0 \ldots \lambda_n} = U_{\lambda_0} \cap \ldots
\cap U_{\lambda_n}$. This is an affine scheme by assumption. Write
$U_{\lambda_0 \ldots \lambda_n} = \Spec(C_{\lambda_0 \ldots \lambda_n})$.
Set
$$
P_{\lambda_0 \ldots \lambda_n} =
P_{\lambda_0} \otimes_A \ldots \otimes_A P_{\lambda_n}
$$
which comes with a canonical surjection onto $C_{\lambda_0 \ldots \lambda_n}$.
Denote the kernel $J_{\lambda_0 \ldots \lambda_n}$ and set
$D_{\lambda_0 \ldots \lambda_n}$
the $p$-adically completed divided power envelope of
$J_{\lambda_0 \ldots \lambda_n}$ in $P_{\lambda_0 \ldots \lambda_n}$
relative to $\gamma$. Let $M_{\lambda_0 \ldots \lambda_n}$ be the
$P_{\lambda_0 \ldots \lambda_n}$-module corresponding
to the restriction of $\mathcal{F}$ to
$\text{Cris}(U_{\lambda_0 \ldots \lambda_n}/S)$ via
Proposition \ref{proposition-crystals-on-affine}.
By construction we obtain a cosimplicial divided power ring $D(*)$
having in degree $n$ the ring
$$
D(n) =
\prod\nolimits_{\lambda_0 \ldots \lambda_n}
D_{\lambda_0 \ldots \lambda_n}
$$
(use that divided power envelopes are functorial and the trivial
cosimplicial structure on the ring $P(*)$ defined similarly).
Since $M_{\lambda_0 \ldots \lambda_n}$ is the ``value'' of $\mathcal{F}$
on the objects $\Spec(D_{\lambda_0 \ldots \lambda_n})$ we see that
$M(*)$ defined by the rule
$$
M(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n}
M_{\lambda_0 \ldots \lambda_n}
$$
forms a cosimplicial $D(*)$-module. Now we claim that we have
$$
R\Gamma(\text{Cris}(X/S), \mathcal{F}) = s(M(*))
$$
Here $s(-)$ denotes the cochain complex associated to a cosimplicial
module (see
Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}).
\medskip\noindent
Hints: The proof of this is similar to the proof of
Proposition \ref{proposition-compute-cohomology} (in particular
the result holds for any module satisfying the assumptions of
that proposition).
\end{remark}
```

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