## Tag `07MN`

Chapter 54: Crystalline Cohomology > Section 54.24: Some further results

Remark 54.24.4 (Alternating 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 quasi-compact 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 a finite affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$ and a total ordering on $\Lambda$. 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 an alternating Čech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

We are going to use the notation introduced in Remark 54.24.3. Denote $\Omega_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed module of differentials of $D_{\lambda_0 \ldots \lambda_n}$ over $A$ compatible with the divided power structure. Let $\nabla$ be the integrable connection on $M_{\lambda_0 \ldots \lambda_n}$ coming from Proposition 54.17.4. Consider the double complex $M^{\bullet, \bullet}$ with terms $$ M^{n, m} = \bigoplus\nolimits_{\lambda_0 < \ldots < \lambda_n} M_{\lambda_0 \ldots \lambda_n} \otimes^\wedge_{D_{\lambda_0 \ldots \lambda_n}} \Omega^m_{D_{\lambda_0 \ldots \lambda_n}}. $$ For the differential $d_1$ (increasing $n$) we use the usual Čech differential and for the differential $d_2$ we use the connection, i.e., the differential of the de Rham complex. We claim that $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet, \bullet}) $$ Here $\text{Tot}(-)$ denotes the total complex associated to a double complex, see Homology, Definition 12.22.3.

Hints: We have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet) $$ by Proposition 54.23.1. The right hand side of the formula is simply the alternating Čech complex for the covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ (which induces an open covering of the final sheaf of $\text{Cris}(X/S)$) and the complex $\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet$, see Proposition 54.21.3. Now the result follows from a general result in cohomology on sites, namely that the alternating Čech complex computes the cohomology provided it gives the correct answer on all the pieces (insert future reference here).

The code snippet corresponding to this tag is a part of the file `crystalline.tex` and is located in lines 4541–4603 (see updates for more information).

```
\begin{remark}[Alternating {\v C}ech complex]
\label{remark-alternating-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 quasi-compact 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 a finite affine open covering
$X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$
and a total ordering on $\Lambda$.
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 an alternating
{\v C}ech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.
\medskip\noindent
We are going to use the notation introduced in
Remark \ref{remark-cech-complex}.
Denote $\Omega_{\lambda_0 \ldots \lambda_n}$
the $p$-adically completed module of differentials of
$D_{\lambda_0 \ldots \lambda_n}$ over $A$ compatible with the divided power
structure. Let $\nabla$ be the integrable connection on
$M_{\lambda_0 \ldots \lambda_n}$ coming from
Proposition \ref{proposition-crystals-on-affine}.
Consider the double complex $M^{\bullet, \bullet}$ with
terms
$$
M^{n, m} =
\bigoplus\nolimits_{\lambda_0 < \ldots < \lambda_n}
M_{\lambda_0 \ldots \lambda_n}
\otimes^\wedge_{D_{\lambda_0 \ldots \lambda_n}}
\Omega^m_{D_{\lambda_0 \ldots \lambda_n}}.
$$
For the differential $d_1$ (increasing $n$) we use the usual
{\v C}ech differential and for the differential $d_2$ we use
the connection, i.e., the differential of the de Rham complex.
We claim that
$$
R\Gamma(\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet, \bullet})
$$
Here $\text{Tot}(-)$ denotes the total complex associated to a
double complex, see
Homology, Definition \ref{homology-definition-associated-simple-complex}.
\medskip\noindent
Hints: We have
$$
R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S),
\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet)
$$
by Proposition \ref{proposition-compare-with-de-Rham}.
The right hand side of the formula is simply the alternating {\v C}ech complex
for the covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$
(which induces an open covering of the final sheaf of $\text{Cris}(X/S)$)
and the complex $\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet$,
see Proposition \ref{proposition-compute-cohomology-crystal}.
Now the result follows from a general result in cohomology on sites,
namely that the alternating {\v C}ech complex computes the cohomology
provided it gives the correct answer on all the pieces (insert future
reference here).
\end{remark}
```

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