# The Stacks Project

## Tag 07MN

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).

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\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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