Remark 60.24.3 (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 $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. Let $X$ be a separated1 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 $U_\lambda = \mathop{\mathrm{Spec}}(C_\lambda )$. 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 $U_{\lambda _0 \ldots \lambda _ n} = \mathop{\mathrm{Spec}}(C_{\lambda _0 \ldots \lambda _ n})$. 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 60.17.4. 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 $\mathop{\mathrm{Spec}}(D_{\lambda _0 \ldots \lambda _ n})$ 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 60.21.1 (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.

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