The Stacks project

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).

[1] 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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