Remark 60.24.4 (07MN): Alternating Čech complex

Remark 60.24.4 (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 $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 $U_\lambda = \mathop{\mathrm{Spec}}(C_\lambda )$. Choose a polynomial algebra $P_\lambda $ over $A$ and a surjection $P_\lambda \to C_\lambda $. Having fixed these choices we can construct an alternating Čech complex which computes $R\Gamma (\text{Cris}(X/S), \mathcal{F})$.

We are going to use the notation introduced in Remark 60.24.3. 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 60.17.4. 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 12.18.3.

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 60.23.1. 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 60.21.3. 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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