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
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
Here \text{Tot}(-) denotes the total complex associated to a double complex, see Homology, Definition 12.18.3.
Hints: We have
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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