I have a naive question about Čech-Alexander complexes in prismatic cohomology (although I suspect that the situation is similar for crystalline cohomology).
They seemed to be introduced as a method of computation of prismatic cohomology $R\Gamma_{\Delta}(\mathfrak{X}/ (A, I), \mathcal{O}_{\Delta})$ for affine formal $(A/I)$-schemes, that is, in the situation when $\mathfrak{X}$ is of the form $\mathfrak{X}=\mathrm{Spf}(R)$.
Can they be used in more general situations? I suppose it is too optimistic to make some global Čech-Alexander complexes (is it? It seems that one can get at least a (cosimplicial) presheaf (e.g. in the étale topology on $\mathfrak{X}$) because of functoriality in $R$), but perhaps somehow indirectly? And if not, are there any other tools to compute prismatic cohomology?
(A similar question can be asked for Čech-Alexander complexes and the crystalline cohomology; in this context, I saw only indirect uses such as for relating crystalline and de Rham cohomology in the affine case, then using different argument in the global case, so I wonder if this is the "standard template" of how the Čech-Alexander complexes are used.)
