2020/11/23 by Masullo, A. M.
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2011.11182
Building on ideas of Berthelot, we develop a crystalline cohomology formalism over divided power rings (A, I0, η) for any ring A, allowing Z-flat A. For a smooth A-scheme Y and a closed subscheme X of Y for which η extends to I0 \mathscrOX, a (quasi-coherent) crystal \mathscrF on (X/A)_\rmcris is equivalent to a specific type of module with integrable A-linear connection over a certain completion DY,η(X)\wedge (called "pd-adic") of the divided power envelope DY,η(X) of Y along X (with divided power structure δ) Our main result, building on ideas of Bhatt and de Jong for Z/(pe)-schemes (where pd-adic completion has no effect), is a natural isomorphism between \rmRΓ((X/A)_\rmcris, \mathscrF) and the Zariski hypercohomology of the pd-adically completed de Rham complex \mathscrF \widehat⊗ \widehatΩ^*_DY,η(X)\wedge/A,δ arising from the module with integrable connection over DY,η(X)\wedge associated to \mathscrF. By a variant of the same methods, we obtain a representative of the complex \mathscrF \widehat⊗ \widehatΩ^*_DY,η(X)\wedge/A,δ in the derived category of sheaves of A-modules on X in terms of a Čech-Alexander construction. When \mathscrF=\mathscrOX/A, our comparison theorem implies that in the derived category of sheaves of A-modules on X, the pd-adic completion of Ω^*_DY,η(X)/A,δ functorially depends only on X. Over Q-algebras A, so pd-adic completion becomes ideal-adic completion, this recovers a result of Hartshorne.