Crystalline cohomology over general bases

Abstract

Building on ideas of Berthelot, we develop a crystalline cohomology formalism over divided power rings $(A, I_0, \eta)$ for any ring $A$, allowing $\mathbf{Z}$-flat $A$. For a smooth $A$-scheme $Y$ and a closed subscheme $X$ of $Y$ for which $\eta$ extends to $I_0 \mathscr{O}_X$, a (quasi-coherent) crystal $\mathscr{F}$ on $(X/A)_{\rm{cris}}$ is equivalent to a specific type of module with integrable $A$-linear connection over a certain completion $D_{Y,\eta}(X)^{\wedge}$ (called "pd-adic") of the divided power envelope $D_{Y,\eta}(X)$ of $Y$ along $X$ (with divided power structure $\delta$) Our main result, building on ideas of Bhatt and de Jong for $\mathbf{Z}/(p^e)$-schemes (where pd-adic completion has no effect), is a natural isomorphism between ${\rm{R}}\Gamma((X/A)_{\rm{cris}}, \mathscr{F})$ and the Zariski hypercohomology of the pd-adically completed de Rham complex $\mathscr{F} \widehat{\otimes} \widehat{\Omega}^*_{D_{Y,\eta}(X)^{\wedge}/A,\delta}$ arising from the module with integrable connection over $D_{Y,\eta}(X)^{\wedge}$ associated to $\mathscr{F}$. By a variant of the same methods, we obtain a representative of the complex $\mathscr{F} \widehat{\otimes} \widehat{\Omega}^*_{D_{Y,\eta}(X)^{\wedge}/A,\delta}$ in the derived category of sheaves of $A$-modules on $X$ in terms of a \v{C}ech-Alexander construction. When $\mathscr{F}=\mathscr{O}_{X/A}$, our comparison theorem implies that in the derived category of sheaves of $A$-modules on $X$, the pd-adic completion of $\Omega^*_{D_{Y,\eta}(X)/A,\delta}$ functorially depends only on $X$. Over $\mathbf{Q}$-algebras $A$, so pd-adic completion becomes ideal-adic completion, this recovers a result of Hartshorne.