  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.
