# Crystalline cohomology over general bases

Research paper by **A. M. Masullo**

Indexed on: **25 Nov '20**Published on: **22 Nov '20**Published in: **arXiv - Mathematics - Algebraic Geometry**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.