The Stacks project

Remark 60.24.14 (Complete comparison). Let $p$ be a prime number. Let $(A, I, \gamma )$ be a divided power ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set $X = Y \times _ S S_0$. Let $\mathcal{F}$ be a finite type crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then

1. there exists a coherent $\mathcal{O}_ Y$-module $\mathcal{F}_ Y$ endowed with integrable connection

   \[ \nabla : \mathcal{F}_ Y \longrightarrow \mathcal{F}_ Y \otimes _{\mathcal{O}_ Y} \Omega _{Y/S} \]

   such that $\mathcal{F}_ Y/p^ e\mathcal{F}_ Y$ is the module with connection over $A/p^ eA$ found in Remark 60.24.11, and

2. we have

   \[ R\Gamma (\text{Cris}(X/S), \mathcal{F}) = R\Gamma (Y, \mathcal{F}_ Y \otimes _{\mathcal{O}_ Y} \Omega ^\bullet _{Y/S}) \]

   in $D(A)$.

Hints: The existence of $\mathcal{F}_ Y$ is Grothendieck's existence theorem (insert future reference here). The isomorphism of cohomologies follows as both sides are computed as $R\mathop{\mathrm{lim}}\nolimits $ of the versions modulo $p^ e$ (see Remark 60.24.10 for the left hand side; use the theorem on formal functions, see Cohomology of Schemes, Theorem 30.20.5 for the right hand side). Each of the versions modulo $p^ e$ are isomorphic by Remark 60.24.11.