The Stacks project

Remark 60.24.11 (Comparison). Let $p$ be a prime number. Let $(A, I, \gamma )$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. Let $Y$ be a smooth scheme over $S$ and set $X = Y \times _ S S_0$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then

1. $\gamma $ extends to a divided power structure on the ideal of $X$ in $Y$ so that $(X, Y, \gamma )$ is an object of $\text{Cris}(X/S)$,

2. the restriction $\mathcal{F}_ Y$ (see Section 60.10) comes endowed with a canonical integrable connection $\nabla : \mathcal{F}_ Y \to \mathcal{F}_ Y \otimes _{\mathcal{O}_ Y} \Omega _{Y/S}$, and

3. 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: See Divided Power Algebra, Lemma 23.4.2 for (1). See Lemma 60.15.1 for (2). For Part (3) note that there is a map, see (60.23.2.1). This map is an isomorphism when $X$ is affine, see Lemma 60.21.4. This shows that $Ru_{X/S, *}\mathcal{F}$ and $\mathcal{F}_ Y \otimes \Omega ^\bullet _{Y/S}$ are quasi-isomorphic as complexes on $Y_{Zar} = X_{Zar}$. Since $R\Gamma (\text{Cris}(X/S), \mathcal{F}) = R\Gamma (X_{Zar}, Ru_{X/S, *}\mathcal{F})$ the result follows.