The Stacks project

Remark 60.24.12 (Perfectness). 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 $X$ be a proper smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules. Then $R\Gamma (\text{Cris}(X/S), \mathcal{F})$ is a perfect object of $D(A)$.

Hints: By Remark 60.24.9 we have

\[ R\Gamma (\text{Cris}(X/S), \mathcal{F}) \otimes _ A^\mathbf {L} A/I \cong R\Gamma (\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)}) \]

By Remark 60.24.11 we have

\[ R\Gamma (\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)}) = R\Gamma (X, \mathcal{F}_ X \otimes \Omega ^\bullet _{X/S_0}) \]

Using the stupid filtration on the de Rham complex we see that the last displayed complex is perfect in $D(A/I)$ as soon as the complexes

\[ R\Gamma (X, \mathcal{F}_ X \otimes \Omega ^ q_{X/S_0}) \]

are perfect complexes in $D(A/I)$, see More on Algebra, Lemma 15.74.4. This is true by standard arguments in coherent cohomology using that $\mathcal{F}_ X \otimes \Omega ^ q_{X/S_0}$ is a finite locally free sheaf and $X \to S_0$ is proper and flat (insert future reference here). Applying More on Algebra, Lemma 15.78.4 we see that

\[ R\Gamma (\text{Cris}(X/S), \mathcal{F}) \otimes _ A^\mathbf {L} A/I^ n \]

is a perfect object of $D(A/I^ n)$ for all $n$. This isn't quite enough unless $A$ is Noetherian. Namely, even though $I$ is locally nilpotent by our assumption that $p$ is nilpotent, see Divided Power Algebra, Lemma 23.2.6, we cannot conclude that $I^ n = 0$ for some $n$. A counter example is $\mathbf{F}_ p\langle x \rangle $. To prove it in general when $\mathcal{F} = \mathcal{O}_{X/S}$ the argument of works. When the coefficients $\mathcal{F}$ are non-trivial the argument of [Faltings-very] seems to be as follows. Reduce to the case $pA = 0$ by More on Algebra, Lemma 15.78.4. In this case the Frobenius map $A \to A$, $a \mapsto a^ p$ factors as $A \to A/I \xrightarrow {\varphi } A$ (as $x^ p = 0$ for $x \in I$). Set $X^{(1)} = X \otimes _{A/I, \varphi } A$. The absolute Frobenius morphism of $X$ factors through a morphism $F_ X : X \to X^{(1)}$ (a kind of relative Frobenius). Affine locally if $X = \mathop{\mathrm{Spec}}(C)$ then $X^{(1)} = \mathop{\mathrm{Spec}}( C \otimes _{A/I, \varphi } A)$ and $F_ X$ corresponds to $C \otimes _{A/I, \varphi } A \to C$, $c \otimes a \mapsto c^ pa$. This defines morphisms of ringed topoi

\[ (X/S)_{\text{cris}} \xrightarrow {(F_ X)_{\text{cris}}} (X^{(1)}/S)_{\text{cris}} \xrightarrow {u_{X^{(1)}/S}} \mathop{\mathit{Sh}}\nolimits (X^{(1)}_{Zar}) \]

whose composition is denoted $\text{Frob}_ X$. One then shows that $R\text{Frob}_{X, *}\mathcal{F}$ is representable by a perfect complex of $\mathcal{O}_{X^{(1)}}$-modules(!) by a local calculation.

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