# The Stacks Project

## Tag 07MY

Remark 51.24.13 (Complete perfectness). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. 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: We know that $K = R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is the derived limit $K = R\mathop{\rm lim}\nolimits K_e$ of the cohomologies over $A/p^eA$, see Remark 51.24.10. Each $K_e$ is a perfect complex of $D(A/p^eA)$ by Remark 51.24.12. Since $A$ is $p$-adically complete the result follows from More on Algebra, Lemma 15.82.4.

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4950–4969 (see updates for more information).

\begin{remark}[Complete perfectness]
\label{remark-complete-perfect}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set
$S = \Spec(A)$ and $S_0 = \Spec(A/I)$. 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 $D(A)$.

\medskip\noindent
Hints: We know that $K = R\Gamma(\text{Cris}(X/S), \mathcal{F})$
is the derived limit $K = R\lim K_e$ of the cohomologies over $A/p^eA$,
see Remark \ref{remark-rlim}.
Each $K_e$ is a perfect complex of $D(A/p^eA)$ by
Remark \ref{remark-perfect}.
Since $A$ is $p$-adically complete the result
follows from
More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-perfect-gives-complete}.
\end{remark}

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