# The Stacks Project

## Tag 07MP

Remark 51.24.5 (Quasi-coherence). In the situation of Remark 51.24.1 assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ the sheaves $R^if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent.

Hints: We have to show that the restrictions to $T'$ are quasi-coherent $\mathcal{O}_{T'}$-modules, where $(U', T', \delta')$ is any object of $\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine. We use the formula (51.24.1.1), the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$ is affine over the base change of $T'$ by $S \to S'$), and Cohomology of Schemes, Lemma 29.4.5 to see that it suffices to show that the sheaves $R^i\tau_{U/T, *}\mathcal{F}_U$ are quasi-coherent. Note that $U \to T_0$ is also quasi-compact and quasi-separated, see Schemes, Lemmas 25.21.15 and 25.21.15.

This reduces us to proving that $R^i\tau_{X/S, *}\mathcal{F}$ is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here $\tau_{X/S}$ is the structure morphism, see Remark 51.9.6. We may work locally on $S$, hence we may assume $S$ affine (see Lemma 51.9.5). Induction on the number of affines covering $X$ and Mayer-Vietoris (Remark 51.24.2) reduces the question to the case where $X$ is also affine (as in the proof of Cohomology of Schemes, Lemma 29.4.5). Say $X = \mathop{\rm Spec}(C)$ and $S = \mathop{\rm Spec}(A)$ so that $(A, I, \gamma)$ and $A \to C$ are as in Situation 51.5.1. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$ as in Section 51.17. Let $(M, \nabla)$ be the module corresponding to $\mathcal{F}$, see Proposition 51.17.4. Applying Proposition 51.21.3 we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is represented by $M \otimes_D \Omega_D^*$. Note that completion isn't necessary as $p$ is nilpotent in $A$! We have to show that this is compatible with taking principal opens in $S = \mathop{\rm Spec}(A)$. Suppose that $g \in A$. Then we conclude that similarly $R\Gamma(\text{Cris}(X_g/S_g), \mathcal{F})$ is computed by $M_g \otimes_{D_g} \Omega_{D_g}^*$ (again this uses that $p$-adic completion isn't necessary). Hence we conclude because localization is an exact functor on $A$-modules.

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4605–4657 (see updates for more information).

\begin{remark}[Quasi-coherence]
\label{remark-quasi-coherent}
In the situation of Remark \ref{remark-compute-direct-image}
assume that $S \to S'$ is quasi-compact and quasi-separated and
that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal
in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$
the sheaves $R^if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent.

\medskip\noindent
Hints: We have to show that the restrictions to $T'$ are quasi-coherent
$\mathcal{O}_{T'}$-modules, where $(U', T', \delta')$ is any object of
$\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine.
We use the formula (\ref{equation-identify-pushforward}),
the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$
is affine over the base change of $T'$ by $S \to S'$), and
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}
to see that it suffices to show that the sheaves
$R^i\tau_{U/T, *}\mathcal{F}_U$ are quasi-coherent.
Note that $U \to T_0$ is also quasi-compact and quasi-separated, see
Schemes, Lemmas \ref{schemes-lemma-quasi-compact-permanence} and
\ref{schemes-lemma-quasi-compact-permanence}.

\medskip\noindent
This reduces us to proving that $R^i\tau_{X/S, *}\mathcal{F}$
is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here
$\tau_{X/S}$ is the structure morphism, see
Remark \ref{remark-structure-morphism}.
We may work locally on $S$, hence we may assume $S$ affine
(see Lemma \ref{lemma-localize}). Induction on the number
of affines covering $X$ and Mayer-Vietoris
(Remark \ref{remark-mayer-vietoris}) reduces the question to
the case where $X$ is also affine (as in the proof of
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}).
Say $X = \Spec(C)$ and $S = \Spec(A)$ so that $(A, I, \gamma)$ and
$A \to C$ are as
in Situation \ref{situation-affine}. Choose a polynomial algebra
$P$ over $A$ and a surjection $P \to C$ as in
Section \ref{section-quasi-coherent-crystals}.
Let $(M, \nabla)$ be the module corresponding to $\mathcal{F}$, see
Proposition \ref{proposition-crystals-on-affine}.
Applying
Proposition \ref{proposition-compute-cohomology-crystal}
we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is represented by
$M \otimes_D \Omega_D^*$. Note that completion isn't necessary
as $p$ is nilpotent in $A$! We have to show that this is compatible
with taking principal opens in $S = \Spec(A)$. Suppose that $g \in A$.
Then we conclude that similarly $R\Gamma(\text{Cris}(X_g/S_g), \mathcal{F})$
is computed by $M_g \otimes_{D_g} \Omega_{D_g}^*$ (again this uses that
$p$-adic completion isn't necessary). Hence we conclude because localization
is an exact functor on $A$-modules.
\end{remark}

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