60.11 Crystals in modules
It turns out that a crystal is a very general gadget. However, the definition may be a bit hard to parse, so we first give the definition in the case of modules on the crystalline sites.
Definition 60.11.1. In Situation 60.7.5. Let $\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X/S}$-modules on $\mathcal{C}$.
We say $\mathcal{F}$ is locally quasi-coherent if for every object $(U, T, \delta )$ of $\mathcal{C}$ the restriction $\mathcal{F}_ T$ is a quasi-coherent $\mathcal{O}_ T$-module.
We say $\mathcal{F}$ is quasi-coherent if it is quasi-coherent in the sense of Modules on Sites, Definition 18.23.1.
We say $\mathcal{F}$ is a crystal in $\mathcal{O}_{X/S}$-modules if all the comparison maps (60.10.0.2) are isomorphisms.
It turns out that we can relate these notions as follows.
Lemma 60.11.2. With notation $X/S, \mathcal{I}, \gamma , \mathcal{C}, \mathcal{F}$ as in Definition 60.11.1. The following are equivalent
$\mathcal{F}$ is quasi-coherent, and
$\mathcal{F}$ is locally quasi-coherent and a crystal in $\mathcal{O}_{X/S}$-modules.
Proof.
Assume (1). Let $f : (U', T', \delta ') \to (U, T, \delta )$ be an object of $\mathcal{C}$. We have to prove (a) $\mathcal{F}_ T$ is a quasi-coherent $\mathcal{O}_ T$-module and (b) $c_ f : f^*\mathcal{F}_ T \to \mathcal{F}_{T'}$ is an isomorphism. The assumption means that we can find a covering $\{ (T_ i, U_ i, \delta _ i) \to (T, U, \delta )\} $ and for each $i$ the restriction of $\mathcal{F}$ to $\mathcal{C}/(T_ i, U_ i, \delta _ i)$ has a global presentation. Since it suffices to prove (a) and (b) Zariski locally, we may replace $f : (T', U', \delta ') \to (T, U, \delta )$ by the base change to $(T_ i, U_ i, \delta _ i)$ and assume that $\mathcal{F}$ restricted to $\mathcal{C}/(T, U, \delta )$ has a global presentation
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_{X/S}|_{\mathcal{C}/(U, T, \delta )} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_{X/S}|_{\mathcal{C}/(U, T, \delta )} \longrightarrow \mathcal{F}|_{\mathcal{C}/(U, T, \delta )} \longrightarrow 0 \]
It is clear that this gives a presentation
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ T \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ T \longrightarrow \mathcal{F}_ T \longrightarrow 0 \]
and hence (a) holds. Moreover, the presentation restricts to $T'$ to give a similar presentation of $\mathcal{F}_{T'}$, whence (b) holds.
Assume (2). Let $(U, T, \delta )$ be an object of $\mathcal{C}$. We have to find a covering of $(U, T, \delta )$ such that $\mathcal{F}$ has a global presentation when we restrict to the localization of $\mathcal{C}$ at the members of the covering. Thus we may assume that $T$ is affine. In this case we can choose a presentation
\[ \bigoplus \nolimits _{j \in J} \mathcal{O}_ T \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O}_ T \longrightarrow \mathcal{F}_ T \longrightarrow 0 \]
as $\mathcal{F}_ T$ is assumed to be a quasi-coherent $\mathcal{O}_ T$-module. Then by the crystal property of $\mathcal{F}$ we see that this pulls back to a presentation of $\mathcal{F}_{T'}$ for any morphism $f : (U', T', \delta ') \to (U, T, \delta )$ of $\mathcal{C}$. Thus the desired presentation of $\mathcal{F}|_{\mathcal{C}/(U, T, \delta )}$.
$\square$
Definition 60.11.3. If $\mathcal{F}$ satisfies the equivalent conditions of Lemma 60.11.2, then we say that $\mathcal{F}$ is a crystal in quasi-coherent modules. We say that $\mathcal{F}$ is a crystal in finite locally free modules if, in addition, $\mathcal{F}$ is finite locally free.
Of course, as Lemma 60.11.2 shows, this notation is somewhat heavy since a quasi-coherent module is always a crystal. But it is standard terminology in the literature.
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