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.

It turns out that we can relate these notions as follows.

**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 $\mathcal{F}|_{\mathcal{C}/(U, T, \delta )}$.
$\square$

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