## 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 $\mathcal{C}$.

1. 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.

2. We say $\mathcal{F}$ is quasi-coherent if it is quasi-coherent in the sense of Modules on Sites, Definition 18.23.1.

3. 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

1. $\mathcal{F}$ is quasi-coherent, and

2. $\mathcal{F}$ 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.

Remark 60.11.4. To formulate the general notion of a crystal we use the language of stacks and strongly cartesian morphisms, see Stacks, Definition 8.4.1 and Categories, Definition 4.33.1. In Situation 60.7.5 let $p : \mathcal{C} \to \text{Cris}(X/S)$ be a stack. A crystal in objects of $\mathcal{C}$ on $X$ relative to $S$ is a cartesian section $\sigma : \text{Cris}(X/S) \to \mathcal{C}$, i.e., a functor $\sigma$ such that $p \circ \sigma = \text{id}$ and such that $\sigma (f)$ is strongly cartesian for all morphisms $f$ of $\text{Cris}(X/S)$. Similarly for the big crystalline site.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07IR. Beware of the difference between the letter 'O' and the digit '0'.