Crystal (mathematics)

Template:For In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by Template:Harvs, who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the crystalline site are analogous to quasicoherent modules over a scheme.

An isocrystal is a crystal up to isogeny. They are $p$-adic analogues of ${𝐐}_l$-adic étale sheaves, introduced by Template:Harvtxt and Template:Harvtxt (though the definition of isocrystal only appears in part II of this paper by Template:Harvtxt). Convergent isocrystals are a variation of isocrystals that work better over non-perfect fields, and overconvergent isocrystals are another variation related to overconvergent cohomology theories.

A Dieudonné crystal is a crystal with Verschiebung and Frobenius maps. An F-crystal is a structure in semilinear algebra somewhat related to crystals.

Crystals over the infinitesimal and crystalline sites

The infinitesimal site $\text{Inf}(X/S)$ has as objects the infinitesimal extensions of open sets of $X$. If $X$ is a scheme over $S$ then the sheaf $O_{X/S}$ is defined by $O_{X/S}(T)$ = coordinate ring of $T$, where we write $T$ as an abbreviation for an object $U\to T$ of $\text{Inf}(X/S)$. Sheaves on this site grow in the sense that they can be extended from open sets to infinitesimal extensions of open sets.

A crystal on the site $\text{Inf}(X/S)$ is a sheaf $F$ of $O_{X/S}$ modules that is rigid in the following sense:

for any map $f$ between objects $T$, $T^{\prime }$; of $\text{Inf}(X/S)$, the natural map from $f^{*}F(T)$ to $F(T^{\prime })$ is an isomorphism.

This is similar to the definition of a quasicoherent sheaf of modules in the Zariski topology.

An example of a crystal is the sheaf $O_{X/S}$.

Crystals on the crystalline site are defined in a similar way.

Crystals in fibered categories

In general, if $E$ is a fibered category over $F$, then a crystal is a cartesian section of the fibered category. In the special case when $F$ is the category of infinitesimal extensions of a scheme $X$ and $E$ the category of quasicoherent modules over objects of $F$, then crystals of this fibered category are the same as crystals of the infinitesimal site.

References

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