In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by Alexander Grothendieck ( 1966a), 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 ${\displaystyle p}$-adic analogues of ${\displaystyle \mathbf {Q} _{l}}$-adic étale sheaves, introduced by Grothendieck (1966a) and Berthelot & Ogus (1983) (though the definition of isocrystal only appears in part II of this paper by Ogus (1984)). 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 ${\displaystyle {\text{Inf}}(X/S)}$ has as objects the infinitesimal extensions of open sets of ${\displaystyle X}$. If ${\displaystyle X}$ is a scheme over ${\displaystyle S}$ then the sheaf ${\displaystyle O_{X/S}}$ is defined by ${\displaystyle O_{X/S}(T)}$ = coordinate ring of ${\displaystyle T}$, where we write ${\displaystyle T}$ as an abbreviation for an object ${\displaystyle U\to T}$ of ${\displaystyle {\text{Inf}}(X/S)}$. 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 ${\displaystyle {\text{Inf}}(X/S)}$ is a sheaf ${\displaystyle F}$ of ${\displaystyle O_{X/S}}$ modules that is rigid in the following sense:

for any map ${\displaystyle f}$ between objects ${\displaystyle T}$, ${\displaystyle T'}$; of ${\displaystyle {\text{Inf}}(X/S)}$, the natural map from ${\displaystyle f^{*}F(T)}$ to ${\displaystyle F(T')}$ 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 ${\displaystyle O_{X/S}}$.

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

Crystals in fibered categories

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

References

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