
Crystal (mathematics)

In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by , 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 and (though the definition of isocrystal only appears in part II of this paper by ). 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 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" → "T" of 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 Inf("X"/"S") is a sheaf "F" of "O""X"/"S" modules that is rigid in the following sense:
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.