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 <math>p</math>-adic analogues of <math>\mathbf{Q}_l</math>-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 <math>\text{Inf}(X/S)</math> has as objects the infinitesimal extensions of open sets of <math>X</math>. If <math>X</math> is a scheme over <math>S</math> then the sheaf <math>O_{X/S}</math> is defined by <math>O_{X/S}(T)</math> = coordinate ring of <math>T</math>, where we write <math>T</math> as an abbreviation for an object <math>U\to T</math> of <math>\text{Inf}(X/S)</math>. 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 <math>\text{Inf}(X/S)</math> is a sheaf <math>F</math> of <math>O_{X/S}</math> modules that is rigid in the following sense:
- for any map <math>f</math> between objects <math>T</math>, <math>T'</math>; of <math>\text{Inf}(X/S)</math>, the natural map from <math>f^* F(T)</math> to <math>F(T')</math> 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 <math>O_{X/S}</math>.
Crystals on the crystalline site are defined in a similar way.
Crystals in fibered categories
In general, if <math>E</math> is a fibered category over <math>F</math>, then a crystal is a cartesian section of the fibered category. In the special case when <math>F</math> is the category of infinitesimal extensions of a scheme <math>X</math> and <math>E</math> the category of quasicoherent modules over objects of <math>F</math>, then crystals of this fibered category are the same as crystals of the infinitesimal site.