Crystal (mathematics)
title: "Crystal (mathematics)" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-geometry"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Crystal_(mathematics)" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 \mathbf{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 \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'; of \text{Inf}(X/S), the natural map from f^* F(T) to 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 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
- (letter to Atiyah, Oct. 14 1963)
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::