An isocrystal is a crystal up to isogeny. They are -adic analogues of -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.
The infinitesimal site has as objects the infinitesimal extensions of open sets of .
If is a scheme over then the sheaf is defined by
= coordinate ring of , where we write as an abbreviation for
an object of . 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 is a sheaf of modules that is rigid in the following sense:
for any map between objects , ; of , the natural map from to 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 .
Crystals on the crystalline site
are defined in a similar way.
In general, if is a fibered category over , then a crystal is a cartesian section of the fibered category. In the special case when is the category of infinitesimal extensions of a scheme and the category of quasicoherent modules over objects of , then crystals of this fibered category are the same as crystals of the infinitesimal site.
Ogus, Arthur (1 December 1984). "F-isocrystals and de Rham cohomology II—Convergent isocrystals". Duke Mathematical Journal. 51 (4). doi:10.1215/S0012-7094-84-05136-6.
Grothendieck, Alexander (1968), "Crystals and the de Rham cohomology of schemes"(PDF), in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L.; et al. (eds.), Dix Exposés sur la Cohomologie des Schémas, Advanced studies in pure mathematics, vol. 3, Amsterdam: North-Holland, pp. 306–358, MR0269663, archived from the original(PDF) on 2022-02-08
Illusie, Luc (1975), "Report on crystalline cohomology", Algebraic geometry, Proc. Sympos. Pure Math., vol. 29, Providence, R.I.: Amer. Math. Soc., pp. 459–478, MR0393034