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.

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.

• 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.
• Berthelot, Pierre (1974), Cohomologie cristalline des schémas de caractéristique p>0, Lecture Notes in Mathematics, Vol. 407, vol. 407, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0068636, ISBN 978-3-540-06852-5, MR 0384804
• Berthelot, Pierre; Ogus, Arthur (1978), Notes on crystalline cohomology, Princeton University Press, ISBN 978-0-691-08218-9, MR 0491705
• Berthelot, P.; Ogus, A. (June 1983). "F-isocrystals and de Rham cohomology. I". Inventiones Mathematicae. 72 (2): 159–199. doi:10.1007/BF01389319.
• Chambert-Loir, Antoine (1998), "Cohomologie cristalline: un survol", Expositiones Mathematicae, 16 (4): 333–382, ISSN 0723-0869, MR 1654786, archived from the original on 2011-07-21
• Grothendieck, Alexander (1966a), "On the de Rham cohomology of algebraic varieties", Institut des Hautes Études Scientifiques. Publications Mathématiques, 29 (29): 95–103, doi:10.1007/BF02684807, ISSN 0073-8301, MR 0199194 (letter to Atiyah, Oct. 14 1963)
• Grothendieck, Alexander (1966b), Letter to J. Tate (PDF), archived from the original (PDF) on 2021-07-21
• 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, MR 0269663, 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, MR 0393034
• Illusie, Luc (1976), "Cohomologie cristalline (d'après P. Berthelot)", Séminaire Bourbaki (1974/1975: Exposés Nos. 453-470), Exp. No. 456, Lecture Notes in Math., vol. 514, Berlin, New York: Springer-Verlag, pp. 53–60, MR 0444668, archived from the original on 2012-02-10, retrieved 2016-08-24
• Illusie, Luc (1994), "Crystalline cohomology", Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Providence, RI: Amer. Math. Soc., pp. 43–70, MR 1265522
• Kedlaya, Kiran S. (2009), "p-adic cohomology", in Abramovich, Dan; Bertram, A.; Katzarkov, L.; Pandharipande, Rahul; Thaddeus., M. (eds.), Algebraic geometry---Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Providence, R.I.: Amer. Math. Soc., pp. 667–684, arXiv:math/0601507, Bibcode:2006math......1507K, ISBN 978-0-8218-4703-9, MR 2483951