A "crystal", says Grothendieck, is characterized by two properties: it is rigid, and it grows. Any sheaf F on Cris(X/S) "grows" over PD thickenings (U,T,δ) of open subsets of X by construction. In order for F to be a crystal, we impose a rigidity which we shall only make precise for sheaves of O_{X/S}-modules. From now on, we shall write u^{−1}for pull-back of a sheaves of sets, and u* for module pull-back (i.e. u^{−1}followed by tensor product with the structure sheaf) – unless there is no danger of confusion.

6.1 Definition. A "crystal" of O_{X/S}-modules is a sheaf