Isocrystal

In algebraic geometry, isocrystals are p-adic analogues of Ql-adic étale sheaves, introduced by Berthelot and Ogus (1983) (though the definition of isocrystal only appears in part II of this paper by Ogus (1984)). The term isocrystal stands for roughly "crystal up to isogeny". 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.
Definition
Suppose that A is a complete discrete valuation ring of characteristic 0 with quotient field k of characteristic p>0 and perfect. An affine enlargement of a scheme X0 over k consists of a torsion-free A-algebra B and an ideal I of B such that B is complete in the I topology and the image of I is nilpotent in B/pB, together with a morphism from...