In algebraic geometry, F-crystals are objects introduced by Mazur (1972) that capture some of the structure of crystalline cohomology groups.
Over the field k, an F-crystal is a free module M of finite rank over the ring W of Witt vectors of k, together with a σ-linear injective endomorphism of M. An F-isocrystal is defined in the same way, except that M is a module for the quotient field K of W rather than W. The Dieudonné–Manin classification theorem was proved by Dieudonné (1955) and Manin (1963).
If the F-isocrystal is a sum of isoclinic pieces with slopes s1 < s2 < ... and dimensions (as Witt ring modules) d1, d2,... then the Newton polygon has vertices (0,0), (x1, y1), (x2, y2),... where the nth line segment joining the vertices has slope sn = (yn−yn−1)/(xn−xn−1) and projection onto the x-axis of length dn = xn − xn−1.
The Hodge polygon of an F-crystal M encodes the structure of M/FM considered as a module over the Witt ring.
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 Spec(B/I) to X0.