F-crystal
Suppose that "k" is a perfect field, with ring of Witt vectors "W" and let "K" be the quotient field of "W", with Frobenius automorphism σ.
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 and . It describes the structure of "F"-isocrystals over an algebraically closed field "k". The category of such "F"-isocrystals is abelian and semisimple, so every "F"-isocrystal is a direct sum of simple "F"-isocrystals. The simple "F"-isocrystals are the modules "E""s"/"r" where "r" and "s" are coprime integers with "r">0. The "F"-isocrystal "E""s"/"r" has a basis over "K" of the form "v", "Fv", "F"2"v"...,"F""r"−1"v" for some element "v", and "F""r""v" = "p""s""v". The rational number "s"/"r" is called the slope of the "F"-isocrystal. Over a non-algebraically closed field "k" the simple "F"-isocrystals are harder to describe explicitly, but an "F"-isocrystal can still be written as a direct sum of subcrystals that are isoclinic, where an "F"-crystal is called isoclinic if over the algebraic closure of "k" it is a sum of "F"-isocrystals of the same slope. The Newton polygon of an "F"-isocrystal encodes the dimensions of the pieces of given slope. If the "F"-isocrystal is a sum of isoclinic pieces with slopes "s"1 < "s"2 < ... and dimensions (as Witt ring modules) "d"1, "d"2... then the Newton polygon has vertices (0,0), ("x"1, "y"1), ("x"2, "y"2)... where the "n"th line segment joining the vertices has slope "s""n" = ("y""n"−"y""n"−1)/("x""n"−"x""n"−1) and projection onto the "x"-axis of length "d""n" = "x""n" − "x""n"−1. The Hodge polygon of an "F"-crystal "M" encodes the structure of "M"/"FM" considered as a module over the Witt ring. More precisely since the Witt ring is a principal ideal domain, the module "M"/"FM" can be written as a direct sum of indecomposable modules of lengths "n"1 ≤ "n"2 ≤ ... and the Hodge polygon then has vertices (0,0), (1,"n"1), (2,"n"1+ "n"2), ... 
While the Newton polygon of an "F"-crystal depends only on the corresponding isocrystal, it is possible for two "F"-crystals corresponding to the same "F"-isocrystal to have different Hodge polygons. The Hodge polygon has edges with integer slopes, while the Newton polygon has edges with rational slopes. 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 "X"0 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 "X"0. A convergent isocrystal over a "k"-scheme "X"0 consists of a module over "B"⊗Q for every affine enlargement "B" that is compatible with maps between affine enlargements . An F-isocrystal (short for Frobenius isocrystal) is an isocrystal together with an isomorphism to its pullback under a Frobenius morphism.
Dieudonné–Manin classification theorem
F-isocrystals are crystals "up to isogeny".