F-crystal
In algebraic geometry, F-crystals are objects introduced by that capture some of the structure of crystalline cohomology groups. The letter F stands for Frobenius, indicating that F-crystals have an action of Frobenius on them. F-isocrystals are crystals "up to isogeny".
F-crystals and F-isocrystals over perfect fields
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.
Dieudonné–Manin classification theorem
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 Es/''r where r'' and s are coprime integers with r>0. The F-isocrystal Es/''r has a basis over K'' of the form v, Fv, F2v,...,Fr−1v for some element v, and Frv = psv. 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
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 s1 < s2 <... and dimensions d1, d2,... then the Newton polygon has vertices,,,... where the nth line segment joining the vertices has slope sn = / and projection onto the x-axis of length dn = xn − xn−1.The Hodge polygon of an ''F''-crystal
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 n1 ≤ n2 ≤... and the Hodge polygon then has vertices,,,...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.
Isocrystals over more general schemes
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 Spec to X''0.A convergent isocrystal over a k-scheme X0 consists of a module over B⊗Q for every affine enlargement B that is compatible with maps between affine enlargements.
An F-isocrystal is an isocrystal together with an isomorphism to its pullback under a Frobenius morphism.