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".
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 ÃÂ.
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 ÃÂ-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 DieudonnéâÂÂManin 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<sub>s/r</sub> where r and s are coprime integers with r>0. The F-isocrystal E<sub>s/r</sub> has a basis over K of the form v, Fv, F<sup>2</sup>v,...,F<sup>râÂÂ1</sup>v for some element v, and F<sup>r</sup>v = p<sup>s</sup>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<sub>1</sub> < s<sub>2</sub> < ... and dimensions (as Witt ring modules) d<sub>1</sub>, d<sub>2</sub>,... then the Newton polygon has vertices (0,0), (x<sub>1</sub>, y<sub>1</sub>), (x<sub>2</sub>, y<sub>2</sub>),... where the nth line segment joining the vertices has slope s<sub>n</sub> = (y<sub>n</sub>âÂÂy<sub>nâÂÂ1</sub>)/(x<sub>n</sub>âÂÂx<sub>nâÂÂ1</sub>) and projection onto the x-axis of length d<sub>n</sub> = x<sub>n</sub> â x<sub>nâÂÂ1</sub>.
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<sub>1</sub> ⤠n<sub>2</sub> ⤠... and the Hodge polygon then has vertices (0,0), (1,n<sub>1</sub>), (2,n<sub>1</sub>+ n<sub>2</sub>), ...
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<sub>0</sub> 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<sub>0</sub>. A convergent isocrystal over a k-scheme X<sub>0</sub> consists of a module over BâÂÂQ 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.