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 ''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

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

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.

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 ''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.

References

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*{{Citation , last1=Ogus , first1=Arthur, authorlink=Arthur Ogus , title=F-isocrystals and de Rham cohomology. II. Convergent isocrystals , doi=10.1215/S0012-7094-84-05136-6 , mr=771383 , year=1984 , journal=Duke Mathematical Journal , issn=0012-7094 , volume=51 , issue=4 , pages=765–850

Algebraic geometry