F-crystal

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title: "F-crystal" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-geometry"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/F-crystal" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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 r0. The F-isocrystal E**s/r has a basis over K of the form v, Fv, F2v,...,F**r−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 2 1, d2,... then the Newton polygon has vertices (0,0), (x1, y1), (x2, y2),... where the nth 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 n1 ≤ n2 ≤ ... and the Hodge polygon then has vertices (0,0), (1,n1), (2,n1+ n2), ...

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 p0 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(B/I) to X0. 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 (short for Frobenius isocrystal) is an isocrystal together with an isomorphism to its pullback under a Frobenius morphism.

References

  • {{citation|title=F-isocrystals |first=Ehud |last=de Shalit|year=2012 |url=http://www.ma.huji.ac.il/~deshalit/new_site/files/F-isocrystals.pdf}}
  • .
  • {{citation|mr=0330169 |last=Mazur|first= B. |author-link=Barry Mazur |title=Frobenius and the Hodge filtration |journal=Bull. Amer. Math. Soc. |volume=78 |issue=5|year=1972|pages= 653–667|doi=10.1090/S0002-9904-1972-12976-8|doi-access=free}}

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algebraic-geometry