In algebraic geometry, F-crystals are objects introduced by Mazur (1972) 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

edit

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

edit

The Dieudonné–Manin classification theorem was proved by Dieudonné (1955) and Manin (1963). 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

edit

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 (as Witt ring modules) d1, d2,... then the Newton polygon has vertices (0,0), (x1, y1), (x2, y2),... where the nth line segment joining the vertices has slope sn = (ynyn−1)/(xnxn−1) and projection onto the x-axis of length dn = xn − xn−1.

The Hodge polygon of an F-crystal

edit

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 n1n2 ≤ ... 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

edit

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(B/I) to X0. A convergent isocrystal over a k-scheme X0 consists of a module over BQ for every affine enlargement B that is compatible with maps between affine enlargements (Faltings 1990).

An F-isocrystal (short for Frobenius isocrystal) is an isocrystal together with an isomorphism to its pullback under a Frobenius morphism.

References

edit
  • Berthelot, Pierre; Ogus, Arthur (1983), "F-isocrystals and de Rham cohomology. I", Inventiones Mathematicae, 72 (2): 159–199, doi:10.1007/BF01389319, ISSN 0020-9910, MR 0700767
  • Crew, Richard (1987), "F-isocrystals and p-adic representations", Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Providence, R.I.: American Mathematical Society, pp. 111–138, doi:10.1090/pspum/046.2/927977, ISBN 9780821814802, MR 0927977
  • de Shalit, Ehud (2012), F-isocrystals (PDF)
  • Dieudonné, Jean (1955), "Lie groups and Lie hyperalgebras over a field of characteristic p>0. IV", American Journal of Mathematics, 77 (3): 429–452, doi:10.2307/2372633, ISSN 0002-9327, JSTOR 2372633, MR 0071718
  • Faltings, Gerd (1990), "F-isocrystals on open varieties: results and conjectures", The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Boston, MA: Birkhäuser Boston, pp. 219–248, MR 1106900
  • Grothendieck, A. (1966), Letter to J. Tate (PDF), archived from the original (PDF) on 2013-01-20, retrieved 2016-08-26.
  • Manin, Ju. I. (1963), "Theory of commutative formal groups over fields of finite characteristic", Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 18 (6): 3–90, doi:10.1070/RM1963v018n06ABEH001142, ISSN 0042-1316, MR 0157972
  • Mazur, B. (1972), "Frobenius and the Hodge filtration", Bull. Amer. Math. Soc., 78 (5): 653–667, doi:10.1090/S0002-9904-1972-12976-8, MR 0330169
  • Ogus, Arthur (1984), "F-isocrystals and de Rham cohomology. II. Convergent isocrystals", Duke Mathematical Journal, 51 (4): 765–850, doi:10.1215/S0012-7094-84-05136-6, ISSN 0012-7094, MR 0771383