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F-crystal

March 17, 2024

For F-crystals where F is a fibered category, see crystal (mathematics).

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 = (yn−yn−1)/(xn−xn−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 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 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 B⊗Q 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).
  • 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

March 17, 2024
crystal, where, fibered, category, crystal, mathematics, algebraic, geometry, objects, introduced, mazur, 1972, that, capture, some, structure, crystalline, cohomology, groups, letter, stands, frobenius, indicating, that, have, action, frobenius, them, isocrys. For F crystals where F is a fibered category see crystal mathematics 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 Contents 1 F crystals and F isocrystals over perfect fields 2 Dieudonne Manin classification theorem 3 The Newton polygon of an F isocrystal 4 The Hodge polygon of an F crystal 5 Isocrystals over more general schemes 6 ReferencesF crystals and F isocrystals over perfect fields editSuppose that k is a perfect field with ring of Witt vectors W and let K be the quotient field of W with Frobenius automorphism s 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 s 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 Dieudonne Manin classification theorem editThe Dieudonne Manin classification theorem was proved by Dieudonne 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 gt 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 editThe 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 lt s2 lt 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 yn yn 1 xn xn 1 and projection onto the x axis of length dn xn xn 1 The Hodge polygon of an F crystal editThe 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 editSuppose that A is a complete discrete valuation ring of characteristic 0 with quotient field k of characteristic p gt 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 B Q 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 editBerthelot 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 Dieudonne Jean 1955 Lie groups and Lie hyperalgebras over a field of characteristic p gt 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 Birkhauser Boston pp 219 248 MR 1106900 Grothendieck A 1966 Letter to J Tate PDF 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 Retrieved from https en wikipedia org w index php title F crystal amp oldid 1208132470, wikipedia, wiki, book, books, library,

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