F-crystal

{{for|F-crystals where F is a fibered category|crystal (mathematics)}} In algebraic geometry, '''F-crystals''' are objects introduced by {{harvtxt|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==

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 {{harvtxt|Dieudonné|1955}} and {{harvtxt|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 ''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''&nbsp;−&nbsp;''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 {{harv|Faltings|1990}}.

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

==References==

*{{Citation | last1=Berthelot | first1=Pierre | author1-link=Pierre Berthelot (mathematician) | last2=Ogus | first2=Arthur |author2-link=Arthur Ogus| title=F-isocrystals and de Rham cohomology. I | doi=10.1007/BF01389319 |mr=700767 | year=1983 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=72 | issue=2 | pages=159–199}} *{{Citation | last1=Crew | first1=Richard | title=Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) | chapter-url=https://books.google.com/books?id=lw1ym4c2crMC&pg=PA117 | publisher=American Mathematical Society | location=Providence, R.I. | series=Proc. Sympos. Pure Math. |mr=927977 | year=1987 | volume=46 | chapter=F-isocrystals and p-adic representations | pages=111–138 | doi=10.1090/pspum/046.2/927977| isbn=9780821814802 }} *{{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 | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Lie groups and Lie hyperalgebras over a field of characteristic p>0. IV | jstor=2372633 | mr=0071718 | year=1955 | journal=American Journal of Mathematics | issn=0002-9327 | volume=77 | issue=3 | pages=429–452 | doi=10.2307/2372633}} *{{Citation | last1=Faltings | first1=Gerd | author1-link=Gerd Faltings | title=The Grothendieck Festschrift, Vol. II | publisher=Birkhäuser Boston | location=Boston, MA | series=Progr. Math. |mr=1106900 | year=1990 | volume=87 | chapter=F-isocrystals on open varieties: results and conjectures | pages=219–248}} *{{citation|last=Grothendieck|first=A.|author-link=Alexander Grothendieck|url=http://www.math.jussieu.fr/~leila/grothendieckcircle/crystals.pdf|title=Letter to J. Tate|year=1966|access-date=2016-08-26|archive-date=2013-01-20|archive-url=https://web.archive.org/web/20130120101555/http://www.math.jussieu.fr/~leila/grothendieckcircle/crystals.pdf|url-status=dead}}. *{{Citation | last1=Manin | first1=Ju. I. |author1-link=Yuri Manin | title=Theory of commutative formal groups over fields of finite characteristic | doi=10.1070/RM1963v018n06ABEH001142 | mr=0157972 | year=1963 | journal=Akademiya Nauk SSSR I Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=18 | issue=6 | pages=3–90}} *{{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}} *{{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}}

Category:Algebraic geometry