| 释义 |
- Definition
- Example
- References
In algebraic geometry, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety in characteristic p. They were introduced by {{harvs|txt||last=Barsotti|authorlink=Iacopo Barsotti|year=1962}} under the name equidimensional hyperdomain and by {{harvs|txt|last=Tate|authorlink=John Tate|year=1967}} under the name p-divisible groups, and named Barsotti–Tate groups by {{harvtxt|Grothendieck|1971}}. Definition{{harvtxt|Tate|1967}} defined a p-divisible group of height h (over a scheme S) to be an inductive system of groups Gn for n≥0, such that Gn is a finite group scheme over S of order pn and such that Gn is (identified with) the group of elements of order pn in Gn+1.More generally, {{harvtxt|Grothendieck|1971}} defined a Barsotti–Tate group G over a scheme S to be an fppf sheaf of commutative groups over S that is p-divisible, p-torsion, such that the points G(1) of order p of G are (represented by) a finite locally free scheme. The group G(1) has rank ph for some locally constant function h on S, called the rank or height of the group G. The subgroup G(n) of points of order pn is a scheme of rank pnh, and G is the direct limit of these subgroups. Example- Take Gn to be the cyclic group of order pn (or rather the group scheme corresponding to it). This is a p-divisible group of height 1.
- Take Gn to be the group scheme pnth roots of 1. This is a p-divisible group of height 1.
- Take Gn to be the subgroup scheme of elements of order pn of an abelian variety. This is a p-divisible group of height 2d where d is the dimension of the Abelian variety.
References- {{Citation | last1=Barsotti | first1=Iacopo | title=Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) | publisher=Librairie Universitaire, Louvain | mr=0155827 | year=1962 | chapter=Analytical methods for abelian varieties in positive characteristic | pages=77–85}}
- {{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Lectures on p-divisible groups | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-06092-5 | doi=10.1007/BFb0060741 | mr=0344261 | year=1972 | volume=302}}
- {{eom|id=p/p071030|title=P-divisible group|first=I.V.|last= Dolgachev}}
- {{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Actes du Congrès International des Mathématiciens (Nice, 1970) | url=http://mathunion.org/ICM/ICM1970.1/ | publisher=Gauthier-Villars | mr=0578496 | year=1971 | volume=1 | chapter=Groupes de Barsotti-Tate et cristaux | pages=431–436}}
- {{Citation | last1=de Jong | first1=A. J. | series=Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998) | url=http://mathunion.org/ICM/ICM1998.2/ | mr=1648076 | year=1998 | journal=Documenta Mathematica | issn=1431-0635 | volume=II | title=Barsotti-Tate groups and crystals | pages=259–265}}
- {{Citation | last1=Messing | first1=William | author1-link=William Messing | title=The crystals associated to Barsotti-Tate groups: with applications to abelian schemes | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0058301 | mr=0347836 | year=1972 | volume=264}}
- {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Séminaire Bourbaki | origyear=1966 | publisher=Société Mathématique de France | location=Paris | mr=1610452 | year=1995 | volume=10 | chapter=Groupes p-divisibles (d'après J. Tate), Exp. 318 | chapterurl=http://www.numdam.org/item?id=SB_1966-1968__10__73_0 | pages=73–86}}
- {{Citation | last1=Tate | first1=John T. | editor1-last=Springer | editor1-first=Tonny A. | title=Proc. Conf. Local Fields( Driebergen, 1966) | publisher=Springer-Verlag | location=Berlin, New York | mr=0231827 | year=1967 | chapter=p-divisible groups.}}
{{DEFAULTSORT:Barsotti-Tate group}} 1 : Algebraic groups |