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''Barsotti–Tate group''

A generalization of the concept of a commutative [[Formal group|formal group]] of finite height. The homomorphism induced by multiplication by the prime number  $  p $
is an epimorphism for a  $  p $-
divisible group.

Let  $  S $
be a [[Scheme|scheme]] and let  $  p $
be a prime number. A  $  p $-
divisible group of height  $  h $
is an inductive system  $  G = ( G _ {n} , i _ {n} ) $
of commutative finite group schemes  $  G _ {n} $
of order  $  p  ^ {nh} $(
cf. [[Group scheme|Group scheme]]) such that the sequences

$$ 
0  \rightarrow  G _ {n}  \rightarrow ^ { {i _ n} }  G _ {n+} 1  \rightarrow ^ { {\phi _ n} } \ 
G _ {n+} 1
$$

are exact (cf. [[Exact sequence|Exact sequence]]; here  $  \phi _ {n} $
is the homomorphism of multiplication by  $  p  ^ {n} $). 
A morphism of  $  p $-
divisible groups is a morphism of the inductive systems. A  $  p $-
divisible group is said to be connected (respectively, étale) if all  $  G _ {n} $
are connected (respectively, étale) group schemes. A connected  $  p $-
divisible group over a field of characteristic  $  p $
is a commutative formal group (considered as the inductive limit of the kernels of  $  \phi _ {n} $, 
i.e. of multiplication by  $  p  ^ {n} $) 
for which multiplication by  $  p $
is an [[Isogeny|isogeny]] [[#References|[6]]]. This fact is generalized to include the case of an arbitrary base scheme  $  S $
on which the homomorphism induced by multiplication by  $  p $
is locally nilpotent [[#References|[4]]]. The category of étale  $  p $-
divisible groups is equivalent to the category of  $  p $-
adic representations of the fundamental group of the scheme  $  S $. 
Each  $  p $-
divisible group  $  G $
over an Artinian scheme  $  S $
contains a maximal connected subgroup  $  G  ^ {0} $, 
which is known as the connected component of the identity, the quotient by which is an étale  $  p $-
divisible group. The dimension of the Lie algebra of any  $  {( G  ^ {0} ) } _ {n} $
is said to be the dimension of the  $  p $-
divisible group  $  G $.

Let  $  A $
be an [[Abelian variety|Abelian variety]] over the field  $  k $
of dimension  $  d $, 
let  $  A ( n) $
be the kernel of the homomorphism of multiplication by  $  p  ^ {n} $
in  $  A $
and let  $  i _ {n} :  A ( n) \rightarrow A ( n + 1 ) $
be a natural inclusion. The inductive system  $  A ( \infty ) = ( A ( n) , i _ {n} ) $
is a  $  p $-
divisible group of height  $  2d $. 
Its connected component of the identity  $  A ( \infty )  ^ {0} $
coincides with the formal completion of  $  A $
along the unit section, while the height of  $  A ( \infty )  ^ {0} $
equals  $  2   \mathop{\rm dim}  A $.

Let  $  G = ( G _ {n} , i _ {n} ) $
be a  $  p $-
divisible group of height  $  h $, 
let  $  {\widehat{G}  } _ {n} $
denote the Cartier dual finite group schemes and let  $  \widehat{i}  _ {n} :  \widehat{G}  _ {n} \rightarrow \widehat{G}  _ {n+} 1 $
be the mapping dual to the mapping of multiplication by  $  p :  G _ {n+} 1 \rightarrow G _ {n} $. 
The system  $  \widehat{G}  = ( \widehat{G}  _ {n} , \widehat{i}  _ {n} ) $
is then a  $  p $-
divisible group of height  $  h $
and is called the Serre dual to the  $  p $-
divisible group  $  G $. 
The sum of the dimensions of  $  G $
and  $  \widehat{G}  $
is equal to  $  h $.

As in formal groups, so, too, in  $  p $-
divisible groups, the concept of a [[Dieudonné module|Dieudonné module]], which plays an important role in the deformation theory of  $  p $-
divisible groups [[#References|[2]]], [[#References|[3]]], [[#References|[4]]] may be introduced.

If  $  S $
is the spectrum of an unequal-characteristic discrete valuation ring  $  A $
with residue field of characteristic  $  p $, 
the structure of a  $  p $-
divisible group is closely connected with the structure of the completion of the algebraic closure of the quotient field  $  K $
of  $  A $, 
considered as a module over the Galois group of the field  $  K $[[#References|[6]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Barsotti,   "Analytic methods for abelian varieties in positive characteristic" , ''Coll. Théorie des Groupes Algébriques (Bruxelles, 1962)'' , Gauthier-Villars  (1962)  pp. 77–85</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,   "Groupes de Barsotti–Tate et cristaux" , ''Proc. Internat. Congress Mathematicians (Nice, 1970)'' , '''1''' , Gauthier-Villars  (1971)  pp. 431–436</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Mazur,   W. Messing,   "Universal extensions and one-dimensional crystalline cohomology" , Springer  (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Messing,   "The crystals associated to Barsotti–Tate groups: with applications to Abelian schemes" , Springer  (1972)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.-P. Serre,   "Groupes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071030/p07103080.png" />-divisibles (d'après J. Tate)" , ''Sem. Bourbaki'' , '''318''' , Benjamin  (1968)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J.T. Tate,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071030/p07103081.png" />-divisible groups"  T.A. Springer (ed.)  et al. (ed.) , ''Proc. Conf. local fields (Driebergen, 1966)'' , Springer  (1967)  pp. 158–183</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Demazure,   "Lectures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071030/p07103082.png" />-divisible groups" , ''Lect. notes in math.'' , '''302''' , Springer  (1972)</TD></TR></table>