Tate-Barsotti group

Me] W. Messing, The crystals associated to Barsotti-Tate groups with applications to abelian schemes. Lecture Notes in Math. 264, Springer-Verlag, 1972. 

IL] L. Illusie, Deformations de groupes de Barsotti-Tate. In Seminaire sur les pinceaux arithmetiques la conjecture de Mordell (ed. L. Szpiro), Asterisque 127, Societe mathematique de France, Paris (1985). 

Gr] A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonne. Seminaire de Mathematiques Superieur 45, Presses de PUniversite de Montreal (1974). 

Definition 1.3. A truncated Barsotti-Tate group of level n on S is a finite flat group scheme G over 5 such that ... 

We will denote by BT(n)s the category of truncated Barsotti-Tate groups of level n on S. Remarks 1.4. 

The kernel of pk on a truncated Barsotti-Tate group of level n with n > k is a truncated level k Barsotti-Tate group (see [Me] II 3.3.11). 

The dual of a truncated Barsotti-Tate group is a truncated Barsotti-Tate group of the same level. 

The scheme X is of finite type over. It is even smooth over n+i because by a theorem of Grothendieck there are no obstructions to deforming a truncated Barsotti-Tate group (see [IL] Theorem 4.4). 

Since the morphism Xo Spec(Fp) is of finite type, the completions of the local rings of Xo at its closed points are Noetherian complete local rings with perfect residue fields. Over these rings we can find a Barsotti-Tate group H with H pk] Go restricted to these rings (see [IL] Theorem 4.4). Hence it follows from Proposition 3.4 that Mx(Go) is locally free over Ox/pkOx of rank h — height(G) at all the closed points of X. ... 

From [IL] Theorem 4.4 we get a truncated Barsotti-Tate group H of level 2 over Sq such that H p] = G. The set of isomorphism classes of all such groups is a homogeneous space under the group ExtJ (G, G ). The set of all isomorphism classes of all M in Af(2)s with M — M is a homogeneous space under M). Hence we can alter our choice of H... 

J2i is an injection still. The multiplicative part of H is zero and it is a truncated Barsotti-Tate group since HM and H are truncated Barsotti-Tate groups (see [BBM] 3.3.9). 

From [IL] Theorem 4.4 we get a truncated Barsotti-Tate group H on S 0 such that i H G. The commutative diagram... 

A p-divisible group scheme or Barsotti-Tate group of corank h is a family of finite abelian group schemes G, of order pf" together with maps i G -> G.+1 such that... 

Barsotti-Tate groups arise in "nature" when one considers the sequence of kernels of multiplication by successive powers of p on an abelian variety. Also, as Grothendieck has observed, there are Barsotti-Tate groups which are naturally associated with the crystalline cohomology of a proper smooth scheme which is defined over a perfect field of characteristic p. Since we do not discuss crystalline cohomology no further mention is made of this example. 

In this paper Barsotti-Tate groups over an arbitrary base scheme S (on which p is locally nilpotent) are studied. Intuitively we can think of these as nicely varying families of Barsotti-Tate groups parametrized by S. The basic theorems are those which allow us to associate various crystals to such groups. Actually in order to construct these crystals we assume that the Barsotti-Tate groups in question are, locally, liftable to infinitesimal neighborhoods. This restriction is surely unnecessary. 

In Chapter I the concept of a Barsotti-Tate group over S is defined, several sorites and several examples are given. This chapter consists... 

Chapter IV is technically the heart of the work. It is here that we construct the crystals alluded to above. The first section constructs for any Barsotti-Tate group G on S a "universal" extension of G by a vector group ... 

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