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Demazure, lectures on p-divisible groups, II.11, p-divisible formal groups

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Let kk be a field of prime characteristic p>0p\gt 0.

Definition

Definition

(pp-divisible group)

A commutative formal kk-group GG is called p-divisible formal k-group or Barsotti-Tate group if it satisfies the following properties:

(pdg1) pid GGp\cdot id_G\to G is an epimorphism.

(pdg2) GG is a pp-torsion group in that G= jker(p jid G)G=\cup_j ker(p^j \cdot id_G)

(pdg3) ker(pid G)ker(p\cdot id_G) is finite.

We have rk(kerpid G)=p hrk(ker \,p \cdot id_G)=p^h, hh\in \mathbb{N}. This hh is called the height ht(G)ht(G) of GG.

Definition

(alternative definition of pp-divisible group)

Let

G 1i 1G 2i 2G 3i 3G_1\stackrel{i_1}{\to}G_2\stackrel{i_2}{\to}G_3\stackrel{i_3}{\to}\cdots

be a codirected diagram of finite k-groups? such that

  1. rk(G j)=p hjrk(G_j)=p^{h j}, hh a fixed integer,

  2. all sequences 0G ji jG j+1p jG j+10\stackrel{}{\to}G_j\stackrel{i_j}{\to}G_{j+1}\stackrel{p^j}{\to}G^{j+1} are exact.

Then colim nG ncolim_n G_n is a pp-divisible group of height hh and ker(p nid G:GG)G nker(p^n id_G:G\to G)\simeq G_n.

Remark

If GG is a p-divisible group in the sense of the first definition, from (pdg1) follows rk(kerp jid G)0p jht(G)rk(ker p^j \cdot id_G)0p^{j\cdot ht (G)}. Since rkrk is multiplicative

0kerp jkerp j+kp jkerp k00\to ker \,p^j\hookrightarrow ker \,p^{j+k}\stackrel{p^j}{\to}ker\, p^k\to 0

is exact.

Definition

(Serre dual? of a pp-divisible group)

Let GG be a pp-divisible group GG. The Serre dual G G^\prime of GG is defined by: let G j:=ker(p jid G)G_j:=ker(p^j id_G) and let p j:G j+1G jp_j:G_{j+1}\to G_j is the map induced by pid Gp id_G. Then we define

G j :=D(G j)G_j^\prime:=D(G_j)
i j :=D(p j):G j G j+1 i_j^\prime:=D(p_j):G^\prime_j\to G^\prime_{j+1}
G :=colim j G j G^\prime:=colim_{j^\prime}G_j^\prime

This is a pp-divisible formal group with ht(G )=ht(G)ht(G^\prime)=ht(G) and we have p j =D(i j)p_j^\prime=D(i_j) and (G ) G(G^\prime)^\prime\simeq G.

Examples

Example

Let p\mathbb{Z}_p denote the ring of p-adic integers, let p\mathbb{Q}_p denote the field of p-adic numbers. The constant formal group ( p/ p) k(\mathbb{Q}_p /\mathbb{Z}_p)_k is a pp-divisible group of height 11.

Conversely any pp-divisible group of height hh is isomorphic to ( p/ p) k h(\mathbb{Q}_p /\mathbb{Z}_p)^h_k.

Example

Let AA be a commutative algebraic k-group, such that pid G:AAp id_G:A\to A is an epimorphism. Then

  1. ker(pid A)ker(p\cdot id_A) is finite.

  2. A(p):= jker(p jid A)A(p):=\cup_j ker(p^j id_A) is a pp-divisible group containing A^ = jker(F jG)\hat A^\circ=\cup_j ker(F^j G).

  3. If A=μ kA=\mu_k we have A(p)= jp jμ k=( p/ p) k A(p)=\cup_j p^j \mu_k=(\mathbb{Q}_p /\mathbb{Z}_p)^\prime_k.

  4. If AA is an abelian variety of dimension gg pid Ap id_A is an epimorphism with rk(herpid G)=p 2grk(her p id_G)=p^{2g} and consequently A(p)A(p) is a pp-divisible group of height 2g2g. This example is further described in chapter V, p-adic cohomology of abelian varieties?, particularly in V.3, structure of the p-divisible group A(p)?.

Proposition

Let GG be a kk-formal group. Then GG is pp-divisible iff the following conditions hold:

  1. π (G)(k¯)( p/ p) r\pi_\circ(G)(\overline k)\simeq (\mathbb{Q}_p /\mathbb{Z}_p)^r, rr finite.

  2. G G^\circ is of finite type, smooth, and ker(V:G (p)G )ker(V:G^{\circ (p)}\to G^\circ) is finite.

Example

Let AA be an algebraic unipotent kk-group, then A^ \hat A^\circ is never pp-divisible unless AA is finite.

Remark

Let GG be pp-divisible. Then we have height(G)=dim(G)+dim(G )height(G)=dim(G)+dim(G^\prime)

Proposition

Let GG be a connected, finite type, smooth formal group. There exist two subgroups HH,K\subseteq Gwith with His is pdivisible,-divisible, p^n K = 0forlarge for large n,, H\cap Kisfinite,and is finite, and G=H+ K$.

Revised on June 3, 2012 16:37:59 by Stephan Alexander Spahn (92.106.37.47)