Definition of the Tate group associated with $p$-divisible groups

Question

Let $A$ be an abelian group, $p$ a prime number and $p_A:A \rightarrow A$ the multiplication by $p$. Let $A\left[p^n\right]$ be the kernel of $p_A$ composed $n$ times with itself. Then Lang, in his algebra p. 50, defines the Tate group $T_p(A)$ associated with the $p$-divisible group $A$ as the inverse limit of the inversely directed family $\left(A\left[p^{n+1}\right]\right)$.

As i understand, $n=0,1,...$ By definition, for $n=0$, the corresponding component of any element of $T_p(A)$ is zero and so for $n=1$ the component of any element of $T_p(A)$ must be inside $A\left[p\right]$. But substituting for $n=1$ at the defining formula for $T_p(A)$ yields $A\left[p^2\right]$.

Can anyone advise me about the correct interpretation of the definition? Thanks :-)

Answer 1

In the inverse family, the map $A[p^{n+1}]\to A[p^n]$ is $p_A$ which is not injective in general. The canonical projection $T_p(A)\to A[p^{n+1}]$ is not injective in general.

Consider the following example with $A=\mathbb Q/\mathbb Z$. Then $A[p^n]=(\frac{1}{p^n}\mathbb Z)/\mathbb Z$. You can use the isomorphisms $\mathbb Z/p^n\mathbb Z\to A[p^n]$, $\bar{a}\mapsto \overline{a/p^n}$, to establish an isomorphism from the inverse family $(A[p^n])_n$ to the usual inverse family $(\mathbb Z/p^n\mathbb Z)_n$. So $T_p(\mathbb Q/\mathbb Z)\simeq \mathbb Z_p$ the additive group of the $p$-adic integers.