Topologies and Completions In my tutorial today we looked at the following question in Atiyah-McDonald:
Let $\alpha_n : \mathbb{Z}_p \to \mathbb{Z}_{p^n}$ be the injection of abelian groups given by $\alpha_n(1) = p^{n-1}$, and let $\alpha : A \to B$ be the direct sum of the $\alpha_n$. Show that the $p$-adic completion of $A$ is just $A$ but the completion of $A$ for the topology induced from the $p$-adic topology on $B$ is the direct product of $\mathbb{Z}_p$. Deduce that $p$-adic completion is not a right-exact functor on the category of all $\mathbb{Z}$-modules. 
The start of the solution that was given has confused me. The tutor started with: Since $pA =0$, the sequence $A/p^nA$ is $$\cdots A \xrightarrow{ \ \ \text{id} \ \ } A \xrightarrow{ \ \ \text{id} \ \ } A,$$ the coherent sequences of which are constant sequences $(a)$, yielding the obvious isomorphism $$\lim_{\longleftarrow} A/p^nA \cong A.$$
I don't understand any component of this part of the proof. How do we obtain such sequences, how do we then determine the coherent sequences and then how is the isomorphism obtained?
 A: As $A:=\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$ it's easy to see that $pA=p\cdot \mathbb{Z}/p\mathbb{Z}=0.$ So now let's consider the inverse system $$\cdots \to A/p^{n+1}A\to A/p^{n}A\to A/p^{n-1}A\to \cdots. $$ What's going on at each step? The ideal $p^{n+1}A$ is a sub-ideal of $p^{n}A$, and the map $A/p^{n+1}A\to A/p^{n}A$ takes an element $[m]_{\text{mod } p^{n+1}}\in A/p^{n+1}A$ to the element $[m]_{\text{mod } p^{n}}\in A/p^{n}A$. However, we can simplify this via some isomorphisms using the fact that $pA=0$. If $pA=0$ then as $p^{n}A\subseteq pA$ we must have $p^{n}A=0.$ So for all $n\in \mathbb{N}$ we have $A/p^n A\cong A$ and the isomorphism $A/p^n A \to A$ is $[m]_{\text{mod } p^{n}}\mapsto [m]_{\text{mod } p}.$ The maps $A/p^{n+1}A\to A/p^{n}A$ are then just $\text{id}_A\colon A\to A$. So that's the first part: we're reduced to the inverse system $$\cdots A \xrightarrow{ \ \ \text{id} \ \ } A \xrightarrow{ \ \ \text{id} \ \ } A.$$
Now we need to compute the coherent sequences to work out what the inverse limit $$\lim_{\longleftarrow} A/p^nA$$ really is. Recall that a coherent sequence $(a_n)$ has the properties $a_n\in A/p^n A$ and $a_n$ is the image of $a_{n+1}$ under the given maps. If we take some element $[m]_{\text{mod }p}\in A=\mathbb{Z}/p\mathbb{Z}$ as our element $a_0$, then we're forced to choose $a_n=\cdots = a_1=a_0=[m]_{\text{mod }p}$ for each $n$ as the maps in our inverse system are just the identity on $A$. In particular, given any coherent sequence $(a_n)$ it is determined by the element $a_0$. Thus, $$\lim_{\longleftarrow} A/p^nA\cong A$$ via the isomorphism $(a_n)\mapsto a_o$.
