Free module over PID and algebraic structure inheritance

Question

Let $O_K$ be a complete discrete valuation ring and $\pi$ is its uniformizer.(This should have been a valuation ring associated to a completed discrete valued field $K$.) Consider $E/K$ finite separable extension. Then $O_E$(integral closure of $O_K$ in $E$) is a f.g. free module of rank $[E:K]$.

The book says "$O_E$ is a free module of rank $[E:K]$. Then $O_E\cong\varprojlim\frac{O_E}{\pi^nO_E}$"

$\textbf{Q:}$ If $O_E$ is free module over $O_K$, then I can pass the inverse limit through its components as tensor commutes with direct sum. I need finiteness condition as direct sum is direct limit which does not commute with inverse limit in genera. So it become completion against $O_K$ and completion of $O_K$ is $O_K$. So I have $O_E\cong\varprojlim\frac{O_E}{\pi^nO_E}$ isomorphic as $O_K$ module. How do I inherit algebraic structure? Note that I have provided a $O_K$ module map. I need a ring map now.

This is related to pg 106 of Taylor, Frohlich Algebraic Number Theory's algebraic proof of uniqueness of absolute value.

My current guess is that first $O_E\cong O_E\otimes_{O_K}O_K$. Since $O_K$ is complete, I know $O_E\otimes_{O_K}O_K\cong O_E\otimes_{O_K}\hat{O_K}$. So $O_E\otimes_{O_K}\hat{O_K}\cong\varprojlim\frac{O_E}{p^nO_E}$. Then inherit ring structre from $O_E\otimes_{O_K}\hat{O_K}$ side.

Answer 1

Turning my comment into a slightly longer answer:

You want to show that there is an isomorphism of rings $O_E\overset{(*)}\to \varprojlim\frac{O_E}{\pi^nO_E}$. While you have a good idea for how to attack this problem, your proposed solution contains a lot of mysterious unspecified isomorphisms, such as $O_E\cong O_K^{[E:K]}$. There are several problems with this:

• The isomorphism $O_E\to O_K^{[E:K]}$ is a module map, as you've noticed - and it's definitely not a ring map in general. (Consider e.g. the module map $\varphi: \mathbb{Z}[i] \to \mathbb{Z}^2$ sending $a+bi$ to $(a,b)$: then $\varphi(i)^2 \neq \varphi(i^2)$.) So, by taking this step, you've already broken the ring structure. You're not going to be able to conclude from this that the ring structure is preserved under your eventual map $(*)$.
• The isomorphism $$O_E\overset{(\dagger)}{\to} O_K^{[E:K]}$$ is not even unique. Likewise, in a later comment, you suggested using isomorphisms $$\left(\frac{O_K}{\pi^n}\right)^{[E:K]}\to \frac{O_K}{\pi^n}\otimes O_K^{[E:K]}\overset{(\ddagger)}{\to}\frac{O_K}{\pi^n}\otimes O_E$$, but this is of course also not unique. In fact, if you pick these isomorphisms at random, you're probably not going to end up with a ring map! The point is that these two isomorphisms need to be related: the ring structure that is broken by the map $(\dagger)$ must be repaired by $(\ddagger)$.

The solution to both of these problems is to write down your isomorphisms explicitly, as functions. (Choose an integral basis of $E/K$, say $x_1, \dots, x_n$, where $n = [E:K]$...) Then, at the end, you can compose them all together and get an exact formula for what your eventual map $(*)$ does to an element of $O_E$. And if you've set it up correctly, and chosen $(\dagger)$ and $(\ddagger)$ correctly, then it will be obvious that $(*)$ is a ring map.