Ring of Witt vectors and Fontaine's deRham period ring

The construction I am interested in is the following:

Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with algebraically closed residue field $k$. Let $K=Frac(V)$ and fix an algebraic closure $\overline{K}$ of $K$, and $\overline{V}$ the integral closure of $V$ in $\overline{K}$. Define $R_{\overline{V}}$ to be the perfection $\lim_{\leftarrow \, Frob} \overline{V}/p\overline{V}$; take the Witt ring $W(R_{\overline{V}})$ one gets a surjective homomorphism $$\theta: W(R_{\overline{V}}) \rightarrow \widehat{\overline{V}}$$ by $\theta([x])= \lim_{\rightarrow \infty} \widetilde{x_m}^{p^m}$ where $x=(x_n)_{n\in \mathbb{N}} \in R_{\overline{V}}$ $\widetilde{x_m}$ a Lifting of $\overline{V}/p\overline{V}$ to $\overline{V}$ and $[\cdot]$ the Teichmüller representative. As is well known the kernel of $\theta$ is generated by a single element $\xi$. From here I am interested in understanding $A_N(\overline{V}):= W(R_{\overline{V}})/\xi^N W(R_{\overline{V}})$.

Now to my question: I want to see what comes out of this construction when I put in $W(k)=V$. At first I thought that it was quite simple because the projection on the first factor $R_{W(k)} \overset{\sim}{\rightarrow} k$ is an isomorphism which yields the isomoprhism $W(R_{W(k)}) \overset{\sim}{\rightarrow} W(k)$ but this does not account for the integral closure being taken in $\overline{K}=\overline{Frac(W(k))}$. So my question boils down to: what is the integral closure $\overline{W(k)}$ of $W(k)$ in $\overline{K}$ and how does above construction work for $\overline{W(k)}$?

• The construction depends only on $\bar K$, so $V$ and $W(k)\subseteq V$ give the same result as their integral closures in $\bar K$ are the same (they both consist of all elements of non-negative valuation). – Piotr Achinger Mar 25 '18 at 12:04
• @PiotrAchinger Thank you very much. I know that the construction only depends on $\overline{K}$, what I dont know is that $V$ $W(k)$ have the same integral closure in $\overline{K}$. I am sure this is very basic but my knowldege on Witt rings comes mainly (solely) from Serre's "Local Fields" which is probably not enough at this point. Could you give me a refeference for a result like this? – Konstantin Mar 25 '18 at 12:47
• This is just transitivity of integral closure: you have $W(k)\subseteq K\subseteq \bar K$ and $V$ is the integral closure of $W(k)$ in $K$. – Piotr Achinger Mar 25 '18 at 13:40
• In any case Serre's "Local Fields" should be sufficient. – Piotr Achinger Mar 25 '18 at 13:41
• @PiotrAchinger I think we are misunderstanding each other: I would like to find out what the above machinery gives me when I put in $W(k)=V$ then $K=Frac(W(k))$ etc. So I am actually looking for the integral closure of $W(k)$ in $\overline{Frac(W(k))}$ not $\overline{Frac(V)}$ – Konstantin Mar 25 '18 at 14:08