topology on the ring of Witt vectors in the theory of period rings of Fontaine For a $p$-adic field $K$ with perfect residue field $k$, we know the standard construction of the ring $R$. I will recall it briefly. It is $\varprojlim_{x \rightsquigarrow x^p} O_{C_K}/pO_{C_K}$, with component-wise addition. It turns out that $R$ is a perfect ring of characteristic $p$ that is also a complete (non-discrete) valuation ring, and let us denote the valuation map by $v_R$. The residue field of $R$ is the algebraic closure $\overline{k}$ of the residue field $k$ of $K$. We can form the ring $W(R)$ of Witt vectors of $R$, and we can form the ring $W(\textrm{fof}(R))$ of Witt vectors of the field of fractions $\textrm{fof}(R)$ of $R$. It is a known fact that $\textrm{fof}(R)$ is algebraically closed. As $\textrm{fof}(R)$ is a perfect field of positive characteristic, $W(\textrm{fof}(R))$ is a d.v.r. with $p$ as a generator for the maximal ideal. But $R$ is not a field, so $W(R)$ is not a d.v.r., but is complete with respect to the topology generated by powers of $p$. But as the construction of $W$ is functorial, there is an inclusion map $W(R) \hookrightarrow W(\textrm{fof}(R))$.
1) What is the topology on $W(R)$? Is it the subspace topology from $W(\textrm{fof}(R))$? I guess not, as the $p$-adic valuation on $W(\textrm{fof}(R))$ "forgets" $v_R$, that is to say every nonzero element in $\textrm{fof}(R)$ and therefore every nonzero element of $R$ becomes a unit in $W(\textrm{fof}(R))$, but not necessarily in $W(R)$.
2)Let $\varepsilon$ be the element of $R$ given by the sequence $(\zeta_{p^n}-1 \mod p)$. Consider the Teichmuller lift $[\varepsilon] \in W(R)$ and set $\pi_\varepsilon := [\varepsilon]-1$. Now the claim is that power series in $\pi_\varepsilon$ with, say, integer coefficients converge in $W(R)$. (This is an unchecked claim in Fontaine's book, on page 79). Since $W(R)$ is not a valuation ring, I thought of showing that $\pi_{\varepsilon}$, or at least a power of it is in $pW(R)$. We observe that $\pi_{\varepsilon}=(\varepsilon-1, \ldots)$ as an element in $W(R)$, but an element of $pW(R)$ has the first coordinate $0$, which in this case, $\varepsilon-1$ is not. What am I missing? Actually I can show that $\pi_{\varepsilon} \in pW(R)$ by making use of the map $\theta: W(R)\to O_{C_K}$, but since this is all so confusing, I would like to know where I went wrong above.
 A: There are (at least) two different topologies on all those rings, and that is the source of your confusion.
Both on $W(R)$ and $W(fof(R))$ one has


*

*the $p$-adic topology, in which e.g. a basis of neighbourhoods of $0$ is given by $(p^n W(R))_n = V_n(W(R))$ resp. $(p^n W(fof(R)))_n = V_n(W(fof(R)))$ where $V_n$ is the Verschiebung on Witt vectors, this all just meaning that the first $n$ components of the respective Witt vectors are $0$;

*the "weak" topology ("topologie faible" in some French sources, e.g. in (Cherbonnier-)Colmez' early papers). This is given by endowing the Witt vectors, which as a set are just $R^{\Bbb N}$ resp $fof(R)^{\Bbb N}$ after all, with the product topology, where in each component one takes the valuation topology of $R$ resp. $fof(R)$.
A basis of neighbourhoods of $0$ is thus given by sets in which "the first finitely many Witt vector coordinates are small w.r.t. $v_R$".
Things to notice:


*

*The p-adic topology can also be described as product topology, if on each component one takes the discrete topology instead of the valuation topology on $R$ resp. $fof(R)$.

*Both the $p$-adic and the weak topology on $W(R)$ agree with the subspace topologies of their respective versions on $W(fof(R))$.

*On the small subset $W(\bar k)$ however, both (!) topologies just induce the plain old $p$-adic topology as subspace topology.

*As regards the second part of your question 1) though, some elements being units in $W(fof(R))$ but not in $W(R)$ has not much to do with any topology. Also, you seem to imply elements of $R$ to be elements of $W(R)$ which makes no sense (maybe you mean their Teichmüller representatives?) Anyway, quite generally for a commutative ring $A$ of characteristic $p$ with a unique maximal ideal $\mathfrak{m}$, the Witt vectors $W(A)$ are a local ring with maximal ideal $m_{W(A)} = \lbrace (x_0, x_1, ...) \in W(A) : x_0 \in \mathfrak{m}\rbrace$. In particular, the units in $W(R)$ are only those elements whose zero-th component satisfies $v_R(x_0) =0$, whereas in the bigger ring $W(fof(R))$, the units are all elements whose zero-th component is $\neq 0$ (more elements have inverses, as it happens with ring extensions).

*All rings under consideration are topological rings w.r.t. to all topologies mentioned, and further, as such are complete.

*For an element $x = (x_0, x_1, ...) \in W(fof(R))$ we have
$$v_R(x_0) > 0 \Leftrightarrow  \lim_{n\to \infty} x^n = 0 \text{ w.r.t. the weak topology}$$
Now $\pi_\epsilon$ is an $x$ like in no. 6 (its $0$-th component is $\epsilon -1$ whose $R$-valuation, if I recall correctly, is $v_R(\epsilon-1) =p/(p-1)$). Then the "unchecked claim" that power series in $\pi_\epsilon$ (with e.g. bounded coefficients $\in W(\bar k)$) indeed do converge (w.r.t. the weak topology) can be shown from nos. 3, 5 and 6 (and maybe the fact that $v_R$ induces an ultrametric).
Note that it is not true that $\pi_\epsilon \in pW(R)$ (as said, its image in $W(R)/pW(R) \simeq R$ is $\epsilon-1$), and whatever proof you believe to have for that must be wrong. Accordingly, w.r.t. the $p$-adic topology, the powers of $\pi_\epsilon$ do not even converge to $0$.
(I wrote my diploma thesis about all this. A detailed discussion and proofs for all the items above are in sections 3.1 and 3.2, alas, it's in German. (I called the weak topology "N-Topologie" there.))
