topology on the ring of Witt vectors in the theory of period rings of Fontaine

Question

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.

Answer 1

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:

1. 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. $fof(R)$.
2. Both the $p$-adic and the weak topology on $W(R)$ agree with the subspace topologies of their respective versions on $W(fof(R))$.
3. On the small subset $W(\bar k)$ however, both (!) topologies just induce the plain old $p$-adic topology as subspace topology.
4. 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).
5. All rings under consideration are topological rings w.r.t. to all topologies mentioned, and further, as such are complete.
6. 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.))