Topology on $p$-adic period rings in an article by Fontaine, part II

Question

This is a follow-up to this question. See that question for background and relevant notation.

In the answer to that question, it is claimed, if I understand the answer correctly, that a basis of neighborhoods of $W(R) \otimes_{W(k)} K = W(R)[1/p]$ is given by sets of the forms $p^{-n}W(\mathfrak{a})+p^NW(R)$ for $\mathfrak{a}$ a non-zero ideal of $R.$ On the other hand, if I consider the following notes, pg. 65, Exercise 4.5, it is claimed that a basis of neighborhoods of $W(R)[1/p]$ is given by sets of the form $$U_{N,\mathfrak{a}} = \bigcup_{n > -N} p^{-n}W(\mathfrak{a}^{p^n})+p^NW(R), N \geq 0,$$ where $\mathfrak{a}$ is a non-zero ideal.

Is it true that the two topologies thus defined are the same? If so, why is this the case? It is easy to see that each $U_{N,A}$ contains a basis element of the form $p^{-n}W(A)+p^NW(R),$ but the other inclusion is not clear to me (is it even true?).

Answer 1

Let me just make a small comment that while the question may seem subtle from the perspective of topological rings, from the perspective of condensed rings everything really is "obvious". The reduction $\mathcal{O}_{\mathbb{C}_p}/p$ is discrete, so there's no question there. Then you just place that discrete ring in the condensed world, and do all the constructions in there. So form the inverse limit over Frobenius $\mathcal{O}_{\mathbb{C}_p}^\flat$, pass to Witt vectors, then invert $p$. This gives a condensed ring: there's nothing to check. On $T$-valued points for a profinite set $T$, it's the analogous period ring construction for the perfectoid algebra $Cont(T;\mathbb{C}_p)$ replacing $\mathbb{C}_p$. From this perspective, the condensed or topological structure on period rings is just a shadow of their structure as a pro-etale sheaf on perfectoid spaces, as exploited in Scholze's old (?!) article on $p$-adic Hodge theory of rigid analytic spaces.

Now, there still remains the question of comparing this condensed structure to the topological structure in Fontaine's definition. Recall that there is a fully faithful functor, $X \mapsto \underline{X}$ with $\underline{X}(T)= Cont(T,X)$, from compactly generated weak Hausdorff topological spaces to condensed sets; this passes to rings as well. The Witt vectors of $\mathcal{O}_{\mathbb{C}_p}^\flat$ is a countable inverse limit of discrete rings, so it is a sequential Hausdorff space and hence compactly generated. Now it suffices to note that multiplication by $p$ map is a closed inclusion (the quotient is $\mathcal{O}_{\mathbb{C}_p}^\flat$ as the latter is perfect), and that the fully faithful functor $X \mapsto \underline{X}$ above sends sequential unions along closed inclusions to sequential unions, by the standard fact in point-set topology that if a compact Hausdorff space is a sequential union of closed subpsaces, then it is equal to one of those closed subspaces.

However, probably in general the topological structure is not so important once you have the condensed structure. So one should probably just be happy with the first paragraph of this answer.

Answer 2

It seems to me that they are different (I had the same question, and was searching online to see if I messed up somewhere).

To see that they are not the same note that for any given $U_{N,\mathfrak{a}}$ and for any $m\gg 0$, the element $p^{-m}[x]$ is a nonzero element of $U_{N,\mathfrak{a}}$ for any nonzero $x\in \mathfrak{a}^{p^m}$. On the other hand, $p^{-n} W(\mathfrak{a})+p^NW(R)\subset p^{-n}W(R)$ does not have an element of this form when $m>n$. Thus this latter set cannot contain a set of the form $U_{N,\mathfrak{a}}$.

In fact, I cannot even figure out why $W(R)[\frac 1p]$ is a topological ring in the Brinon-Conrad topology... In the linked exercise, it is claimed that $U_{N,\mathfrak{a}}\cdot U_{N,\mathfrak{a}}\subset U_{N,\mathfrak{a}}$. But I don't think this is true. Let $\varpi\in A$ be a pseudouniformizer. Letting $N=0$ and $\mathfrak{a}=(\varpi)$, I believe that \begin{equation} p^{-n}W(\mathfrak{a}^{p^n})+p^NW(R) \end{equation} is the set of all elements that can be written as \begin{equation} p^{-n}[x_{n}]+p^{-n+1}[x_{n-1}]+\dots + p^{-1}[x_{1}]+a \end{equation} where $a\in W(R)$, and $x_{i}\in (\varpi^{p^i})$ for $i=n,\dots,1$.

If this is the case, then for any $\ell>0$, the element $p^{-\ell}[\varpi^{p^\ell}]$ is in $U_{N,\mathfrak{a}}$. But $p^{-\ell}[\varpi^{p^\ell}]\cdot p^{-\ell}[\varpi^{p^\ell}] = p^{-2\ell}[\varpi^{2p^\ell}]$. This is not of the form $p^{-2\ell}[x_{2\ell}]$ for some $x_{2\ell}\in (\varpi^{p^{2\ell}})$ since $2p^\ell<p^{2\ell}$ for all large $\ell$. So I do not think the claim is true.

(In a similar manner, I don't think the product of any two opens $U_{N,\mathfrak{a}}\cdot U_{M,\mathfrak{b}}$ is contained in $U_{0,(\varpi)}$: for any fixed $\alpha,\beta>0$, we have $(\alpha +\beta) p^\ell<p^{2\ell}$ when $\ell\gg 0$.)

Edit: Now I am also a bit worried that $p^{-n}W(\mathfrak{a})+p^NW(R)$ is not a basis for the inductive limit topology. If this were the case, then since multiplication by $p$ should be a homeomorphism on $W(R)[\frac 1p]$, we can multiply this set by say $p^{2n}$ to get an open subset $p^nW(\mathfrak a)+p^{N+2n}W(R)$. But if this were open in $W(R)[\frac 1p]$, then the inclusion $W(R)\subset W(R)[\frac 1p]$ would not be continuous. Open sets of $W(R)$ contain some $W(\mathfrak a)+p^NW(R)$ which always contain some nonzero Teichmuller element $[x]$ where $x\in \mathfrak a$ is nonzero, while the above set does not contain nonzero Teichmuller elements for $n>0$.

I think that the inductive limit topology should have a neighborhood basis of $0$ given by sets of the form \begin{equation} \bigcup_{n>-N} p^{-n}W(\mathfrak a_n)+p^NW(R) \end{equation} where the $\mathfrak a_n\subset R$ are open ideals. This differs from the Brinon-Conrad topology because the ideals $\mathfrak{a}_n$ can go to $0$ as slowly as necessary to get around the issue raised above.