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Fix a $p$-adic field $K$ with perfect residue field $k.$ Let $\mathcal{C}$ be the completion of the algebraic closure of $K,$ and let $$R = \varprojlim \mathcal{C}/p,$$ where the transition maps in the inverse limit are given by the Frobenius. One can then show that $R$ is a perfect valuation ring (of characteristic $p.$) We can then form $W(R),$ the ring of Witt vectors of $R.$

There is a natural topology on $W(R),$ making it into a topological ring, a basis of neighborhoods is given by $p^N W(R)+ W(I)$ for $N\geq 0$ and $I$ a non-zero ideal of $R.$

In this article, pg. 536, Fontaine uses the topology I just defined to define a topology on $W(R)[1/p] = K \otimes_{W(k)} W(R),$ by using what he calls the "tensor product topology". He then claims that the topology one get from the tensor product topology is the "same" (up to identifications under isomorphisms) as the one obtained by taking the topology coming from the inductive limit $$ \cdots \rightarrow W(R) \rightarrow W(R) \rightarrow \cdots$$ where transfer maps are multiplication by $p.$

My questions are the following:
1) What is precisely this tensor product topology? Naively, I would say that it is the topology on $K \otimes_{W(k)} W(R) \cong W(R)[1/p]$ where a basis of neighborhoods are given by $$(p^N W(R)+ W(I)) \otimes_{W(k)} K + W(R) \otimes_{W(k)} p^n \mathcal{O}_K$$ (where $\mathcal {O}_K$ is the valuation ring of $K).$ What makes me think this can not be a basis of neighborhoods comes from the fact that it seems to me (maybe erroneously) that $p^NW(R) \otimes_{W(k)} K \cong W(R)[1/p].$ Thus, it seems to me that Fontaine must have some other sort of topology in mind for this tensor product, or I am making a silly mistake. For example, what is a basis of neighborhoods for the tensor product topology? Is it part of a more general construction?
2. Why does the topology of the tensor product and inductive limit coincide?

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Inductive limit topologies are always nasty to describe: Basically you can't do better than their very definition. Namely, a subset $U\subset W(R)[1/p]$ is an open neighborhood of $0$ if and only if for all $n$ the intersection $U\cap p^{-n} W(R)$ is open. In particular, this means that it must contain $p^N W(R)$ for some $N$, but also $p^{-n} W(\mathfrak a_n)$ for some open ideal $\mathfrak a_n\subset R$. So a basis of open neighborhoods of $0$ is given by the subsets

$$p^N W(R) + \sum_{n\geq 0} p^{-n} W(\mathfrak a_n)$$

for varying $N\geq 0$ and open ideals $\mathfrak a_n\subset R$, $n\geq 0$. Note that such subsets are never contained in $p^{-n} W(R)$ for any $n$...

As Dustin notes in the other question, passing to the condensed world is helpful here, as then inductive limits are completely naive (they are just inductive limits on $S$-valued points for any profinite $S$), and in fact there is no question about which condensed structure to put on $W(R)$ or $W(R)[1/p]$, and it is clear that $W(R)\otimes_{W(k)} K = W(R)[1/p]$ as condensed rings. (Also, as everything lies in the essential image of the fully faithful functor from compactly generated topological spaces to condensed sets, there is no loss of information in passing to the condensed world.)

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  • $\begingroup$ Dear Peter, Thank you for this fascinating reply! Just three questions (maybe I should post these as separate question?) : 1. Can one construct all the period rings, such as B_{dr}, B_{st}, using the condensed perspective? $\endgroup$
    – Dedalus
    Jun 27 at 7:22
  • $\begingroup$ 2. In Fontaine’s theory you have the notion of a representation being cristalline / semi-stable / de Rham. Basically, if I recall correctly, a representation V of the Galois group G_k of a p-adic number field is semi-stable if the natural map (B_{st} \otimes_{Q_p} V)^{G_k} \to B_{st} \otimes_{Q_p} V is an isomorphism. If one just follows one’s nose, I would say that this should translate directly to the condensed world. Is this true? Namely, one embeds G_k as a condensed group, and one just does all the constructions there. $\endgroup$
    – Dedalus
    Jun 27 at 7:23
  • $\begingroup$ 3. Is there a source for this more modern perspective, or is it all forthcoming? I would love to be able to ignore topology :) $\endgroup$
    – Dedalus
    Jun 27 at 7:23
  • $\begingroup$ You're welcome! 1) Yes; just take their algebraic definition and do it internally in condensed rings. 2) Yes, everything translates directly. 3) I don't think anything has to be written, as nothing deep happens. I think you should be able to translate yourself, maybe referring to the lecture notes on my webpage for the basics. (But note that in my first p-adic Hodge theory paper, in hindsight I'm already discussing period rings from the condensed perspective, so maybe that's a little helpful.) $\endgroup$ Jun 27 at 19:17
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What you are missing is that it is using the topology of K as a W(k) module (not as a topological field), so basis for K is like $p^{-n}W(k)$ for $n\in\mathbb{Z}$. So the basis of W(R)[1/p] is rather consisted of things like $p^{-n}$-scaled $p^{N}W(R)+W(I)$. Topology of inductive limit coincides with tensor product because inductive limit and tensor product commute and the inductive limit mentioned above is just W(R) tensored with the inductive limit W(k)->W(k)->... which is precisely expressing the basis of neighborhoods of 0 of K we chose, $p^{-n}W(k)$.

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  • $\begingroup$ Why is $K$ a free $W(k)$-module? If $K = \mathbb{Q}_p,$ is this really true? $\endgroup$
    – Dedalus
    Sep 5 '19 at 11:29
  • $\begingroup$ Also, maybe I am being dense, but even if say the basis is like $p^{-n}W(k),$ does this really effect the argument that says that $(p^N W(R) +W(I)) \otimes_{W(k)} K \cong W(R)[1/p]?$ This seems to be only a statement about the topology on $W(R).$ $\endgroup$
    – Dedalus
    Sep 5 '19 at 11:54
  • $\begingroup$ (The argument for $(p^NW(R) +W(I)) \otimes_{W(k)} K \cong W(R)[1/p]$ being that $K$ contains $p^{-N},$ so its imag in $W(R)[1/p]$ must be everything. ) $\endgroup$
    – Dedalus
    Sep 5 '19 at 11:56
  • $\begingroup$ (Small correction to the previous comment. Of course the argument for the isomorphism should be that $K$ contains $p^{-n}$ for all $n \geq 0.$ $\endgroup$
    – Dedalus
    Sep 5 '19 at 12:17
  • $\begingroup$ @Dedalus I put that word without much care. Thanks $\endgroup$
    – GTA
    Sep 5 '19 at 14:34
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I think a good reference is Schneider's book Nonarchimedean Funtional Analysis, but my answer is simply an expanded version of GTA's comments.

In Chapter IV, §14, A and B you find a description of two possible "tensor product topologies" with which you can endow the tensor product of two locally convex vector spaces: in your setting, $W(R)$ is only a $W(k)$-module, but this makes almost no difference if you follow the proofs. The projective limit topology has as basis of neighborhoods the products $p^{-n}W(k)\otimes p^NW(R)\otimes W(I)$ by definition, because $K$ has, as a basis of neighborhoods, the $p^{-n}W(k)$. That the two coincide is Proposition 17.6, which holds in your case because your spaces are complete (for $W(R)$ this is again on page 536 of the paper by Fontaine); you need to use the definition of "locally convex final topology" given in Schneider's book, Chapter I, § 5, E to realise that the inductive limit topology given by multiplication by $p$ is the inductive tensor product topology.

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  • $\begingroup$ Thank you for the reference. I posted a follow-up mathoverflow.net/questions/339967/… where I tried to compare two topologies which should be ”the same”, but I am not sure what to make of this. If you had the time, I would be most grateful if you looked into it since this has been bugging me for some time. $\endgroup$
    – Dedalus
    Sep 6 '19 at 10:12
  • $\begingroup$ Also, do you mean $p^{-n}W(k),$ or do you mean $W(k)[1/p^n]$ above? If the first, I do not see how the $U_{N, \mathfrak{a}}$ that Brinon and Conrad (see the link in the question I linked to) define can be contained in a product of neighborhoods as above... $\endgroup$
    – Dedalus
    Sep 6 '19 at 10:34
  • $\begingroup$ I have corrected the $[1/p^n]$ into $p^{-n}$, you are right. As for your other question, I've seen it but for the time being I don't really have an answer. $\endgroup$ Sep 6 '19 at 14:03
  • $\begingroup$ Thanks nonetheless. It seems to me as if the two topologies are not the same, and yet, Fontaine uses the "inductive limit topology " for constructing $B_{DR},$ while Brinon-Conrad uses the topology in the question I linked. So probably I am making a very trivial mistake, but these things seem to be quite subtle. $\endgroup$
    – Dedalus
    Sep 6 '19 at 15:16

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