Topological structure on higher dimensional local fields

Question

Let $F$ be a $n$-dimensional local field. If $n=0$ or $1$, the topological structure on $F$ was well-known, however if $n>1$ i.e, $F$ is a higher dimensional local field, I don't know something nice topological structure on $F$. Matthew Morrow introduced so-called "higher topology" on the higher dimensional local fields in his survey https://arxiv.org/abs/1204.0586, but this "higher topology" does not provide the structure as topological field with $F$. Indeed he described that any fixed element $\alpha\in F$, multiplication $$\alpha \times\colon F\longrightarrow F ~;~ \beta \longmapsto \alpha\beta$$ are all continuous map in higher topology. Unfortunately, this property is weaker than the definition of topological ring. So we want some topological structures as that addition and multiplication $$ +,\times\colon F\times F\longrightarrow F $$ are continuous, and compatible with that of residue fields. I.e, In these topologies, for the ring of integer $\mathscr{O}_{F}$ of $F$ with relative topology, the canonical surjection $\mathscr{O}_{F}\longrightarrow F_{n-1}$ should be continuous and open morphism, where $F_{n-1}$ is the residue field of $F$ equipped with this topology.

I think that the attempt to give $F$ to such a topology have so for been unsuccessful up on now. I know Fesenko, Parshin and Camara are challenging this experiments. but these does not also seem to work.

Question. Are there exist the topological structures on higher dimensional local fields satisfying some properties as above? or do you know about some related research?

Answer 1

It cannot be done. Alexei Parshin has proven a concrete No-Go result:

There is no topology on a 2-local field such that simultaneously

• it is a topological ring (i.e. addition and multiplication are continuous)
• if you restrict the topology to the top ring of integers $\mathcal{O}$, and then under the quotient map $\mathcal{O}\twoheadrightarrow \mathcal{O}/\mathfrak{m}$ the quotient space topology agrees with the usual topology of the 1-local first residue field.

And this stays true (of course) for n-local fields for any n>=2.

So, it's simply impossible to have all these things.

There are several different approaches to work around this:

• you can work with sequential topological spaces (but note that limits (or colimits... I forgot which...sorry) in this category are incompatible to when you form them in plain topological spaces, so this does not give you genuine topological rings; you only get ring objects in sequential spaces)
• you can work in a version of topological algebra, where you only demand continuity in each factor individually (check out Yekutieli's semi-topological algebra)

• you can work in iterated ind-pro categories (also known as n-Tate categories). This is the approach proposed by Kato himself in the "Existence theorem for higher local fields" paper.

  These approaches are each a little different, and might work well to varying extents. Feel free to add your own approach. Maybe using Clausen--Scholze condensed mathematics should be thrown on this problem.... I don't know. Just a shot in the dark.

  A more detailed survey of all the above approaches (also explaining Parshin's No-Go argument) is provided in https://arxiv.org/pdf/1510.05597.pdf in Section 1.