What are the potential applications of perfectoid spaces to homotopy theory?

Question

This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about perfectoid spaces. Of course, a general answer is that homotopy theorists know by now to pay attention whenever number theorists get excited about something, but I'm looking for specific examples.

Here are a few examples I know of:

1. $THH$ and $TC$ are intimately related to the $A\Omega$ cohomology theory introduced by Bhatt-Morrow-Scholze. This is my favorite example.

2. Perfectoid spaces feature heavily in Scholze-Weinstein's paper on moduli of $p$-divisible groups, and homotopy theorists really like $p$-divisible groups. I don't know if there are specific ideas yet on how to apply their work in homotopy theory, though.

3. I've talked to some people interested in applying them to the study of Lubin-Tate space and the Gross-Hopkins period map.

4. There's some speculation that it could help with $TAF$.

I don't know much about any of these examples past the first, so please feel free to expand on these examples in addition to providing new ones. I'd also be happy to hear about applications of related things like $p$-adic Hodge theory in general or the pro-étale site, even if they don't explicitly mention perfectoid spaces.

Answer 1

Good question!

Actually, it seems unlikely that perfectoid methods per se play a key role in homotopy theory. The reason is that perfectoid things are "infinitely ramified", but there are theorems to the effect that many objects of interest in algebraic topology do not admit any ramified covers. For example, for a $K(n)$-local $E_\infty$-ring $A$, the ring $\pi_0 A$ can never be a ramified extension of $\mathbb Z_p$. For $n=1$, this follows from $\pi_0 A$ carrying a canonical structure of a $\delta$-ring.

On the other hand, it seems that the more recent prismatic ideas have a chance of being more directly of interest. One point of overlap is that both $K(1)$-local $E_\infty$-rings and prismatic things are very closely linked to $\delta$-rings. Another point of overlap is that computations in prismatic cohomology are often done in some form via analysis of Drinfeld's stack $\Sigma$, and some kind of descent. This can often be mimicked in algebraic topology by using relative $\mathrm{THH}$ and some kind of Adams spectral sequence. See for example the work of Liu--Wang computing $\mathrm{TC}$ of rings of integers of $p$-adic fields (reproving the results of Hesselholt--Madsen). Another application of prismatic cohomology is the work of Bhatt--Clausen--Mathew showing that $L_{K(1)}K(\mathbb Z/p^n\mathbb Z)$ vanishes (since reproved by Land--Mathew--Meier--Tamme by other means -- and better, as their result applies to all chromatic heights). The idea of using relative $\mathrm{THH}$ has also (in a slightly different context) been used by Hahn--Wilson to study redshift.

So this is all in the spirit of 1) in your question. By the way, here's a weird conjecture about the relation between prisms and algebraic topology. [Edit: This conjecture is wrong. See Jacob Lurie's comments below.] Recall that perfect prisms $(A,I)$ are equivalent to perfectoid rings $R=A/I$, and for such the $p$-completed $\mathrm{THH}$ defines a cyclotomic spectrum concentrated in even degrees with $\pi_\ast \mathrm{THH}(R;\mathbb Z_p) = I^{\ast/2}/I^{\ast/2+1}$. I conjecture that this process can be extended to all prisms, functorially defining a cyclotomic spectrum (concentrated in even degrees) with $\pi_\ast = I^{\ast/2}/I^{\ast/2+1}$ for any prism $(A,I)$. For example, for a Breuil--Kisin prism $(\mathfrak S=W(k)[[u]],I)$ with $\mathcal O_K=\mathfrak S/I$, this ought to give the relative $\mathrm{THH}(\mathcal O_K/\mathbb S[u];\mathbb Z_p)$.

Such a construction would induce some interesting arithmetic structures on any prism. For example, by taking $S^1$-fixed points, one should get an even $E_\infty$-ring with $\pi_0 = A$, and $\pi_2$ some invertible $A$-module; the latter should be the Breuil--Kisin twist $A\{1\}$. Moreover, this even $E_\infty$-ring gives a $1$-dimensional formal group over $A$. Such a natural $1$-dimensional formal group over any prism has actually been constructed by Drinfeld (but I still haven't fully understood its significance).

Regarding 2), 3) and 4), I don't really know. I think a key open question is the computation of the Picard group of the $K(n)$-local category, which is related to the $\mathcal O_D^\times$-equivariant line bundles on the Lubin--Tate space. It's not inconceivable that perfectoid methods can be helpful here!

Answer 2

Edit Sep 2023: the ideas below have now been developed more thoroughly in my new paper Prisms and Tambara functors I

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As Peter points out, it is more reasonable to look for connections between prisms and homotopy theory. I'll answer my own queston with some recent observations/speculations on the relation between prisms and equivariant homotopy theory, most of which are recorded in my paper A slice refinement of Bökstedt periodicity.

• Let $A=\mathbb Z[q]$, acted on by Adams operations $\psi^k(q)=q^k$, and write $B^\bullet$ for the multiplicative monoid underlying a semiring $B$. Then $n \mapsto ([n]_q, \psi^n)$ defines a monoid map $$\operatorname{End}({\mathbb T}) \ge \mathbb N^\bullet \to A^\bullet \rtimes \operatorname{End}_{\mathrm{Ring}}(A).$$ For an oriented prism $(A,d)$ we can do something similar with the submonoid $p^{\mathbb N}\le\mathbb N^\bullet$ by sending $p^k\mapsto (d\dotsm\phi^{k-1}(d), \phi^k)$.

• The prism condition $\delta(d)\in A^\times$ is equivalent to $FV=p$ (Corollary 3.28), which is a special case of the Mackey functor condition $\operatorname{res}(\operatorname{tr}(x)) = \sum_{g\in G/H} g\cdot x$.

• $q$-divided powers give lifts of the Norm map (Propositions 3.32 and 3.33), which is part of the Tambara functor structure of $\mathrm{THH}$. Conversely, the fact that $q$-divided powers descend to a map $W_1(R) \to W_2(R)$ is essentially equivalent to the "existence of higher $q$-divided powers" lemma (Lemma 16.7 of Prisms and prismatic cohomology)

• In view of the previous point, the fact that the Norm scales slice filtration is closely related to a lemma needed for the convergence of the $q$-logarithm (Remark 1.4 of my paper and Proposition 4.9 of The $p$-completed cyclotomic trace in degree 2)

So this suggests it might be possible to define a notion of $G$-prism for a compact Lie group $G$, recovering prisms for $G=\mathbb Q_p/\mathbb Z_p$, presumably related to Dress-Siebeneicher's $G$-Witt vectors, such that $\pi^G_0$ of a nice $G$-$E_\infty$-ring spectrum is a $G$-prism.