Suppose $R$ is a perfectoid ring in mixed characteristic, and $R'$ its characteristic-$p$ tilt. Scholze's results on tilting say that the étale theories over $R$ and $R'$ are equivalent in an almost sense (i.e. considering the category of extensions in the sense of "almost mathematics"). I want to know whether this is true of other rigid invariants. The question I'm interested in specifically is this: consider the categories $\operatorname{Mod}^a(R), \operatorname{Mod}^a(R')$ of almost modules over $R, R'$. Is it true that (making suitable definitions) the periodic cyclic homology of these two categories over $\mathbb{Z}$ is isomorphic, or related by a spectral sequence? 


More generally, I'm looking for interesting tilting statements visible on the level of comparing the *categories* $\operatorname{Mod}^a(R)$ and $\operatorname{Mod}^a(R')$ (possibly with monoidal structure).



periodic cyclic homology and tilting in the sense of Scholze
Of course, there cannot be a direct relation at the categorical level: After all, one category is $R$-linear while the other is $R^\flat$-linear (I write $R^\flat=R'$ for the tilt, as usual).


On the other hand, what is certainly true is that if $\pi^\flat\in R^\flat$ is some element with $\pi=(\pi^\flat)^\sharp\in R$ dividing $p$, then $R/\pi\cong R^\flat/\pi^\flat$, and the resulting categories of (almost) modules are equivalent.


Sometimes one can then use excision sequences to reduce questions on $K$-theory or (topological) cyclic homology to the tilt. For an example, see Lemma 1.3.7 in Hesselholt-Nikolaus' survey paper on [Topological cyclic homology](https://arxiv.org/abs/1905.08984), where they show that $K(\mathcal O\_C)\to K(C)$ is a $p$-adic equivalence by reducing to the same assertion for $K(\mathcal O\_C^\flat)\to K(C^\flat)$, by identifying both fibres with $K(\mathcal O\_C/\pi)\cong K(\mathcal O\_{C^\flat}/\pi^\flat)$. However, the individual terms here are very different: $K(C)$ is after profinite completion the same as $\mathrm{ku}$ (which is actually reproved there as Corollary 1.3.8), while $K(C^\flat)$ is after $p$-adic completion the Eilenberg-MacLane spectrum $H\mathbb Z\_p$. These $p$-adic completions of $K$-theory are in all cases equal to topological cyclic homology here.


I would expect the passage from modules to almost modules to not change the picture much, so I would expect invariants of $\mathrm{Mod}^a(R)$ and $\mathrm{Mod}^a(R^\flat)$ to be different. What seems more reasonable is to compare relative variants, like those associated to perfect complexes of $R$-modules supported on the special fibre, or a corresponding almost variant. However, as far as I know, nothing general of this sort is known.


