periodic cyclic homology and tilting in the sense of Scholze 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).
 A: 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, 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.
A: As a complement to Peter's answer, let me try to address the precise question of whether the periodic cyclic homologies of the almost categories are equivalent.  The answer is "no".  First, what is the definition of the periodic cyclic homology of the almost category?  This is not obvious, but Efimov has shown the way (though perhaps his work post-dates your question?). Let me start by recalling his theory, which he calls "continuous K-theory".  For more details of what I'm writing you can see Marc Hoyois' nice exposition at http://www.mathematik.ur.de/hoyois/papers/efimov.pdf .
First, any invariant of (idempotent-complete $\mathbb{Z}$-linear, but let me not carry these adjectives around) small stable $\infty$-categories, for example periodic cyclic homology, can equivalently be viewed as an invariant of compactly generated stable $\infty$-categories, by taking compact objects.  If we want to take care of functoriality we should posit that we only consider those functors between compactly generated $\infty$-categories which preserve the compact objects; equivalently, and this will be better for what's coming next, those functors whose right adjoints also preserve colimits.
Now, there is a substantial broadening of the notion of compactly generated table $\infty$-category due to Lurie, the notion of a dualizable stable $\infty$-category.  Every dualizable stable $\infty$-category $\mathcal{C}$ is the kernel of a localization functor between compactly generated stable $\infty$-categories, where again the right adjoint to the localization functor is required to preserve all colimits; and actually this is a precise characterization of the dualizable stable $\infty$-categories.
This leads to Efimov's fantastic result (previsaged to some extent by Tamme's work on excision, see https://arxiv.org/abs/1703.03331): if $E$ is a localizing invariant, then there is a unique functorial extension of $E$ to a localizing invariant of dualizable stable $\infty$-categories.  This tells you what $E(\mathcal{C})$ has to be, if $\mathcal{C}$ is presented as the kernel of a localization functor between compactly generated guys as above: just the fiber of $E$ applied to that localization functor.
The almost category provides a good example, because it more or less explicitly arises as the kernel of such a localization functor, namely $\operatorname{Mod}(\mathcal{O}_C)^a$ is the kernel of the base-change functor $\operatorname{Mod}(\mathcal{O}_C)\rightarrow \operatorname{Mod}(k)$ where $k$ is the residue field.  The weird property of the residue field that makes this base-change functor a localization is that the derived tensor product of $k$ with itself over $\mathcal{O}_C$ is $k$ again.  This is what makes almost mathematics run.
Summing up, what this shows is that the perodic cyclic homology of the almost category is basically the same as the periodic cyclic homology of the usual category of modules.  They only differ by the periodic cyclic homology of the residue field, which is anyway the same for $\mathcal{O}_C$ and its tilt.
You also ask if something weaker could be true, that these invariants of $R$ and its tilt are "related by a spectral sequence".  I guess there's a bit a freedom in interpreting what this means, but I suppose with appropriate definitions and interpretations the answer should be "yes".  More precisely, if $\pi$ and $\pi^\flat$ are as in Peter's answer, then the associated gradeds for the $\pi$-adic and $\pi^\flat$ adic filtrations are isomorphic (provided, let's say, that $\pi$ and $\pi^\flat$ are non-zerodivisors).  If we agree to replace periodic cyclic homology with its "continuous" variant (now not in the sense of Efimov but in the sense of the inverse limit of the periodic cyclic homologies of the quotients by the stages in the $\pi$-adic filtration... though this is also conjecturally a special case of the Efimov construction, applied to the so-called nuclear modules which Peter and I have defined), then it's easy to imagine that there should be a spectral sequence converging from the invariant for the associated graded to the invariant for the ring, so that indeed the two things may well be related by two spectral sequences with the same $E_2$-page.
