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I have been looking Bhatt-Morrow-Scholze's paper:

https://arxiv.org/pdf/1802.03261.pdf

and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this as linear over the free $E_\infty$ ring generated by the commutative monoid $\mathbb{N}$, $\mathbb{S}[\mathbb{N}]$ (as B-M-S do in Section 11 of their paper). Do the homotopy groups of relative topological periodic cyclic homology $\pi_* TP(C/\mathbb{S}[\mathbb{N}])$ carry something like a flat connection over $\mathbb{Z}_p[[z]]$?

I couldn't find anything like this in the paper, but one might naively hope this is the case by analogy with the Getzler connection on the usual (relative) cyclic homology as well as the analogy between topological cyclic homology and crystalline cohomology.

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  • $\begingroup$ I am not familiar with Getzler's (or Kaledin's) Gauß–Manin connection, but the case for $\operatorname{THH}$ can be reduced to $\operatorname{HH}$. The point is that there is a map $\mathbb Z\to\operatorname{THH}(\mathbb F_p)$ of $\mathbb T$-equivariant commutative ring spectra, which gives rise to a map $\operatorname{HH}(\mathbb Z[t]/\mathbb Z)\to\operatorname{THH}(\mathbb F_p[t])$ of $\mathbb T$-equivariant commutative ring spectra. (PS: this argument does not work in mixed char.) I would write more details when I have access to a computer, if there is still no answer then. $\endgroup$
    – Z. M
    Commented Jan 25 at 10:03
  • $\begingroup$ @Z.M It would be great to hear more when you have a chance! $\endgroup$
    – Daniel Pomerleano
    Commented Jan 25 at 22:35
  • $\begingroup$ In the meanwhile, you could glimpse at my another answer which explains how to identify the $\mathbb T$-equivariant spectrum $\operatorname{THH}(\mathbb F_p[t])$ (and I gave references there). $\endgroup$
    – Z. M
    Commented Jan 26 at 5:12

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