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Let $TP$ be periodic topological cyclic homology. What is $\pi_* TP(\mathbb{Z}_p)$?

(i) I know that $\pi_* TP(\mathbb{F}_p) \cong \mathbb{Z}_p[v^{\pm 1}]$ with $v$ in degree $-2$ by IV.4.8 of Nikolaus-Scholze.

(ii) I know that $\pi_n THH(\mathbb{Z}_p)$ is $\mathbb{Z}_p$ for $n=0$, $\mathbb{Z}_p/m$ for $n=2m-1 \geq 0$, and zero otherwise. I tried to write down the Tate spectral sequence for $\pi_* TP(\mathbb{Z}_p)$ but it seems very complicated. I think Tsalidis and Bokstedt-Madsen calculated the homotopy groups of $TP(\mathbb{Z}_p)/p$ but I don't know how to get $\pi_* TP(\mathbb{Z}_p)$ from this.

(iii) A related question is what is the Breuil-Kisin twisted prismatic cohomology $\Delta_{\mathbb{Z}_p}\{i\}$?

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The calculation of $\pi_* TP(\mathbb{F}_p) = \pi_* THH(\mathbb{F}_p)^{tS^1} = \pi_* \widehat{\mathbb{H}}(S^1, THH(\mathbb{F}_p))$ (the notation has changed over the years) was first published by Hesselholt-Madsen (Topology, 1997). The calculation of $\pi_* TP(\mathbb{Z})/p$ for odd primes $p$ was published by B"okstedt-Madsen (conference proceedings, 1994 and 1995) and, independently, by Tsalidis (Amer J. Math, 1997). They also calculated $\pi_* TC(\mathbb{Z})/p$. Comparison with known spectra related to topological $K$-theory allowed a determination of $\pi_* TC(\mathbb{Z})_p$, see Rognes (Math. Proc. Camb. Philos Soc., 1993, Corollary 3). I also made the corresponding calculations of $\pi_* TP(\mathbb{Z})/2$, $\pi_* TC(\mathbb{Z})/2$ and $\pi_* TC(\mathbb{Z})_2$ (Journal of Pure and Applied Algebra, 1999). There is some discussion of the additive extensions in the Tate spectral sequence for $\pi_* TP(\mathbb{Z})_2$ in Theorem 1.9 on page 231 of one of those JPAA papers, but the general picture appears to be complicated. My former PhD student Knut Berg determined (ca. 2013) the continuous mod $2$ homology of $TP(\mathbb{Z})$ as an $A_*$-comodule algebra, where $A_*$ is the dual Steenrod algebra, as well as the Adams $E_2$-term and $d_2$-differentials. This gives some other information about $\pi_* TP(\mathbb{Z})_2$ than the Tate spectral sequence, but a complete picture of the later differential pattern is missing.

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  • $\begingroup$ I heard that new computations are done in Antieau–Krause–Nikolaus. They succeeded to compute the absolute prismatic cohomology in terms of Breuil–Kisin cohomology via Adams spectral sequence, and this should give computational information about $\operatorname{TP}(\mathcal O_K)$. $\endgroup$
    – Z. M
    Commented Jun 21, 2022 at 9:11

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