Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

Complex orientations and $\mathrm{TP}$ of complete DVRs

Pages: 319 – 330

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a16


Gabriel Angelini-Knoll (Institut für Mathematik, Freie Universität Berlin, Germany)


Let $L$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$. We show that periodic topological cyclic homology of $\mathcal{O}_L$, over the base $\mathbb{E}_\infty$-ring $\mathbb{S}_{W(\mathbb{F}_q)} [z]$ carries a $p$-height one formal group law $\operatorname{mod}(p)$ that depends on an Eisenstein polynomial of $L$ over $\mathbb{Q}_p$ for a choice of uniformizer $\varpi \in \mathcal{O}_L$.


periodic topological cyclic homology, complex orientation

2010 Mathematics Subject Classification

11S70, 19D55, 55N22, 55Q51

Received 16 December 2021

Received revised 6 May 2022

Accepted 6 May 2022

Published 26 April 2023