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

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 $\varpi \in \mathcal{O}_L$.

Keywords

periodic topological cyclic homology, complex orientation

2010 Mathematics Subject Classification

11S70, 19D55, 55N22, 55Q51

Full Text (PDF format)

Received 16 December 2021

Received revised 6 May 2022

Accepted 6 May 2022

Published 26 April 2023