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Topological Cyclic Homology of Rings of Integers in p-adic Fields via Syntomic Cohomology

Author: Wei Yang · Added: July 26, 2026

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Article date: July 2026.

Abstract

Let $K/\mathbb{Q}_p$ be a finite extension, with valuation ring $\mathcal{O}_K$. Taking the calculation of the BMS syntomic complexes $\mathbb{Z}_p(n)(\mathcal{O}_K)$ from the companion note as arithmetic input, we compute

from the Bhatt—Morrow—Scholze motivic filtration. We prove degreewise strong convergence, identify the two-step filtration in every positive even stem with the Moore coefficient sequence, and resolve all hidden additive extensions. For odd $p$ the Moore spectrum has exponent $p$, so the extensions split.

At $p=2$, the relation $2\operatorname{id}_{\mathbb{S}/2}=i\eta q$ reduces the problem to the integral product of the Hopf class with the unique two-torsion class in an odd stem. The image of $\eta$ is $-1\in\widehat{\mathcal{O}_K^\times}_{\,2}$, and the obstruction is the quaternion symbol $\{-1,-1\}_K$. Thus $\pi_{2m}\mathrm{TC}(\mathcal{O}_K;\mathbb{F}_2)\cong\mathbb{Z}/4$ exactly when both $m$ and $[K:\mathbb{Q}_2]$ are odd. No relative-THH descent calculation is used.