THH(Z) and the image of J

TL;DR

This paper establishes deep equivalences between topological Hochschild homology and the image of J spectrum at odd primes, providing new insights into algebraic K-theory, topological cyclic homology, and crystalline-de Rham comparisons.

Contribution

It proves new equivalences of cyclotomic spectra related to THH and the image of J, offering fresh perspectives and calculations in algebraic K-theory and topological cyclic homology.

Findings

Equivalence of THH(Z) and the trivialized j_p spectrum at p

New calculations in K(1)-localized algebraic K-theory

Refinement of the noncommutative crystalline-de Rham comparison

Abstract

Let $p$ be an odd prime number and $\mathrm{j}_p$ the $p$-complete connective image of J spectrum. We establish an equivalence of cyclotomic $\mathbb{E}_\infty$-rings $\mathrm{THH}(\mathbb{Z})^{\wedge}_p \simeq \mathrm{sh}(\mathrm{j}_p^{\mathrm{triv}})$ and an equivalence of $\mathbb{E}_\infty$-rings $\mathrm{TP}(\mathbb{Z})^{\wedge}_p \simeq \mathrm{j}_p^{\mathrm{t}\mathrm{S}^1}$. We also record a few applications of this: a new perspective, with some new information, on the description of $\mathrm{TC}(\mathbb{Z})^{\wedge}_p$ as a spectrum; height $1$ analogues of the fiber squares of Antieau-Mathew-Morrow-Nikolaus, resulting in new calculations in $\mathrm{K}(1)$-localized algebraic K-theory; and a proof of a slight refinement of the noncommutative crystalline-de Rham comparison result of Petrov-Vologodsky.