THH(Z) and the image of J

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 $\mathrm{THH}(\mathbb{Z}{)}_p^{\wedge }\simeq \mathrm{sh}({\mathrm{j}}_p^{\mathrm{triv}})$ and an equivalence of ${\mathbb{E}}_{\infty }$-rings $\mathrm{TP}(\mathbb{Z}{)}_p^{\wedge }\simeq {\mathrm{j}}_p^{\mathrm{t}{\mathrm{S}}^1}$. We also record a few applications of this: a new perspective, with some new information, on the description of $\mathrm{TC}(\mathbb{Z}{)}_p^{\wedge }$ 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.

arXiv commentv2: additional detail in §2.2, and other minor revisions