THH(Z) and the image of J
Sanath K. Devalapurkar, Arpon Raksit
TL;DR
This work establishes a height-1 description of THH and TC for the integers at odd primes by relating THH(bZ)^∧_p to the shifted trivial cyclotomic spectrum on j_p and TP(bZ)^∧_p to the S^1-Tate fixed points j_p^{tS^1}. It further refines the understanding of TC(bZ)^∧_p, giving a canonical decomposition involving j_p, its shift, and a KU_p- or ell_p- summand, and derives a height-1 analogue of Antieau–Mathew–Morrow–Nikolaus-type results for K(1)-local TC and K-theory. The paper also proves a noncommutative crystalline–de Rham comparison at p, showing TP(C ⊗ F_p)^∧_p ≃ HP(C/Z_p)^∧_p for dualizable Z_p-linear categories, with Z_p^{tS^1}-linearity, and discusses a refinement that fails for p=2. In addition, it outlines a prismatization program connecting THH(Z)^∧_p to arithmetic stacks via prismatization and the even filtration, and sketches future work on stacks and motivic filtrations at all primes. These results provide new computational tools for TC(Z), K(1)-local K-theory, and noncommutative p-adic comparisons, with implications for higher chromatic phenomena and arithmetic geometry.
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.