THH(Z) and the image of J

2025/05/04 by Sanath K. Devalapurkar, Arpon Raksit, Devalapurkar, Sanath K. +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · #Sexual Differentiation and Disorders

paper · pdf · doi:10.48550/arxiv.2505.02218

Abstract

Let p be an odd prime number and j_p the p-complete connective image of J spectrum. We establish an equivalence of cyclotomic 𝔼_∞-rings THH(ℤ)^\wedge_p ≃ sh(j_p^triv) and an equivalence of 𝔼_∞-rings TP(ℤ)^\wedge_p ≃ j_p^tS1. We also record a few applications of this: a new perspective, with some new information, on the description of TC(ℤ)^\wedge_p as a spectrum; height 1 analogues of the fiber squares of Antieau-Mathew-Morrow-Nikolaus, resulting in new calculations in K(1)-localized algebraic K-theory; and a proof of a slight refinement of the noncommutative crystalline-de Rham comparison result of Petrov-Vologodsky.

Citations

• Notes on Tate cohomology
• Bökstedt periodicity and quotients of DVRs
• Nilpotence and descent in equivariant stable homotopy theory

Cited by

• Notes on Tate cohomology

Related

• K-theoretic Tate-Poitou duality and the fiber of the cyclotomic trace
• The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum
• Topological Hochschild homology of the image of j
• Tate-valued Characteristic Classes
• Topological Hochschild Homology of K/p as a K_p^\wedge module