A chromatic vanishing result for TR

TL;DR

This paper proves a vanishing result for telescopically localized TR, impacting the understanding of certain spectra in algebraic K-theory and related areas, with implications for Morava K-theory and Thom spectra.

Contribution

It establishes a new vanishing theorem for T(k)-local TR on specific E₁-rings, connecting TR with the spectrum of curves on K-theory and recent categorical results.

Findings

T(k)-local TR vanishes on connective L_n^{p,f}-acyclic E₁-rings for 1 ≤ k ≤ n

Implications for connective Morava K-theory and Thom spectra y(n)

Utilizes the relationship between TR and the spectrum of curves on K-theory

Abstract

In this note, we establish a vanishing result for telescopically localized $\mathrm{TR}$. More precisely, we prove that $T(k)$-local $\mathrm{TR}$ vanishes on connective $L_n^{p,f}$-acyclic $\mathbf{E}_1$-rings for every $1 \leq k \leq n$ and deduce consequences for connective Morava K-theory and the Thom spectra $y(n)$. The proof relies on the relationship between $\mathrm{TR}$ and the spectrum of curves on K-theory together with fact that algebraic K-theory preserves infinite products of additive $\infty$-categories which was recently established by C\'{o}rdova Fedeli.