A chromatic vanishing result for TR

Abstract

In this note, we establish a vanishing result for telescopically localized topological restriction homology TR. More precisely, we prove that T ( k ) T(k) -local TR vanishes on connective L n p , f L_n^{p,f} -acyclic E 1 \mathbb {E}_1 -rings for every 1 ≤ k ≤ n 1 \leq k \leq n and deduce consequences for connective Morava K-theory and the Thom spectra y ( n ) y(n) . The proof relies on the relationship between 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 Córdova Fedeli [Topological Hochschild homology of adic rings, Ph.D. thesis, University of Copenhagen, 2023].