Semiadditive Power Operations in Chromatic Height $1$
Abstract
We recast the theory of power operations in chromatic height $1$ using higher semiadditivity. In particular, we show that if the height $1$ cyclotomic extension of a commutative algebra R is Galois, then power operations on R-algebras are exactly δ-ring operations. We also show that semiadditive power operations on chromatically localized algebraic K-theory agree with power operations defined using polynomial functoriality. As an application, we deduce that the map from the K-theory of a ℚ-linear symmetric monoidal category to its K(1)-localization is a δ-ring homomorphism on the level of underlying spaces. It is well known that operations on the K-theory of ℚ-linear symmetric monoidal categories are λ-ring operations, and since δ-rings are `p-typical' λ-rings, this is an instance of the redshift philosophy.
Related
- Some chromatic height computations for homotopy fixed points of iterated topological Hochschild homology
- WAMS Operations in modern power systems: A median expectation-based state estimation approach for grid resilience towards cyber attacks
- Motivic Cp-power operations in MGL
- Integrating Data Center Flexibility into Power System Operations: A Framework for Temporal Redispatch
- The Segal conjecture for large objects and semiadditive lifts
BibTeX
@misc{myers2026semiadditive,
title = {{Semiadditive Power Operations in Chromatic Height $1$}},
author = {Beckham Myers},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34992},
eprint = {2610.34992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34992},
}
RIS
TY - UNPB
TI - Semiadditive Power Operations in Chromatic Height $1$
AU - Beckham Myers
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34992
UR - https://arxiv.org/abs/2610.34992
AB - We recast the theory of power operations in chromatic height $1$ using higher semiadditivity. In particular, we show that if the height $1$ cyclotomic extension of a commutative algebra R is Galois, then power operations on R-algebras are exactly δ-ring operations. We also show that semiadditive power operations on chromatically localized algebraic K-theory agree with power operations defined using polynomial functoriality. As an application, we deduce that the map from the K-theory of a ℚ-linear symmetric monoidal category to its K(1)-localization is a δ-ring homomorphism on the level of underlying spaces. It is well known that operations on the K-theory of ℚ-linear symmetric monoidal categories are λ-ring operations, and since δ-rings are `p-typical' λ-rings, this is an instance of the redshift philosophy.
ER -