Chromatic Purity of Dualizable Categories
Abstract
We develop the chromatic theory of dualizable stable categories, with chromatic purity for continuous K-theory at its center. We generalize the chromatic purity theorem for algebraic K-theory to continuous K-theory of dualizable stable categories and reformulate it as the purity of H-unital rings. Using categorical completion theory, we construct chromatic fracture squares and obtain a refinement of chromatic purity: for every dualizable stable category 𝒞, the T(n)⊕ T(n−1)-completion map induces an equivalence KT(n)cont(𝒞)xrightarrow≃KT(n)cont(NucT(n)⊕ T(n−1)(𝒞)). We also establish chromatic descent for continuous K-theory of dualizable homotopy fixed points and prove a categorical nuclear refinement of descent. This framework allows us to lift redshift bounds, Tate vanishing, and blueshift from spectra to dualizable stable categories. We show that the T(n)-completion of a rigid symmetric monoidal stable category is equivalent to the dualizable limit of module categories over a tower of type n generalized Moore spectra. As applications, we prove that both nuclear solid and nuclear gaseous module categories satisfy chromatic redshift.
Related
- The coanalyticity of the reduced K-theory on modules over free tensor algebras of dualizable t-categories
- Arithmetic purity of strong approximation for toric varieties with constant global sections
- The linear positive torsor presentation and arithmetic purity of strong approximation
- Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties
- Arithmetic purity of strong approximation for complete intersections of large dimension
BibTeX
@misc{jin2026chromatic,
title = {{Chromatic Purity of Dualizable Categories}},
author = {Yifan Jin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26497},
eprint = {2609.26497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26497},
}
RIS
TY - UNPB
TI - Chromatic Purity of Dualizable Categories
AU - Yifan Jin
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26497
UR - https://arxiv.org/abs/2609.26497
AB - We develop the chromatic theory of dualizable stable categories, with chromatic purity for continuous K-theory at its center. We generalize the chromatic purity theorem for algebraic K-theory to continuous K-theory of dualizable stable categories and reformulate it as the purity of H-unital rings. Using categorical completion theory, we construct chromatic fracture squares and obtain a refinement of chromatic purity: for every dualizable stable category 𝒞, the T(n)⊕ T(n−1)-completion map induces an equivalence K_T(n)^cont(𝒞)xrightarrow≃K_T(n)^cont(Nuc_T(n)⊕ T(n−1)(𝒞)). We also establish chromatic descent for continuous K-theory of dualizable homotopy fixed points and prove a categorical nuclear refinement of descent. This framework allows us to lift redshift bounds, Tate vanishing, and blueshift from spectra to dualizable stable categories. We show that the T(n)-completion of a rigid symmetric monoidal stable category is equivalent to the dualizable limit of module categories over a tower of type n generalized Moore spectra. As applications, we prove that both nuclear solid and nuclear gaseous module categories satisfy chromatic redshift.
ER -