Note

Localizing Motives and Selmer K-Theory

Author: Wei Yang · Added: September 25, 2026

Article date: September 25, 2026.

Abstract

We construct a quotient of localizing motives by requiring the $\mathbb{A}^1$ -localized unit to be $KU$ -local. Mapping from the unit defines an invariant of small stable categories with a canonical map to Selmer $K$ -theory. We compute the fiber of this map and prove that it is an equivalence on connective ring spectra and on $\mathbb{Q}$ -linear stable categories, as well as after $p$ -completion on $\mathbb{Z}[1/p]$ -linear stable categories. For every prime $p$ , the comparison fails on some small $\mathbb{F}_p$ -linear stable category, which can be realized by a single nonconnective algebra. The obstruction is the noncompactness of a representing motive, together with the continuity of $\operatorname{TC}/p$ . At chromatic heights at least two, the comparison is the localized cyclotomic trace; the resulting calculation for $KU_p^\wedge$ distinguishes the two invariants even after $p$ -completion.