Bockstein operations and AD algebras with unbounded torsion in $K_1$

Abstract

Eilers showed that for AD algebras of real rank zero with bounded torsion in $\mathrm{K}_1$, the coefficient transformations $κ$ are redundant in the classification by ordered scaled total $K$-theory. In this paper we treat the unbounded torsion case and prove that, in contrast, $κ$ becomes necessary. Specifically, we construct two non-isomorphic unital AD algebras of real rank zero, $E_0$ and $E_1$, such that their ordered scaled total $K$-theory invariants agree when the $κ$-maps are forgotten, i.e., \[ \bigl( \underline{\mathrm{K}}(E_0), \underline{\mathrm{K}}(E_0)_+, [1_{E_0}] \bigr)_{\underline{\mathrm{K}}_{\langleκ\rangle}} \cong \bigl( \underline{\mathrm{K}}(E_1), \underline{\mathrm{K}}(E_1)_+, [1_{E_1}] \bigr)_{\underline{\mathrm{K}}_{\langleκ\rangle}} \] but are not isomorphic under the full $Λ$-module structure: \[ \bigl( \underline{\mathrm{K}}(E_0), \underline{\mathrm{K}}(E_0)_+, [1_{E_0}] \bigr)_Λ \not\cong \bigl( \underline{\mathrm{K}}(E_1), \underline{\mathrm{K}}(E_1)_+, [1_{E_1}] \bigr)_Λ .\] This completes the picture for the necessity of all three operations $ρ$, $β$, and $κ$ in this context.