AI 中文总结
该研究针对K₁中具无界挠的AD代数，证明有序标度总K理论分类中系数变换κ必要，构造了两个不变量在忽略κ时一致但含完整Λ模结构时不同构的代数，完善了相关运算必要性的结论。
AI 中文摘要
Eilers证明，对于实秩零且K₁中具有有界挠的AD代数，在有序标度总K理论分类中，系数变换κ是冗余的。本文处理无界挠情形，证明相反，κ变得必要。具体而言，我们构造两个非同构的实秩零单位AD代数E₀和E₁，当忽略κ映射时，它们的有序标度总K理论不变量一致，即(underline{K}(E₀), underline{K}(E₀)₊, [1_{E₀}])_{underline{K}_{⟨κ⟩}} ≅ (underline{K}(E₁), underline{K}(E₁)₊, [1_{E₁}])_{underline{K}_{⟨κ⟩}}；但在完整Λ模结构下不同构，即(underline{K}(E₀), underline{K}(E₀)₊, [1_{E₀}])_{Λ} ≇ (underline{K}(E₁), underline{K}(E₁)₊, [1_{E₁}])_{Λ}。这完善了该语境下ρ、β、κ三种运算必要性的整体结论。
英文摘要
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.
