We prove that for any ring $𝐑$ of Krull dimension not greater than 1 and $n\ge 3$, the group ${\mathrm{E}}_n\left(𝐑[X,X^{-1}]\right)$ acts transitively on ${\mathrm{Um}}_n\left(𝐑[X,X^{-1}]\right)$. In particular, we obtain that for any ring $𝐑$ with Krull dimension not greater than 1, all finitely generated stably free modules over $𝐑[X,X^{-1}]$ are free. All the obtained results are proved constructively.

In this paper, we shall discuss possible theories of defining equivariant singular Bott-Chern classes and corresponding uniqueness property. By adding a natural axiomatic characterization to the usual ones of equivariant Bott-Chern secondary characteristic classes, we will see that the construction of Bismut’s equivariant Bott-Chern singular currents provides a unique way to define a theory of equivariant singular Bott-Chern classes. This generalizes J. I. Burgos Gil and R. Liţcanu’s discussion...