On the chain equivalence of projective chain complexes (Q1933300)

This article concerns equivalences between perfect complexes over an Invariant Basis Number ring \(R\). They are viewed in \({{\mathsf C}^b}({\mathcal{P}_R})\), the category of bounded chain complexes of finitely generated projective \(R\)-modules whose morphisms are degree preserving chain maps. Two complexes \({\mathbf P}\) and \({\mathbf Q}\) in \({{\mathsf C}^b}({\mathcal{P}_R})\) are quasi-isomorphic if there exists a chain map between them which induces an isomorphism in homology. It is known that quasi-isomorphic complexes are homotopy equivalent. The author studies the existence of a stronger form of equivalence between such complexes. In this article, the complexes \({\mathbf P}\) and \({\mathbf Q}\) are said to be stably equivalent if there exist acyclic complexes of free \(R\)-modules \({\mathbf F}\) and \({\mathbf G}\) so that the complexes \({\mathbf P} \oplus {\mathbf F}\) and \({\mathbf Q} \oplus {\mathbf G}\) are isomorphic in \({{\mathsf C}^b}({\mathcal{P}_R})\). The main result of this paper states that two quasi-isomorphic complexes \({\mathbf P}\) and \({\mathbf Q}\) are stably equivalent if and only if there exist free modules \(F_i\) and \(G_i\) and an isomorphism \(P_i \oplus F_i \cong Q_i \oplus G_i\) in each degree \(i\).