Preservation of quasi-isomorphisms of complexes
Let \(A\) be a commutative ring, \(X \rightarrow Y\) a quasi-isomorphism of (unbounded) complexes of \(A\)-modules, and \(Z\) another complex. The author gives some conditions implying that \(\Hom(Z,X) \rightarrow \Hom(Z,Y)\), \(\Hom(Y,Z) \rightarrow \Hom(X,Z)\) or \(Z\otimes X \rightarrow Z \otimes Y\) are quasi-isomorphisms, improving some results existing in literature. For instance, the following theorem is proved: let \(C_\lrcorner (A)\) (resp. \(C_\llcorner (A)\)) be the category of complexes of \(A\)-modules \((T_i,d_i)\) (with differential of degree \(-1\)) such that \(d_i=0\) for an infinite decreasing (resp. increasing) sequence of integers \(i\). If \(Z\in C_\lrcorner (A)\), \(X,Y\in C_\llcorner (A)\) and \(\mathrm{Ext}_A^{\geq 1}(Z_j,X_k)=0=\mathrm{Ext}_A^{\geq 1}(Z_j,Y_k)\) for all \(j,k\), then any quasi-isomorphism \(X \rightarrow Y\) induces a quasi-isomorphism \(\Hom(Z,X) \rightarrow \Hom(Z,Y)\). Some applications are given.
scientific article; zbMATH DE number 6143608
scientific article