Isomorphism of complexes and lifts (Q2717172)

This paper deals with the problem of lifting an isomorphism of complexes NEWLINE\[NEWLINE\varphi:{\mathbf F}\otimes R/H\to{\mathbf C}\otimes R/HNEWLINE\]NEWLINE to an isomorphism \(\varphi:{\mathbf F}\to{\mathbf C}\), where \((R,M)\) is a commutative, noetherian local ring, \({\mathbf F}\) is an exact complex of free \(R\)-modules and \({\mathbf C}\) is a complex of free \(R\)-modules.NEWLINENEWLINENEWLINEThe authors produce an ideal \(K\) with the following property: There is a \(t\) such that any isomorphism \(\varphi:{\mathbf F} \otimes R/K^t\to {\mathbf C}\otimes R/K^t\) can be lifted to an isomorphism \(\widetilde \varphi:{\mathbf F}\to {\mathbf C}\), which coincides with \(\varphi\) modulo some ideal \(H\subset\sqrt K\). When the length of \(F\) is one, \(K\) is also the largest possible ideal with that property.NEWLINENEWLINENEWLINEThe authors also examine the more general situation of \(\sigma\)-homomorphisms, where \(\sigma\) varies in a subgroup \({\mathcal G}\) of the automorphism group of \(R\); more particularly, they consider the case \(R=k[[X_1, \dots, X_n]]\), \({\mathcal G}=\) group of automorphisms of \(R\), which is particularly useful in the context of isomorphisms of singularities.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].