Existence of totally reflexive modules via Gorenstein homomorphisms (Q412568)

Defined by Auslander in 1967, totally reflexive modules were intensively studied. Their existence is a well-known fact since every projective module is trivially totally reflexive. It is interesting to study the case of modules which are non-trivial totally reflexive. Let \(R\) and \(S\) be two local rings, \(I\) be a Gorenstein ideal of \(S\) and \(S\rightarrow S/I\cong R\) a local ring homomorphism. The author shows that, if there is a particular lifting of \(R\) to a Gorenstein ring, then there are non-trivial totally acyclic complexes over \(R\). This result generalizes the one obtained by \textit{L. L. Avramov, V. N. Gasharov} and \textit{I. V. Peeva} [Publ. Math., Inst. Hautes Étud. Sci. 86, 67--114 (1997; Zbl 0918.13008)].