A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring
The author proves a characterization of the regular ones among the Noetherian local rings \(R\) by a surprisingly weak homological condition on the syzygy modules in a minimal free resolution of the residue class field \(k\) of \(R\): if there exists a surjective homomorphism from a direct sum of syzygy modules of \(k\) onto a module \(M\neq 0\) of finite projective dimension, then \(R\) is regular. There is also a dual theorem in which \(M\) is of finite injective dimension with non-zero socle, the surjection is replaced by an injection, and the syzygy modules are exchanged with their counterparts in a minimal injective resolution. The paper is dedicated to the memory of Maurice Auslander, and the author attributes a slightly more special case of the main result to him.
scientific article; zbMATH DE number 896562