A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring Original Research Article

Let R be a commutative noetherian local ring with residue field k. We introduce two new invariants of R-modules and compute them for the module k. As a consequence, we show, when R is nonregular, that no direct sum of the syzygy modules of k surjects onto a nonzero module of finite projective dimension. A dual statement is also true: no direct sum of the cosyzygy modules of k (in an injective resolution) contains a nonzero module of finite injective dimension with nonzero scale.