Syzygies of associated graded modules

Source abstract

Given a finitely generated module M over a Noetherian local ring R, we give a characterization for the first syzygy of the associated graded module Gm(M) to be equigenerated. As an application of this, we identify a complex of free Gm(R)-modules arising from given free resolution of M over R, which is a resolution of Gm(M) if and only if Gm(M) is a pure Gm(R)-module. We also give several applications of the purity of Gm(M). Our results demonstrate that while not all algebraic properties of a module carry over to its associated graded module, the purity of the minimal free resolution of Gm(M) ensures that several important invariants are inherited. In addition, we provide sufficient conditions for Cohen–Macaulayness and purity of Gm(M), and we provide a local version of the Herzog–Kühl equations.