Syzygies of associated graded modules

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 $G_{\mathfrak{m}}(M)$ to be equigenerated. As an application of this, we identify a complex of free $G_{\mathfrak{m}}(R)$-modules, arising from given free resolution of $M$ over $R$, which is a resolution of $G_{\mathfrak{m}}(M)$ if and only if $G_{\mathfrak{m}}(M)$ is a pure $G_{\mathfrak{m}}(R)$-module. We also give several applications of the purity of $G_{\mathfrak{m}}(M)$. 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 $G_{\mathfrak{m}}(M)$ ensures that several important invariants are inherited. In addition, we provide sufficient conditions for Cohen-Macaulayness and purity of $G_{\mathfrak{m}}(M)$, and provide a local version of the Herzog-Kühl equations.