Syzygies of associated graded modules
H. Ananthnarayan, Manav Batavia, Omkar Javadekar
TL;DR
This work investigates how the syzygies of a finitely generated module $M$ over a Noetherian local ring relate to the syzygies of its associated graded module $G_{\mathfrak m}(M)$. It provides a criterion for when the first syzygy of $G_{\mathfrak m}(M)$ is equigenerated and constructs a pure complex $\mathbb{F}^*_{\bullet}$ from a free resolution $\mathbb{F}_{\bullet}$ of $M$, showing that $G_{\mathfrak m}(M)$ has a pure $G_{\mathfrak m}(R)$-resolution iff $\mathbb{F}^*_{\bullet}$ resolves $G_{\mathfrak m}(M)$. The paper then develops a comprehensive framework linking purity to inheritance of invariants, establishes a local Herzog–Kühl-type equation set, and derives sufficient conditions for purity and Cohen–Macaulayness of $G_{\mathfrak m}(M)$, with consequences for the original ring $R$ and for modules of finite projective dimension. Overall, it clarifies when passing to the associated graded module preserves key homological data and provides practical criteria for identifying Cohen–Macaulay and pure scenarios in local settings.
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 $G_{\mathfrak{m}}(M)$ 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.