Syzygies of associated graded modules

TL;DR

This paper characterizes when the first syzygy of an associated graded module is equigenerated, explores conditions for purity and Cohen-Macaulayness of the associated graded module, and relates these properties to free resolutions.

Contribution

It provides new criteria for the purity and Cohen-Macaulayness of associated graded modules and links free resolutions of modules to their associated graded counterparts.

Findings

Characterization of equigenerated first syzygy of associated graded modules

Conditions for purity and Cohen-Macaulayness of associated graded modules

A local version of Herzog-Kühl equations

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-K\"uhl equations.