Componentwise regularity (I)

TL;DR

This paper introduces componentwise regularity, explores its properties, and extends Buchberger's criterion and Green's crystallization theorem within this new framework, providing tools for analyzing polynomial modules.

Contribution

It defines componentwise regularity and generalizes Buchberger's criterion and Green's theorem using this concept, offering new methods for studying polynomial modules.

Findings

Established a criterion analogous to Buchberger's for weight orders.

Proved a stronger version of Green's crystallization theorem.

Identified a necessary condition relating Betti numbers and componentwise linearity.

Abstract

We define the notion of componentwise regularity and study some of its basic properties. We prove an analogue, when working with weight orders, of Buchberger's criterion to compute Gr\"obner bases; the proof of our criterion relies on a strengthening of a lifting lemma of Buchsbaum and Eisenbud. This criterion helps us to show a stronger version of Green's crystallization theorem in a quite general setting, according to the componentwise regularity of the initial object. Finally we show a necessary condition, given a submodule $M$ of a free one over the polynomial ring and a weight such that $in(M)$ is componentwise linear, for the existence of an $i$ such that $\beta_i(M)=\beta_i(in(M))$.