Asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules

TL;DR

This paper establishes linear bounds on the Castelnuovo-Mumford regularity of multigraded modules formed by products of powers of ideals, providing asymptotic control over their complexity.

Contribution

It proves the existence of uniform linear bounds for the regularity of multigraded modules involving products of ideals, extending previous understanding to a multigraded setting.

Findings

Existence of linear bounds on regularity for multigraded modules.

Bounds depend linearly on the sum of exponents of ideals.

Applicable to modules over Noetherian standard graded algebras.

Abstract

Let $A$ be a Noetherian standard $\mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,\ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $\mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k'$ such that \[ \mathrm{reg}(I_1^{n_1} \cdots I_t^{n_t} M) \leq (n_1 + \cdots + n_t) k + k' \quad\mbox{for all }~n_1,\ldots,n_t \in \mathbb{N}. \]