On the radical of multigraded modules

TL;DR

This paper introduces a functor that generalizes the radical operation from monomial ideals to multigraded modules, linking it to Alexander duality and Auslander-Reiten theory.

Contribution

It defines a new functor for multigraded modules that extends radical concepts and explores its connections to duality and translation functors.

Findings

Generalizes properties shared by monomial ideals and their radicals

Establishes connections between the functor and Alexander duality

Links the functor to Auslander-Reiten translate

Abstract

We define a functor $\rr^\ast$ from the category of positively determined modules to the category of squarefree modules which plays the role of passing from a monomial ideal to its radical. By using this functor, we generalize several results on properties that are shared by a monomial ideal and its radical. Moreover, we study the connection of $\rr^\ast$ to the Alexander duality and Auslander-Reiten translate functor.