Topological Constructions for Multigraded Squarefree Module

TL;DR

This paper develops topological methods to analyze multigraded squarefree modules over polynomial rings, generalizing existing techniques for squarefree monomial ideals to broader module contexts.

Contribution

It introduces a construction of cochain complexes for multigraded squarefree modules and interprets their homological invariants via topological computations, extending prior ideal-based approaches.

Findings

Provides a topological framework for homological invariants of multigraded modules

Generalizes methods from squarefree monomial ideals to modules

Enables new computational approaches for module invariants

Abstract

Let $R=\Bbbk[x_1,\..., x_n]$ and $M=R^s/I$ a multigraded squarefree module. We discuss the construction of cochain complexes associated to $M$ and we show how to interpret homological invariants of $M$ in terms of topological computations. This is a generalization of the well studied case of squarefree monomial ideals.