On the Rank of Multigraded Differential Modules

TL;DR

This paper establishes a lower bound on the rank of multigraded differential modules over polynomial rings, linking the module's rank to the dimension of its homology and providing new insights into their structure.

Contribution

It introduces a lower bound on the rank of free multigraded differential modules over polynomial rings, based on the dimension of their homology.

Findings

Rank of differential modules is at least 2^d when homology is non-zero and finite dimensional.

Defines Betti number for differential modules and relates it to module rank.

Provides theoretical bounds connecting homology properties to module structure.

Abstract

A $\mathbb{Z}^d$-graded differential $R$-module is a $\mathbb{Z}^d$-graded $R$-module $D$ equipped with an endomorphism, $\delta$, that squares to zero. For $R=k[x_1,\ldots,x_d]$, this paper establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology $H(D)=\mathrm{ker}(\delta)/\mathrm{im}(\delta)$ of $D$ is non-zero and finite dimensional over $k$ then there is an inequality $\mathrm{rank}_R D \geqslant 2^d$.