On the Rank of Multigraded Differential Modules

Abstract

A ${\mathbb{Z}}^{\mathit{d}}$-graded differential R-module is a ${\mathbb{Z}}^{\mathit{d}}$-graded R-module D with a morphism $\mathit{\delta }:\mathit{D}\to \mathit{D}$ such that ${\mathit{\delta }}^2=0$. For $\mathit{R}=\mathit{k}[{\mathit{x}}_1,\dots ,{\mathit{x}}_{\mathit{d}}]$, 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 $ker\mathit{\delta }/im\mathit{\delta }$ of D is nonzero and finite dimensional over k, there is an inequality ${rk}_{\mathit{R}}\mathit{D}\ge 2^{\mathit{d}}$.