On the Rank of Multigraded Differential Modules

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 $R=k[x_1,\dots ,x_d]$, 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 $H(D)=\ker (\delta )/\mathrm{im}(\delta )$ of $D$ is non-zero and finite dimensional over $k$ then there is an inequality ${\mathrm{rank}}_{R}D⩾2^d$.

arXiv comment21 pages