On the Rank of Multigraded Differential Modules

A Zd-graded differential R-module is a Zd-graded R-module D with a morphism δ:D→D such that δ2=0. For R=k[x1,…,xd], 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 kerδ/imδ of D is nonzero and finite dimensional over k, there is an inequality rkRD≥2d.