Topological Constructions for Multigraded Squarefree Module

Let $R=\mathbb{k}[x_1,\dot{\mathrm{.}}\mathrm{.},x_n]$ and $M=R^s\mathrm{/}I$ 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.