Free modules of a multigraded resolution from simplicial complexes

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Let R= [x₁,..., xₘ] be a polynomial ring in m variables over with the standard Zᵐ grading and L a multigraded Noetherian R-module. When is a field, Tchernev has an explicit construction of a multigraded free resolution called the T-resolution of L over R. Despite the explicit canonical description, this method uses linear algebraic methods, which makes the structure hard to understand. This paper gives a combinatorial description for the free modules, making the T-resolution clearer. In doing so, we must introduce an ordering on the elements. This ordering identifies a canonical generating set for the free modules. This combinatorial construction additionally allows us to define the free modules over Z instead of a field. Moreover, this construction gives a combinatorial description for one component of the differential. An example is computed in the first section to illustrate this new approach.