Free modules of a multigraded resolution from simplicial complexes

Abstract

Let $R=\Bbbk [x_1,..., x_m]$ be a polynomial ring in $m$ variables over $\Bbbk$ with the standard $\mathbb{Z}^m$ grading and $L$ a multigraded Noetherian $R$-module. When $\Bbbk$ 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 $\mathbb{Z}$ 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.