Free modules of a multigraded resolution from simplicial complexes

TL;DR

This paper provides a combinatorial framework to understand the multigraded free resolution of modules over polynomial rings, clarifying the structure of T-resolutions and extending their applicability beyond fields.

Contribution

It introduces a combinatorial description of the free modules in T-resolutions, making their structure more transparent and applicable over the integers.

Findings

Provides a canonical generating set for free modules in T-resolutions.

Extends the combinatorial description of T-resolutions over z, not just fields.

Includes an explicit example illustrating the new approach.

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.