Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x1,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T
• (Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ϕ over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T
• (ϕ) that is a resolution of Kerϕ. We also show that the length of T
• (ϕ) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ϕ.