Representations of matroids and free resolutions for multigraded modules

Abstract

Let K be a field, let R=K[x_1,..., x_m] be a polynomial ring with the standard Z^m-grading (multigrading), let L be a Noetherian multigraded R-module, and let F: E --> G 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(F, 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 f over K of a matroid M a canonical finite complex of finite dimensional K-vector spaces T(f) that is a resolution of Ker(f). We also show that the length of T(f) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map f.