Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M . Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley-Reisner ideals of M and so-called elongations of M . Generalizing Greene’s theorem from coding theory, we show that ...

Let ∆ be a simplicial complex and I∆ its Stanley–Reisner ideal. We write ∆ for the exterior algebraic shifted complex of ∆ and ∆ for a combinatorial shifted complex of ∆. It will be proved that for all i and j one has the inequalities βii+j(I∆e) ≤ βii+j(I∆c) on the graded Betti numbers of I∆e and I∆c . In addition, the bad behavior of graded Betti numbers of I∆c will be studied.

When studying a graded module $M$ over the Cox ring of smooth projective toric variety $X$, there are two standard types resolutions commonly used to glean information: free and vector bundle its sheafification. Each approach comes with own challenges. There is geometric information that fail encode, while can resist study using algebraic combinatorial techniques. Recently, Berkesch, Erman, Smi...

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex. The description is in terms of the homology of the canonical Alexander dual complex. As applications we are able to prove for monomial ideals and j = 1 a conjecture of J. Herzog giving lower boun...

In this article we associate to every lattice ideal IL,ρ ⊂ K[x1, . . . , xm] a cone σ and a graph Gσ with vertices the minimal generators of the Stanley-Reisner ideal of σ. To every polynomial F we assign a subgraph Gσ(F ) of the graph Gσ. Every expression of the radical of IL,ρ, as a radical of an ideal generated by some polynomials F1, . . . , Fs gives a spanning subgraph of Gσ, the ∪ s i=1Gσ...