On simple A-multigraded minimal resolutions

TL;DR

This paper studies simple multigraded minimal free resolutions of semigroup-graded ideals, introducing the gcd-complex to determine Betti numbers and characterizing the indispensable resolution for certain lattice ideals.

Contribution

It introduces the gcd-complex for analyzing Betti numbers and characterizes when the Koszul complex is the indispensable resolution for lattice ideals.

Findings

Homology of gcd-complex determines Betti numbers.

Every A-homogeneous minimal free resolution can be made simple.

Koszul complex is the indispensable resolution for certain lattice ideals.

Abstract

Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(\bf b)$ for a degree $\mathbf{b}\in \A$. We show that the homology of $D_{gcd}(\bf b)$ determines the $i$-Betti numbers of degree $\bf b$. We discuss the notion of an indispensable complex of $R/I$. We show that the Koszul complex of a complete intersection lattice ideal $I$ is the indispensable resolution of $R/I$ when the $A$-degrees of the elements of the generating $R$-sequence are incomparable.