Minimal resolutions of lattice ideals

TL;DR

This paper constructs canonical and simplified minimal free resolutions for lattice ideals over various fields, using combinatorial lattice path methods and generalizing sylvan resolutions for monomial ideals.

Contribution

It introduces a canonical minimal free resolution for lattice ideals with a combinatorial differential and a simpler non-canonical resolution generalizing sylvan resolutions.

Findings

Canonical resolution constructed over fields of characteristic 0 or large positive primes.

Differential described as a sum over lattice paths with combinatorial weights.

Generalization of sylvan resolutions to lattice modules.

Abstract

A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial description as a sum over lattice paths in $\mathbb{Z}^n$ of weights that come from sequences of faces in simplicial complexes indexed by lattice points. Over a field of any characteristic, a non-canonical but simpler resolution is constructed by selecting choices of higher-dimensional analogues of spanning trees along lattice paths. These constructions generalize sylvan resolutions for monomial ideals by lifting them equivariantly to lattice modules.