Koszul complexes and relative homological algebra of functors over posets

TL;DR

This paper introduces a method using Koszul complexes to compute relative Betti diagrams of vector space-valued functors over posets, avoiding explicit resolutions.

Contribution

It establishes conditions under which grading functors yields explicit Koszul complexes that determine relative Betti diagrams.

Findings

Koszul complexes can compute relative Betti diagrams without explicit resolutions

Grading functors leads to explicit Koszul complexes

Homology dimensions give the relative Betti diagrams

Abstract

Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.