Minimal resolutions of toric substacks by line bundles

TL;DR

The paper develops minimal resolutions of certain toric substack sheaves using combinatorial and homological tools, enhancing understanding of their structure.

Contribution

It introduces a method to construct minimal resolutions of pushforward sheaves of toric substacks via line bundles, with explicit differentials and combinatorial descriptions.

Findings

Constructed minimal resolutions as strong deformation retracts.

Provided a canonical combinatorial description of differentials.

Utilized homological perturbation lemma and Moore-Penrose inverses.

Abstract

We construct minimal resolutions of pushforwards of structure sheaves of toric substacks of smooth toric stacks by line bundles as strong deformation retracts of cellular resolutions constructed by Hanlon, Hicks and Lazarev. We also provide a canonical and combinatorial description of the differentials of such minimal resolutions. Two key ingredients are the homological perturbation lemma and the Moore-Penrose inverses.