Subdividing simplicial virtual resolutions with homology

TL;DR

This paper introduces new tools for analyzing virtual resolutions of monomial ideals via simplicial complexes, focusing on homology removal and acyclicity to better understand their structure.

Contribution

It provides a characterization of simplicial virtual resolutions through acyclicity and offers methods to identify minimal complexes and eliminate homology.

Findings

Characterization of virtual resolutions by acyclicity of subcomplexes

Technique for removing homology from simplicial virtual resolutions

Description of minimal supporting simplicial complexes

Abstract

While sporadic examples of virtual resolutions with homology have been constructed, their occurrence is not well understood or controlled. Our results build a new set of tools for studying virtual resolutions of monomial ideals as arising from simplicial complexes, including characterizing them by the acyclicity of certain induced subcomplexes. Using this characterization, we give a description of minimal simplicial complexes supporting virtual resolutions as well as a technique for removing homology from simplicial virtual resolutions.