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Subdividing simplicial virtual resolutions with homology

Eric Nathan Stucky, Jay Yang

TL;DR

This work investigates when monomial ideals on toric varieties admit virtual resolutions that contain homology, by encoding the data in labeled simplicial complexes and linking the algebra to simplicial homology through $H_i(F_{\Delta})_{\alpha} \cong \widetilde{H}_{i-1}(\Delta_m;\mathbf{k})$. A central acyclicity criterion shows $F_{\Delta}$ is a virtual resolution iff there exists $d$ with all induced subcomplexes $\Delta_m$ (for $m\in B^{d}$) acyclic, enabling explicit construction of virtual resolutions with or without higher homology. The authors classify minimal labeled complexes capable of supporting homology, proving that the smallest such examples come from bipyramids $\diamondsuit^{c-1}$ and giving a complete description for the case of products of projective spaces. They then develop subdivisions, in particular virtual-compatible subdivisions, that can reduce homology under concrete topological and divisibility conditions, and show that for $X=\mathbb{P}^n\times\mathbb{P}^k$ with $\Delta=\diamondsuit^k$ one can obtain free (length-preserving) virtual resolutions from existing virtual resolutions. The results yield practical tools to control homology in virtual resolutions and provide structural insight into which modules admit short, homology-free virtual resolutions in toric settings.

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.

Subdividing simplicial virtual resolutions with homology

TL;DR

This work investigates when monomial ideals on toric varieties admit virtual resolutions that contain homology, by encoding the data in labeled simplicial complexes and linking the algebra to simplicial homology through . A central acyclicity criterion shows is a virtual resolution iff there exists with all induced subcomplexes (for ) acyclic, enabling explicit construction of virtual resolutions with or without higher homology. The authors classify minimal labeled complexes capable of supporting homology, proving that the smallest such examples come from bipyramids and giving a complete description for the case of products of projective spaces. They then develop subdivisions, in particular virtual-compatible subdivisions, that can reduce homology under concrete topological and divisibility conditions, and show that for with one can obtain free (length-preserving) virtual resolutions from existing virtual resolutions. The results yield practical tools to control homology in virtual resolutions and provide structural insight into which modules admit short, homology-free virtual resolutions in toric settings.

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.
Paper Structure (5 sections, 18 theorems, 36 equations, 6 figures)

This paper contains 5 sections, 18 theorems, 36 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Virtual Resolutions from Simplicial Complexes
  3. Minimal Simplicial Complexes with Homology
  4. Subdivisions and Homology
  5. Products of Projective Spaces

Key Result

Theorem 1.1

Fix a smooth toric variety $X$ with Cox ring $S$ and irrelevant ideal $B$. For a labeled simplicial complex $(\Delta,\ell)$, its associated chain complex $F_\Delta$ is a virtual resolution if and only if for each of the subcomplexes there exists some $d\geq 0$ such that $\widetilde{H}_i(\Delta_m;\mathbf{k})=0$ for all monomials $m\in B^{d}$ and integers $i\geq 0$.

Figures (6)

  • Figure A: A labeled simplicial complex giving a virtual resolution with homology
  • Figure B: The labeled simplicial complex obtained by a virtual compatible subdivision of the labeled simplicial complex from Figure \ref{['fig:intro-virtual']}
  • Figure C: A simplicial complex with vertices labeled by the names to be used throughout the section
  • Figure D: This figure depicts a labeled simplicial complex $(\Delta,\ell)$ and its subdivision $(\Delta',\ell')$ which introduces new homology, despite being a virtual compatible subdivision.
  • Figure E: This figure depicts a labeled simplicial complex $(\Delta,\ell)$ and example of a subdivision $(\Delta', \ell')$ applied to a case where Proposition \ref{['subdivision-reduces-homology-part1']} applies.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Example 1.2
  • Definition 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1: bps-monomial-resolutions*Construction 2.1
  • Definition 2.2
  • Theorem 2.3: bps-monomial-resolutions
  • ...and 39 more