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PreprintCC BY-NC-ND 4.0arXiv

Pairs of lattice complements with complementary homology

  • Sara Faridi
  • Dharm Veer
  • Volkmar Welker
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Abstract

In this paper we study whether there is a global to local principle for the homology of order complexes of finite lattices. We explore whether the non-vanishing of the homology of a lattice in degree a+b−2 forces the existence of a pair of lattice-complements whose open intervals below have non-vanishing homology in degrees a and b respectively. We consider the same question also in the language of multigraded free resolutions. We provide positive evidence for the existence of such pairs and give counterexamples to strengthenings of the question.

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Cite

BibTeXRIS
BibTeX
@misc{faridi2026pairs,
  title = {{Pairs of lattice complements with complementary homology}},
  author = {Sara Faridi and Dharm Veer and Volkmar Welker},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26397},
  eprint = {2609.26397},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26397},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Pairs of lattice complements with complementary homology
AU  - Sara Faridi
AU  - Dharm Veer
AU  - Volkmar Welker
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26397
UR  - https://arxiv.org/abs/2609.26397
AB  - In this paper we study whether there is a global to local principle for the homology of order complexes of finite lattices. We explore whether the non-vanishing of the homology of a lattice in degree a+b−2 forces the existence of a pair of lattice-complements whose open intervals below have non-vanishing homology in degrees a and b respectively. We consider the same question also in the language of multigraded free resolutions. We provide positive evidence for the existence of such pairs and give counterexamples to strengthenings of the question.
ER  -