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PreprintarXiv licencearXiv

Graded Betti numbers of graded Möbius algebras of uniform matroids

  • Louiza Fouli
  • Sean Grate
  • Selvi Kara
  • Adam LaClair
  • Jason McCullough
  • Vinh Nguyen
  • Aleksandra Sobieska
  • Prajwal Udanshive
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Abstract

Graded Möbius algebras were a key tool in the proof of the Dowling-Wilson Top Heavy Conjecture. They are commutative algebras whose Hilbert functions recover the Whitney numbers of the second kind, i.e. the number of flats of a given rank. The graded Betti numbers of the defining ideal of a graded Möbius algebra refine the Hilbert function and describe its minimal free resolution. In this paper we derive precise formulas for the graded Betti numbers of the defining ideals of graded Möbius algebra for any uniform matroid. We also study when the graded Möbius algebra of an arbitrary matroid is linearly presented.

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Cite

BibTeXRIS
BibTeX
@misc{fouli2026graded,
  title = {{Graded Betti numbers of graded Möbius algebras of uniform matroids}},
  author = {Louiza Fouli and Sean Grate and Selvi Kara and Adam LaClair and Jason McCullough and Vinh Nguyen and Aleksandra Sobieska and Prajwal Udanshive},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03827},
  eprint = {2609.03827},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03827},
}
RIS
TY  - UNPB
TI  - Graded Betti numbers of graded Möbius algebras of uniform matroids
AU  - Louiza Fouli
AU  - Sean Grate
AU  - Selvi Kara
AU  - Adam LaClair
AU  - Jason McCullough
AU  - Vinh Nguyen
AU  - Aleksandra Sobieska
AU  - Prajwal Udanshive
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03827
UR  - https://arxiv.org/abs/2609.03827
AB  - Graded Möbius algebras were a key tool in the proof of the Dowling-Wilson Top Heavy Conjecture. They are commutative algebras whose Hilbert functions recover the Whitney numbers of the second kind, i.e. the number of flats of a given rank. The graded Betti numbers of the defining ideal of a graded Möbius algebra refine the Hilbert function and describe its minimal free resolution. In this paper we derive precise formulas for the graded Betti numbers of the defining ideals of graded Möbius algebra for any uniform matroid. We also study when the graded Möbius algebra of an arbitrary matroid is linearly presented.
ER  -