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PreprintCC BY 4.0arXiv

Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities

  • V. H. Jorge Perez
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Abstract

In this paper, we explore mixed Buchsbaum--Rim multiplicities of finite-colength submodules of free modules over Noetherian local rings. We extend the Hilbert-polynomial construction to submodules in arbitrary homogeneous degrees and prove a product formula for the resulting coefficients. We then examine module analogues of Teissier's coefficient inequalities and the Minkowski inequality. Explicit examples show that the coefficientwise power inequalities, log-convexity, and the product-root inequality can all fail; several failures occur over regular local rings. We also prove that equality of integral closures forces equality of all mixed Buchsbaum--Rim multiplicities, while a one-dimensional example shows that the converse fails for arbitrary modules. A converse is recovered when the module closures are the integral closures of modules of the form IF for m-primary ideals I. The general converse problem in dimension at least two is stated at the end.

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Cite

BibTeXRIS
BibTeX
@misc{perez2026counterexamples,
  title = {{Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities}},
  author = {V. H. Jorge Perez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13888},
  eprint = {2610.13888},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13888},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Counterexamples to Minkowski-type inequalities and the Rees converse for modules via mixed Buchsbaum-Rim Multiplicities
AU  - V. H. Jorge Perez
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13888
UR  - https://arxiv.org/abs/2610.13888
AB  - In this paper, we explore mixed Buchsbaum--Rim multiplicities of finite-colength submodules of free modules over Noetherian local rings. We extend the Hilbert-polynomial construction to submodules in arbitrary homogeneous degrees and prove a product formula for the resulting coefficients. We then examine module analogues of Teissier's coefficient inequalities and the Minkowski inequality. Explicit examples show that the coefficientwise power inequalities, log-convexity, and the product-root inequality can all fail; several failures occur over regular local rings. We also prove that equality of integral closures forces equality of all mixed Buchsbaum--Rim multiplicities, while a one-dimensional example shows that the converse fails for arbitrary modules. A converse is recovered when the module closures are the integral closures of modules of the form IF for m-primary ideals I. The general converse problem in dimension at least two is stated at the end.
ER  -