On a question of Cid-Ruiz: positive results and counterexamples

Abstract

In this paper we give positive criteria and counterexamples to a question of Cid-Ruiz on whether intrinsic polar multiplicities detect reductions of modules. Over an excellent regular local surface with infinite residue field, we compute the entire polar sequence of every finite-colength submodule of a free module, showing that its only variable entry is the order of the determinant ideal. This computation yields classes for which equality of polar multiplicities characterizes reduction, while also producing counterexamples in every positive rank, including integrally closed modules with identical intrinsic Hilbert functions and arbitrarily large relative Buchsbaum--Rim multiplicity. In arbitrary dimension, we prove that, under equidimensionality and a sharp dimension bound on the kernel of the induced map of associated graded Rees algebras, equality of the sums of the polar multiplicities characterizes reduction.

Non-freezing of the proximity digraph and non-pseudostable convergence in heterogeneous Hegselmann--Krause models: Counterexamples to conjectures and results of Mirtabatabaei and Bullo
Peter Hegarty, Damiano Ognissanti, Edvin Wedin
arXiv21 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematics Dynamical systemsarXiv
A positive answer to Barsotti and Carman's question on unit groups of Burnside rings of fusion systems
Chunbo Zhou, Heguo Liu
arXiv5 Nov 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
On an open question about the equivalence of monotone maximality and surjectivity condition in CAT(0) spaces: Negative and partially positive answers
Parin Chaipunya
arXiv18 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
The Converse Problem for the Morley Tetrahedron: Counterexamples, Conjectures, and Partial Results
Quang Hung Tran
arXiv6 Oct 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematics Geometry & topologyarXiv
Unique Normal Forms: Proof-Graph Counterexamples, Infinite-Carrier Certificates, and Confluence Repair
Moses Rahnama
HAL1 Jul 2026PreprintLicence not statedLogic & foundationsHAL

Cite

BibTeX RIS

BibTeX

@misc{freitas2026question,
  title = {{On a question of Cid-Ruiz: positive results and counterexamples}},
  author = {T. H. Freitas and V. H. Jorge Pérez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35281},
  eprint = {2610.35281},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35281},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On a question of Cid-Ruiz: positive results and counterexamples
AU  - T. H. Freitas
AU  - V. H. Jorge Pérez
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35281
UR  - https://arxiv.org/abs/2610.35281
AB  - In this paper we give positive criteria and counterexamples to a question of Cid-Ruiz on whether intrinsic polar multiplicities detect reductions of modules. Over an excellent regular local surface with infinite residue field, we compute the entire polar sequence of every finite-colength submodule of a free module, showing that its only variable entry is the order of the determinant ideal. This computation yields classes for which equality of polar multiplicities characterizes reduction, while also producing counterexamples in every positive rank, including integrally closed modules with identical intrinsic Hilbert functions and arbitrarily large relative Buchsbaum--Rim multiplicity. In arbitrary dimension, we prove that, under equidimensionality and a sharp dimension bound on the kernel of the induced map of associated graded Rees algebras, equality of the sums of the polar multiplicities characterizes reduction.
ER  -

Abstract

Related

Cite