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PreprintCC BY 4.0arXiv

A Counterexample to the Huneke-Wiegand Conjecture

  • Son Pham
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Abstract

We construct a symmetric numerical semigroup Γ of multiplicity $56$, embedding dimension $26$, and Frobenius number $181$ such that, over every field k, the one-dimensional Gorenstein local domain R=k[tΓ]m has a nonprincipal ideal I=(t56,t70)R for which I⊗RHomR(I,R) is torsion-free. This gives a counterexample to the Huneke-Wiegand conjecture in its general Gorenstein formulation. The proof reduces to an explicit finite sumset identity, interpreted through the Huneke-Iyengar-Wiegand colon criterion and Leamer's tensor-torsion formula. We also identify the tensor product with the trace ideal of I, compute EndR(I)=R[t101,t107] and its conductor, and show that the counterexample persists under completion. A separate graded tensor-graph calculation and exact, standard-library Python verifiers accompany the paper. We explain how the example differs from the field-extension construction of Christensen, Gerko, and Iyengar.

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Cite

BibTeXRIS
BibTeX
@misc{pham2026counterexample,
  title = {{A Counterexample to the Huneke-Wiegand Conjecture}},
  author = {Son Pham},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07615},
  eprint = {2609.07615},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07615},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Counterexample to the Huneke-Wiegand Conjecture
AU  - Son Pham
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07615
UR  - https://arxiv.org/abs/2609.07615
AB  - We construct a symmetric numerical semigroup Γ of multiplicity $56$, embedding dimension $26$, and Frobenius number $181$ such that, over every field k, the one-dimensional Gorenstein local domain R=k[t^Γ]_m has a nonprincipal ideal I=(t^56,t^70)R for which I⊗_RHom_R(I,R) is torsion-free. This gives a counterexample to the Huneke-Wiegand conjecture in its general Gorenstein formulation. The proof reduces to an explicit finite sumset identity, interpreted through the Huneke-Iyengar-Wiegand colon criterion and Leamer's tensor-torsion formula. We also identify the tensor product with the trace ideal of I, compute End_R(I)=R[t^101,t^107] and its conductor, and show that the counterexample persists under completion. A separate graded tensor-graph calculation and exact, standard-library Python verifiers accompany the paper. We explain how the example differs from the field-extension construction of Christensen, Gerko, and Iyengar.
ER  -