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PreprintCC BY 4.0arXiv

An overring characterization of Gorenstein Prüfer domains

  • Hwankoo Kim
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Abstract

We prove that a commutative integral domain is Gorenstein Prüfer if and only if every overring is Gorenstein flat over the original domain. It is enough to test overrings that are finitely generated as algebras. The proof combines finite duality with a conductor-quotient argument. Under the overring hypothesis, the conductor quotient of a proper overring generated by two elements as a module is an IF-ring, and degree-one duality lifts first syzygies of finitely presented quotient modules to totally reflexive modules. This gives a bound of one for the Gorenstein flat dimension of every module annihilated elementwise by conductor powers. Quadratic integral elements provide such conductors at nonnormal localizations, while normal localizations give principal-ideal neighborhoods. Constructible compactness yields a finite cover, and a bounded complex of localizations combines these bounds over the original domain. Consequently every finitely generated ideal is Gorenstein flat and the domain is coherent. Examples distinguish finite-type from module-finite tests and Gorenstein flatness from ordinary flatness.

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Cite

BibTeXRIS
BibTeX
@misc{kim2026overring,
  title = {{An overring characterization of Gorenstein Prüfer domains}},
  author = {Hwankoo Kim},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15454},
  eprint = {2610.15454},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15454},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An overring characterization of Gorenstein Prüfer domains
AU  - Hwankoo Kim
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15454
UR  - https://arxiv.org/abs/2610.15454
AB  - We prove that a commutative integral domain is Gorenstein Prüfer if and only if every overring is Gorenstein flat over the original domain. It is enough to test overrings that are finitely generated as algebras. The proof combines finite duality with a conductor-quotient argument. Under the overring hypothesis, the conductor quotient of a proper overring generated by two elements as a module is an IF-ring, and degree-one duality lifts first syzygies of finitely presented quotient modules to totally reflexive modules. This gives a bound of one for the Gorenstein flat dimension of every module annihilated elementwise by conductor powers. Quadratic integral elements provide such conductors at nonnormal localizations, while normal localizations give principal-ideal neighborhoods. Constructible compactness yields a finite cover, and a bounded complex of localizations combines these bounds over the original domain. Consequently every finitely generated ideal is Gorenstein flat and the domain is coherent. Examples distinguish finite-type from module-finite tests and Gorenstein flatness from ordinary flatness.
ER  -