Rigid Ideals in Gorenstein Rings of Dimension One

Abstract

Motivated by a conjecture of Huneke and Wiegand concerning torsion in tensor products of modules over local rings, we investigate the existence of ideals I in a one-dimensional Gorenstein local ring R satisfying \(\text {Ext}^{1}_{R}(I,I)= 0\).

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Acknowledgements

We thank Adam Boocher, Olgur Celikbas, and Graham Leuschke for helpful comments during the preparation of this work. We also are grateful to various referees for their detailed comments on an earlier version of this paper.

Funding

This article is based on work supported by the National Science Foundation under Grant No. 0932078000, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall semester of 2012. The first author was partially supported by NSF grant DMS-1460638; the second author partly supported by NSF grant DMS-1700985; the third author partly supported by Simons Collaboration Grants 209213 and 426885.

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Correspondence to Craig Huneke.

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Huneke, C., Iyengar, S.B. & Wiegand, R. Rigid Ideals in Gorenstein Rings of Dimension One. Acta Math Vietnam 44, 31–49 (2019). https://doi.org/10.1007/s40306-018-00315-0

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Keywords

  • Complete intersection ring
  • Gorenstein ring
  • Rigid module
  • Tensor product
  • Torsion

Mathematics Subject Classification (2010)

  • 13D07
  • 13C14
  • 13C99